# Remark on Remnant and Residue Entropy with GUP

Hui-Ling Li<EMAIL_ADDRESS>College of Physics Science and Technology,
Shenyang Normal University, Shenyang 110034, China  Wei Li  College of Physics
Science and Technology, Shenyang Normal University, Shenyang 110034, China
Yi-Wen Han  School of Computer Science and Information Engineering, Chongqing
Technology and Business University, Chongqing 400070, China

###### Abstract

Abstract:

In this article, close to the Planck scale, we discuss on the remnant and
residue entropy from a Rutz-Schwarzschild black hole in the frame of Finsler
geometry. Employing the corrected Hamilton-Jacobi equation, the tunneling
radiation of a scalar particle is presented, and the revised tunneling
temperature and revised entropy are also found. Taking into account
generalized uncertainty principle (GUP), we analyze the remnant stability and
residue entropy based on thermodynamic phase transition. In addition, the
effects of the Finsler perturbation parameter, GUP parameter and angular
momentum parameter on remnant and residual entropy are also discussed.

Keywords: generalized uncertainty principle, residue entropy, phase
transition, remnant

PACS: 04.70.-s, 04.70.Dy, 97.60.Lf

##  I Introduction

Although a complete self-consistent theory of quantum gravity has not been
established, it is an effective way to understand the behavior of gravity by
combining various models of quantum gravitational effects. In various quantum
gravitation models, such as string theory, loop quantum gravity and non-
commutative geometry, it is believed that there exists a minimum observable
length, and this minimum observable length should have the order of the Planck
scale. Considering a minimum observable length, a Hilbert space representation
of quantum mechanics has been formulated by Kempf et al. in Ref. [1], which
plays an important role on modifications of general relativity and black hole
physics for research programs in recent years. Furthermore, Nozari and Etemad
generalized the seminal work of Kempf et al. to the case that there is also a
maximal particles’ momentum, which resolved some shortcomings such as an
infinite amount of energy for a free test particle, and provided several novel
and interesting features 2 . Especially, in very recently, Nozari et al. have
addressed the origin of natural cutoff, including a minimal measurable length,
and suggested that quantum gravity cutoffs are global (topological) properties
of the symplectic manifolds 3 . Now, one of the research directions of quantum
gravity is to construct new theoretical models from the minimum observable
scale, including Double Special Relativity (DSR) 4 , Gravity’s Rainbow 5 ; 6 ,
Modified Dispersion Relation (MDR) 7  and Generalized Uncertainty Principle
(GUP) 1 ; 8 ; 9 ; 10  and so on. Due to the problems of semi-classical
tunneling radiation, the quantum gravitational effect caused by GUP is
considered to study the quantum tunneling radiation of black holes which leads
to a very interesting result: GUP can prevent a black hole from evaporating
completely, leaving remnant.

Recently, based on GUP, people began to study the quantum tunneling and
remnant of black hole. Combining the GUP with the minimum observed length into
the tunneling method of Parikh and Wilczek, Nozari and Mehdipour 11  have
successfully derived the quantum tunneling radiation of scalar particles from
a Schwarzschild black hole. They pointed out that GUP has an important
correction to the final state of black hole evaporation, and the black hole
cannot evaporate completely. Following that, the possible effects of natural
cutoffs as a minimal length, a maximal momentum, and a minimal momentum on the
quantum tunneling have been completely discussed 12 . Later, by utilizing the
deformed Hamilton-Jacobi equations, Benrong et al. pointed out that breaking
covariance in quantum gravity effective models is a key for a black hole to
have the remnant left in the evaporation 13 . In addition, in the frame of
improved exponential GUP 14 , the remnant of a Schwarzschild black hole has
been given at the end of the evaporation process 15 . On the other hand,
considering the quantum gravitational effect under the influence of GUP,
adopting the fermion tunneling method and the modified Dirac equation, Chen
and Wang etc. 16 ; 17 ; 18  elaborated on the fermion tunneling of black holes
and black rings. It is shown that GUP may slow down the temperature of the
black hole increase, and prevent the black hole from evaporating completely,
which leads to the existence of the minimum non-zero mass, that is, the black
hole remnant. Subsequently, the quantum tunneling radiation and remnant of
Gödel black hole and high dimensional Myers-Perry black hole have also been
deeply studied 19 ; 20 ; 21 . Above research proves that the black hole
remnant can exist, but related issues of the stability of remnant and residue
entropy have been comparatively less discussed for a Finsler black hole.

Although Einstein’s general relativity described by Riemannian geometry is one
of the most successful gravitational theories, it still has some problems in
explaining the accelerating expansion of the universe and establishing a
complete theory of quantum gravity. One has considered that the difficulties
caused by general relativity may have something to do with the mathematical
tools it uses. So, people try to establish a modified gravitational theory
described by Finsler geometry. Finsler geometry is the most general
differential geometry, which regards Riemannian geometry as its special case,
and it is just Riemannian geometry without quadratic restriction 22 . In
recent years, the application of Finsler geometry in black hole physics has
gradually aroused people’s interest. People began to construct field equations
of all kinds of Finsler spacetime. With the study of Finsler field equation,
people try to construct various solutions of Finsler black hole 22 ; 23 ; 24 ;
25 . The study of these black hole solutions makes us understand the physical
properties of Finsler spacetime more deeply. In this paper, based on GUP, we
take a simple Finsler black hole as an example, and investigate the remnant
and residue entropy from a Rutz-Schwarzschild black hole.

##  II Tunneling radiation of a scalar particle based on GUP

In this section, considering the effect of GUP, applying with the corrected
Hamilton-Jacobi equation, we will focus on investigating the scalar particle’s
tunneling radiation from a Rutz-Schwarzschild black hole. Under the frame of
Finsler geometry, Rutz constructed the generalized Einstein field equation and
derived a Finsler black hole solution. The metric (Rutz-Schwarzschild black
hole) is given by 24

$\displaystyle ds^{2}=-\left(1-\frac{{2M}}{r}\right)(1-\varepsilon
d\Omega/dt)dt^{2}+\left(1-\frac{{2M}}{r}\right)^{-1}dr^{2}+r^{2}d\Omega^{2}.$
(1)

Here $\varepsilon\ll 1$ is the Finsler perturbation parameter, and the line
element reduces to the Schwarzschild metric when $\varepsilon=0$. The solution
is a non-Riemannian solution, and the time component of the metric depends on
the tangent vector $d\Omega/dt$. It is obvious that the correction term
$\varepsilon d\Omega/dt$ remains while the mass vanishes and still exists when
$r\rightarrow\infty$. The metric is very different from the Schwarzschild
black hole, and it is very interesting to discuss on the scalar particle’s
tunneling radiation based on GUP.

