#  Gradient flow of Einstein-Maxwell theory and Reissner-Nordström black holes

Davide De Biasio,${}^{\text{a}}$ Julian Freigang,${}^{\text{a}}$ Dieter
Lüst${}^{\text{a,b}}$ and Toby Wiseman${}^{\text{c}}$

###### Abstract

Ricci flow is a natural gradient flow of the Einstein-Hilbert action. Here we
consider the analog for the Einstein-Maxwell action, which gives Ricci flow
with a stress tensor contribution coupled to a Yang-Mills flow for the Maxwell
field. We argue that this flow is well-posed for static spacetimes with pure
electric or magnetic potentialsand show it preserves both non-extremal and
extremal black hole horizons. In the latter case we find the flow of the near
horizon geometry decouples from that of the exterior. The Schwarzschild black
hole is an unstable fixed point of Ricci flow for static spacetimes. Here we
consider flows of the Reissner-Nordström (RN) fixed point. The magnetic RN
solution becomes a stable fixed point of the flow for sufficient charge.
However we find that the electric RN black hole is always unstable.
Numerically solving the flow starting with a spherically symmetric
perturbation of a non-extremal RN solution, we find similar behaviour in the
electric case to the Ricci flows of perturbed Schwarzschild, namely the
horizon shrinks to a singularity in finite time or expands forever. In the
magnetic case, a perturbed unstable RN solution has a similar expanding
behaviour, but a perturbation that decreases the horizon size flows to a
stable black hole solution rather than a singularity. For extremal RN we solve
the near horizon flow for spherical symmetry exactly, and see in the electric
case two unstable directions which flow to singularities in finite flow time.
However, even turning these off, and fixing the near horizon geometry to be
that of RN, we numerically show that the flows appear to become singular in
the vicinity of its horizon.

${}^{\text{a}}$ Max–Planck–Institut für Physik, Werner–Heisenberg–Institut,
Föhringer Ring 6, 80805 München, Germany

${}^{\text{b}}$Arnold Sommerfeld Center for Theoretical Physics,
Ludwig Maximilians Universität München,

Theresienstrasse 37, 80333 München, Germany

${}^{\text{c}}$Theoretical Physics Group, Blackett Laboratory, Imperial
College London
London SW7 2AZ, UK

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 Einstein-Maxwell theory
  3. 3 Gradient flow of the Einstein-Maxwell action
  4. 4 Static EM flows of black hole spacetimes
    1. 4.1 Static non-extremal black hole spacetimes
    2. 4.2 Conserved charges under EM flow
    3. 4.3 Static extremal black hole spacetimes
      1. 4.3.1 Ricci tensor term
      2. 4.3.2 DeTurck diffeomorphism term
      3. 4.3.3 The electric Maxwell field
      4. 4.3.4 The magnetic Maxwell field
      5. 4.3.5 Putting the terms together and the near horizon limit
  5. 5 Spherically symmetric flows about non-extremal Reissner-Nordström
    1. 5.1 Linear perturbations about RN
    2. 5.2 Magnetic non-extremal Reissner-Nordström linear stability
    3. 5.3 Electric non-extremal Reissner-Nordström linear stability
    4. 5.4 End point of unstable magnetic and electric flows
  6. 6 Spherically symmetric flows about extremal Reissner-Nordström
    1. 6.1 Near horizon Electric flow
    2. 6.2 Near horizon Magnetic flow
    3. 6.3 Flows of the full extremal spacetimes
  7. 7 Summary

##  1 Introduction

Ricci flow [1, 2] may be regarded as a natural gradient flow of the Euclidean
signature Einstein-Hilbert action with respect to the DeWitt metric [3, 4],
and provides a way to study both the stability of saddle points in the
canonical ensemble, as well as the global structure of the space of solutions.
An example is gravity in a box with spherical spatial symmetry and fixed
boundary radius. Describing this at finite temperature using a Euclidean
continuation, the saddle points of the action are the small and large black
holes, as well as ‘hot’ flat space, and being Ricci flat, these are fixed
points of Ricci flow. The small black hole is always thermodynamically
unstable in the canonical ensemble [5] and there is an analog of the Hawking-
Page phase transition between the large black hole and hot flat space [6].
This thermodynamic instability of the small black hole results in a negative
mode of the Lichnerowicz operator [7, 8, 9, 10, 11], which drives an
instablity under Ricci flow [3]. A very interesting recent study of the Wick
rotation from Lorentzian signature find that mode stability about a fixed
point in the semiclassical Euclidean path integral is directly related to
stability under Ricci flow [12].

This instability implies a relevant deformation of the small black hole that
generates two flows away from it. In [3] it was found these flows connect to
the other two stable fixed points, the large black hole, and hot flat space,
the latter requiring a surgery to resolve a singularity along the flow. This
is intruigingly qualitatively similar to the behaviour of real time evolution
of a small black hole surrounded by a bath of radiation at its Hawking
temperature. Removing the box by taking it to infinite size, the small black
hole tends to the usual Schwarzschild solution and the large black hole is
removed with the box. Since the unstable mode is localized about the horizon,
the instability under Ricci flow persists. One of the two flows from
Schwarzschild then again tends to flat spacetime (after resolving a
singularity at finite flow time that changes the spacetime topology). The
other ‘eats up’ spacetime indefinitely as there is now no large black hole for
it to flow to.

These Ricci flows of the Schwarzschild black hole can be thought of as RG flow
of a sigma model [2]. However it is also possible that the off-shell
geometries that the flow passes through may have significance in the
gravitational path integral [3]. In the context of string theory
phenomenology, Ricci flow has also been suggested as a tool to refine and
extend the so-called Swampland distance conjecture [13, 14]. In particular,
pushing the intuition behind the AdS conjecture [15] even further, it allows a
generalization of infinite distance moduli space limits (which are expected to
be accompanied by infinite towers of light states) to the case of
displacements in the metric components.

The above discussion was for Euclidean signature. In fact Ricci flow is not
well defined for Lorentzian spacetimes in general. However, restricting to
static or stationary metrics it is well posed and preserves smooth non-
extremal Killing horizons [3, 16, 17], the surface gravity and angular
velocities of these being constant along the flow. Furthermore, the Euclidean
negative mode discussed above preserves the Euclidean $U(1)$ isometry, and
hence can be Wick rotated to an instability of the Ricci flow of static black
hole spacetimes. Thus the unstable flows of Euclidean Schwarzschild can be
Wick rotated to give flows of static spacetimes away from Lorentzian
Schwarzschild.

Motivated by these considerations it is interesting to consider flow of a
gravitational system with matter. Here we will consider the simple setting of
gravity coupled to a Maxwell field. There is a natural family of flows that
are gradient flows of the Einstein-Maxwell action with respect to a natural
metric on the superspace. These flows couple the metric and Maxwell vector
potential. The flow of the metric is similar to Ricci flow but with an extra
contribution from the Maxwell stress tensor. The flow of the Maxwell field is
given by the Yang-Mills flow in the background of the metric. We call this
combined gradient flow the Einstein-Maxwell flow (EM flow). The non-trivial
fixed points of the flow are then the Reissner-Nordström (RN) solutions.

Firstly we introduce the flow in section 3, showing that when it is restricted
to static metrics with either electric or magnetic potentials, then it is
well-posed in the sense that it can be made parabolic in character. We
emphasize that general potentials with both electric and magnetic parts give
rise to Poynting energy flux, and their backreaction is therefore not
compatible with static symmetry even though the electric and magnetic fields
are static (see [18] for a discussion on this). Hence here, with the
restriction to static metrics, we will only consider purely electric or purely
magnetic Maxwell fields. In section 4 we show that this flow of static metrics
and Maxwell potentials preserves both non-extremal and extremal horizons. 111
In [19] Ricci flow was discussed for geometries with extremal horzions where
the near horizon geometry was a solution of the Einstein equations, so was
fixed in flow time and consequently could be viewed as a Dirichlet boundary
condition. Here we consider the more general case where the horizon geometry
flows as well.  We find a new feature of the flows of spacetimes with extremal
horizons is that, as one might expect, the flow of the near horizon geometry
decouples from the flow of the exterior spacetime. 222 For an excellent review
of extremal horizons and their near horizon geometry see [20]. Both flows
preserve the surface gravity of the solution in the non-extremal setting. We
show that the electric flow preserves the electric potential difference, but
argue it does not preserve the electric charge. Conversely we show that the
magnetic flow does preserve the magnetic charge.

In section 5 we consider the stability of the RN fixed point for non-extremal
solutions in the spherically symmetric sector using numerical methods both for
the case of electric and magnetic charge. Negative modes of the (non-extremal)
Euclidean magnetic RN solution were studied by Monteiro and Santos [21]. They
considered the negative mode of Reissner-Nordström and found that precisely
where the specific heat at fixed (magnetic) charge becomes positive, the
negative mode disappears. These Euclidean negative modes can be Wick rotated
to give an unstable mode of the static magnetic EM flow we consider here. The
expectation is then that RN is a stable fixed point of the magnetic EM flow
for sufficient magnetic charge, and this is indeed what we see. We then
consider the non-extremal RN fixed point of the electric EM flow, which we
recall preserves the electric potential difference. Now an interesting fact is
that in the grand canonical ensemble, where we fix temperature and electric
potential, the RN solution has an unstable thermodynamics up to extremality.
One might then think this suggests that RN will be an unstable fixed point of
the electric EM flow all the way up to extremal electric charge. However due
to there being no simple Wick rotation to Euclidean solutions in this electric
setting, we know of no formal correspondence between the static flow
stability, Euclidean stability or thermodynamics. Nonetheless our numerical
analysis of the linear behaviour of the electric EM flow about RN indicates a
complicated structure of unstable modes which indeed appear to persist to the
extremal limit. In particular we find more unstable modes as charge increases,
rather than less as in the magnetic case.

Having discussed the linear stability of non-extremal RN for the electric and
magnetic flows we then discuss where the unstable modes of these non-extremal
RN solutions flow to by simulating the spherically symmetric non-linear flow
equations numerically. In the electric case, for all electric charge the RN
fixed point is unstable and we find qualitatively similar flow behaviour as
for the uncharged Schwarzschild fixed point under Ricci flow. Perturbing the
unstable electric RN fixed point to initially reduce the horizon size results
in a flow that shrinks the horizon to a zero size singularity in finite flow
time, whereas initially expanding the horizon gives a flow that ‘eats up’
spacetime, apparently indefinitely. One might expect similar behaviour to the
uncharged case, as the flow preserves the surface gravity and electric
potential, and with these fixed values there is only one infilling RN
solution. In the magnetic case, for large charge the RN solution is linearly
stable under the magnetic EM flow, and we find large perturbations flow back
to it. However for the small charge case, the RN fixed point is unstable,
there also exists a stable fixed point with smaller horizon size and the same
magnetic charge and surface gravity (which are both preserved by the magnetic
EM flow). We find that perturbing the unstable fixed point so that the horizon
initially decreases in size, gives a flow that asymptotes to this smaller
stable solution. An initial perturbation that expands the horizon results in a
flow that again ’eats up’ spacetime, as in the electric and uncharged cases.

Finally in section 6 we study the extremal RN fixed points, and here we are
able to analytically solve the full non-linear near horizon flow for spherical
symmetry. As expected, we find the magnetic extremal near horizon geometry is
stable, and the electric one is unstable, and we elucidate the various
singularities that these flows tend towards. Again restricting to spherical
symmetry, we perform numerical simulation of the flow of the full extremal
black hole spacetimes, and again find stability in the magnetic case, where
initial configurations that are non-linear deformations of extremal RN,
preserving extremality, flow back to the fixed point. In the electric case if
the near horizon geometry is perturbed our near horizon analysis already
implies the spacetime flows away from RN. However one might not perturb the
near horizon geometry, which is then fixed under the flow, and then wonder
whether extremal RN is stable under deforming the exterior spacetime. Our
simulations show that it is still unstable, and furthermore we find that a
curvature singularity appears to develop just outside the horizon under this
flow.

##  2 Einstein-Maxwell theory

This paper will be concerned with a natural gradient geometric flow of the
static Einstein-Maxwell system. Having derived the flow and discussed its
well-posedness, we then focus on its natural fixed points, the Reissner-
Nordström (RN) solutions, both non-extremal and extremal, with a view to
determining their stability.

Therefore to begin our discussion we will briefly review the 4-dimensional
Einstein-Maxwell system and its RN black hole solutions. We will also
introduce convenient coordinates for the later discussion. The theory contains
the metric $g_{\mu\nu}$ and gauge field $A_{\mu}$ and classical solutions
solve the Einstein-Maxwell equations,

$\displaystyle
R_{\mu\nu}=2F_{\mu\alpha}F_{\nu}^{~{}~{}\alpha}-\frac{1}{2}F^{2}g_{\mu\nu}\;,\quad\nabla^{\mu}F_{\mu\nu}=0$
(2.1)

which derive from the action,

$\displaystyle S=\frac{1}{16\pi G}\int d^{4}x\sqrt{-g}\left(R-F^{2}\right)\,.$
(2.2)

We note that we have chosen to use a gravitational normalization for the
Maxwell field. The static vacuum solutions to this are the Reissner-Nordström
black holes. The metric may be written as,

$\displaystyle ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2}\;,\quad
f(r)=\left(1-\frac{r_{+}}{r}\right)\left(1-\frac{r_{-}}{r}\right)$ (2.3)

with $d\Omega^{2}=d\theta^{2}+\sin^{2}{\theta}d\phi^{2}$, and the magnetically
and electrically charged solutions have gauge potentials,

$\displaystyle A_{mag}=-\sqrt{r_{+}r_{-}}\cos{\theta}d\phi\;,\quad
A_{elec}=\sqrt{r_{+}r_{-}}\left(\frac{1}{r_{+}}-\frac{1}{r}\right)dt$ (2.4)

respectively. 333 One may also source the RN solution with a mixed
electromagnetic solution of the form
$A=\alpha\left(\frac{1}{r_{+}}-\frac{1}{r}\right)dt+\beta\cos{\theta}d\phi$
for appropriate $\alpha$, $\beta$. Here, as stated earlier, we will only be
concerned with purely electric or magnetic solutions as generally mixed fields
will produce a Poynting energy flux whose backreaction is not compatible with
the static symmetry of the metric. In the case of these spherical solutions a
static solution with both electric and magnetic charge is allowed since the
spherical symmetry ensures the electric and magnetic fields are parallel,
being radially directed, and hence the Poynting vector vanishes.  The mass $M$
and magnetic or electric charge $Q$ (in appropriate units) of the solutions
are,

$\displaystyle 2GM=r_{+}+r_{-}\;,\quad Q=\sqrt{r_{+}r_{-}}$ (2.5)

so that $GM\geq|Q|$. We may also consider the negatively charged solutions by
reversing the signs of the gauge potentials above, but here just consider the
positively charged case for simplicity. Regarding $r_{+}$ as the outer horizon
position implies $r_{+}>r_{-}$.

The solutions obey a First Law, $dM=TdS+\mu dQ$, where the Hawking
temperature, $T$, Bekenstein-Hawking entropy $S$, and potential $\mu$
associated to the charge, $Q$, of the non-extremal solutions are given by,

$\displaystyle T=\frac{\kappa}{2\pi}=\frac{r_{+}-r_{-}}{4\pi
r_{+}^{2}}\;,\quad S=\frac{\pi
r_{+}^{2}}{G}\;,\quad\mu=\frac{1}{G}\sqrt{\frac{r_{-}}{r_{+}}}$ (2.6)

where $\kappa$ is the surface gravity. We note that for the electric RN, one
can see from writing $A_{elec}=-\psi\,dt$ that $G\mu$ corresponds to the
potential difference from the horizon to that at infinity, so
$\psi_{r_{+}}-\psi_{\infty}$, noting that the potential at the horizon
vanishes. The extremal limit corresponds to taking $r_{-}\to r_{+}$, so that
the surface gravity vanishes, and corresponds to taking the charge $|Q|\to
GM$. The specific heat capacity at constant charge, $c_{Q}$, and capacitance
at fixed temperature, $\epsilon_{T}$, are;

$\displaystyle c_{Q}=T\left.\frac{\partial S}{\partial
T}\right|_{Q}=-\frac{2\pi
r_{+}^{2}}{G}\frac{\left(r_{+}-r_{-}\right)}{\left(r_{+}-3r_{-}\right)}\;,\quad\epsilon_{T}=\left.\frac{\partial
Q}{\partial\mu}\right|_{T}=\frac{Gr_{+}(r_{+}-3r_{-})}{r_{+}-r_{-}}$ (2.7)

respectively. We note that $c_{Q}$ is negative for small charges, and becomes
positive for $r_{+}\geq r_{-}>\frac{1}{3}r_{+}$, so for charges
$|Q|>\frac{\sqrt{3}GM}{2}=Q_{crit}$, and that $\epsilon_{T}$ has precisely the
opposite sign to $c_{Q}$. If we work at fixed temperature $T$ and charge $Q$,
the relevant ensemble is the canonical ensemble, and as discussed in [21, 22]
then thermodynamic stability is simply governed by positivity of $c_{Q}$. Thus
at fixed temperature and charge, the canonical ensemble is unstable for
$|Q|<Q_{crit}$, but becomes stable for sufficient charges $|Q|>Q_{crit}$. If
we instead choose to work at fixed temperature $T$ and potential $\mu$, then
we must use the grand canonical ensemble, and the condition for stability,
namely that the corresponding Gibbs potential has a local minimum, is then
given by positivity of _both_ $c_{Q}$ as well as $\epsilon_{T}$ (see for
example [23]). Now we see from their explicit forms that $c_{Q}$ and
$\epsilon_{T}$ have opposite sign, and hence the grand canonical ensemble is
unstable for all values of temperature and potential.