By taking into account the effect of GUP, in a curved spacetime, the revised
Hamilton-Jacobi equation for the motion of scalar particles can be expressed
as 26

$\displaystyle
g^{00}\left({\partial_{0}S+eA_{0}}\right)^{2}+\left[{g^{kk}\left({\partial_{k}S+eA_{k}}\right)^{2}+m^{2}}\right]\times\left\\{{1-2\beta\left[{g^{jj}\left({\partial_{j}S}\right)^{2}+m^{2}}\right]}\right\\}=0.$
(2)

Here $\beta=\beta_{0}l_{p}^{2}/\hbar^{2}=\beta_{0}/M_{p}^{2}c^{2}$,
$\beta_{0}\left({\leq 10^{34}}\right)$ is a dimensionless constant, $l_{P}$
and $M_{P}$ are Planck length and Planck mass. Adopting rational
approximation, according to the line element and the modified Hamilton-
Jacobian equation, we can get the following motion equation of a scalar
particle

$\displaystyle-\left[\left(1-\frac{2M}{r}\right)\left(1-\varepsilon
d\Omega/dt\right)\right]^{-1}\left({\partial_{t}S}\right)^{2}-\left[{\left({1-\frac{2M}{r}}\right)\left({\partial_{r}S}\right)^{2}+\frac{1}{{r^{2}}}\left({\partial_{\theta}S}\right)^{2}}\right.$
$\displaystyle+\left.{\frac{1}{{r^{2}\sin^{2}\theta}}\left({\partial_{\varphi}S}\right)^{2}+m^{2}}\right]\left\\{{1-2\beta\left[{\left({1-\frac{2M}{r}}\right)\left({\partial_{r}S}\right)^{2}}\right.}\right.$
$\displaystyle+\left.{\left.{\frac{1}{{r^{2}}}\left({\partial_{\theta}S}\right)^{2}+\frac{1}{{r^{2}\sin^{2}\theta}}\left({\partial_{\varphi}S}\right)^{2}+m^{2}}\right]}\right\\}=0.$
(3)

Setting $S=-\omega t+W\left(r\right)+\Theta\left({\theta,\varphi}\right)$,
with $\omega$ standing for the energy of a scalar particle, and inserting the
action $S$ into Eq. (3), we have

$\displaystyle
2\beta\left({1-\frac{2M}{r}}\right)^{2}\left({\partial_{r}W(r)}\right)^{4}-\left({1-\frac{2M}{r}}\right)\left({\partial_{r}W(r)}\right)^{2}$
$\displaystyle+\left[{\left.{\left({1-\frac{2M}{r}}\right)\left({\partial_{r}W(r)}\right)^{2}+\frac{1}{{r^{2}}}\left({\partial_{\theta}\Theta(\theta,\varphi)}\right)^{2}+\frac{1}{{r^{2}\sin^{2}\theta}}\left({\partial_{\varphi}\Theta(\theta,\varphi)}\right)^{2}}\right]}\right.$
$\displaystyle\times\left[{4\beta\left({1-\frac{2M}{r}}\right)}{\left({\partial_{r}W(r)}\right)^{2}-1}\right]+\omega^{2}\left[{\left({1-\frac{2M}{r}}\right)\left({1-\varepsilon
d\Omega/dt}\right)}\right]^{-1}=-\lambda$ (4)

and

$\displaystyle{\rm{2}}\beta\left[{\left.{\frac{1}{{r^{2}}}\left({\partial_{\theta}\Theta(\theta,\varphi)}\right)^{2}+\frac{1}{{r^{2}\sin^{2}\theta}}\left({\partial_{\varphi}\Theta(\theta,\varphi)}\right)^{2}}\right]{\rm{=}}\lambda}\right.,$
(5)

For the Eq. (5), the mode of angular momentum of a tunneling particle is
associated with its components $\partial_{\theta}\Theta$ and
$\partial_{\varphi}\Theta$ as follows 27

$\displaystyle\frac{1}{{r^{2}}}\left({\partial_{\theta}S}\right)^{2}+\frac{1}{{r^{2}\sin^{2}\theta}}\left({\partial_{\varphi}S}\right)^{2}=L^{2}.$
(6)

Eqs. (5) and (6) yield

$\displaystyle 2\left({L^{2}}\right)^{2}=\frac{\lambda}{\beta}.$ (7)

Formula (7) shows that $\lambda$ is related to the angular momentum $L$ of the
scalar particle, and it is pointed out that the angular part will have an
effect on the tunneling radiation of the scalar particle. Thus, the equation
(4) can change to

$\displaystyle
2\beta\left({1-\frac{2M}{r}}\right)^{2}\left({\partial_{r}W(r)}\right)^{4}+\left({4m^{2}\beta+\sqrt{8\beta\lambda}-1}\right)$
$\displaystyle\times\left({1-\frac{2M}{r}}\right)\left({\partial_{r}W(r)}\right)^{2}+\left({2m^{2}\beta+\sqrt{8\beta\lambda}-1}\right)m^{2}$
$\displaystyle+\omega^{2}\left[{\left({1-\frac{2M}{r}}\right)\left({1-\varepsilon
d\Omega/dt}\right)}\right]^{-1}-\sqrt{\frac{\lambda}{2\beta}}+\lambda=0.$ (8)

The equation (8) is solved and the higher-order term of $\beta$ is ignored,
then $W(r)$ is taken as

$\displaystyle W(r)_{\pm}=$
$\displaystyle\pm\int{\frac{{\sqrt{\left({1-2M/r}\right)\left({1-\varepsilon
d\Omega/dt}\right)\left({m^{2}(1-2\beta
m^{2})-\lambda+\sqrt{\lambda/2\beta}}\right)+\omega^{2}}}}{{\sqrt{\left({1-\varepsilon
d\Omega/dt}\right)}\left({1-2M/r}\right)}}}$ (9)
$\displaystyle\times\left\\{{1+\beta\left[{m^{2}+\omega^{2}\left[{\left({1-\frac{2M}{r}}\right)\left({1-\varepsilon
d\Omega/dt}\right)}\right]^{-1}}\right]+\frac{{\beta\lambda}}{2}}\right\\}dr,$

where $+(-)$ represents the solution of the outgoing (ingoing) wave. The
Laurent series is expanded at the event horizon $r=r_{+}$, and the solution of
Eq. (9) is obtained by using the contour integral

$\displaystyle W\left({r_{+}}\right)_{\pm}=\pm i\frac{{2\pi
M\omega}}{{\sqrt{1-\varepsilon
d\Omega/dt}}}\left[{1+\frac{1}{2}\beta\left({3m^{2}+\lambda+\frac{{4\omega^{2}}}{{1-\varepsilon
d\Omega/dt}}}\right)+\sqrt{\frac{{\beta\lambda}}{8}}}\,\right]+{\rm{\texttt{real
part}}}.$

Applying the invariant tunneling rate under the canonical transformation 28 ;
29 , considering the influence of the time part on the tunneling rate, through
the Kruskal coordinate $(T,R)$, that is $T=e^{\kappa r^{*}}\sinh\left({\kappa
t}\right)$ and $R=e^{\kappa r^{*}}\cosh\left({\kappa t}\right)$, we get the
contribution of the extra imaginary part of the time segment and have
${\mathop{\rm Im}\nolimits}\,\,\omega
t_{out\left({in}\right)}=-{{\pi\omega}\mathord{\left/{\vphantom{{\pi\omega}{2\kappa}}}\right.\kern-1.2pt}{2\kappa}}$.
As a result, the total tunneling rate of a scalar particle passing through the
event horizon of Rutz-Schwarzschild black hole is as follows

$\displaystyle\Gamma$ $\displaystyle\propto$
$\displaystyle\exp\left\\{{\left[{{\mathop{\rm Im}\nolimits}\left({\omega
t_{out}}\right)+{\mathop{\rm Im}\nolimits}\left({\omega
t_{in}}\right)-{\mathop{\rm Im}\nolimits}\oint{Pdr}}\right]}\right\\}$ (11)
$\displaystyle=$
$\displaystyle\exp\left\\{{-4\pi\frac{{2M\omega}}{{\sqrt{1-\varepsilon
d\Omega/dt}}}\left[{1+\frac{1}{2}\beta\left({3m^{2}+\lambda+\frac{{4\omega^{2}}}{{1-\varepsilon
d\Omega/dt}}}\right)+\sqrt{\frac{{\beta\lambda}}{8}}}\right]}\right\\}.$

Compared with the Boltzmann factor
$\Gamma=\exp\left(-{{\omega\mathord{\left/{\vphantom{\omega
T}}\right.\kern-1.2pt}T}}\right)$, the corrected tunneling temperature of the
black hole is

$\displaystyle T$ $\displaystyle=$ $\displaystyle\frac{{\sqrt{1-\varepsilon
d\Omega/dt}}}{{8\pi
M}}\left[{1+\frac{1}{2}\beta\left({3m^{2}+\lambda+\frac{{4\omega^{2}}}{{1-\varepsilon
d\Omega/dt}}}\right)+\sqrt{\frac{{\beta\lambda}}{8}}}\,\right]^{-1}$ (12)
$\displaystyle=$ $\displaystyle
T_{H}\left[{1+\frac{1}{2}\beta\left({3m^{2}+\lambda+\frac{{4\omega^{2}}}{{1-\varepsilon
d\Omega/dt}}}\right)+\sqrt{\frac{{\beta\lambda}}{8}}}\,\right]^{-1}.$

Here $T_{H}=\sqrt{1-\varepsilon d\Omega/dt}/8\pi M$ is the Hawking
temperature. According to the first law of black hole thermodynamics, the
entropy of Rutz-Schwarzschild black hole is calculated as

$\displaystyle S=\int{\frac{{dM}}{T}}=\int{\frac{{8\pi
M}}{{\sqrt{1-\varepsilon
d\Omega/dt}}}}\left[{1+\frac{1}{2}\beta\left({3m^{2}+\lambda+\frac{{4\omega^{2}}}{{1-\varepsilon
d\Omega/dt}}}\right)+\sqrt{\frac{{\beta\lambda}}{8}}}\,\right]dM.$ (13)

From Eqs. (11)—(13), we obviously find that the scalar particle’s tunneling
rate, tunneling temperature and the entropy are all not only dependent on the
black hole mass $M$, tunneling scalar particle’s mass $m$ and energy $\omega$,
but also dependent on Finsler perturbation parameter $\varepsilon$, GUP
parameter $\beta$ and angular momentum parameter $\lambda$.

##  III Remnant and entropy based on thermodynamics phase transition

On the basis of the above scalar particle’s tunneling radiation, considering
the GUP, now we focus on discussing on the remnant and entropy at the end of
evaporation. Since all the tunneling particles at the event horizon can be
regarded as massless, the mass of scalar particles is no longer considered in
the following process. According to the uncertainty relation $\Delta
p\geq{\hbar\mathord{\left/{\vphantom{\hbar{\Delta
x}}}\right.\kern-1.2pt}{\Delta x}}$ and the lower limit of tunneling particle
energy 30 ; 31  $\omega\geq{\hbar\mathord{\left/{\vphantom{\hbar{\Delta
x}}}\right.\kern-1.2pt}{\Delta x}}$, near the event horizon, the uncertainty
of the position can be taken as the radius of the black hole 30 ; 31 , that is
$\Delta x\approx r_{BH}=r_{+}$. Consequently, the tunneling temperature of the
black hole evaporating to Planck scale is

$\displaystyle T$ $\displaystyle=$ $\displaystyle\frac{{\sqrt{1-\varepsilon
d\Omega/dt}}}{{8\pi
M}}\left\\{{1+\frac{1}{2}\beta\left({\lambda+\frac{{4\omega^{2}}}{{1-\varepsilon
d\Omega/dt}}}\right)+\sqrt{\frac{{\beta\lambda}}{8}}}\right\\}^{-1}$ (14)
$\displaystyle=$ $\displaystyle
T_{H}\left\\{{1-\frac{1}{2}\beta\left({\lambda+\frac{{4\hbar^{2}}}{{\left({1-\varepsilon
d\Omega/dt}\right)r_{+}^{2}}}}\right)-\sqrt{\frac{{\beta\lambda}}{8}}}\right\\}.$

It can be seen that the modified tunneling temperature near Planck scale is
related to the properties of the background spacetime of Finsler black hole,
the energy of a tunneling particle and the parameter of quantum gravitational
effect. When the radius of the black hole satisfies

$\displaystyle r_{+}<\sqrt{\frac{{8\beta\hbar^{2}}}{{\left({1-\varepsilon
d\Omega/dt}\right)\left({4-2\beta\lambda-\sqrt{2\beta\lambda}}\right)}}}\,,$
(15)

the revised tunneling temperature $T<0$ violates the third law of
thermodynamics. This means that, by considering the effect of GUP, the
evaporation will stop when the tunneling temperature is infinitely close to
absolute zero, which leads to a minimum radius, namely

$\displaystyle r_{i}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{{8\beta\hbar^{2}}}{{\left({1-\varepsilon
d\Omega/dt}\right)\left({4-2\beta\lambda-\sqrt{2\beta\lambda}}\right)}}}$ (16)
$\displaystyle=$
$\displaystyle\ell_{p}\sqrt{\frac{{8\hbar^{2}\beta_{0}}}{{\left({1-\varepsilon
d\Omega/dt}\right)\left({4\hbar^{2}-2\lambda\beta_{0}\ell_{p}^{2}-\ell_{p}\hbar\sqrt{2\lambda\beta_{0}}}\right)}}}\,.$

The expression (16) is represented by the mass. By using the relation between
event horizon and mass $r_{+}=2M$, near the Planck scale, the tunneling
temperature can be expressed as

$\displaystyle T=\frac{{\sqrt{1-\varepsilon d\Omega/dt}}}{{8\pi
GM}}\left\\{{1-\frac{1}{2}\beta\left({\lambda+\frac{{\hbar^{2}}}{{\left({1-\varepsilon
d\Omega/dt}\right)G^{2}M^{2}}}}\right)-\sqrt{\frac{{\beta\lambda}}{8}}}\right\\}.$
(17)

In order to ensure the temperature $T\geq 0$, the mass of the black hole
satisfies