The work of [10, 11] related the negative mode of the Euclidean continuation
of Schwarzschild to its negative specific heat capacity, and hence
thermodynamic instability. The magnetic RN solution can also be analytically
continued to a smooth Euclidean solution in the same manner. Monteiro and
Santos [21] have shown it also has a Euclidean negative mode for charges where
the specific heat at constant charge is negative, and hence the canonical
ensemble is unstable, and this mode disappears as the charge is increased to
$|Q|=Q_{crit}$ (see also [24]). However for electric RN, one fixes $A_{elec}$
at infinity, and therefore this is naturally related to the ensemble with
fixed electric potential. This is therefore associated to the unstable grand
canonical ensemble. However the static Lorentzian electric RN solution cannot
straightforwardly be analytically continued to a (real) Euclidean solution,
due to the electric gauge field whose potential would naively become
imaginary. Interestingly, however, we will find later that the stability of
the static EM flows about RN do reflect these behaviours in the sense that the
flow of the magnetically charged fixed point becomes stable for sufficient
charge, but that of the electric fixed point does not. This is particularly
interesting in light of the recent links between the stability of the
canonical ensemble for Schwarzschild in a box and its stability under Ricci
flow [12], and suggests a generalization which includes charge.

We will find our geometric flows preserve surface gravity of a horizon, and
hence we will be interested in the set of solutions at a fixed temperature in
order to understand possible end-points of the flow. Furthermore, we will see
that the magnetic flow preserves magnetic charge, and the electric flow
preserves the electric potential difference between the horizon and infinity.
From the expressions above it follows that for magnetic RN solutions, at a
given magnetic charge there is a maximum temperature,

$\displaystyle T_{max}=\frac{1}{6\sqrt{3}\pi|Q|}\;.$ (2.8)

This arises since for fixed charge there is a minimum mass, given by the BPS
bound $M\geq|Q|$, and a unique solution exists for all greater masses. Then
for $M\to|Q|$ the temperature vanishes, but for very large $M\gg|Q|$ the
temperature also becomes small, so by continuity in between there is a
maximum. Assuming one takes a temperature $T<T_{max}$ there are then two
magnetic RN solutions. In analogy with Hawking-Page we may think of these as a
small and large black hole. Now the smaller one, with mass
$M<\frac{2|Q|}{\sqrt{3}G}$, is thermodynamically stable (with no Euclidean
negative mode), and the larger one has a greater horizon size and mass
$M>\frac{2|Q|}{\sqrt{3}G}$, and is thermodynamically unstable (with a
Euclidean negative mode). Given a charge, then increasing the temperature to
its critical value, $T_{max}$, these two solutions merge to a single
marginally stable solution. For greater temperatures there are no RN black
hole spacetimes. For electric RN solutions, then fixing the electric potential
the situation is different. For a given potential there exists a single black
hole for every temperature, and these solutions are always thermodynamically
unstable.

When considering a RN fixed point of the flow, we will find it convenient to
choose units so that $r_{+}=1$, so that the outer horizon is a unit sphere,
and the exterior spacetime is $r>1$. In this case, a useful radial coordinate
we will use later is $\rho$, related to $r$ by,

$\displaystyle r=\frac{1}{1-\rho^{2}}$ (2.9)

which compactifes the infinite range $r\in[1,\infty)$ to the interval
$\rho\in[0,1)$. Then the RN metric (in these units, so $r_{+}=1$) becomes,

$\displaystyle
ds^{2}=-\rho^{2}F(\rho)dt^{2}+4r(\rho)^{4}\frac{d\rho^{2}}{F(\rho)}+r(\rho)^{2}d\Omega^{2}\;,\quad
F(\rho)=1-r_{-}+r_{-}\rho^{2}$ (2.10)

with the magnetic Maxwell potential as above, and the electric one being,

$\displaystyle A_{elec}=\sqrt{r_{-}}\rho^{2}dt\,.$ (2.11)

Taking $r_{-}\to 0$ yields the uncharged Schwarzschild solution, whereas
taking $r_{-}\to 1$ gives the extremal RN black holes. In this extremal limit
we introduce a new coordinate,

$\displaystyle\rho^{\prime}=\rho^{2}$ (2.12)

so that again $\rho^{\prime}\in[0,1)$, and then the extremal metric is,

$\displaystyle ds^{2}=-\rho^{\prime
2}dt^{2}+r(\rho^{\prime})^{4}\frac{d\rho^{\prime 2}}{\rho^{\prime
2}}+r(\rho^{\prime})^{2}d\Omega^{2}\;,\quad r=\frac{1}{1-\rho^{\prime}}$
(2.13)

now with either an electric or magnetic potential,

$\displaystyle A_{elec}=\rho^{\prime}dt\;,\quad A_{mag}=-\cos{\theta}d\phi$
(2.14)

for positively charged solutions. Taking $\rho^{\prime}\to 0$, so that $r\to
1$, we recognise the near horizon geometry as $AdS_{2}\times S^{2}$.

##  3 Gradient flow of the Einstein-Maxwell action

Given an action $S$ which is a functional of some fields $\Psi^{A}$, then the
variation $\delta S/\delta\Psi^{A}$ is a covector on the space of fields, the
‘superspace’. As discussed in [3], then given a metric on superspace,
$G_{AB}$, which is a functional of the fields $\Psi^{A}$, this may be
converted to a vector, and a gradient flow is defined as,

$\displaystyle\frac{d}{d\lambda}\Psi^{A}=G^{AB}\frac{\delta
S}{\delta\Psi^{B}}\;.$ (3.1)

The simplest such local superspace metrics will only depend on the fields and
not their derivatives. If $G_{AB}$ is a positive definite metric, then this is
a gradient descent of the action, which monotonically decreases along the
flow. Different choices of superspace metric will yield different flow
equations.

Let us consider the Einstein-Hilbert action in this context,

$\displaystyle S_{EH}=\frac{1}{16\pi G}\int d^{4}x\sqrt{-g}R$ (3.2)

We may associate $\Psi^{A}$ above as the metric $g_{\mu\nu}(x)$, and the
abstract index $A$ as enumerating both the metric components and coordinate
position. A natural metric on superspace is given by the DeWittk metric (using
the notation of [12]),

$\displaystyle(h_{\mu\nu},\tilde{h}_{\alpha\beta})=\frac{1}{32\pi G}\int
d^{4}x\sqrt{-g}h_{\mu\nu}G^{\mu\nu\alpha\beta}\tilde{h}_{\alpha\beta}$ (3.3)

where $h_{\mu\nu}$ and $\tilde{h}_{\alpha\beta}$ are two perturbations of the
metric $g_{\mu\nu}$, and,

$\displaystyle
G^{\mu\nu\alpha\beta}=\frac{1}{2}\left(g^{\mu\alpha}g^{\nu\beta}+g^{\mu\beta}g^{\nu\alpha}+kg^{\mu\nu}g^{\alpha\beta}\right)$
(3.4)

for real $k$. Then rearranging the gradient flow equation (3.1) as,
$G_{AB}(d\Psi^{A}/d\lambda)\delta\Psi^{B}=\delta S$, with $\delta S$ given by
the variation $\delta\Psi^{B}$, explicitly gives,

$\displaystyle\left(\frac{d}{d\lambda}g_{\mu\nu},\delta
g_{\alpha\beta}\right)=\delta S_{EH}=-\frac{1}{16\pi G}\int
d^{4}x\sqrt{-g}\delta
g_{\alpha\beta}\left(R^{\alpha\beta}-\frac{1}{2}g^{\alpha\beta}R\right)\;.$
(3.5)

We will focus on the case $k=-1$, and then this gives the flow equation as,

$\displaystyle\frac{d}{d\lambda}g_{\mu\nu}(x)=-2R_{\mu\nu}$ (3.6)

which is simply the Ricci flow. An important point is that since the DeWitt-1
metric is not positive definite, the Einstein-Hilbert action does not
generally vary monotonically along Ricci flows. A monotone functional is
famously given by the Perelmann entropy [25].

Fixed points of Ricci flow are ones so that $\dot{g}_{\mu\nu}=0$ where
$\dot{}=d/d\lambda$, and are thus Ricci flat metrics. However one may also
consider solutions to the Ricci soliton equation,

$\displaystyle R_{\mu\nu}=\nabla_{(\mu}\xi_{\nu)}$ (3.7)

to be geometric fixed points of the flow, since for these we have
$\dot{g}_{\mu\nu}=-\mathrm{Lie}_{\xi}g_{\mu\nu}$. For non-vanishing $\xi$ the
metric is clearly varying, but the geometry it represents is not. Thus a Ricci
soliton represents a geometric fixed point of the flow, so the geometry
doesn’t flow, but the coordinates it is presented in do.

Ricci flow is itself diffeomorphism invariant. By performing a flow time
dependent diffeomorphism,

$\displaystyle x^{\mu}\to
x^{\mu}+v^{\mu}(\lambda,x)\;,\quad\frac{d}{d\lambda}v^{\mu}=\xi^{\mu}$ (3.8)

then we generate a new flow of the metric,

$\displaystyle\frac{d}{d\lambda}g_{\mu\nu}=-2R_{\mu\nu}+\mathrm{Lie}_{\xi}g_{\mu\nu}$
(3.9)

where we may write $\mathrm{Lie}_{\xi}g_{\mu\nu}=2\nabla_{(\mu}\xi_{\nu)}$.
The flows $g_{\mu\nu}(\lambda)$ for the Ricci flow and the above flow are the
same geometrically, but the two metrics will differ by a coordinate
transformation at any given $\lambda$, this transformation depending on the
flow time.

Due to its diffeomorphism invariance, Ricci flow is not well-posed as a p.d.e.
for initial data at some starting flow time $\lambda$. However, following
DeTurck [26], we may make the choice that,

$\displaystyle\xi^{\mu}=g^{\alpha\beta}\left(\Gamma^{\mu}_{~{}~{}\alpha\beta}-\bar{\Gamma}^{\mu}_{~{}~{}\alpha\beta}\right)$
(3.10)

given a smooth fixed (i.e. flow time independent) reference connection
$\bar{\Gamma}^{\mu}_{~{}~{}\alpha\beta}$ on the manifold. Then the principle
part of the flow is,

$\displaystyle\frac{d}{d\lambda}g_{\mu\nu}=_{PP}g^{\alpha\beta}\partial_{\alpha}\partial_{\beta}g_{\mu\nu}$
(3.11)

and for Riemannian metrics this is parabolic. For Lorentzian signature metrics
this is generally not well-posed, as timelike perturbations ‘anti-diffuse’.
However following [16, 17] we may restrict to the space of static or
stationary metrics, and then the flow is indeed parabolic, and thus well-
posed.

Let us take the reference connection $\bar{\Gamma}^{\mu}_{~{}~{}\alpha\beta}$
to be the Levi-Civita connection of a static reference metric,
$\bar{g}_{\mu\nu}$. Then the flow obviously preserves staticity of the metric
– starting with a static metric, it will remain static. If the metric is
static we may generally write it locally as,

$\displaystyle g_{\mu\nu}(x)=\left(\begin{array}[]{cc}-N(x^{k})&0\\\
0&h_{ij}(x^{k})\end{array}\right)$ (3.14)

with $N\geq 0$ and $\det(h_{ij})>0$, using coordinates $x^{\mu}=(t,x^{k})$
adapted to the static symmetry. Note that in these static coordinates $N$ may
vanish at black hole horizons, but away from these it should be positive. We
may always choose coordinates locally so that the spatial metric $h_{ij}$ is a
smooth Riemannian metric. Then since $g_{\mu\nu}$ has no explicit time
dependence, the principle part becomes,

$\displaystyle\frac{d}{d\lambda}g_{\mu\nu}=_{PP}h^{ij}\partial_{i}\partial_{j}g_{\mu\nu}$
(3.15)

which is indeed a parabolic diffusion-like flow for the metric components on
the curved space $h_{ij}$.

We now consider gradient flow of the Einstein-Maxwell action,

$\displaystyle S=\frac{1}{16\pi G}\int d^{4}x\sqrt{-g}\left(R-F^{2}\right)$
(3.16)

with $F_{\mu\nu}=2\partial_{[\mu}A_{\nu]}$. Now schematically our field space
$\Psi^{A}$ is composed of the metric $g_{\mu\nu}(x)$ together with the gauge
field $A_{\mu}(x)$. A natural superspace metric is then given by

$\displaystyle(h_{\mu\nu},\tilde{h}_{\alpha\beta})=\frac{1}{32\pi G}\int
d^{4}x\sqrt{-g}h_{\mu\nu}G_{(g)}^{\mu\nu\alpha\beta}\tilde{h}_{\alpha\beta}\;,\quad(u_{\mu},\tilde{u}_{\alpha})=\frac{1}{16\pi
G}\int d^{4}x\sqrt{-g}u_{\mu}G_{(A)}^{\mu\alpha}\tilde{u}_{\alpha}$

where, as above, $h_{\mu\nu}$, $\tilde{h}_{\alpha\beta}$ are metric
perturbations, and $u_{\mu}$, $\tilde{u}_{\alpha}$ are perturbations to the
gauge field $A_{\mu}$, and we take,

$\displaystyle G_{(A)}^{\mu\alpha}=\frac{4}{\tau}g^{\mu\alpha}\;,\quad
G_{(g)}^{\mu\nu\alpha\beta}=\frac{1}{2}\left(g^{\mu\alpha}g^{\nu\beta}+g^{\mu\beta}g^{\nu\alpha}-g^{\mu\nu}g^{\alpha\beta}\right)$
(3.18)

for a constant $\tau$, and further we take $(h_{\mu\nu},u_{\alpha})=0$ so that
the metric and gauge perturbations are orthogonal. We may regard the parameter
$\tau$ as determining the speed of the flow of the gauge field relative to
that of the metric, and later we will focus on the case $\tau=1$. We see that
this superspace metric is constructed only from the metric (so doesn’t
involved the gauge field) and that its restriction to the metric is the
DeWitt-1 superspace metric. Then we obtain the gradient flow equation,

$\displaystyle\frac{d}{d\lambda}g_{\mu\nu}=-2R_{\mu\nu}+4F_{\mu\alpha}F_{\nu}^{~{}\alpha}-F^{2}g_{\mu\nu}\;,\quad\frac{d}{d\lambda}A_{\mu}=\tau\nabla^{\alpha}F_{\alpha\mu}\;.$
(3.19)

We will call this the Einstein-Maxwell (or EM) flow, and its fixed points,
$\dot{g}_{\mu\nu}=0=\dot{A}_{\mu}$, are solutions of the Einstein-Maxwell
equations. This flow is both diffeomorphism and gauge invariant. Thus we may
consider the analog of Ricci solitons, so field configurations where the
geometry and gauge field flow only up to diffeomorphisms and gauge
transformations,

$\displaystyle\dot{g}_{\mu\nu}=-\mathrm{Lie}_{\xi}g_{\mu\nu}\;,\quad\dot{A}_{\mu}=-\mathrm{Lie}_{\xi}A_{\mu}-\partial_{\mu}\Lambda$
(3.20)

with $\xi$ generating the flow dependent diffeomorphism, and $\Lambda$ the
gauge transformation of the Maxwell field along the flow. We may explicitly
modify the flow to add such a flow time dependent diffeomorphism $\xi$ and
gauge transformation $\Lambda$, as,

$\displaystyle\frac{d}{d\lambda}g_{\mu\nu}=\mathrm{Lie}_{\xi}g_{\mu\nu}-2R_{\mu\nu}+4F_{\mu\alpha}F_{\nu}^{~{}\alpha}-F^{2}g_{\mu\nu}\;,\;\frac{d}{d\lambda}A_{\mu}=\mathrm{Lie}_{\xi}A_{\mu}+\partial_{\mu}\Lambda+\tau\nabla^{\alpha}F_{\alpha\mu}$
(3.21)

to give the same flow, up to diffeomorphisms and gauge transformations. The
analog of Ricci solitons can then be thought of as the fixed points of this
flow.

As for Ricci flow, the Einstein-Maxwell flow is not parabolic, even when
restricted to static field configurations. In a linearization of the flow, the
two derivative terms vanish on linear perturbations that are diffeomorphisms,
or gauge transformations of the gauge field. In order to obtain a well-posed
flow, we take the modified flow above in equation (3.21) and choose $\xi$ as
the DeTurck vector in equation (3.10), and further take,

$\displaystyle\Lambda=\tau\nabla^{\alpha}A_{\alpha}\,.$ (3.22)

Then choosing the reference metric to be static, the flow truncates to the
space of static metrics and gauge fields. The principle part of the flow on
static metrics and gauge fields then becomes,

$\displaystyle\frac{d}{d\lambda}g_{\mu\nu}=_{PP}h^{ij}\partial_{i}\partial_{j}g_{\mu\nu}\;,\quad\frac{d}{d\lambda}A_{\mu}=_{PP}\tau
h^{ij}\partial_{i}\partial_{j}A_{\mu}$ (3.23)

and since $h_{ij}$ is a smooth Riemannian metric, the character of the flow is
governed by the components of $g_{\mu\nu}$ and $A_{\mu}$ diffusing on the
spatial geometry $h_{ij}$, and hence is parabolic assuming that we choose
$\tau>0$.