$\displaystyle M\geq\sqrt{\frac{{2\beta\hbar^{2}}}{{\left({1-\varepsilon
d\Omega/dt}\right)\left({4-2\beta\lambda-\sqrt{2\beta\lambda}}\right)G^{2}}}}\,,$
(18)

which implies the minimum mass of a black hole

$\displaystyle M_{min}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{{2\beta\hbar^{2}}}{{\left({1-\varepsilon
d\Omega/dt}\right)\left({4-2\beta\lambda-\sqrt{2\beta\lambda}}\right)G^{2}}}}$
(19) $\displaystyle=$ $\displaystyle
M_{p}\sqrt{\frac{{2\beta_{0}\hbar^{2}}}{{c^{2}\left({1-\varepsilon
d\Omega/dt}\right)\left({4-2\lambda\beta_{0}/M_{p}^{2}c{{}^{2}}-\sqrt{2\lambda\beta_{0}/M_{p}^{2}c{{}^{2}}}}\right)}}}\,.$

The value is the Rutz-Schwarzschild black hole’s remnant, that is
$M_{res}=M_{min}$. In order to explain the problem better, we can analyze heat
capacity $C$, which is

$\displaystyle C=T\left({\frac{{\partial S}}{{\partial
T}}}\right)=T\frac{{\partial S}}{{\partial M}}\frac{{\partial M}}{{\partial
T}}=T\frac{{\partial S}}{{\partial M}}\left({\frac{{\partial T}}{{\partial
M}}}\right)^{-1}=\frac{N}{F}.$ (20)

Here $S$ is the modified black hole entropy, at the Planck scale it can be
rewritten as

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int{\frac{{dM}}{T}}=\int{\frac{{8\pi GM}}{{\sqrt{1-\varepsilon
d\Omega/dt}}}}\left\\{{\left({1+\frac{1}{2}\beta\lambda+\sqrt{\frac{{\beta\lambda}}{8}}}\right)+\left({\frac{{\beta\hbar^{2}}}{{2\left({1-\varepsilon
d\Omega/dt}\right)G^{2}M^{2}}}}\right)}\right\\}dM$ (21) $\displaystyle=$
$\displaystyle\frac{{4\pi GM^{2}}}{{\sqrt{1-\varepsilon
d\Omega/dt}}}\left({1+\frac{1}{2}\beta\lambda+\sqrt{\frac{{\beta\lambda}}{8}}}\right)+\frac{{4\pi\beta\hbar^{2}}}{{G\left({1-\varepsilon
d\Omega/dt}\right)^{{3\mathord{\left/{\vphantom{32}}\right.\kern-1.2pt}2}}}}\ln
M,$

and the coefficients $N$ and $F$ are, respectively,

$\displaystyle N=\left[{2G^{2}M^{2}\left({1-\varepsilon
d\Omega/dt}\right)\left({1-\frac{1}{2}\beta\lambda-\sqrt{\frac{{\beta\lambda}}{8}}}\right)-\beta\hbar^{2}}\right]\times\left[{16\pi
G^{3}M^{4}\sqrt{1-\varepsilon d\Omega/dt}}\right]$ (22)

and

$\displaystyle F=$ $\displaystyle\left[{2G^{2}M^{2}\left({1-\varepsilon
d\Omega/dt}\right)\left({1+\frac{1}{2}\beta\lambda+\sqrt{\frac{{\beta\lambda}}{8}}}\right)+\beta\hbar^{2}}\right]$
(23)
$\displaystyle\times\left[{3\beta\hbar^{2}-2G^{2}M^{2}\left({1-\frac{1}{2}\beta\lambda-\sqrt{\frac{{\beta\lambda}}{8}}}\right)\left({1-\varepsilon
d\Omega/dt}\right)}\right].$

From the expression (20), we find

$\displaystyle
M=M_{Cr}=M_{p}\sqrt{\frac{{6\beta_{0}\hbar^{2}}}{{c^{2}\left({1-\varepsilon
d\Omega/dt}\right)\left({4-2\lambda\beta_{0}/M_{p}^{2}c{{}^{2}}-\sqrt{2\lambda\beta_{0}/M_{p}^{2}c{{}^{2}}}}\right)}}}\,,\,\,\,C\to
0\,,$ (24) $\displaystyle M\to
M_{min}=M_{res}=M_{p}\sqrt{\frac{{2\beta_{0}\hbar^{2}}}{{c^{2}\left({1-\varepsilon
d\Omega/dt}\right)\left({4-2\lambda\beta_{0}/M_{p}^{2}c{{}^{2}}-\sqrt{2\lambda\beta_{0}/M_{p}^{2}c{{}^{2}}}}\right)}}},\,\,\,C\to
0\,.$

The expression (25) is consistent with the expression (19). From above
equations we can see that, for the Rutz-Schwarzschild black hole, considering
the tunneling of scalar particles, when the black hole evaporates to the
Planck scale, the phase transition takes place, leaving a stable remnant.
Then, based on the expressions (14), (17), (20) and (21), we draw the
following thermodynamic curves 1—5, and further analyze the entropy and
remnant in detail. We set $\gamma=\varepsilon d\Omega/dt$ and
$M_{p}=c=\hbar=1$ in all of the following drawings for research convenience.

As Figure1 and Figure 2 show, in a large range of radius and mass, for the
cases $\beta=0$ and $\beta=1$, the tunneling temperatures tend to be
consistent. But as the radius (or mass) closes to the critical value $r_{cr}$
(or $M_{cr}$), these curves become markedly different. The tunneling
temperature reaches the maximum value at $r_{cr}$ (or $M_{cr}$) for the case
of $\beta=1$, then it decreases with the decrease of radius (or mass).
Finally, the tunneling temperature tends to zero when the radius (or mass)
attains to the minimum value $r_{i}$ (or $M_{res}$) for the case of $\beta=1$,
which leads to remnant. We can also explain it through the Figure 3.

Figure 1: The figure shows the temperature $T$ versus the radius of event
horizon $r_{+}$ for varying $\beta$. Above blue dashed curve is $\gamma=0.1$
and $\beta=0$. Below orange curve is $\gamma=0.1$ and $\beta=1$. Figure 2: The
figure shows the temperature $T$ versus the mass $M$ for varying $\beta$.
Above purple dashed curve is $\gamma=0.1$ and $\beta=0$. Below black curve is
$\gamma=0.1$ and $\beta=1$.

In Figure 3, the heat capacity monotonically increases with the mass
increasing for the case of $\beta=0$, and it is always negative. However, for
the case of $\beta=1$, we find that, at $M=M_{cr}$, heat capacity is
divergent, which means the existence of phase transition. That is the black
hole undergoes a phase transition from $C<0$ (unstable hole) to $C>0$ (stable
hole), which also corresponds to the phase transitions in Figure1 and Figure 2
at the critical value $r_{cr}$ and $M_{cr}$. Furthermore, we can see that the
mass tends to minimum value (namely $M_{min}=M_{res}$) when $C\rightarrow 0$.
In addition, we also note that the same problem is treated in noncommutative
geometry in Ref. [32] (see Figure 7). The result shows that the heat capacity
tends to zero when the mass approaches to a certain value, but some unusual
feature is also presented.

Figure 3: The figure shows the heat capacity $C$ versus the mass $M$ for
varying $\beta$. The blue dashed curve is $\gamma=0.1$ and $\beta=0$. The
orange curve is $\gamma=0.1$ and $\beta=1$.