We have written this static flow for a general gauge potential $A_{\mu}$
compatible with the static symmetry. However, as emphasized earlier, in
general a static potential will have a stress tensor that is only stationary,
but not static due to a Poynting energy flux being generated when both
(static) electric and magnetic fields are present. Thus here we take the
Maxwell potential to be either purely electric, $A=-\psi dt$ (where the gauge
function $\Lambda=\tau\nabla^{\alpha}A_{\alpha}$ vanishes), or purely magnetic
so $A=A_{i}dx^{i}$. These electric or magnetic forms are consistently
preserved by the flow for the DeTurck choice of $\xi$ and the gauge choice in
equation (3.22).

From this point on, we will term the parabolic Einstein-Maxwell flow in (3.21)
of a static spacetime, with either magnetic or electric Maxwell field, and
DeTurck vector $\xi_{\mu}$ and gauge choice $\Lambda$ as in equation (3.22),
as the “magnetostatic or electrostatic Einstein-Maxwell (EM) flow”.

##  4 Static EM flows of black hole spacetimes

We now consider the EM flow on static black hole spacetimes. While the flow is
parabolic away from horizons, it isn’t clear that it will preserve the smooth
structure of a horizon, and this is what we aim to demonstrate. Further in the
extremal case we will see the nice property that the flow of the near horizon
geometry decouples from the flow of the exterior of the horizon as one might
expect.

###  4.1 Static non-extremal black hole spacetimes

The static EM flow of static non-extremal black holes is very simple to
consider, being similar to the discussion for Ricci flow. In fact for Ricci
flow one may simply perform a Euclidean continuation of a static metric,
identifying the period of the Euclidean time coordinate so that the Euclidean
manifold becomes smooth at the horizon. Since Ricci flow of this Euclidean
geometry preserves the $U(1)$ isometry associated to the static symmetry, then
it is obvious that the smoothness of the horizon is preserved.

However in the case of the EM flow a gauge field with electric potential
component cannot generally be analytically continued to Euclidean time. Hence
we will use a different argument, that of [16, 17], which exploits the
similarity between a static Killing horizon and the origin of polar
coordinates. Following [16, 17] the most general smooth static symmetric
metric with a Killing horizon with surface gravity $\kappa$ (with respect to
the Killing vector $\partial/\partial t$), can be written in coordinates
$x^{\mu}=(t,r,x^{a})$ adapted to the static symmetry and horizon as,

$\displaystyle ds^{2}=-r^{2}Vdt^{2}+U(dr+rU_{a}dx^{a})^{2}+h_{ab}dx^{a}dx^{b}$
(4.1)

where $r=0$ is the horizon, and $V,U,U_{a},h_{ab}$ are all smooth functions of
$r^{2}$ and of $x^{a}$, with $V,U>0$, and $h_{ab}$ is a Riemannian metric, and
at the horizon there is the additional condition that,

$\displaystyle\left.\left(V-\kappa^{2}U\right)\right|_{r=0}=0\,.$ (4.2)

Then making the coordinate transformation $\alpha=r\sinh\kappa t$,
$\beta=r\cosh\kappa t$ (analgous to that going from polar to Cartesians
coordinates) one finds a metric tensor with smooth components and non-
vanishing determinant.

The most general symmetric two tensor that is also smooth on the Killing
horizon takes a similar form, although with different component functions $V$,
$U$, $U_{a}$ and $h_{ab}$ to the metric. These must obey the same smoothness
conditions, so be smooth functions of $r^{2}$ and of $x^{a}$, and also satisfy
(4.2), but need not have the positivity of $V$, and $h_{ab}$ need not be a
Riemannian metric.

Now consider a covector $\omega$. The most general form compatible with the
symmetry $\mathrm{Lie}_{\partial/\partial t}\omega=0$, and with smoothness at
the Killing horizon can be written as,

$\displaystyle\omega=r^{2}\Phi\,dt+rWdr+\omega_{a}dx^{a}$ (4.3)

where now $\Phi$, $W$ and the components $\omega_{a}$ are again smooth
functions of $r^{2}$ and of $x^{a}$. We may see this by writing a general
smooth vector in the coordinates $\alpha,\beta$ as,

$\displaystyle\omega=\omega_{\alpha}d\alpha+\omega_{\beta}d\beta+\omega_{a}dx^{a}$
(4.4)

where $\omega_{\alpha}$, $\omega_{\beta}$, $\omega_{a}$ are smooth functions
of $\alpha$, $\beta$ and $x^{a}$. Then transforming to $t$ and $r$ coordinates
we obtain,

$\displaystyle\omega=\kappa r\left(\omega_{\alpha}\cosh{\kappa
t}+\omega_{\beta}\sinh\kappa t\right)dt+\left(\omega_{\alpha}\sinh\kappa
t+\omega_{\beta}\cosh\kappa t\right)dr+\omega_{a}dx^{a}\;.$ (4.5)

In order to have the static symmetry we see that these components should have
no explicit $t$ dependence. Noting that a smooth function $f$ of $\alpha$ and
$\beta$ such that $\partial_{t}f=0$ must be a function of
$\beta^{2}-\alpha^{2}=r^{2}$, then we see the behaviour of $\omega_{\alpha}$
and $\omega_{\beta}$ is constrained to go as,

$\displaystyle\omega_{\alpha}=-\alpha
W(r^{2},x^{a})+\frac{1}{\kappa}\beta\Phi(r^{2},x^{a})\;,\quad\omega_{\beta}=-\frac{1}{\kappa}\alpha\Phi(r^{2},x^{a})+\beta
W(r^{2},x^{a})$ (4.6)

for $\Phi(r^{2},x^{a})$, $W(r^{2},x^{a})$ smooth functions of $r^{2}$ and
$x^{a}$, which yields the form in equation (4.3) above.

Then a gauge field, $A$, which is purely electric, compatible with the static
symmetry and smooth on the Killing horizon will take the form,

$\displaystyle A=r^{2}\Phi\,dt$ (4.7)

(after a gauge transformation to remove any spatial components). In the purely
magnetic case, it will take the form,

$\displaystyle A=rWdr+\omega_{a}dx^{a}\;.$ (4.8)

We take the reference metric to be some smooth spacetime with a coinciding
Killing horizon at $r=0$ with the same surface gravity, so,

$\displaystyle\bar{g}_{\mu\nu}dx^{\mu}dx^{\nu}=-r^{2}\bar{V}dt^{2}+\bar{U}(dr+r\bar{U}_{a}dx^{a})^{2}+\bar{h}_{ab}dx^{a}dx^{b}$
(4.9)

so that $\bar{V}$, $\bar{U}$, $\bar{U}_{a}$ and $\bar{h}_{ab}$ obey the same
conditions as the corresponding component functions of the metric. Being
static, then staticity with respect to $\partial/\partial t$ is preserved by
the flow. Likewise the electric or magnetic form of the gauge potential is
also preserved.444We note that there is no mixing between these electric and
magnetic components since they transform differently under the static $t\to-t$
symmetry.

Translating to $\alpha$ and $\beta$ coordinates, all the tensors are smooth in
the coordinates $\alpha$, $\beta$, $x^{a}$, and compatible with the symmetry
$\partial/\partial t$. Now consider the flow equations. The tensors derived
from the metric, reference metric and gauge field, the curvature and field
strengths and the various Lie derivative and gauge transform terms, will also
be smooth and respect the symmetry. Combining these into the ‘righthand’ sides
of the flow equations will again yield a smooth symmetric two tensor and a
covector. Since these respect the static symmetry, and are smooth in $\alpha$,
$\beta$, then returning to $t,r,x^{a}$ coordinates they will take the smooth
forms discussed above, and thus the smooth form of the metric and gauge field
will be preserved by the flow, and in particular its constant surface gravity
$\kappa$ with respect to $\partial/\partial t$ will be preserved by the flow.

###  4.2 Conserved charges under EM flow

Since black holes can carry charges, we may wonder whether any charges or
potentials are conserved by the static EM flow. Let us consider firstly the
electric flow in asymptotically flat spacetimes. We then take the asymptotic
boundary condition that the metric (and reference metric) tend to Minkowski
and the non-vanishing component of the vector field, $A^{t}$, tends to a
constant, $\Phi_{\infty}$. Thus we have,

$\displaystyle ds^{2}\to-
dt^{2}+dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}{\theta}d\phi^{2}\right)\;,\quad
A^{t}\to\Phi_{\infty}$ (4.10)

as $r\to\infty$. We may interpret $\Phi_{\infty}$ as the electric potential
difference of the spacetime, since we have seen that $A^{t}\to 0$ at black
hole horizons. Thus we see that for black hole spacetimes the electric static
EM flow preserves the electric potential difference, $\mu$, since it is fixed
by the asymptotic boundary conditions.

One may ask, does it also preserve the charge? Working to higher order in
powers of $1/r$, given a potential of the form,

$\displaystyle
A^{t}=\Phi_{\infty}+\frac{1}{r}f(\theta,\phi)+O\bigl{(}\frac{1}{r^{2}}\bigr{)}$
(4.11)

then the electric charge is,

$\displaystyle Q=\frac{1}{4\pi}\int_{S^{2}_{\infty}}\star F=\frac{1}{4\pi}\int
d\theta d\phi\sin\theta f(\theta,\phi)$ (4.12)

where the integral is taken over the 2-sphere at infinite radius. Naively it
appears that this electric charge is also fixed in flow time, since one may
simply show that,

$\displaystyle\dot{A}^{t}=O\left(\frac{1}{r^{2}}\right)$ (4.13)

and thus the leading $1/r$ term in $A_{t}$ is not affected by the flow. This
is similar to the observation in [27] that the ADM mass of a static spacetime
is invariant under Ricci flow, since again the leading asymptotic fall-off is
not corrected.

While technically correct, we argue that this is not physically the correct
picture. A counter example in our context that we discussed earlier is that we
may Ricci flow from the Schwarzschild black hole to flat spacetime (via a
topology change), so clearly we should physically expect that the mass
changes.555One might be tempted to say that this is associated to the surgery
required to change topology, but this is not the case as the surgery is local
to the region in the interior where the singularity develops as the horizon
shrinks, and does not affect the asymptotics at all. The resolution is that
there are different ways to compute charges or masses during a flow. Charges
are defined far away from the system of interest, and thus there are two
limits we are concerned with, large radius and large flow times. Depending on
the ordering of these limits we may obtain a definition of charge that doesn’t
vary with flow time, or one that does. If we define charge at fixed finite
flow time and infinite radius, we don’t allow the charge to vary in a finite
flow time as information from the interior of the system cannot propagate out
to infinity. On the other hand, if we define charges on a very large but
formally finite sphere, these charges may vary at sufficiently late times in
the flow. This is analogous to the discussion in [27] where although ADM mass
is fixed during Ricci flow, quasi-local masses do evolve.

This may be precisely illustrated with a simple example given by the following
exact solution to diffusion in flat Euclidean 3d space,

$\displaystyle
F(\lambda,\vec{x})=\frac{1}{|\vec{x}|}\int_{0}^{\infty}d\omega\tilde{F}(\omega)e^{-\omega\lambda}\sin\left(|\vec{x}|\sqrt{\omega}\right)$
(4.14)

where $\vec{x}$ are the usual Euclidean coordinates and $\lambda$ is diffusion
flow time, and $\tilde{F}(\omega)$ is the integral transform of the initial
data at $\lambda=0$. We consider an initial profile given by,

$\displaystyle\tilde{F}(\omega)=\left\\{\begin{array}[]{cc}\frac{1}{\pi\omega}&\omega<1/L\\\
0&\omega\geq 1/L\end{array}\right.$ (4.17)

and then the $p$-th flow time derivative at $\lambda=0$ is,

$\displaystyle\left.\frac{d^{p}}{d\lambda^{p}}F(\lambda,\vec{x})\right|_{\lambda=0}=\left\\{\begin{array}[]{cc}\frac{1}{r}+O\left(\frac{1}{r^{2}}\right)&p=0\\\
-\frac{1}{r^{2}}\frac{2(-1)^{p}\cos\left(r/\sqrt{L}\right)}{\pi\sqrt{L}^{2p-1}}+O\left(\frac{1}{r^{3}}\right)&p>0\;.\end{array}\right.$
(4.20)

Since $F(0,\vec{x})\sim 1/r$, but
$\left.\frac{d^{p}}{d\lambda^{p}}F(\lambda,\vec{x})\right|_{\lambda=0}\sim
1/r^{2}$ for any $p>0$, we see that the asymptotic $\sim 1/r$ behaviour
persists unchanged at finite flow time, so that $F(\lambda,\vec{x})\sim 1/r$.
However it is quite evident that pointwise, so at any fixed position
$\vec{x}$, we have $F(\lambda,\vec{x})\to 0$ as $\lambda\to\infty$. In
diffusion we know that it will take a diffusion time $\lambda\sim r^{2}$ to
effect a change on the scale $r$. Thus while formally at finite flow time the
$1/r$ coefficient will remain unchanged, if we actually look at some very
large, but finite fixed radius, and ask how the diffusion field looks at late
flow times, it will tend to zero, as will all its spatial derivatives.
Formally for any point $\vec{x}$,

$\displaystyle\lim_{\lambda\to\infty}\partial_{i_{1}}\partial_{i_{2}}\ldots\partial_{i_{n}}F(\lambda,\vec{x})=0$
(4.21)

for any number of derivatives $n\geq 0$.

Let us return to the context of the electric static EM flow. The charge is
fixed for finite flow times if we compute it from the formula above (ie. from
the $1/r$ fall-off). However, if instead we compute it as the charge inside
some very large radius $R$,

$\displaystyle Q=\frac{1}{4\pi}\int_{S^{2}_{r=R}}\star F$ (4.22)

this will closely approximate the charge computed with the original definition
for early times, $\lambda\ll R^{2}$, but then will change and deviate from
this definition if we wait sufficiently long flow times, $\lambda\sim R^{2}$.
With this more physical definition of charge, we conclude that for the
electric static EM flow the electric potential is fixed as a boundary
condition, and the charge will vary with flow time.

The converse holds for the magnetic static EM flow. Then asymptotically we
have the boundary condition that,

$\displaystyle A_{a}\to A_{a}(\theta,\phi)$ (4.23)

for large $r$, where $x^{a}=\\{\theta,\phi\\}$ and $A_{a}(\theta,\phi)$ is a
fixed covector field on the 2-sphere at infinity. We may regard this as a
‘Dirichlet’ boundary condition for the magnetic potential as $r\to\infty$. The
magnetic charge is now computed as,

$\displaystyle Q_{mag}=\frac{1}{4\pi}\int_{S^{2}_{\infty}}F=\frac{1}{4\pi}\int
d\theta d\phi\left(\partial_{\theta}A_{\phi}-\partial_{\phi}A_{\theta}\right)$
(4.24)

and thus is fixed by the asymptotic data $A_{a}$. As it is the leading
behaviour of $A_{a}$ that determines this magnetic charge in the large $r$
limit, we emphasize that this remains fixed independent of whether we evaluate
the charge integral strictly in the infinite radius limit, or instead at some
very large but finite radius $R$.

Thus in summary, the electrostatic EM flow preserves the electric potential as
a boundary condition (but not electric charge), and the magnetostatic EM flow
preserves the magnetic charge as a boundary condition. Both flows preserve the
surface gravity. We will return to this point later when drawing an analogy
between the behaviour of these flows starting with an RN black hole, and the
thermodynamic stability of the RN solution.

###  4.3 Static extremal black hole spacetimes

Already for Ricci flow it is far less obvious that the flow is compatible with
preserving a static smooth extremal horizon.

We begin our discussion considering the flow of the metric. Then we will
consider adding the gauge field, which is either purely electric or purely
magnetic. Following [19] a general static extremal horizon can be written as,

$\displaystyle
ds^{2}=-r^{2}e^{T}dt^{2}+e^{R}\left(\frac{dr}{r}+r\omega_{a}dx^{a}\right)^{2}+\gamma_{ab}dx^{a}dx^{b}$
(4.25)

where $r=0$ is the horizon and crucially $T$, $R$, $\omega_{a}$ and
$\gamma_{ab}$ are smooth functions of $r$ and $x^{a}$ there, and further $T$
and $R$ obey, ì

$\displaystyle(T-R)|_{r=0}=0\;,\quad\partial_{r}(T-R)|_{r=0}=\psi$ (4.26)

where $\psi$ is a constant. The near horizon geometry is then given by the
limit as $r\to 0$,

$\displaystyle
ds^{2}_{NH}=-e^{T}\left(r^{2}dt^{2}+\frac{dr^{2}}{r^{2}}\right)+\gamma_{ab}dx^{a}dx^{b}$
(4.27)

which is a (warped) product of AdS2 with the horizon 2-geometry $\gamma_{ab}$.