In Figure 4 and Figure 5, the crucial difference is shown for the two curves.
To clearly see the difference between them, we can refer to Figure 5. The
traditional entropy is zero as the mass becomes zero for the case of
$\beta=0$. However, by considering GUP, the entropy is also revised for the
case of $\beta=1$, and there exits minimum $S_{min}$ (namely residual entropy
$S_{res}=S_{min}$) at the remnant, which means the entropy is no longer zero
in the final stages of black hole evaporation.

Figure 4: The figure shows the entropy $S$ versus the mass $M$ for varying
$\beta$ on a large scale of entropy. The blue dashed curve is $\gamma=0.1$ and
$\beta=0$. The orange curve is $\gamma=0.1$ and $\beta=1$. Figure 5: The
figure shows the entropy $S$ versus the mass $M$ for varying $\beta$ within a
small scale of entropy. The blue dashed curve is $\gamma=0.1$ and $\beta=0$.
The orange curve is $\gamma=0.1$ and $\beta=1$.

To see clearly the Finsler perturbation parameter and angular momentum
parameter effect on remnant and residual entropy, we present the following
tables for different $\gamma$ and $\lambda$ when $\beta=1$. From Tables 1 and
Tables 2, we can find that remnant and residual entropy increase as Finsler
perturbation parameter increase, and they are also increase as angular
momentum parameter increase.

Table 1: Remnant and residual entropy with $\lambda=0.001$ for different $\gamma$ $\gamma$ | $M_{res}$ | $S_{res}$
---|---|---
0.1 | 0.749748 | 3.29388
0.01 | 0.714856 | 1.90766
0.0001 | 0.711309 | 1.76666
Table 2: Remnant and residual entropy with $\gamma=0.1$ for different $\lambda$ $\lambda$ | $M_{res}$ | $S_{res}$
---|---|---
0.5 | 1.054090 | 22.8521
0.1 | 0.814124 | 7.17324
0.01 | 0.760867 | 3.95553

In addition , when $\beta=0$ and $\varepsilon=0$ (namely $\gamma=0$), the
results reduce to that of Schwarzschild black hole without GUP . In this case,
the black hole can radiate continuously until the mass and entropy decrease to
zero. In the case of $\beta=1$ and $\varepsilon=0$, the results reduce to that
of Schwarzschild black hole with GUP. From Eq. (21) and (25), we can see that,
the remnant and the residue entropy in the frame of Finsler geometry are
larger than those of Schwarzschild black hole. Moreover , we also see that the
influence of Finsler parameter on remnant and residue entropy from Table 1.

##  IV Discussion on remnant based on Parikh-Wilczek tunneling

Based on GUP, using Parikh-Wilczek tunneling method, we continue to explore
the quantum tunneling and remnant of a Rutz-Schwarzschild black hole in the
frame of Finsler geometry. The Rutz-Schwarzschild metric in the Painleve
coordinate system can be rewritten as

$\displaystyle ds^{2}=-\left(1-\frac{{2M}}{r}\right)(1-\varepsilon
d\Omega/dt)dt^{\prime 2}+2\sqrt{\frac{{2M}}{r}(1-\varepsilon
d\Omega/dt)}dt^{\prime}dr+dr^{2}+r^{2}d\Omega^{2},$ (26)

which is obtained by the coordinate transformation

$\displaystyle dt\rightarrow
dt^{\prime}-\frac{\sqrt{2M/r}}{1-2M\sqrt{1-\varepsilon d\Omega/dt}/r}dr.$ (27)

We can derive the radial null geodesics

$\displaystyle\dot{r}\equiv\frac{dr}{dt^{\prime}}=\sqrt{1-\varepsilon
d\Omega/dt}\left(\pm 1-\sqrt{\frac{2M}{r}}\,\right),$ (28)

where $+(-)$ represents the solution of the outgoing (ingoing) geodesics. Now,
we consider the influence of GUP with a minimal length on quantum tunneling.
The expression GUP with a minimal length is 12

$\displaystyle\Delta x\Delta p\geq\hbar\left[1+\beta(\Delta p)^{2}\right],$
(29)

and the generalized energy 11  is

$\displaystyle\omega=E(1+\beta E^{2}),$ (30)

where $\beta=\alpha^{2}l_{p}^{2}$. From the GUP expression, the revised
commutation relation between the radial coordinate and the conjugate momentum
becomes

$\displaystyle[r,p_{r}]=i\hbar(1+\beta p_{r}^{2}).$ (31)

In the classical limit, above commutation relation can be replaced by the
following poisson bracket

$\displaystyle\left\\{r,p_{r}\right\\}=1+\beta p_{r}^{2}.$ (32)

In order to obtain the imaginary part of the action, we use the following
deformed Hamiltonian equation

$\displaystyle\dot{r}=\left\\{r,H\right\\}=\left\\{r,p_{r}\right\\}\left.\frac{dH}{dp_{r}}\right|_{r}.$
(33)

In the framework of Parikh and Wilczek’s tunneling radiation, the imaginary
part of the action can be given by

$\displaystyle
ImI=Im\int_{r_{in}}^{r_{out}}p_{r}dr=Im\int_{r_{in}}^{r_{out}}\int_{0}^{p_{r}}p^{\prime}_{r}dr.$
(34)

According to the work in Ref. [12], the Hamiltonian is $H=M-\omega^{\prime}$,
$p_{r}^{2}\approx\omega^{\prime 2}$, $p_{r}\approx\omega^{\prime}$. Thus, by
eliminating the momentum in the favor of the energy in Eq. (34), we have

$\displaystyle{\mathop{\rm Im}\nolimits}I$ $\displaystyle=$
$\displaystyle{\mathop{\rm
Im}\nolimits}\int_{M}^{M-\omega}{\int_{r_{in}}^{r_{out}}{\frac{{1+\beta\omega^{\prime
2}}}{{\dot{r}}}}}drdH^{\prime}$ (35) $\displaystyle=$
$\displaystyle{\mathop{\rm
Im}\nolimits}\int_{M}^{M-\omega}{\int_{r_{in}}^{r_{out}}{\frac{{1+\beta\omega^{\prime
2}}}{{\left({1-\varepsilon
d\Omega/dt}\right)\left({1-\frac{{2(M-\omega^{\prime})}}{r}}\right)}}}}drd(M-\omega^{\prime})$
$\displaystyle=$ $\displaystyle\frac{{4\pi M\omega}}{{\left({1-\varepsilon
d\Omega/dt}\right)}}\left[{1-\frac{\omega}{{2M}}+\beta\omega^{2}\left({\frac{1}{3}-\frac{\omega}{{4M}}}\right)}\right].$

The tunneling rate with effect of GUP with a minimal length is therefore

$\displaystyle\Gamma\sim\exp\left({-2{\mathop{\rm
Im}\nolimits}S}\right)=\exp\left[{-\frac{{8\pi ME}}{{\left({1-\varepsilon
d\Omega/dt}\right)}}+\frac{{4\pi E^{2}}}{{\left({1-\varepsilon
d\Omega/dt}\right)}}-\frac{{32\pi ME^{3}}}{{3\left({1-\varepsilon
d\Omega/dt}\right)}}}\right.$
$\displaystyle\left.{\begin{array}[]{*{20}c}{}\hfil&{}\hfil\\\
\end{array}+\frac{{10\pi\beta E^{4}}}{{\left({1-\varepsilon
d\Omega/dt}\right)}}+{\rm O}\left({\beta^{2}}\right)}\right]=\exp(\Delta S),$
(37)

where $\Delta S$ is the difference in black hole entropies before and after
emission 11 ; 12 ; 33 . It is shown that Parikh-Wilczek’s procedure is also
valid for the Rutz-Schwarzschild metric. When $\beta=0$ and $\varepsilon=0$,
the result reduces to the result of Schwarzschild black hole, which is the
same with that derived by Parikh and Wilczek 33 . Note that, for a
Schwarzschild black hole, with Parikh-Wilczek tunneling mechanism, the quantum
tunneling of massless particles from black hole horizon has been studied based
on GUP 11 ; 12 . It is obvious that our result coincides with the one in Ref.
[12] when $\varepsilon=0$. We find that the above equation (36) gives
tunneling rate’s correction which caused by the varied background spacetime
and GUP. When $\beta=0$, the tunneling temperature

$\displaystyle T=\left(1+\frac{E}{{2M}}\right)T_{H}$ (38)

is higher than the original temperature $T_{H}$. This predicts the varied
spacetime can accelerate the evaporation and the remnant does not arise. when
GUP is taken into account, the tunneling temperature

$\displaystyle T=\left(1+\frac{E}{{2M}}-\frac{4}{3}\beta
E^{2}+\frac{5}{{4M}}\beta E^{3}\right)T_{H}$ (39)

is lower than the original Hawking temperature $T_{H}$, which implies that GUP
can slow down the increase of the Hawking temperature and stop the black hole
from completely evaporating. As a result, a remnant of the black hole is left
at the end of evaporation.

##  V Conclusion

In conclusion, employing the tunneling radiation of a scalar particle and
generalized uncertainty principle, we study the remnant and residue entropy
from a Rutz-Schwarzschild black hole based on black hole thermodynamic.
Firstly, based on the Hamilton-Jacobi equation revised by GUP, we present the
modified tunneling temperature-uncertainty relation and modified entropy-
uncertainty relation by using quantum tunneling method. Then, using the
generalized uncertainty principle, we calculate the remnant and residual
entropy after evaporation when the black hole reaches the Planck scale.
Finally, based on black hole thermodynamic phase transition, a detailed
analysis of whether there is a stable remnant and residual entropy in the
final stage of evaporation is given, that is, the thermodynamic stability of a
black hole is related to its thermal capacity and temperature. In order to
ensure the thermal stability of a black hole, when the black hole evaporates
to the Planck scale, the black hole with negative heat capacity must be
transformed into a black hole with positive heat capacity. As a result, at
remnant, the modified tunneling temperature and heat capacity tend to zero,
and modified entropy reaches the minimum value, which imply the black holes
are in thermal equilibrium with the outside environment. In addition, the
effects of the Finsler perturbation parameter and angular momentum parameter
on remnant and residual entropy are also discussed.

The emergence of Finsler black hole solution infuses new vitality into general
relativity and puts forward a new research and development idea for black hole
physics theory. In this paper, we consider a simple Finsler black hole. For
the Rutz-Schwarzschild black hole, we investigate the remnant and residue
entropy based on the scalar particles’ tunneling radiation via semi-classical
Hamilton-Jacobi method. In view of the size of the Finsler perturbation
parameter $\varepsilon<3.15\times 10^{-4}s$ in geometrical units 24 , the
small physical quantity is ignored and we approximately adopt Hamilton-Jacobi
equation based on GUP to study tunneling radiation from horizon. Strictly
speaking, we should apply new Hamilton-Jacobi equation in the framework of
complicated Finsler geometry to discuss the related problems. Up to now,
Finsler gravity theory has not been established completely. Here, adopting the
approximate method we only to understand simply some quantum results of
Finsler gravity. We hope that our research can provide some enlightenments for
the thermodynamic evolution of black hole physics.

We also note that, the expression of GUP is not unique, which gives rise to
different correction on tunneling radiation. Therefore, the remnant and
residue entropy should also be distinct from that in the frameworks of other
GUPS. We are going to discuss the related issues in further.

## Acknowledgements

This work is supported in part by the National Natural Science Foundation of
China (Grant No. 11703018) and Natural Science Foundation of Liaoning
Province, China (Grant No. 20180550275).

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QMUL-PH-21-02

Relativistic membrane solutions in AdS4

David Vegh

Centre for Research in String Theory, School of Physics and Astronomy
Queen Mary University of London, 327 Mile End Road, London E1 4NS, UK

email<EMAIL_ADDRESS>

In this note we discuss various classical membrane solutions in AdS4
spacetime: simple embeddings given by polynomials in ambient space, solutions
with non-linear waves, and piecewise linear solutions.

###### Contents

  1. 1 Introduction
    1. 1.1 Coordinate systems
    2. 1.2 Action and equation of motion
  2. 2 Simple membrane solutions in AdS4
    1. 2.1 Linear surfaces
    2. 2.2 Quadric surfaces
    3. 2.3 Cubic surfaces
  3. 3 Waves traveling in one direction
  4. 4 Attaching patches
  5. 5 Trivalent vertex on the boundary
  6. 6 Crossing shockwaves
  7. 7 Collision of three shockwaves
  8. 8 Discussion

##  1 Introduction

The dynamics of classical Nambu-Goto strings in (2+1)-dimensional anti-de
Sitter (AdS) spacetime can be described in many different ways. The system is
integrable [1, 2, 3] which allows for an exact discretization of the equation
of motion [4, 5, 6, 7, 8, 9]. Time-evolution can be computed by solving
discrete equations which is advantageous because numerical errors do not
arise. The corresponding string embedding is a piecewise linear segmented
string.

A natural question is whether such constructions can be generalized to higher
dimensional objects, for instance, to membranes moving in AdS4. Even though in
this case integrability techniques are not readily available, the membrane
does ‘inherit’ certain solutions which are higher-dimensional analogs of
segmented strings. The purpose of this paper is to study these (and some
other) solutions.

In the remainder of the Introduction, we describe the AdS geometry and the
membrane action along with its equation of motion. In Section 2, we give a few
simple membrane embeddings. The characterizing feature of these is that they
are given by polynomial equations in ambient space (linear, quadric, cubic).
Section 3 studies membrane solutions with non-linear waves traveling in one
dimension. In Section 4 we describe how to attach linear patches. The
attaching curve is a shockwave on the membrane where the membrane normal
vector jumps. Section 5 and 6 discuss valence three and four vertices that may
arise when different shockwaves cross. Section 7 discusses the collision of
three shockwaves on the brane.