Suppose we flow the metric by a symmetric tensor $A_{\mu\nu}$ so that,

$\displaystyle\frac{d}{d\lambda}g_{\mu\nu}=A_{\mu\nu}\;.$ (4.28)

Then we should have that the form above is preserved. In order for this to
hold the tensor $A_{\mu\nu}$ must have an analogous form to that of the
metric,

$\displaystyle A_{\mu\nu}=\left(\begin{array}[]{ccc}-r^{2}\hat{A}_{tt}&0&0\\\
&\frac{1}{r^{2}}\hat{A}_{rr}&\hat{A}_{a}\\\
&&\hat{A}_{ab}+r^{2}\hat{A}_{a}\hat{A}_{b}\end{array}\right)$ (4.32)

so that the component functions
$\hat{A}_{tt},\hat{A}_{rr},\hat{A}_{a},\hat{A}_{ab}$ are smooth in $r$ and
$x^{a}$. Then to preserve the metric smoothness conditions (4.26) we further
require,

$\displaystyle\left.\left(\hat{A}_{tt}-\hat{A}_{rr}\right)\right|_{r=0}=0\;,\quad\left.\partial_{r}(\hat{A}_{tt}-\hat{A}_{rr})\right|_{r=0}=\psi\left.\hat{A}_{tt}\right|_{r=0}\;.$
(4.33)

This tensor is then also smooth on the Lorentzian spacetime. Linearly
combining such tensors satisfying (4.33) with the same value of $\psi$ again
yields smooth symmetric tensors (also with this value of $\psi$), and
contracting their indices with the metric gives a smooth function of $r$ and
$x^{a}$. The near horizon limit of this smooth tensor exists and is given as,

$\displaystyle
A^{NH}_{\mu\nu}=\left(\begin{array}[]{ccc}-r^{2}A^{NH}(x^{a})&0&0\\\
&\frac{1}{r^{2}}A^{NH}(x^{a})&0\\\ &&A^{NH}_{ab}(x^{a})\end{array}\right)$
(4.37)

where the components are defined from those of the original tensor at $r=0$
as,

$\displaystyle A^{NH}(x^{a})=\left.\hat{A}_{tt}(r,x^{a})\right|_{r=0}\;,\quad
A^{NH}_{ab}(x^{a})=\left.\hat{A}_{ab}(r,x^{a})\right|_{r=0}$ (4.38)

and may be thought of as a scalar and symmetric 2-tensor respectively, defined
on the 2-dimensional near horizon spatial geometry $\gamma_{ab}$.

The static EM flow updates both the metric and gauge field by the sum of
several terms, and we now consider these individually, showing that smoothness
is preserved.

####  4.3.1 Ricci tensor term

We now show that the Ricci tensor of this geometry is smooth. This implies
that flowing the metric by the Ricci tensor preserves smoothness. This
analysis is simplified by noting that we may use residual coordinate freedom
to choose $\omega_{a}$ to vanish in the metric (4.25). Explicitly we may write
(4.25) as,

$\displaystyle
ds^{2}=-r^{2}e^{T}dt^{2}+e^{R}\left(1-r^{2}w_{a}w^{a}e^{R}\right)\frac{dr^{2}}{r^{2}}+{\gamma}^{\prime}_{ab}\left(dx^{a}+e^{R}w^{a}dr\right)\left(dx^{b}+e^{R}w^{b}dr\right)$
(4.39)

where ${\gamma}^{\prime}_{ab}=\gamma_{ab}+e^{R}r^{2}w_{a}w_{b}$ and $w^{a}$ is
defined by $w_{a}=\gamma^{\prime}_{ab}w^{b}$. We now take a new coordinate
$y^{a}(r,x)$ which obeys the linear differential equation,

$\displaystyle\partial_{r}y^{a}=e^{R}w^{b}\partial_{b}y^{a}\;.$ (4.40)

This may be solved by taking boundary conditions $y^{a}=x^{a}$ at $r=0$ and
then integrating in $r$ starting from the surface $r=0$. Since the components
of $e^{R}w^{a}$ are smooth in $r$ and $x^{a}$ then this will have smooth
solutions and the inverse transformation $y^{a}\to x^{a}$ will also exist (at
least locally near $r=0$). 666We are grateful to James Lucietti for discussion
on this point. These coordinates then yield a metric of our desired form,

$\displaystyle
ds^{2}=-r^{2}e^{T}dt^{2}+e^{R^{\prime}}\frac{dr^{2}}{r^{2}}+\gamma^{\prime\prime}_{ab}dy^{a}dy^{b}\;,\quad\gamma^{\prime\prime}_{ab}=\gamma^{\prime}_{cd}\left.\frac{\partial
x^{c}}{\partial y_{a}}\right|_{r}\left.\frac{\partial x^{d}}{\partial
y_{b}}\right|_{r}\;$ (4.41)

where $e^{R^{\prime}}=e^{R}\left(1-r^{2}w_{a}w^{a}e^{R}\right)$. An important
fact we will use later is that since $y^{a}\simeq x^{a}$ near $r=0$, the near
horizon metric is unchanged by this transformation. Furthermore the near
horizon form of any other smooth tensor is also unchanged.

Now proceeding we will assume we have taken the above coordinates so that our
extremal metric can be written as,

$\displaystyle
ds^{2}=-r^{2}e^{T}dt^{2}+e^{R}\frac{dr^{2}}{r^{2}}+\gamma_{ab}dx^{a}dx^{b}$
(4.42)

without the off-diagonal $w_{a}$ terms. Then by direct calculation one can
obtain the Ricci tensor,

$\displaystyle R_{\mu\nu}=\left(\begin{array}[]{ccc}-r^{2}\hat{R}_{tt}&0&0\\\
&\frac{1}{r^{2}}\hat{R}_{rr}&\hat{R}_{a}\\\
&&\hat{R}_{ab}+r^{2}\hat{R}_{a}\hat{R}_{b}\end{array}\right)\;.$ (4.46)

We define $K_{ab}=\partial_{r}\gamma_{ab}$ and $K=\gamma^{ab}K_{ab}$. For
fixed $r$ then $\gamma_{ab}$ is a 2-metric, whose covariant derivative we
denote as $\tilde{\nabla}$, and its Ricci tensor we write as
$R^{(\gamma)}_{ab}$. Moreover we decompose the following expressions
covariantly over this $r$-dependent 2-geometry $\gamma_{ab}$ and emphasise
that Latin indices are raised/lowered with respect to $\gamma_{ab}$. Then we
may write,

$\displaystyle\hat{R}_{tt}$ $\displaystyle=$ $\displaystyle
e^{T-R}\left(e^{R}Q_{0}+rQ_{1}+r^{2}Q_{2}\right)$ $\displaystyle\hat{R}_{rr}$
$\displaystyle=$ $\displaystyle\left(e^{R}W_{0}+rQ_{1}+r^{2}W_{2}\right)$
$\displaystyle\hat{R}_{a}$ $\displaystyle=$
$\displaystyle\frac{1}{r}\left(U^{0}_{a}+rU^{1}_{a}\right)$
$\displaystyle\hat{R}_{ab}+r^{2}\hat{R}_{a}\hat{R}_{b}$ $\displaystyle=$
$\displaystyle e^{-R}\left(e^{R}V^{0}_{ab}+rV^{1}_{ab}+r^{2}V^{2}_{ab}\right)$
(4.47)

where,

$\displaystyle Q_{0}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\tilde{\nabla}^{2}T-\frac{1}{4}(\tilde{\nabla}T)^{2}-\frac{1}{4}\tilde{\nabla}_{a}T\tilde{\nabla}^{a}R-e^{-R}$
$\displaystyle W_{0}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\tilde{\nabla}^{2}R-\frac{1}{4}(\tilde{\nabla}R)^{2}-\frac{1}{4}\tilde{\nabla}_{a}T\tilde{\nabla}^{a}R-e^{-R}$
$\displaystyle U^{0}_{a}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\tilde{\nabla}_{a}(T-R)$ (4.48)

and

$\displaystyle Q_{1}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}\partial_{r}T+\frac{1}{2}\partial_{r}R-\frac{1}{2}K$
$\displaystyle U^{1}_{a}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\partial_{r}\tilde{\nabla}_{a}T-\frac{1}{4}\partial_{r}T\tilde{\nabla}_{a}(T-R)+\frac{1}{4}K\tilde{\nabla}_{a}R$
(4.49)
$\displaystyle\qquad+\frac{1}{4}K_{ab}\tilde{\nabla}^{b}(T-R)-\frac{1}{2}\tilde{\nabla}_{a}K+\frac{1}{2}\tilde{\nabla}^{b}K_{ab}$

and where,

$\displaystyle Q_{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\partial^{2}_{r}T-\frac{1}{4}(\partial_{r}T)^{2}+\frac{1}{4}(\partial_{r}T)(\partial_{r}R)-\frac{1}{4}(\partial_{r}T)K$
$\displaystyle W_{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\partial^{2}_{r}T-\frac{1}{4}(\partial_{r}T)^{2}+\frac{1}{4}(\partial_{r}T)(\partial_{r}R)+\frac{1}{4}(\partial_{r}R)K-\frac{1}{2}\partial_{r}K-\frac{1}{4}K_{ab}K^{ab}$

and then finally,

$\displaystyle V^{0}_{ab}$ $\displaystyle=$ $\displaystyle
R^{(\gamma)}_{ab}-\frac{1}{2}\tilde{\nabla}_{a}\tilde{\nabla}_{b}T-\frac{1}{2}\tilde{\nabla}_{a}\tilde{\nabla}_{b}R-\frac{1}{4}\tilde{\nabla}_{a}T\tilde{\nabla}_{b}T-\frac{1}{4}\tilde{\nabla}_{a}R\tilde{\nabla}_{b}R$
$\displaystyle V^{1}_{ab}$ $\displaystyle=$ $\displaystyle-K_{ab}$
$\displaystyle V^{2}_{ab}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\partial_{r}K_{ab}+\frac{1}{4}KK_{ab}-\frac{1}{4}K_{ab}\partial_{r}(T-R)+\frac{1}{4}\gamma_{ab}\left(K^{cd}K_{cd}-K^{2}\right)\;.$

Firstly one can check that $\hat{R}_{a}$ is in fact a smooth function of $r$,
even though naively it appears to go as $O(1/r)$, due to the fact that $T=R$
at $r=0$ so $U^{0}_{a}$ vanishes at $r=0$. The other components
$\hat{R}_{tt}$, $\hat{R}_{rr}$, $\hat{R}_{ab}$ are then clearly smooth in $r$
and $x^{a}$. Secondly one can explicitly confirm that the smoothness
conditions (4.33) indeed hold, by virtue of the behaviour of the metric
functions. Furthermore we see that the near horizon geometry of the Ricci
tensor is given by,

$\displaystyle R^{NH}_{\mu\nu}=\left(\begin{array}[]{ccc}-r^{2}R^{NH}&0&0\\\
&\frac{1}{r^{2}}R^{NH}&0\\\ &&R^{NH}_{ab}\end{array}\right)$ (4.55)

where,

$\displaystyle R^{NH}$ $\displaystyle=$ $\displaystyle
e^{T}\left(-\frac{1}{2}\tilde{\nabla}^{2}T-\frac{1}{2}(\tilde{\nabla}T)^{2}-e^{-T}\right)$
$\displaystyle R^{NH}_{ab}$ $\displaystyle=$ $\displaystyle
R^{(\gamma)}_{ab}-\tilde{\nabla}_{a}\tilde{\nabla}_{b}T-\frac{1}{2}\tilde{\nabla}_{a}T\tilde{\nabla}_{b}T\;.$
(4.56)

Clearly the near horizon geometry only depends on the intrinsic geometry and
not on any $r$ derivatives at the horizon $r=0$. Thus we see that the Ricci
tensor term in our RN flow equation for the metric indeed preserves the
extremal horizon structure.

####  4.3.2 DeTurck diffeomorphism term

Now we consider the diffeomorphism term in the metric flow equation,
$\nabla_{(\mu}\xi_{\nu)}$. In order to keep the metric static and preserve
$g_{tr}=g_{ta}=0$ during the flow, the diffeomorphism vector field must have
$\xi^{t}=0$, and the remaining components cannot depend on $t$. Additionally
for the diffeomorphism to preserve the extremal horizon we find it must take
the form,

$\displaystyle\xi^{\mu}=\left(0,r^{2}f,v^{a}\right)$ (4.57)

where $f$ and $v^{a}$ are smooth functions of $r$ and $x^{a}$. The vanishing
$\xi^{r}$ component at $r=0$ implies that the position of the horizon is not
changed in the flow, as one would expect. Perhaps less obvious is that this
component must vanish quadratically in $r$ as the following expressions show,

$\displaystyle\nabla_{(\mu}\xi_{\nu)}=\left(\begin{array}[]{ccc}-r^{2}\hat{A}_{tt}&0&0\\\
&\frac{1}{r^{2}}\hat{A}_{rr}&\hat{A}_{a}\\\
&&\hat{A}_{ab}+r^{2}\hat{A}_{a}\hat{A}_{b}\end{array}\right)$ (4.61)

where we have,

$\displaystyle\hat{A}_{tt}$ $\displaystyle=$ $\displaystyle
e^{T}\left(\frac{1}{2}v^{a}\tilde{\nabla}_{a}T+rf+\frac{1}{2}r^{2}f\partial_{r}T\right)$
$\displaystyle\hat{A}_{rr}$ $\displaystyle=$ $\displaystyle
e^{R}\left(\frac{1}{2}v^{a}\tilde{\nabla}_{a}R+rf+r^{2}\left(\frac{1}{2}f\partial_{r}R+\partial_{r}f\right)\right)$
$\displaystyle\hat{A}_{a}$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{R}\tilde{\nabla}_{a}f+\frac{1}{2}\gamma_{ab}\partial_{r}v^{b}$
$\displaystyle\hat{A}_{ab}+r^{2}\hat{A}_{a}\hat{A}_{b}$ $\displaystyle=$
$\displaystyle\tilde{\nabla}_{(a}v_{b)}+\frac{r^{2}}{2}fK_{ab}\;.$ (4.62)

In particular $\hat{A}_{a}$ would be singular, going as $\sim 1/r$ if
$\xi^{r}\sim r$ rather than quadratically in $r$.777 We note that while a
general behaviour $\xi^{r}=rf(r,x^{a})$ for $f$ smooth in its arguments would
be singular, it seems possible to have $\xi^{r}=rc+r^{2}f(r,x^{a})$, with $c$
a constant. However as we discuss shortly, the DeTurck vector component
$\xi^{r}$ does not have such a linear in $r$ leading behaviour. Thus, we drop
this contribution.  One can check that this indeed obeys the smoothness
conditions (4.33). The near horizon form is then,

$\displaystyle(\nabla_{(\mu}\xi_{\nu)})^{NH}=\left(\begin{array}[]{ccc}-r^{2}A^{NH}&0&0\\\
&\frac{1}{r^{2}}A^{NH}&0\\\ &&A^{NH}_{ab}\end{array}\right)$ (4.66)

with,

$\displaystyle
A^{NH}=\left.\frac{1}{2}e^{T}v^{a}\tilde{\nabla}_{a}T\right|_{r=0}\;,\quad
A^{NH}_{ab}=\left.\tilde{\nabla}_{(a}v_{b)}\right|_{r=0}$ (4.67)

and we see this only depends on $v^{a}$ and not on $f$.

We may check that the DeTurck choice of diffeomorphism indeed gives the above
form for $\xi^{\mu}$ in (4.57). Recall that,

$\displaystyle\xi^{\mu}=g^{\alpha\beta}\left(\Gamma^{\mu}_{~{}\alpha\beta}-\bar{\Gamma}^{\mu}_{~{}\alpha\beta}\right)$
(4.68)

with the reference connection being the Levi-Civita connection of a smooth
static reference metric,

$\displaystyle
d\bar{s}^{2}=-r^{2}e^{\bar{T}}dt^{2}+e^{\bar{R}}\left(\frac{dr}{r}+r\bar{\omega}_{a}dx^{a}\right)^{2}+\bar{\gamma}_{ab}dx^{a}dx^{b}$
(4.69)

where $\bar{T}$, $\bar{R}$, $\bar{\omega}_{a}$ and $\bar{\gamma}_{ab}$ obey
the same restrictions as their counterparts in the actual metric. In
particular, we also need to have $\psi=\bar{\psi}$ in order to obtain a smooth
extremal horizon when adding up tensors composed from the actual and the
reference metric. We emphasize that while we have chosen to use our coordinate
freedom to eliminate the $drdx^{a}$ terms in the metric for these
calculations, this generally will not eliminate those terms in the reference
metric.

One then finds the DeTurck vector field indeed has the form required for a
diffeomorphism that preserves the extremal horizon, as in equation (4.57). It
may explicitly be checked that $\xi^{t}=0$ and that the component $\xi^{r}$
goes as,

$\displaystyle\xi^{r}=re^{-R}\left(\frac{e^{\bar{T}-\bar{R}}}{e^{T-R}}-1\right)+O(r^{2})\;.$
(4.70)

This naively looks to only vanish as $\sim r$ rather than $\sim r^{2}$ at the
horizon. However due to the smoothness of the metric and reference metric, and
hence $T-R=\bar{T}-\bar{R}=0$ at $r=0$, this expression actually goes as
$O(r^{2})$, as in equation (4.57). Finally one finds that,

$\displaystyle\begin{split}\xi^{a}&=-\frac{1}{2}\gamma^{ab}\tilde{\nabla}_{b}(T+R)+\frac{1}{2}e^{\bar{T}-T}\bar{\gamma}^{ab}\tilde{\nabla}_{b}\bar{T}\\\
&\quad+\frac{1}{2}e^{\bar{R}-R}\bar{\gamma}^{ab}\tilde{\nabla}_{b}\bar{R}+\gamma^{bc}\left(\tilde{\Gamma}^{a}_{~{}bc}-\bar{\tilde{\Gamma}}^{a}_{~{}bc}\right)+O(r)\end{split}$
(4.71)

where $\tilde{\Gamma}^{a}_{~{}bc}$ and $\bar{\tilde{\Gamma}}^{a}_{~{}bc}$ are
the (Levi-Civita) connections of the horizon 2-metric $\gamma_{ab}$, and
reference metric $\bar{\gamma}_{ab}$, respectively, and indices are
raised/lowered w.r.t. $\gamma_{ab}$.

Hence on the horizon $r=0$ the 2d vector inducing the near horizon
diffeomorphism is,

$\displaystyle
v^{a}=e^{\bar{T}-T}\bar{\gamma}^{ab}\tilde{\nabla}_{b}\bar{T}-\gamma^{ab}\tilde{\nabla}_{b}T+\gamma^{bc}\left(\tilde{\Gamma}^{a}_{~{}bc}-\bar{\tilde{\Gamma}}^{a}_{~{}bc}\right)\;.$
(4.72)

Having discussed the Ricci tensor and DeTurck diffeomorphism terms for a
static extremal horizon, and shown they give smooth tensors, we now see that
DeTurck-Ricci flow will preserve the smooth structure of a static extremal
horizon. Moreover we may take the near horizon limit of the flow, to obtain a
geometric flow of the near horizon geometry. This near horizon flow will be
independent of the details of the geometry away from the horizon, depending
only on the intrinsic geometry of the horizon itself.

Now we turn to the remaining parts in the static EM flow. These are the
Maxwell field and its flow, and also its backreaction terms in the metric flow
equation. We will consider the purely electric and magnetic cases separately.

####  4.3.3 The electric Maxwell field

A purely electric potential is preserved under the EM flow. We find that
smoothness for the Maxwell field on the static extremal horizon geometry
implies that a pure electric potential takes the form,

$\displaystyle A=r\Phi(r,x^{a})dt$ (4.73)

where the potential function $\Phi$ is smooth in $r$ and $x^{a}$. Note that
the electric potential vanishes at the horizon which is necessary for the
vector field $A^{\mu}$ to have finite norm. Further, as we will see, in order
for the field strength $F_{\mu}^{~{}\alpha}F_{\nu\alpha}$ term in the flow to
preserve the extremal metric we require,

$\displaystyle\left.\tilde{\nabla}_{a}\Phi\right|_{r=0}=0$ (4.74)

so that $\Phi$ is constant over the horizon. The right-hand side of the
Maxwell field equation in the static EM flow (for a diffeomorphism of the form
(4.57)) can be decomposed as,

$\displaystyle\nabla^{\mu}F_{\mu t}+\mathrm{Lie}_{\xi}A_{t}$ $\displaystyle=$
$\displaystyle rQ_{0}+r^{2}e^{-R}Q_{1}+r^{3}e^{-R}Q_{2}$ $\displaystyle Q_{0}$
$\displaystyle=$
$\displaystyle\tilde{\nabla}^{2}\Phi-\frac{1}{2}\tilde{\nabla}^{a}(T-R)\tilde{\nabla}_{a}\Phi+v^{a}\tilde{\nabla}_{a}\Phi$
$\displaystyle Q_{1}$ $\displaystyle=$ $\displaystyle
2\partial_{r}\Phi+\frac{1}{2}\Phi\left(K-\partial_{r}(T+R)\right)+e^{R}f\Phi$
$\displaystyle Q_{2}$ $\displaystyle=$
$\displaystyle\partial_{r}^{2}\Phi+\frac{1}{2}\left(K-\partial_{r}(T+R)\right)\partial_{r}\Phi+e^{R}f\partial_{r}\Phi$
(4.75)

with the $r$\- and $a$-components vanishing. Note that these terms preserve
the form of the electric potential. In particular the condition (4.74) is
preserved as it implies that $Q_{0}=O(r)$. Hence the value of $\Phi$ on the
horizon will be constant over the 2-geometry, and remain unchanged in flow
time. Then taking the near horizon limit, the electric gauge potential is just
given by,

$\displaystyle A^{NH}=r\Phi^{NH}dt$ (4.76)

for constant $\Phi^{NH}=\Phi(0,x^{a})$, and is fixed along the flow. It thus
generates a constant radially directed electric field, $F^{NH}_{tr}$, on the
horizon. The charge enclosed by the horizon is then simply proportional to its
horizon area.

We note that the flow gauge transformation term $\partial_{\mu}\Lambda$ is
trivial here because $\Lambda\sim\nabla_{\alpha}A^{\alpha}$ vanishes for a
static electric potential. Finally the Maxwell field backreacts on the metric
flow via the symmetric tensor $F_{\mu~{}~{}}^{~{}\alpha}F_{\nu\alpha}$ and its
trace. We find,

$\displaystyle
F_{\mu}^{~{}\alpha}F_{\nu\alpha}=\left(\begin{array}[]{ccc}-r^{2}\hat{A}_{tt}&0&0\\\
&\frac{1}{r^{2}}\hat{A}_{rr}&\hat{A}_{a}\\\
&&\hat{A}_{ab}+r^{2}\hat{A}_{a}\hat{A}_{b}\end{array}\right)$ (4.80)

where,

$\displaystyle\hat{A}_{tt}$ $\displaystyle=$
$\displaystyle-\tilde{\nabla}_{a}\Phi\tilde{\nabla}^{a}\Phi-e^{-R}\partial_{r}(r\Phi)\partial_{r}(r\Phi)$
$\displaystyle\hat{A}_{rr}$ $\displaystyle=$
$\displaystyle-e^{-T}\partial_{r}(r\Phi)\partial_{r}(r\Phi)$
$\displaystyle\hat{A}_{a}$ $\displaystyle=$
$\displaystyle-\frac{1}{r}e^{-T}\partial_{r}(r\Phi)\tilde{\nabla}_{a}\Phi$
$\displaystyle\hat{A}_{ab}+r^{2}\hat{A}_{a}\hat{A}_{b}$ $\displaystyle=$
$\displaystyle-e^{-T}\tilde{\nabla}_{a}\Phi\tilde{\nabla}_{b}\Phi\;.$ (4.81)

Note we see the previously claimed condition (4.74) is required for
$(\hat{A}_{tt}-\hat{A}_{rr})$ to vanish on the horizon, and one can check then
that the remaining smoothness constraint in equation (4.33) holds.

Since $F_{\mu}^{~{}\alpha}F_{\nu\alpha}$ is smooth, its trace with respect to
the metric will be too, and so will be the backreaction term
$F_{\mu}^{~{}\alpha}F_{\nu\alpha}-\frac{1}{4}g_{\mu\nu}F^{2}$ in the metric
equation of the EM flow.

####  4.3.4 The magnetic Maxwell field

Now we turn to the purely magnetic case. We assume our magnetic Maxwell field
to take the following form,

$\displaystyle A=b_{r}(r,x^{a})dr+b_{a}(r,x^{a})dx^{a}$ (4.82)

where $b_{r}$ and $b_{a}$ are smooth functions in $r$ and $x^{a}$. This form
is preserved by the flow equations, which are given by,

$\displaystyle\tau\nabla^{\mu}F_{\mu r}+\mathrm{Lie}_{\xi}A_{r}$
$\displaystyle=$ $\displaystyle Q_{0}+rQ_{1}+r^{2}Q_{2}$ $\displaystyle Q_{0}$
$\displaystyle=$
$\displaystyle\tau\tilde{\nabla}^{a}F_{ar}+\frac{\tau}{2}\tilde{\nabla}^{a}(T-R)F_{ar}+v^{a}\tilde{\nabla}_{a}b_{r}+b_{a}\partial_{r}v^{a}$
$\displaystyle Q_{1}$ $\displaystyle=$ $\displaystyle 2b_{r}f$ $\displaystyle
Q_{2}$ $\displaystyle=$ $\displaystyle f\partial_{r}b_{r}+b_{r}\partial_{r}f$
(4.83) $\displaystyle\tau\nabla^{\mu}F_{\mu a}+\mathrm{Lie}_{\xi}A_{a}$
$\displaystyle=$ $\displaystyle Q_{0}+re^{-R}Q_{1}+r^{2}e^{-R}Q_{2}$
$\displaystyle Q_{0}$ $\displaystyle=$
$\displaystyle\tau\tilde{\nabla}^{b}F_{ba}+\frac{\tau}{2}\tilde{\nabla}^{b}(T+R)F_{ba}+\mathrm{Lie}_{v}b_{a}$
$\displaystyle Q_{1}$ $\displaystyle=$ $\displaystyle 2\tau F_{ra}$
$\displaystyle Q_{2}$ $\displaystyle=$
$\displaystyle-\tau\partial_{r}F_{ra}+\frac{\tau}{2}\partial_{r}(T-R)F_{ra}+\frac{\tau}{2}KF_{ra}+\tau\gamma^{bc}K_{ba}F_{cr}$
(4.84) $\displaystyle+e^{R}f\partial_{r}b_{a}+e^{R}b_{r}\tilde{\nabla}_{a}f$

with the $t$-component being trivial. The gauge fixing term
$\partial_{\mu}\Lambda$ is determined by $\Lambda$ which takes the form,

$\displaystyle\Lambda=\tau\nabla_{\mu}A^{\mu}$ $\displaystyle=$
$\displaystyle\tau\left(\Lambda_{0}+re^{-R}\Lambda_{1}+r^{2}e^{-R}\Lambda_{2}\right)$
$\displaystyle\Lambda_{0}$ $\displaystyle=$
$\displaystyle\tilde{\nabla}_{a}b^{a}+\frac{1}{2}\tilde{\nabla}^{a}(T+R)b_{a}$
$\displaystyle\Lambda_{1}$ $\displaystyle=$ $\displaystyle 2b_{r}$
$\displaystyle\Lambda_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\partial_{r}(T-R)+K)b_{r}+\partial_{r}b_{r}$ (4.85)

so we see this is a smooth function. It is then clear that the covector given
by its gradient, $\partial_{\mu}\Lambda$, then preserves the form of the gauge
field (4.82) with its contribution to the flow. The term
$F_{\mu~{}~{}}^{~{}\alpha}F_{\nu\alpha}$ gives,

$\displaystyle
F_{\mu}^{~{}\alpha}F_{\nu\alpha}=\left(\begin{array}[]{ccc}-r^{2}\hat{A}_{tt}&0&0\\\
&\frac{1}{r^{2}}\hat{A}_{rr}&\hat{A}_{a}\\\
&&\hat{A}_{ab}+r^{2}\hat{A}_{a}\hat{A}_{b}\end{array}\right)$ (4.89)

where,

$\displaystyle\hat{A}_{tt}$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle\hat{A}_{rr}$ $\displaystyle=$ $\displaystyle
r^{2}\gamma^{ab}F_{ra}F_{rb}$ $\displaystyle\hat{A}_{a}$ $\displaystyle=$
$\displaystyle\gamma^{bc}F_{rb}F_{ac}$
$\displaystyle\hat{A}_{ab}+r^{2}\hat{A}_{a}\hat{A}_{b}$ $\displaystyle=$
$\displaystyle\gamma^{cd}F_{ac}F_{bd}+r^{2}e^{-R}F_{ra}F_{rb}$ (4.90)

with $F_{ra}=\partial_{r}b_{a}-\tilde{\nabla}_{a}b_{r}$ and
$F_{ab}=\tilde{\nabla}_{a}b_{b}-\tilde{\nabla}_{b}b_{a}$. Because we have
$\hat{A}_{tt}=0$, and $\hat{A}_{rr}\sim O(r^{2})$, the smoothness conditions
in (4.33) are trivially fulfilled. Hence the backreaction of the magnetic
potential in the metric flow preserves smoothness of the extremal horizon.

The near horizon limit of the magnetic gauge potential
$A=b_{r}(r,x^{b})dr+b_{a}(r,x^{b})dx^{a}$ at $r=0$ becomes the 2-vector
potential,

$\displaystyle A^{NH}=b^{NH}_{a}(x^{b})dx^{a}\;,\quad
b^{NH}_{a}(x^{b})=b_{a}(0,x^{b})$ (4.91)

defined over the 2-geometry $\gamma_{ab}$ of the horizon.

####  4.3.5 Putting the terms together and the near horizon limit

We have seen above that each term in the static EM flow equations preserves
the smoothness of both the extremal horizon and the gauge field. Recall that
to simplify the task of computing these terms we have chosen coordinates where
the off-diagonal terms, $w_{a}$, vanish. However we emphasize that having
shown smoothness in these simpler coordinates, then guarantees smoothness in
more general coordinates where the off-diagonal terms are present.

We will now consider the near horizon limit of the flow. We explicitly showed
above that the transformation to remove these off-diagonal terms does not
affect the near horizon form of smooth tensors. Thus having computed the near
horizon forms in coordinates where $w_{a}$ vanishes gives their form for any
(smooth) choice of coordinates with terms $w_{a}$.

Using the results above we may take the near horizon limit of the EM flow so
that for $r=0$ we have in the electric case,

$\displaystyle\frac{d}{d\lambda}T$ $\displaystyle=$
$\displaystyle\tilde{\nabla}^{2}T+v^{\prime
a}\tilde{\nabla}_{a}T+2e^{-2T}\left(e^{T}-(\Phi^{NH})^{2}\right)$
$\displaystyle\frac{d}{d\lambda}\gamma_{ab}$ $\displaystyle=$
$\displaystyle-2R^{(\gamma)}_{ab}+2\tilde{\nabla}_{(a}v^{\prime}_{b)}+\tilde{\nabla}_{a}T\tilde{\nabla}_{b}T+2\gamma_{ab}e^{-2T}(\Phi^{NH})^{2}$
(4.92)

where

$\displaystyle v^{\prime
a}=e^{\bar{T}-T}\bar{\gamma}^{ab}\tilde{\nabla}_{b}\bar{T}+\xi^{(\gamma)a}\;,\quad\xi^{(\gamma)a}=\gamma^{bc}\left(\tilde{\Gamma}^{a}_{~{}bc}-\bar{\tilde{\Gamma}}^{a}_{~{}bc}\right)$
(4.93)

generates a diffeomorphism along the near horizon flow, and has a contribution
from the near horizon 2d DeTurck vector, $\xi^{(\gamma)a}$. We note that,
following our discussion above, the near horizon electric potential
$\Phi^{NH}$ is simply a constant that doesn’t change with flow time. We
further note that $v^{\prime a}$ differs from the previous $v^{a}$ above in
equation (4.72).

In the magnetic case the near horizon limit of flow takes the following form,

$\displaystyle\frac{d}{d\lambda}T$ $\displaystyle=$
$\displaystyle\tilde{\nabla}^{2}T+v^{\prime
a}\tilde{\nabla}_{a}T+2e^{-T}-(F^{NH})^{2}$
$\displaystyle\frac{d}{d\lambda}\gamma_{ab}$ $\displaystyle=$
$\displaystyle-2R^{(\gamma)}_{ab}+2\tilde{\nabla}_{(a}v^{\prime}_{b)}+\tilde{\nabla}_{a}T\tilde{\nabla}_{b}T+4F^{NH}_{ac}F_{b}^{NH~{}c}-\gamma_{ab}(F^{NH})^{2}$
$\displaystyle\frac{d}{d\lambda}b^{NH}_{a}$ $\displaystyle=$
$\displaystyle\tau\tilde{\nabla}^{b}F^{NH}_{ba}+\tilde{\nabla}_{a}\Lambda^{NH}+\mathrm{Lie}_{v^{\prime}}b^{NH}_{a}$
(4.94)

where $F^{NH}_{ab}=\tilde{\nabla}_{a}b^{NH}_{b}-\tilde{\nabla}_{b}b^{NH}_{a}$,
$\Lambda^{NH}=\tau\tilde{\nabla}^{a}b_{a}$, and $v^{\prime a}$ is as defined
above in the electric case. The two derivative terms on the right-hand side of
both the electric and the magnetic metric flow equations have precisely the
correct form to give a parabolic flow,

$\displaystyle\frac{d}{d\lambda}T=\gamma^{ab}\partial_{a}\partial_{b}T+\ldots\;,\quad\frac{d}{d\lambda}\gamma_{ab}=\gamma^{cd}\partial_{c}\partial_{d}\gamma_{ab}+\ldots$
(4.95)

where $\ldots$ represent lower derivative terms. Thus $T$ and the metric
components obey coupled diffusion equations governed by the near horizon
(inverse) metric $\gamma^{ab}$ which is Riemannian, and hence give parabolic
flows. We note that it is the 2d near horizon limit of the DeTurck vector that
ensures this parabolic flow for the near horizon metric.

For the electric flow the electric potential remains constant. In the magnetic
case we see that the principle part of the potential flow is,

$\displaystyle\frac{d}{d\lambda}b_{a}=\tau\gamma^{cd}\partial_{c}\partial_{d}b_{a}+\ldots$
(4.96)

so that it also yields a parabolic flow, due to the gauge fixing term
$\tilde{\nabla}_{a}\Lambda^{NH}$ in equation (4.3.5).

Away from the horizon we know that the EM flow has a parabolic character. It
also preserves the smooth static extremal horizon structure. Thus we may
construct this near horizon flow and then regard its solution as Dirichlet
data for the parabolic flow in the exterior of the horizon. This gives a
heuristic argument for well-posedness in the presence of an extremal horizon.
It would be interesting to develop a rigorous mathematical proof of well-
posedness.

##  5 Spherically symmetric flows about non-extremal Reissner-Nordström

Taking the metric to be electric or magnetic Reissner-Nordström, and taking
the reference metric to be the same, gives a fixed point of the EM flow, where
$\dot{g}_{\mu\nu}=\dot{A}_{\mu}=0$. The first question we now address is to
understand the stability of these two fixed points if we initially perturb the
metric and Maxwell field from the Reissner-Nordström form. For fixed points
that are unstable we further explore where the instability flows to. In order
to study this we restrict to the spherically symmetric setting. We expect, but
emphasize we have no formal argument, that if the spherically symmetric sector
of perturbations is stable, then the fixed point will generally be stable.