###  1.1 Coordinate systems

The canonical immersion of AdS4 into the ambient ${\mathbb{{R}}}^{2,3}$ linear
space is obtained by considering the hyperboloid given by the equation

$X\cdot X\equiv-
X_{-1}^{2}-X_{0}^{2}+X_{1}^{2}+X_{2}^{2}+X_{3}^{2}=-R^{2}\qquad\qquad
X\in{\mathbb{{R}}}^{2,3}.$ (1)

Henceforth, we consider AdS spacetimes of unit radius and thus we set $R=1$.
Global AdS wraps this surface infinitely many times and can be recovered by
considering the universal cover of the hyperboloid: the angle on the $X_{-1}$,
$X_{0}$ plane has to be ‘unwrapped’ and this non-compact direction is then
identified with global AdS time. The metric on AdS4 is the induced metric in
${\mathbb{{R}}}^{2,3}$ and its isometry group is $SO(2,3)$. In the following,
vectors in the ambient (or embedding) space will be denoted by capital
letters.

In the following, we will also use coordinates which only cover the Poincaré
patch. They will be called Poincaré coordinates and can be defined by

$(t,\,x,\,y,\,z)=\left({X_{0}\over X_{3}-X_{-1}},\ {X_{1}\over X_{3}-X_{-1}},\
{X_{2}\over X_{3}-X_{-1}},\ {1\over X_{3}-X_{-1}}\right).$

The metric written in Poincaré coordinates is clearly conformally flat

$ds^{2}={-dt^{2}+dx^{2}+dy^{2}+dz^{2}\over z^{2}}$ (2)

Gravity ‘acts’ in the $z$ direction. Massive objects never reach the boundary
at $z=0$. They fall ‘down’ in the positive $z$ direction.

Another coordinate system $\\{\tau,r,\theta,\phi\\}$ is obtained by the
transformation

$X=\left(\begin{array}[]{c}\cosh r\cos\tau\\\ \cosh r\sin\tau\\\ \sinh
r\sin\theta\sin\phi\\\ \sinh r\sin\theta\cos\phi\\\ \sinh
r\cos\theta\end{array}\right)$ (3)

As mentioned before, $\tau$ is the (non-compact) global time direction. It is
easy to see that this parametrization satisfies (1). In these coordinates, the
metric becomes

$ds^{2}=-\cosh^{2}r\,{d\tau}^{2}+{dr}^{2}+\sinh^{2}r\left({d\theta}^{2}+{d\phi}^{2}\sin^{2}\theta\right)$
(4)

For purposes of visualization, a somewhat nicer radial coordinate is
$\rho\in[0,1)$ defined by

$\rho=\sqrt{\cosh r-1\over\cosh r+1}$

The metric in terms of this coordinate is

$ds^{2}=-\left({1+\rho^{2}\over
1-\rho^{2}}\right)^{2}\,d\tau^{2}+{4\over(1-\rho^{2})^{2}}\left(d\rho^{2}+\rho^{2}\,d\Omega^{2}\right)$

where $d\Omega^{2}\equiv{d\theta}^{2}+{d\phi}^{2}\sin^{2}\theta$. A timeslice
of global AdS is then just a ball of unit radius. We will call these variables
global coordinates.

###  1.2 Action and equation of motion

Our starting point will be the (2+1)-dimensional membrane action coupled to a
background three-form field $C$

$S=-\int d^{3}\sigma\sqrt{-g}+\kappa\int
d^{3}\sigma\,\partial_{a}Y^{M}\partial_{b}Y^{N}\partial_{c}Y^{L}\epsilon^{abc}C_{MNL}$
(5)

Since we will be considering classical solutions only, for simplicity, we have
rescaled the action such that the brane tension is set to one. Here
$\sigma^{\mu}$ and $Y^{M}$ are coordinates on the worldvolume and AdS4,
respectively. The induced metric on the brane is denoted $g_{\mu\nu}$. We
choose a background three-form field such that its field strength $dC$ is the
volume form of AdS4. This choice preserves all the symmetries of the target
space. $\kappa$ parametrizes the strength of the coupling of the membrane to
the pull-back of the three-form.

If we use Poincaré coordinates for AdS, we have $\\{Y^{M}\\}=\\{t,x,y,z\\}$.
Worldvolume coordinates can be chosen such that
$\\{\sigma^{1},\sigma^{2},\sigma^{3}\\}=\\{t,x,y\\}$. The embedding will be
characterized by a single function $z(t,x,y)$ which gives the brane position
in the $z$ (radial) direction. On the Poincaré patch, the antisymmetric three-
form can be chosen

$C={dt\wedge dx\wedge dy\over z^{3}}.$ (6)

The equation of motion is

$(1+z_{,\alpha}z^{,\alpha}){z_{,\beta}}^{,\beta}-z^{,\alpha}z^{,\beta}z_{,\alpha\beta}+{3\over
z}(1+z_{,\alpha}z^{,\alpha})\pm{3\kappa\over
z}(1+z_{,\alpha}z^{,\alpha})^{3\over 2}=0$ (7)

where $\alpha,\beta\in\\{t,x,y\\}$, comma indicates partial derivative (e.g.
$z_{,\alpha\beta}\equiv\partial_{\alpha}\partial_{\beta}z$), and indices are
lowered/raised using a flat metric $\eta=\textrm{diag}(-1,1,1)$. The first two
terms would be the equation of motion in Minkowski target space. In AdS4,
these are modified by the presence of the third term. The sign of the last
term depends on whether the surface is in a normal position or ‘upside-down’.
(This can happen when the solution is a double-valued function
$z=z_{\pm}(t,x,y)$. In this case, the sign will be different for the two
branches.)

In the following, we will be interested in solutions to this non-linear
partial differential equation.

Figure 1:  Linear solution with $\kappa=0$. The shape of the brane at any
given time on the Poincaré patch is a hemisphere. The equator lies on the
boundary of AdS. The radius $R$ of the hemisphere changes in time according to
$R^{2}=R_{0}^{2}+(t-t_{0})^{2}$.

##  2 Simple membrane solutions in AdS4

In this section we are going to discuss a few simple solutions to the membrane
equation of motion in (7). These solutions can be defined in the
${\mathbb{{R}}}^{2,3}$ ambient space by a polynomial equation. A
characterizing feature is the degree of the (homogeneous) polynomial. In the
following subsections, we list classes of such solutions according to their
degree.

###  2.1 Linear surfaces

The simplest solution lies in a linear subspace in the ambient space,

$X\cdot N=\lambda$ (8)

where $X,N\in{\mathbb{{R}}}^{2,3}$. Here $X$ is a point in the embedding, $N$
is a fixed unit-length vector and $\lambda$ is a constant. This surface has to
be intersected with the hyperboloid in (1). The resulting three-dimensional
subspace is the worldvolume of the membrane.

It is easy to see that on the Poincaré patch (8) translates into either (i)
planes at a fixed angle

$z(t,x,y)={x\over\lambda},$ (9)

or (ii) shrinking and expanding spheres

$z(t,x,y)=\lambda+\sqrt{1+t^{2}-x^{2}-y^{2}+\lambda^{2}}.$ (10)

and their shifted, rotated, and boosted cousins.