We write a general smooth static spherically symmetric metric as,

$\displaystyle
ds^{2}=-\rho^{2}T(\lambda,\rho)dt^{2}+4r(\rho)^{4}A(\lambda,\rho)d\rho^{2}+r(\rho)^{2}S(\lambda,\rho)d\Omega^{2}$
(5.1)

where $d\Omega^{2}=d\theta^{2}+\sin^{2}{\theta}d\phi^{2}$, $\rho$ is the
compactified radial coordinate introduced in section 2 related to the usual
coordinate as $r=1/(1-\rho^{2})$. Then $T,A,S$ specify the geometry and will
depend on flow time $\lambda$. Following the discussion in section 4.1, they
should be smooth functions of $\rho^{2}$ at the horizon $\rho=0$. Flat
spacetime in this radial variable $\rho$ takes the form,

$\displaystyle
ds^{2}_{flat}=-dt^{2}+4r(\rho)^{4}d\rho^{2}+r(\rho)^{2}d\Omega^{2}$ (5.2)

and thus we require $T,A,S\to 1$ as $\rho\to 1$ for our metric above to impose
the boundary condition that it is asymptotically flat. Please note that (5.2)
only covers flat space for $r\geq 1$.

Further we require that,

$\displaystyle\left.\left(T-4\kappa^{2}A\right)\right|_{\rho=0}=0\,.$ (5.3)

Then this spacetime has a smooth non-extremal horizon at $\rho=0$ with surface
gravity $\kappa$, and as discussed above, this surface gravity is preserved by
the EM flow. Choosing,

$\displaystyle T=F\;,\quad A=F^{-1}\;,\quad S=1\;,\quad
F(\rho)=1-r_{-}+r_{-}\rho^{2}$ (5.4)

then yields the RN metric from section 2. We now make this choice for the
reference metric.

Due to static spherical symmetry the most general gauge field must take the
form,

$\displaystyle A=\rho^{2}\Phi(\lambda,\rho)dt+\rho
A_{\rho}(\lambda,\rho)d\rho+A^{(\Omega)}\;,\quad
A^{(\Omega)}=A_{\theta}d\theta+A_{\phi}d\phi$ (5.5)

where $\Phi$ and $A_{\rho}$ depend on flow time and are smooth functions of
$\rho^{2}$. Here $A^{(\Omega)}$ is a one-form living on the 2-sphere (so only
having $d\theta$ and $d\phi$ components). In order for the field strength of
$A^{(\Omega)}$ to preserve the static spherical symmetry, its components
$A_{\theta}$ and $A_{\phi}$ must have no time or radial dependence. Then the
only choice compatible with spherical symmetry is the potential,
$A^{(\Omega)}=-q_{B}\cos{\theta}d\phi$ for a constant $q_{B}$ (or an $SO(3)$
rotation of this). As mentioned previously, in spherical symmetry we may
consistently have both an electric and magnetic field, as both the electric
and magnetic field are radial so there is no Poynting energy flux. However,
following our discussion above we will restrict to either consider the purely
electric or the magnetic cases. Noting that the radial component is pure
gauge, thus we restrict to the forms,

$\displaystyle A_{elec}=\rho^{2}\Phi(\lambda,\rho)dt\;,\quad
A_{mag}=-q_{B}\cos{\theta}d\phi\;.$ (5.6)

with $q_{B}$ a constant. We observed previously that in the magnetic case,
magnetic charge is fixed along the flow by boundary conditions, and here we
see it is given by $q_{B}$. In the electric case we require $\Phi(\rho)\to\mu$
as $\rho\to 1$, and then $(G\mu)$ gives the electric potential difference
between the horizon and infinity (since the potential $A_{t}\sim\rho^{2}$
vanishes at the horizon).

We see there is a fundamental difference between the electrostatic and
magnetostatic EM flows, namely that in the magnetic case there is no dynamics
associated to the gauge field, whereas in the electric case the potential
$\Phi(\rho)$ flows. Is this fixed form in the magnetic case consistent with
the EM flow equations? This magnetic potential obeys
$\nabla^{\mu}F_{\mu\nu}=0$ for any choice of metric functions $T$, $A$ and
$S$. The additional terms in the Maxwell part of the EM flow,
$\partial_{\mu}\Lambda$ and $\mathrm{Lie}_{\xi}A_{\mu}$, both vanish in this
magnetic case, since $\Lambda=\tau\nabla_{\alpha}A^{\alpha}$ vanishes, and
$\xi$ is radially directed (taking the RN reference metric above) but
$A_{\mu}$ has no radial dependence. Thus indeed the magnetic form is held
invariant under the flow.

In the electric case we may solve $\nabla^{\mu}F_{\mu\nu}=0$ on the static
spherically symmetric metric above by taking,

$\displaystyle\Phi(\rho)=-\frac{q}{\rho^{2}}\left(\int_{0}^{\rho}d\rho^{\prime}\,\rho^{\prime}\,\frac{\sqrt{T(\rho^{\prime})A(\rho^{\prime})}}{S(\rho^{\prime})}\right)$
(5.7)

which then has charge $q$. However, clearly if we deform the geometry,
changing $T$, $A$, $S$ as will happen as the metric flows, then this will no
longer yield vanishing $\nabla^{\mu}F_{\mu\nu}$, and hence the electric
potential will flow. Thus we see that the dynamics of electric and magnetic
gauge potentials are quite different under the EM flow.

###  5.1 Linear perturbations about RN

We now consider linear perturbations to both these fixed points. To proceed we
expand around the fixed point RN geometry and look for EM flow solutions of
the form,

$\displaystyle g_{\mu\nu}=g^{(RN)}_{\mu\nu}+\epsilon e^{\Omega\lambda}\delta
g_{\mu\nu}\;,\quad A_{\mu}=A^{(RN)}_{\mu}+\epsilon e^{\Omega\lambda}\delta
A_{\mu}$ (5.8)

linearizing in $\epsilon$, where $\Omega$ is a complex constant, and $\delta
g_{\mu\nu}$ and $\delta A_{\mu}$ preserve the static spherical symmetry. We
expect all such linear ‘mode’ solutions with this exponential flow time
dependence provide a basis for the general flow solution, and we may regard
finding these solutions as an eigenvalue problem, where $\Omega$ is the
eigenvalue. Mode solutions with $\mathrm{Re}(\Omega)\leq 0$ are stable in flow
time, and unstable modes have $\mathrm{Re}(\Omega)>0$. The existence of such
unstable modes would indicate the RN fixed point is an unstable fixed point of
the static EM flow, in the sense that a generic initial perturbation to the RN
fixed point will contain some component of the unstable mode, and this will
take the flow away from the fixed point as it exponentially grows in flow
time.

We will choose the reference metric to be the same RN solution as that of the
fixed point metric, and hence $\xi^{\mu}=0$ at the fixed point. In both the
electric and magnetic cases we also have $\Lambda=0$ for the RN solution with
the gauge potentials discussed previously. Thus perturbing the RN solution as
above, we will have,

$\displaystyle\xi^{\mu}=\epsilon
e^{\Omega\lambda}\delta\xi^{\mu}\;,\quad\Lambda=\epsilon
e^{\Omega\lambda}\delta\Lambda\;.$ (5.9)

We may then separate such eigenmodes into those with
$\delta\xi^{\mu}=\Lambda=0$, and those with $\delta\xi^{\mu},\Lambda\neq 0$.
The former case will be an eigenmode on the original Einstein-Maxwell flow
equation (3.19), whilst the latter is an eigenmode up to a flow time dependent
diffeomorphism and gauge transformation.

More explicitly we take,

$\displaystyle T(\rho)$ $\displaystyle=$
$\displaystyle\left(1-r_{-}+r_{-}\rho^{2}\right)\left(1+\epsilon
e^{\Omega\lambda}\delta T(\rho)\right)$ (5.10) $\displaystyle A(\rho)$
$\displaystyle=$
$\displaystyle\left(1-r_{-}+r_{-}\rho^{2}\right)^{-1}\left(1+\epsilon
e^{\Omega\lambda}\delta A(\rho)\right)$ (5.11) $\displaystyle S(\rho)$
$\displaystyle=$ $\displaystyle 1+\epsilon e^{\Omega\lambda}\delta S(\rho)$
(5.12)

and in the electric case, we take the electric potential as,

$\displaystyle\Phi(\rho)$ $\displaystyle=$ $\displaystyle\sqrt{r_{-}}+\epsilon
e^{\Omega\lambda}\delta\Phi(\rho)$ (5.13)

while in the magnetic case, as discussed above, the gauge potential is simply
fixed, with magnetic charge $q_{B}=\sqrt{r_{-}}$.

We now proceed to study the unstable modes of the flow equations numerically
over the interval $\rho\in[0,1)$. The functions, $\delta T,\delta A,\delta S$,
and in the electric case $\delta\Phi$ too, are all smooth functions of
$\rho^{2}$ (so even functions) on $\rho\in[0,1)$. We then discretize the
system of equations in terms of the function $\delta T,\delta A,\delta S$ and
$\delta\Phi$ using even pseudospectral differencing [28] which automatically
imposes the correct even behaviour at the horizon $\rho=0$. We further impose
the asymptotically flat boundary condition that $\delta T=\delta A=\delta
S=\delta\Phi=0$ at $\rho=1$. Having discretised, our flow equation is then
schematically of the form,

$\displaystyle Ov=\Omega v$ (5.14)

where due to the discretization $O$ and $v$ are a matrix and vector
respectively, and $v$ represents the values of the functions $\delta T,\delta
A,\delta S$, and in the electric case $\delta\Phi$, at the discrete points in
the interior of the interval $[0,1]$. If we take an even discretization of
$[0,1]$ with $N$ points, not including the boundary at $\rho=1$, then $O$ is
an $N\times N$ matrix, and $v$ is an $N$-vector. We find the mode solutions by
numerically determining the eigenvalues (ie. the values of $\Omega$) and
eigenfunctions of the matrix $O$. At any fixed resolution only the lower lying
eigenvalues represent eigenmodes of the continuum flow, with large eigenvalue
modes being discretization artefacts. Since the negative modes we will find
lie at modest values of $|\Omega|$, it is straightforward to find them with
relatively low resolution. We may check from the wavefunctions, and by
increasing resolution that they are indeed continuum modes, and also assess
the numerical error in the eigenvalues we determine. We find that already 100
points gives excellent accuracy for the eigenvalues of these negative modes
(to better than one part in $10^{9}$).

###  5.2 Magnetic non-extremal Reissner-Nordström linear stability

We now consider the eigenfunctions of the flow linearized about magnetic RN.
Since the gauge field has no dynamics, the constant $\tau$ plays no role. The
work of Monteiro and Santos [21] shows that Euclidean magnetic RN has a
negative mode for $|Q|<\sqrt{3}GM/2$, which for us implies $r_{-}<1/3$. This
mode involves the metric, and not the gauge potential which is simply the
unperturbed magnetic solution $A_{mag}=-\sqrt{r_{-}}\cos{\theta}d\phi$.
Continuing back to Lorentzian signature, these negative modes precisely yield
$\delta\xi=0$ unstable eigenmodes of the linearized static EM flow equations
for the metric, with real $\Omega>0$, signifying an exponentially growing
instability. The fact that $\delta\xi=0$ for these modes originates from the
fact that the tensor and vector perturbation decouple in this magnetic case
[21]. One might naively then think that the magnetic RN solution becomes a
stable fixed point for larger charges and up to extremality. However, it isn’t
clear that unstable modes cannot exist outside the sector $\delta\xi=0$, as
this has not previously been studied. We emphasize that while such modes would
not have a clear interpretation in terms of the Euclidean action studied in
[21], if present they would certainly affect the stability of the fixed point
of the static EM flow and that is our concern here.

In figure 1 we show the unstable mode determined numerically against varying
charge. We find that there is only one unstable mode for $r_{-}<1/3$, and we
find none for $r_{-}>1/3$. We explicitly check the numerical wavefunctions of
the unstable mode for charges where it exists and see that it is compatible
with $\delta\xi=0$ in the continuum. Also plotted in that figure is the
maximum value of the perturbation to the Ricci scalar. For the unstable mode
of Schwarzschild, so for $r_{-}=0$ this vanishes as the Euclidean negative
mode is traceless. We see it is non-vanishing for finite charge, but vanishes
again when the unstable mode disappears at the critical $r_{-}=1/3$, since as
we will argue shortly, at that point the mode is a static perturbation of the
Einstein-Maxwell equations and so must have vanishing Ricci scalar. In figure
2 we plot the wavefunctions $\delta T$, $\delta A$, $\delta S$ for $r_{-}=0$,
$r_{-}=1/6$ and $r_{-}=1/3$, normalized so that $\delta S=1$ at the horizon.
This data is for $N=400$ points, and we find $|\xi|<2\times 10^{-6}$ on the
interval in all these cases.

Figure 1:  The left panel shows the eigenvalue, $\Omega$, of the one unstable
mode for the flow of the non-extremal magnetically charged RN solution against
$r_{-}$ which controls its charge. For $r_{-}=0$ this is the Gross-Perry-Yaffe
negative mode of Schwarzschild. For charges $0<r_{-}<1/3$ this unstable mode
persists, disappearing at $r_{-}=1/3$, when it becomes a zero mode which is
generated by a perturbation within the RN family of solutions. For greater
charges the fixed point is stable (within this spherically symmetric setting).
$\xi$ remains zero for all these unstable modes, and in the right panel the
maximum value of the perturbation to the Ricci scalar for the mode is shown.

Figure 2:  The numerically generated wavefunctions, $\delta T$ (Red), $\delta
A$ (Blue) and $\delta S$ (Purple) for the unstable mode are plotted against
the radial coordinate $\rho$ for the unstable mode for $r_{-}=0$ (top left),
$r_{-}=1/6$ (top right) and $r_{-}=1/3$ (bottom). The instability becomes a
static perturbation for $r_{-}=1/3$, which corresponds to a perturbation
within the RN family, and may be computed independently. This is compared by
means of dashed curves in the bottom plot, the agreement providing a check of
our numerically constructed eigenfunction.

Thus we conclude that the only unstable mode in the static spherically
symmetric sector is the continuation of the Euclidean mode described by
Monteiro and Santos, and having $\delta\xi=0$. It is possible that one exists
without spherical symmetry, but we believe it would be unlikely.

Finally we address what the perturbation mode is at the critical charge,
$r_{-}=1/3$. This was not discussed explicitly in [21]. Since it has
$\Omega=0$ then it is a static perturbation of the RN solution. Given the
uniqueness of RN this static perturbation must be generated by a deformation
in the family of RN solutions. Indeed we can confirm that this is the case.
For $r_{-}=1/3$ we can perturb the RN parameters $r_{\pm}$ and also perform a
coordinate transformation as follows,

$\displaystyle r_{+}=1+\epsilon\delta r_{+}\;,\quad
r_{-}=\frac{1}{3}+\epsilon\delta r_{-}\;,\quad\rho\to\rho\left(1+\epsilon
f(\rho)\right)\;.$ (5.15)

Further we wish to preserve the charge, $Q=\sqrt{r_{+}r_{-}}$ which implies,

$\displaystyle\delta r_{-}=-\frac{1}{3}\delta r_{+}\;.$ (5.16)

This deformation within the RN magnetic solution family, together with the
coordinate transformation, then induces a perturbation to our fixed point,

$\displaystyle\begin{split}T&=\left(1-\frac{1}{3}(1-\rho^{2})\right)\left(1+\epsilon\delta
T\right)\;,\quad
A=\left(1-\frac{1}{3}(1-\rho^{2})\right)^{-1}\left(1+\epsilon\delta
A\right)\;,\\\ S&=1+\epsilon\delta S\\\ \end{split}$

with,

$\displaystyle\delta T=\frac{\frac{2}{3}(1-\rho^{2})\delta
r_{+}+2\left(1-\frac{1}{3}(1-2\rho^{2})\right)f(\rho)}{1-\frac{1}{3}(1-\rho^{2})}\;,\quad\delta
S=2\delta r_{+}+\frac{4\rho^{2}f(\rho)}{1-\rho^{2}}$ (5.17)

and a similar expression for $\delta A$. We take $\delta T$, $\delta A$,
$\delta S$ from our numerically constructed zero mode, which recall we have
normalized so that $\delta S=1$ at the horizon. Since for our perturbation
above, $\delta S=2\delta r_{+}$ at $\rho=0$, thus this perturbation is
generated by a deformation of the RN with $\delta r_{+}=1/2$. The coordinate
transformation is determined by the condition that $\xi=0$, which implies that
$f$ obeys,

$\displaystyle
f^{\prime\prime}+\frac{3+5\rho^{2}}{\rho(1-\rho^{2})}f^{\prime}+\frac{8(3+4\rho^{2}-\rho^{4})}{2-3\rho^{2}+\rho^{6}}f=-\frac{4\delta
r_{+}}{2+\rho^{2}}\;.$ (5.18)

We require the behaviour that $f\sim f_{0}+O(\rho^{2})$, so $f$ gives a smooth
metric perturbation at $\rho=0$ (the other behaviour $f\sim 1/\rho^{2}$ gives
a non-smooth metric perturbation). One finds $f\sim(1-\rho)$ as $\rho\to 1$,
and we require $f^{\prime}(1)=\delta r_{+}$ in order to have $\delta A=\delta
S=0$ at $\rho=1$ which is imposed by our boundary conditions. One may solve
this o.d.e. with these boundary conditions numerically, and taking $\delta
r_{+}=1/2$ to match to our numerically determined zero mode, we obtain
$f_{0}\simeq-0.393$, and a numerical form of $f(\rho)$ on $\rho\in[0,1)$. From
this we can then deduce the form of $\delta T$, $\delta A$ and $\delta S$ from
the above expressions, and we may check these match those of the numerically
determined $\Omega=0$ mode. Indeed we find they match precisely up to
numerical error. These wavefunctions, determined from $f$ above are shown in
figure 2 as the dashed curves, and those of the numerical static mode are
shown as thin solid lines. We see they lie on top of each other, confirming
this mode is indeed a perturbation within the RN family.

###  5.3 Electric non-extremal Reissner-Nordström linear stability

Now we consider the fixed point given by the electrically charged RN solution.
As discussed above, there is no decoupling of the metric flow from that of the
gauge field electric potential. In particular one cannot perturb the metric
and consistently have the gauge field fixed – it will be forced to flow.
Furthermore since the linearized flow will depend explicitly on the parameter
$\tau$, this may influence the presence of unstable modes. For simplicity we
will focus on the case of $\tau=1$ here. In this electric case there is no
straightforward Euclidean continuation, and therefore no prior results from
considering Euclidean negative modes.

For $\tau=1$ it appears that at least one unstable mode is present up until
extremality. Perhaps more surprisingly new unstable modes appear at specific
values of the electric charge. At the numerical resolutions we use it is
difficult to see the behaviour very near extremality, but it appears that an
increasing number of unstable modes appear in the limit that $r_{-}\to 1$. In
figure 3 we plot the eigenvalues with positive real part, noting that in fact
we find all these eigenvalues are purely real. Their wavefunctions are real
and we have also confirmed that they are compatible with a smooth continuum
form upon increasing resolution. All the unstable modes here, except in the
zero charge $r_{-}=0$ case, have non-vanishing $\delta\xi$, and non-trivial
gauge potential $\delta\Phi$. We have checked that they all have a non-
vanishing perturbation to the Ricci scalar, confirming that they are not
simply a pure diffeomorphism on the metric, but physically change the geometry
as well as the gauge field.

Figure 3:  Figure showing the eigenvalues, $\Omega$, for the unstable modes
for the flow of the electrically charged RN solution against $r_{-}$ for
$r_{-}\leq 0.999$ (and for Einstein-Maxwell flow parameter $\tau=1$). The GPY
negative mode at $r_{-}=0$ continues to an unstable mode of the electrically
charged solutions. However additional negative modes appear as the charge is
increased. For $r_{-}\leq 0.999$ up to three unstable modes may exist, but
very near extremality, for $0.999<r_{-}<1$ (not shown in the plot) more
unstable modes appear. It therefore appears that electrically charged RN is an
unstable fixed point of the EM flow near extremality, unlike the magnetically
charged RN solution which is stable.

In figure 4 and 5 we show the wavefunctions for values
$r_{-}=0.5,0.9,0.99,0.999$ of the two most unstable modes present for those
values (the second unstable mode only emerges for $r_{-}>0.854$). We note that
the number of nodes increases with each new branch of unstable modes. In the
limit $r_{-}\to 1$ the wavefunctions appear to tend only to have non-trivial
$\delta S$ with the other components and gauge field apparently vanishing in
this limit. We also observe that the curvature of the function $\delta S$ at
the horizon appears to increase as $r_{-}\to 0$, with $\delta S$ becoming
increasingly localized at the horizon $\rho=0$, indicating the perturbation
may not be smooth in the limit of extremality.

Figure 4:  Figure showing the most unstable mode wavefunctions for flow of
perturbations of electrically charged RN for charges $r_{-}=1/2$ (top left),
$r_{-}=0.9$ (top right), $r_{-}=0.99$ (bottom left) and $r_{-}=0.999$ (bottom
right). The wavefunction $\delta T$ is shown in red, $\delta A$ in blue,
$\delta S$ in purple, and the potential perturbation $\delta\Phi$ is in light
blue. The modes are normalized to have $\delta S=1$ at the horizon, and we see
$\delta S$ dominates the near horizon behaviour as extremality is approached,
and becomes increasingly localized there.

Figure 5:  Similar to the previous figure, the wavefunctions for the second
most unstable mode are shown for perturbations of electrically charged RN, for
$r_{-}=0.9$ (top left), $r_{-}=0.99$ (top right) and $r_{-}=0.999$ (bottom).
We see the wavefunctions have one node, and again $\delta S$ grows relative to
the other component functions at the horizon as extremality is approached.

Finally we comment that the emergence of new unstable modes with increasing
charge indicates that at special values of charge, there are static modes, so
linear perturbations that don’t flow. This occurs in the magnetic case when
the unstable mode disappears, but there $\delta\xi=0$ and at the critical
charge this zero mode is just a perturbation tangent to the RN space of
solutions as discussed above. Here however $\delta\xi\neq 0$, and we believe
that these zero modes are not associated to perturbations that are tangent to
the RN solution. Instead they should be tangent to new branches of ‘soliton’
solutions of the Einstein-Maxwell system that presumably merge with the RN
solutions at particular values of charge (depending on the parameter $\tau$).
We have not attempted to directly construct these soliton solutions as non-
linear solutions to the Einstein-Maxwell soliton equation, but it should be
possible to do so.

###  5.4 End point of unstable magnetic and electric flows

In the case of the uncharged asymptotically flat Schwarzschild solution in
static spherical symmetry, the single unstable mode of Ricci flow generates
two flows, one that leads the horizon to shrink to zero size in finite flow
time, and the other that expands the horizon ‘eating up’ the whole spacetime
[3]. In both the magnetic case for subcritical charge and in the electric case
for all charges we have instabilities and we now ask where these flow.

In order to address this we solve the non-linear flow described at the start
of this section 5. We evolve the metric functions $T,A,S$ in the ansatz above,
and for the electric case also the potential $\Phi$. We discretize the $\rho$
coordinate on the interval $[0,1]$ using sixth order finite differencing. We
impose a Neumann boundary condition for the functions $T,A,S$ and $\Phi$ at
$\rho=0$ in order to enforce the smooth horizon boundary condition (which
requires these functions to be smooth in $\rho^{2}$). The asymptotic flatness
is imposed as a Dirichlet boundary condition, $T,A,S\to 1$ at $\rho=1$. In the
electric case the potential $\Phi\to\sqrt{r_{-}}$ at $\rho=1$, fixing the
potential difference to the horizon. We evolve the parabolic system using
second order accurate Crank-Nicholson differencing in flow time. The resulting
implicit system is solved using an iterative method. We have checked that the
code correctly converges to the continuum. The results presented here are for
$N=200$ spatial points, and a flow time step of $1\times 10^{-5}$ or less. We
have also checked that the horizon smoothness and the value of the surface
gravity are correctly preserved by the numerical flows. We reiterate that we
have only considered the flow parameter $\tau=1$ here.

We discuss the electrostatic EM flow first. We begin by perturbing the RN
fixed point, taking initial data,

$\displaystyle T=1-(1-\rho^{2})r_{-}\;,\quad A=1/T\;,\quad S=1\pm
0.01\times(1-\rho^{2})^{4}\;,\quad\Phi=\sqrt{r_{-}}$ (5.19)

so we are perturbing only the sphere part of the metric via the function $S$.
Whilst obviously finite, this deformation to $S$ is sufficiently small in
amplitude that its initial evolution is well described by linear perturbation
theory about the RN solution. It preserves the boundary conditions, and
depending on the sign either initially decreases ($-$ sign) or increases ($+$
sign) the horizon size. We expect it to have overlap with the linear unstable
mode (or modes) of the electric RN fixed point, and thus to generate flows
driven by these unstable modes.

In figure 6 we show the evolution of the geometry by plotting the invariants
$\sqrt{-g_{tt}}=\rho\sqrt{T}$ against
$\sqrt{g_{\theta\theta}}=r(\rho)\sqrt{S}$ at a sequence of flow times
$\lambda$. These are invariants in the sense that they transform as scalars
under a gauge transformation $\rho\to F(\rho)$. The curve of $\sqrt{-g_{tt}}$
against $\sqrt{g_{\theta\theta}}$ is then gauge invariant and illustrates how
$g_{tt}$ varies with areal radius. In the figure we show the two evolutions
started with the negative and positive initial perturbations to the horizon
radius for a reasonably large charge, corresponding to $r_{-}=0.8$. We see the
perturbation that initially shrinks the horizon flows in finite flow time to a
singularity (here occuring at $\lambda\simeq 3.9$), whereas the perturbation
that initially expands the horizon leads to it growing in flow time in an
apparently unbounded manner. In the latter case the horizon accelerates in its
expansion, and we can only follow it numerically for a finite period, before
time gradients become too large to resolve accurately.

This behaviour is qualitatively the same behaviour as for the uncharged
Schwarzschild solution. We see precisely the same qualtitative behaviour for
smaller and larger charges, although as one approaches near extremal charges
it becomes more challenging to perform the simulations. For large charges
where linear theory tells us multiple unstable modes exist, presumably one may
seed the initial flows by these different unstable modes to generate different
flows. However we expect that without fine tuning it is the dominant unstable
mode that determines the final behaviour, and we may regard our perturbation
above which is not chosen to coincide with the unstable modes, but just
overlap them, as giving this generic evolution. It would be interesting to
explore whether more exotic behaviours could be found in the case of large
charges where there are multiple unstable modes which may be tuned initially
in the seed perturbation.

This similar qualitative behaviour to the uncharged case is perhaps to be
expected. Our electric EM flow preserves surface gravity and electric
potential difference to the horizon. At fixed electric potential and surface
gravity there is only one infilling RN solution, which is precisely the
initially perturbed fixed point. Thus, as in the uncharged case, there is no
natural end state solution to flow to, except flat spacetime (with a constant
electric potential). While we haven’t explored this in detail here, one may
resolve the singularity formed by the shrinking horizon case, and presumably
this then flows to flat spacetime after the topology is suitably changed, as
happens in the uncharged case. We may understand the other growing horizon as
the flow ‘searching’ for a larger stable black hole. If one puts the flow in a
box, then as in the uncharged case, a new large black hole solution exists
[29], and this expanding horizon flow would presumably settle down to that
solution.

Figure 6:  Figure showing curves of $\sqrt{-g_{tt}}$ plotted against
$\sqrt{g_{\theta\theta}}$ at constant intervals of electric EM flow time
$\Delta\lambda$ and for a small initial perturbation of an electric RN
solution with reasonably large charge corresponding to $r_{-}=0.8$ (recall
$r_{+}=1$). The lefthand frame shows the flow where the horizon is initially
reduced in size, and we see that it flows to a singularity at $\lambda\simeq
3.9$ (here $\Delta\lambda=0.05$). The righthand frame shows the situation
where the horizon is perturbed initially to be larger than that of RN, and we
see it apparently expands forever (here the curves are shown for flow
intervals $\Delta\lambda=0.25$). The geometry of the RN fixed point is shown
by the red dashed curve, and the arrows indicate the sense of change of the
geometry for increasing flow time. Both these behaviours are qualitatively
similar to the Schwarzschild fixed point under Ricci flow. We see analogous
behaviour for all charges of the electric RN fixed point.

Now let us discuss the case of the magnetic flow, and recall that the magnetic
charge is preserved by the flow. Analogous plots of $\sqrt{-g_{tt}}$ against
$\sqrt{g_{\theta\theta}}$ are shown in figure 7 for two situations, one below
the critical charge and one above. Above the critical charge our linear
analysis has shown no negative modes, and indeed we see that large initial
deformations of the RN solution flow back to it. The figure shows the example
of $r_{-}=0.4$ (recall that $r_{-}>1/3$ gives charges larger than the critical
one) for an initially large deformation,

$\displaystyle T=1-(1-\rho^{2})r_{-}\;,\quad A=1/T\;,\quad S=1\pm
0.5\times(1-\rho^{2})^{4}\;.$ (5.20)

The plot shows both the positive and negative initial deformations on the same
axes, and both flow back to the RN fixed point. We observe this same behaviour
for all charges greater than the critical one, so $r_{-}>1/3$, and for a
variety of non-linear initial deformations to the geometry.

The case of sub-extremal charge, so $r_{-}<1/3$, is much more interesting. In
this case we expect from [21] one unstable mode of the flow, and this is
confirmed by our linear analysis. There is now an important difference from
the electric case. While there, at fixed surface gravity and electric
potential there is one solution, the perturbed fixed point, now in the
magnetic case where magnetic charge is conserved, one has an additional fixed
point. This has the same surface gravity and charge as the unstable RN fixed
point with $r_{+}=1$, this second solution now having $r_{+}<1$ and being a
stable fixed point. Thus for an initial perturbation that reduces the horizon
size, rather than flowing to zero size and a singularity, another option is to
flow to this second stable fixed point with its smaller horizon, but the same
charge. This is precisely what we find occurs in our simulations. For an
initial perturbation similar to that in the electric case (although now there
is no potential), so,

$\displaystyle T=1-(1-\rho^{2})r_{-}\;,\quad A=1/T\;,\quad S=1\pm
0.01\times(1-\rho^{2})^{4}$ (5.21)

we observe that the flow with the initial RN deformed to have a smaller
horizon flows to the second RN fixed point. This is shown in the figure 7 for
the case $r_{-}=0.2$, but we see the same behaviour for all initial charges
$r_{-}<1/3$ that we have studied. The initial unstable fixed point has horizon
size governed by having $r_{+}=1$, and the stable end point then has $r_{-}$
given by the other (positive) root of the cubic,

$\displaystyle(1-r_{-})r_{+}^{3}-r_{+}^{2}+r_{-}=0\;.$ (5.22)

Unlike the uncharged or electric cases, where with a change of topology that
removes the horizon we may have an infilling flat spacetime fixed point (with
constant potential in the electric case), in this magnetic setting where the
magnetic charge is fixed by the asymptotic boundary conditions, this is not
possible. With a flat spacetime topology there is no way to infill the
geometry smoothly to a give a fixed point, and yet carry the magnetic charge.
Thus it intuitively makes sense that in this magnetic case we see the horizon
shrinking flow go to the stable charged black hole, rather than choose to
shrink the horizon to zero size and a singularity, which whilst it could be
resolved, could not now flow to a flat spacetime fixed point due to the flow
preserving magnetic charge. Another way to understand this different behaviour
is that the charge enclosed at any radius is the same, as there is no charged
matter, so the horizon carries the fixed magnetic charge. When the horizon
shrinks, the backreaction from the fixed charge becomes larger, and its
contribution to the stress tensor would diverge if the horizon shrinks to zero
size. Thus this trapped magnetic flux acts to prevent the horizon shrinking.

As we see in the figure 7, the flow that initially increases the horizon size
of the unstable fixed point, appears to continue to do so in an accelerated
manner as in the uncharged or electric cases. We see this same behaviour for
all $r_{-}<1/3$ as we perturb the unstable RN fixed point in a manner
initially expanding the horizon. As mentioned above for the electric case,
presumably if we placed this system in a spherical box, a new stable ‘large’
black hole fixed point would exist, and this horizon expanding flow would
asymptote to this. In the asymptotically flat setting we consider here, the
flow expands the horizon forever, always searching, but not finding, a large
stable black hole to settle on.

Figure 7:  Figure showing curves of $\sqrt{-g_{tt}}$ against
$\sqrt{g_{\theta\theta}}$ at constant intervals now for the magnetic EM flow.
The top left and right hand frames are similar to those of the previous
figure, being for an unstable magnetic RN solution with $r_{-}=0.2$ perturbed
initially to have smaller horizon size (left) or larger horizon size (right).
The latter gives the same expanding horizon behaviour as that of the uncharged
or electric cases – here $\Delta\lambda=0.5$. The red dashed curve again shows
the geometry of the initial RN fixed point that is perturbed. In the initially
contracting case here a new stable magnetic RN solution exists for the same
surface gravity and charge, and the flow asymptotes to this – here it has
$r_{+}=0.64$ and we show the new stable fixed point geometry using longer red
dashes. The flow curves in this case are plotted for
$\lambda=0,10,12,\ldots,22,24,30,50$ and $1000$. The lower frame shows flows
from a non-linear deformation of a fixed point with larger charge $r_{+}=0.4$,
so it is stable. Two flows are shown, one with larger horizon initially (blue
curves) and one with smaller (purple curves). Both quickly return to the
stable fixed point (shown by the red dashed curve) under the flow (for both
curves are shown at intervals $\Delta\lambda=0.5$). Arrows indicate the sense
of change of the geometry under increasing flow time.

Finally we show in figure 8 that as the magnetic charge is decreased to zero,
we recover the shrinking flow to a zero sized singularity at the horizon. We
plot the metric function $S$ at the horizon, $\rho=0$, as a function of flow
time, which gives the squared radius of the horizon sphere. The plot shows the
horizon flow for the same initial metric perturbation as in (5.21) with
various charges, including the zero charge case. As discussed in [3], for zero
charge this develops a singularity, here at approximately $\lambda\sim 6.7$
where the horizon shrinks to zero radius. The largest charge shown is that
where $r_{-}=0.2$ as in figure 7 and we see the horizon settles down to the
radius of the corresponding stable black hole with that same charge and
surface gravity (so $r_{+}=0.64$). We see that for smaller charges the flow of
the black holes approach the correspondingly smaller stable black hole
solutions determined by (5.22). The curve of $S$ against flow time tends to
that of the uncharged case, up to the point where the horizon stops shrinking
and settles on the smaller stable horizon.

Figure 8:  Figure showing curves of $S$ evaluated at the horizon, $\rho=0$,
against flow time for magnetic flows of perturbed magnetic RN which initially
decrease the horizon size. This shows how the (square of the) horizon radius
flows. The purple curve is the uncharged case, where the horizon shrinks to a
singularity at finite flow time, $\lambda\sim 6.7$. The other curves show
charges given by $r_{-}=0.025,0.05,0.1$ and $0.2$. We see that for all these
charges the horizon asymptotes to the stable one for that charge and surface
gravity (the values of $S$ for the stable black hole are shown as red dashed
lines and are computed from equation (5.22)). As the charge is decreased, the
curves become more similar to the uncharged case until the point that the
horizon settles on that of the smaller stable black hole solution.

##  6 Spherically symmetric flows about extremal Reissner-Nordström

For static extremal black hole spacetimes, with pure electric or magnetic
charge, we have argued above that we may perform static EM flows, and further
that the flow of the near horizon geometry decouples from the exterior.
Following our discussion of the stability of static EM flows in the
spherically symmetric case about non-extremal RN, it is natural to extend this
to consider the extremal case. To do this we consider the general spherically
symmetric metric using the coordinates introduced earlier in equation (2.13),
taking,

$\displaystyle ds^{2}=-\rho^{\prime
2}Tdt^{2}+r(\rho^{\prime})^{4}A\frac{d\rho^{\prime 2}}{\rho^{\prime
2}}+r(\rho^{\prime})^{2}Sd\Omega^{2}\;.$ (6.1)

From our earlier discussion, in order to have a smooth extremal horizon
located at $\rho^{\prime}=0$ we should have $T,A,S$ being smooth functions of
$\rho^{\prime}$ and further satisfying,

$\displaystyle\left.\left(T-A\right)\right|_{\rho^{\prime}=0}=0\;,\quad\left.\frac{\partial_{\rho^{\prime}}\left(T-A\right)}{T}\right|_{\rho^{\prime}=0}-4=\psi^{\prime}$
(6.2)

for some constant $\psi^{\prime}$. Now considering the static EM flow, $T,A,S$
become also functions of flow time $\lambda$. For the pure magnetic case, we
simply have,

$\displaystyle A_{mag}=-\cos{\theta}d\phi$ (6.3)

as this is preserved by the flow. In the electric case, instead, we take,

$\displaystyle A_{elec}=\rho^{\prime}\Phi dt$ (6.4)

which is smooth provided $\Phi$ is a smooth function of $\rho^{\prime}$ and
flow time. One may check explicitly that the spherically symmetric static EM
flow equations preserve the smoothness conditions (6.2), following our general
analysis earlier. From our discussion in section 2, using our choice of units,
the extremal RN fixed point is given by $T=A=S=1$, with $\Phi=1$ in the
electric case, so that $\psi^{\prime}=-4$.

Firstly we may explicitly solve the near horizon non-linear flow equations in
both the electric and magnetic cases. Specializing our general discussion in
section 4.3 to our spherically symmetric ansatz, the near horizon form of the
metric is,

$\displaystyle ds^{2}_{NH}=-T_{0}\left(\rho^{\prime
2}dt^{2}+\frac{d\rho^{\prime 2}}{\rho^{\prime 2}}\right)+S_{0}d\Omega^{2}$
(6.5)

and in the electric case,

$\displaystyle A^{NH}_{elec}=\rho^{\prime}\Phi_{0}dt$ (6.6)

with $T_{0}$, $S_{0}$ being functions of only flow time. As discussed in
section 4.3, for the electric near horizon flow the potential $\Phi_{0}$ is a
constant in flow time. As one might expect from the non-extremal case above,
solving this near horizon flow we will find the magnetic RN solution is a
stable fixed point of the magnetic flow, but the electric solution is
unstable.

In either the magnetic or electric case we may fix the near horizon metric to
be that of RN, and then consider the flow in the exterior. We will simulate
these flows numerically and fully non-linearly for flow parameter $\tau=1$. In
the magnetic case the flows approach magnetic RN at late times. However in the
electric case, even with the horizon fixed to be that of extremal RN, we see
generic initial data – even that corresponding to small initial perturbations
– seem to develop to a singularity.

###  6.1 Near horizon Electric flow

The electrostatic near horizon flow is determined by the flow equations,

$\displaystyle T_{0}^{\prime}(\lambda)=2-2\frac{\Phi_{0}^{2}}{T_{0}}\;,\quad
S_{0}^{\prime}(\lambda)=-2+\frac{2\Phi_{0}^{2}S_{0}(\lambda)}{T_{0}(\lambda)^{2}}\;.$
(6.7)

The RN near horizon fixed point is given by
$T_{0}(\lambda)=S_{0}(\lambda)=\Phi_{0}$, and we have chosen units
$\Phi_{0}=1$. We may expand about this fixed point perturbatively and solve
(6.7) giving,

$\displaystyle\begin{split}T_{0}(\lambda)&=1+\epsilon e^{2\lambda}a\;,\quad
S_{0}(\lambda)=1+\epsilon e^{2\lambda}\left(b-4\lambda
a\right)\;,\quad\Phi_{0}=1\end{split}$ (6.8)

to first order in $\epsilon$, where $a$ and $b$ are integration constants.
Note that we do not perturb $\Phi_{0}$ as it is constant in flow time, and a
perturbation would simply correspond to changing the charge of the fixed point
solution, which by a change of units can be mapped back to one. We thus see
the fixed point is unstable, and there are two relevant deformations of it,
parameterized by $\epsilon a$ and $\epsilon b$. Starting with the RN near
horizon solution as $\lambda\to-\infty$, adding these relevant deformations
will then flow the near horizon metric away from near horizon RN for finite
$\lambda$.

The simplest class of non-linear solutions which tend to the RN solution for
$\lambda\to-\infty$, is that corresponding to $T_{0}$ being constant,
$T_{0}(\lambda)=1$. The flow equation for $S_{0}$ consequently linearizes, so
that the non-linear solution for $S_{0}$ is simply,

$\displaystyle S_{0}(\lambda)=1+\epsilon\,e^{2\lambda}\;.$ (6.9)

for a constant $\epsilon$. We see that for $\epsilon<0$ the horizon sphere
shrinks to zero size at finite flow time $\lambda$. Alternatively for
$\epsilon>0$ it expands forever. We thus see that the extremal RN black hole
is an unstable fixed point of the RN flow, with behaviour reminiscient of the
non-extremal Schwarzschild black hole under Ricci flow [3].

The general near horizon solution has non-constant $T_{0}$, and is given by,

$\displaystyle T_{0}(\lambda)$ $\displaystyle=$
$\displaystyle\left(1+P_{\pm}(\lambda)\right)$ $\displaystyle S_{0}(\lambda)$
$\displaystyle=$
$\displaystyle\frac{\left(1+\left(c-4(\lambda-\lambda_{0})\right)P_{\pm}(\lambda)+P_{\pm}(\lambda)^{2}\right)}{1+P_{\pm}(\lambda)}$
(6.10)

where we have defined,

$\displaystyle P_{\pm}(\lambda)=W_{0}\left(\pm
e^{2(\lambda-\lambda_{0})}\right)$ (6.11)

with $W_{0}(x)$, the Lambert W function that gives the principle solution of
$y$ for the relation $x=ye^{y}$. The constant $\lambda_{0}$ arises from
translation invariance of the equations in $\lambda$, and $c$ is the remaining
constant of integration that determines the different behaviours.
Asymptotically as $\lambda\to-\infty$ we have,

$\displaystyle P_{\pm}(\lambda)\to\pm e^{2(\lambda-\lambda_{0})}$ (6.12)

and the function $W_{0}(x)$ exists for all $x>-1/e$ and is monotonically
increasing. At $x=-1/e$ it takes the value minus one.

Figure 9:  Analytic behaviour of the near horizon electric flow for small
perturbations of the RN fixed point with $\Phi_{0}=1$. The various colors
correspond to different choices for $c$ and the branches $P_{\pm}$ and are
detailed in the main text. The arrows point in the direction of the flow,
while the black dot highlights the fixed point. The near horizon magnetic flow
can be obtained by reversing the arrows and swapping $S_{0}$ and $T_{0}$.

The analytic behaviour of the near horizon flow is captured in Figure (9). The
sign in the definition (6.12) of $P_{\pm}\left(\lambda\right)$ defines two
distinct branches for evolutions away from the fixed point. For these, the
various trajectories in the $\left(S_{0},T_{0}\right)$ plane are then labelled
by the value of $c$. For the $P_{+}\left(\lambda\right)$ branch (shown as blue
curves in the figure), then $S_{0}\to 0$ in finite flow time with $T_{0}$
remaining finite. For the $P_{-}\left(\lambda\right)$ branch with $c>0$ (shown
as orange curves) the same behaviour occurs. However, taking instead $c<0$
(shown as red curves) then $S_{0}\to\infty$ and $T_{0}\to 0$ in a finite flow
time. In the special case $c=0$ for the $P_{-}\left(\lambda\right)$ branch
(shown as the black curve) which separates these two behaviours, both $T_{0}$
and $S_{0}$ go to zero at the same finite flow time. Finally taking the limit
$c\to\pm\infty$ for the $P_{\pm}\left(\lambda\right)$ branches results in the
two $T_{0}=1$ behaviours in (6.9) above, one with $S_{0}$ that shrinks to zero
in finite flow time (purple in the figure) and the other with growing $S_{0}$
(green in the figure) so that the horizon expands forever. The only behaviour
which exists for all flow times is this last one.

The Ricci scalar of the near horizon geometry, which is the Ricci scalar of
the full spacetime restricted to $\rho=0$, is simply given by,

$\displaystyle\left.R\right|_{\rho=0}=-\frac{2}{T_{0}}+\frac{2}{S_{0}}$ (6.13)

For all finite values of $c$ then either $S_{0}\to 0$ at a finite flow time,
with finite $T_{0}$, or alternatively $T_{0}\to 0$ with $S_{0}\to\infty$ at a
finite flow time. In both cases the Ricci scalar diverges, showing that the
geometry encounters a curvature singularity at finite flow time. The special
case $c=0$ where $T_{0}\to 0$ and $S_{0}\to 0$ could allow a cancellation
between the two terms in the Ricci scalar, but calculation reveals it is still
singular. The only exception to a finite flow time singularity is the flow
$T_{0}=1$ with expanding horizon, where the flow exists for all time, and the
scalar curvature tends to $\left.R\right|_{r=0}\to-2/T_{0}$. We note that any
solution to the Einstein-Maxwell equations has vanishing Ricci scalar (since
the stress tensor is traceless), and hence the asymptotic geometry that this
flow tends to cannot be the near horizon geometry to an Einstein-Maxwell
solution due to its non-vanishing Ricci scalar.

###  6.2 Near horizon Magnetic flow

Now we consider the magnetic case where the near horizon flow equations are,

$\displaystyle T_{0}^{\prime}(\lambda)=2-2\frac{T_{0}}{S_{0}^{2}}\;,\quad
S_{0}^{\prime}(\lambda)=-2+\frac{2}{S_{0}(\lambda)}\;.$ (6.14)

Then introducing functions $\bar{T}_{0}\left(\lambda\right)\equiv
S_{0}\left(-\lambda\right)$ and $\bar{S}_{0}\left(\lambda\right)\equiv
T_{0}\left(-\lambda\right)$, we derive the flow equations for these,

$\displaystyle\bar{T}_{0}^{\prime}\left(\lambda\right)=2-\frac{2}{\bar{T}_{0}}\;,\quad\bar{S}_{0}^{\prime}\left(\lambda\right)=-2+2\frac{\bar{S}_{0}}{\bar{T}_{0}^{2}}$
(6.15)

which are precisely the near horizon electric equations (6.7) taking
$S_{0}\to\bar{S}_{0}$ and $T_{0}\to\bar{T}_{0}$. Hence, the near horizon
magnetic flow is equivalent to the electric one, after exchanging $T_{0}$ and
$S_{0}$, and flipping the sign of the flow time $\lambda$. This implies that
the unstable near horizon RN fixed point that lies in the infinite past of the
electric near horizon flow translates into a non-linearly stable fixed point
of the magnetic near horizon flow.

###  6.3 Flows of the full extremal spacetimes

Finally we numerically investigate the non-linear spherically symmetric flows
starting with a deformation of the extremal electric and magnetic RN solution
that preserves extremality. Again we consider only the case $\tau=1$. We solve
the full non-linear flow for the metric functions $T,A,S$, and in the electric
case $\Phi$ too, discretizing in the spacetime coordinates using sixth order
finite differencing as for the non-extremal case discussed earlier. We employ
a second order accurate Crank-Nicholson differencing in flow time, and solve
the implicit system at each time step using an iterative solver. The code
correctly reproduces the continuum limit, and the results we present here are
for $N=200$ spatial points, and for a flow time step of $5\times 10^{-5}$. In
the non-extremal setting we needed a boundary condition at the horzion for our
metric functions. However in the extremal case the flow equations are simply
imposed at the horizon, and as we have discussed above, decouple from the
exterior flow as they involve no radial derivatives. While we have the
analytic solution for any initial data for the near horizon flow, it is
convenient here to simply evolve the near horizon flow numerically. We then
check it correctly reproduces the analytic solutions.

Our expectation is that deformations of the magnetic RN solution should return
to it under the flow, and indeed this is what we see. An example is shown in
figure 10 where we start with a non-linear deformation of the magnetic
solution, initially deforming the metric functions as,

$\displaystyle T=A=1+2(1-\rho^{\prime 2})^{4}\;,\quad
S=1-\frac{3}{4}(1-\rho^{\prime 2})^{4}\;.$ (6.16)

We note that these are compatible with smoothness of the horizon, as in
equation (6.2), and preserve $\psi^{\prime}=-4$. We already know that the near
horizon flow, ie. the behaviour at $\rho^{\prime}=0$, must return to the
magnetic near horizon RN geometry. In figure 10 we display the geometric
invariant $\sqrt{-g_{tt}}$ plotted against $\sqrt{g_{\theta\theta}}$. These
curves are shown at intervals of flow time $\Delta\lambda=0.25$, and we see
they asymptote back towards the magnetic RN fixed point. This behaviour is
generic for other non-linear deformations we have implemented, provided they
are not too large (when they can potentially give rise to singularities away
from the horizon). Thus at least in the spherically symmetric setting, the
extremal magnetic RN solution appears to be a stable fixed point of the non-
linear flow for the full extremal horizon spacetime.

Figure 10:  Figure showing curves of $\sqrt{-g_{tt}}$ plotted against
$\sqrt{g_{\theta\theta}}$ at constant intervals ($\Delta\lambda=0.25$) of flow
time, for a non-linear initial deformation of the extremal magnetic RN
solution. The RN fixed point is shown in dashed red, and we see the initally
deformed geometry asymptotes back towards extremal RN as the flow proceeds.

Now let us consider the electrically charged RN fixed point. Our near horizon
analysis already indicates it is unstable, and in figure 11 we follow two
flows where the fixed point is initially perturbed by taking,

$\displaystyle S=1\pm 0.01\times(1-\rho^{\prime 2})^{2}$ (6.17)

with $T=A=\Phi=1$. From our near horizon analysis for the negative sign above
we know the horizon behaviour is then $T=1$ and $S=1-0.01e^{2\lambda}$ at
$\rho^{\prime}=0$, and so the horizon sphere will shrink to zero size at time
$\lambda=2.30$. For the positive case the horizon behaviour is $T=1$ and
$S=1+0.01e^{2\lambda}$ and so the horizon sphere exponentially expands in flow
time. Simulating the full flows we again plot the invariant $\sqrt{-g_{tt}}$
against $\sqrt{g_{\theta\theta}}$ for both cases in the figure 11. We indeed
see these behaviours at the horizon, with the geometry away from the horizon
responding but remaining smooth. In the expanding case the simulation is run
until flow time $\lambda\simeq 5$ when the function $S$ becomes too large
($S\sim O(10^{2})$) at the horizon to properly resolve gradients there. In the
case where the sphere shrinks to zero size we expect that one can resolve the
singularity, as done in [3], to continue the flow through to flat spacetime.

Figure 11:  Similar figure to the previous one, but now for flows that begin
with a small perturbation of the electrically charged extremal RN solution.
The perturbation is subtracted and added to initially decrease and increase
the horizon size, giving the left and right frames plotted for flow time
intervals $\Delta\lambda=0.1$ and $0.25$ respectively. The flow behaviour at
the extremal horizon decouples and follows the near horizon flow, leading to a
collapse in finite flow time or exponential horizon expansion. We see the
exterior of the geometry follows this near horzion behaviour in a smooth
manner. In the collapsing case the singularity occurs at $\lambda=2.30$. In
the expanding case curves are shown for flow times up to $\lambda=5$, when
gradients near the horizon become difficult to numerically resolve.

Finally we can consider the case where we do not initially perturb the horizon
geometry. We show here the example deformation,

$\displaystyle S=1\pm 0.5\times\rho^{\prime 4}(1-\rho^{\prime 2})^{2}$ (6.18)

with $T=A=\Phi=1$, and again note this preserves smoothness of the extremal
horizon with $\psi^{\prime}=-4$. The near horizon analysis then implies
$T=S=1$ for all flow time at the horizon $\rho^{\prime}=0$. The plethora of
unstable modes of the non-extremal flow suggests that while the horizon
geometry is pinned to that of the extremal electric RN, the exterior spacetime
may be unstable. Indeed this appears to be the case. In figure 12 we plot the
same invariants as in the previous figures for both these flows, now zooming
in on the region near the horizon. The horizon size indeed remains fixed, and
the flow away from the horizon remains smooth for some time. We show the flow