The equation of motion in (7) is satisfied if

$\lambda=\pm{\kappa\over\sqrt{1-\kappa^{2}}}$ (11)

For $\kappa=0$ this gives $\lambda=0$ and $N$ is the constant normal vector of
the membrane. For $\kappa\neq 0$, $N$ may be called a generalized normal
vector.

The spheres are centered on the boundary if $\kappa=0$. An example is shown in
Figure 1. The three-dimensional picture is a timeslice of the Poincaré patch.
At any given instance the shape of the membrane is a hemisphere. The black
circle (the equator) lies on the $z=0$ boundary of AdS.

###  2.2 Quadric surfaces

Certain quadric surfaces (a.k.a. ‘hypercycles’) in the ambient space also
satisfy the equation of motion. In the following, we give examples for such
curves in $SO(2,3)$ frames in which the defining equations are the simplest.

  * •

There are degree-2 curves which in a frame have the form

$X_{1}^{2}+X_{2}^{2}=\lambda$ (12)

where $\lambda>0$. On the Poincaré patch we get

$z(t,x,y)=\sqrt{x^{2}+y^{2}\over\lambda}$ (13)

This is a nothing but a static cone. It can be thought of as a ‘fat string’
which hangs from the boundary point at $x=y=z=0$. In global coordinates, the
brane has a cigar shape as seen in Figure 2.

The embedding satisfies the equation of motion if $\lambda$ can be set such
that

$\kappa=-{\lambda+{1\over 3}\over\sqrt{\lambda(1+\lambda)}}$ (14)

Note that there is no surface corresponding to $\kappa=0$.

  * •

Another curve is given by

$X_{-1}^{2}+X_{2}^{2}=\lambda$ (15)

For the embedding on the Poincaré patch we obtain two solutions

$z_{\pm}(t,x,y)=\sqrt{t^{2}-x^{2}-y^{2}-1-2\lambda\pm
2\sqrt{(1+\lambda)x^{2}+\lambda(1-t^{2}+y^{2}+\lambda)}}$ (16)

which satisfy the equation of motion if $\lambda$ is set such that the
equation

$\kappa=\mp{\lambda+{1\over 3}\over\sqrt{\lambda(1+\lambda)}}$ (17)

is satisfied. Assuming a fixed positive sign in (7), here minus sign should be
taken for the $z_{+}$ solution and plus sign for $z_{-}$.

  * •

Yet another curve is given by

$X_{-1}^{2}+X_{0}^{2}=\lambda$ (18)

The Poincaré embedding is

$z_{\pm}(t,x,y)=\sqrt{t^{2}-x^{2}-y^{2}-1-2\lambda\pm
2\sqrt{-(1+\lambda)t^{2}+\lambda(1+x^{2}+y^{2}+\lambda)}}$ (19)

Examples for these solutions are plotted in Figure 3.

Figure 2:  A quadric membrane solution on a timeslice of global AdS4. The
(yellow) membrane connects two antipodal points on the boundary of AdS (i.e.
the surface of the ball). For such a static solution to exist, the coupling to
the three-from field has to be non-zero. This provides an extra force on the
membrane and keeps it from collapsing.

Figure 3:  Quadric membrane solutions on the Poincaré patch. The top of the
bounding boxes are the $z=0$ boundary planes of AdS.

Figure 4:  A timeslice of the cubic solution in global AdS. The (yellow)
membrane rotates about the vertical axis as a rigid body.

###  2.3 Cubic surfaces

A cubic solution to the EOM with $\kappa=0$ is given by the (Wick-rotated)
Cartan-Münzner polynomial

$X_{3}^{3}-3\sqrt{3}X_{-1}X_{0}X_{2}+{3\sqrt{3}\over
2}X_{1}(X_{-1}^{2}-X_{0}^{2})-{3\over
2}X_{3}\left[X_{-1}^{2}+X_{0}^{2}+2(X_{1}^{2}+X_{2}^{2})\right]=0$ (20)

In global AdS coordinates the embedding is given by111The equation is linear
in $R\equiv\tanh^{2}r$. Using this variable as a new radial coordinate, we can
parametrize membranes via $R=R(\tau,\theta,\phi)$. The equation of motion with
no coupling to the three-form turns out to be  $\displaystyle
0=4\sin^{3}(\theta)\left(\partial_{t}^{2}R+2\right)R^{4}-2\sin(\theta)\partial_{\phi}R\partial_{\theta}R\partial_{\theta}\partial_{\phi}R-\sin(\theta)R\left(\left(\partial_{t}^{2}R+8\right)\partial_{\phi}R^{2}-2\partial_{t}R\partial_{t}\partial_{\phi}R\partial_{\phi}R\right.$
$\displaystyle\left.+\partial_{t}R^{2}\left(\partial_{\phi}^{2}R+\sin^{2}(\theta)\partial_{\theta}^{2}R\right)+\sin^{2}(\theta)\partial_{\theta}R^{2}\left(\partial_{t}^{2}R+8\right)+\sin(\theta)\partial_{\theta}R\partial_{t}R\left(\cos(\theta)\partial_{t}R-2\sin(\theta)\partial_{t}\partial_{\theta}R\right)\right)$
$\displaystyle-2\sin(\theta)R^{3}\left(2\partial_{\phi}^{2}R+\sin(\theta)\left(\sin(\theta)\left(\partial_{t}R^{2}+2\partial_{\theta}^{2}R+2\partial_{t}^{2}R+8\right)+2\cos(\theta)\partial_{\theta}R\right)\right)$
$\displaystyle+2\sin(\theta)R^{2}\left(2\partial_{\phi}R^{2}+2\partial_{\phi}^{2}R+\sin(\theta)\left(\sin(\theta)\left(3\partial_{t}R^{2}+2\left(\partial_{\theta}R^{2}+\partial_{\theta}^{2}R\right)\right)+2\cos(\theta)\partial_{\theta}R\right)\right)$
$\displaystyle+\sin(\theta)\partial_{\theta}R^{2}\left(\partial_{\phi}^{2}R+\sin(\theta)\cos(\theta)\partial_{\theta}R\right)+\partial_{\phi}R^{2}\left(\sin(\theta)\partial_{\theta}^{2}R+2\cos(\theta)\partial_{\theta}R\right)+8\sin^{3}(\theta)R^{5}$
It can be checked that the cubic solution does indeed satisfy this equation.

$2\cos 3\theta\tanh^{2}r-3\cos\theta-3\sqrt{3}\sin\theta\sin(2\tau-\phi)=0$
(21)

Since $\tau$ and the angle $\phi$ only appear through the combination
$2\tau-\phi$, the surface rotates with a constant angular velocity like a
rigid body. This is depicted in Figure 4. Figure 5 shows the time-evolution of
the solution on the Poincaré patch. The membrane touches the boundary along a
curve which contains two cusps.

Figure 5:  Time-evolution of the cubic membrane solution on the Poincaré
patch. The top of the bounding box is the boundary of AdS. Here the membrane
has two cusps.

Figure 6:  Traveling waves on a brane embedded into the Poincaré patch. The
picture shows a timeslice. The top of the bounding box is the AdS boundary.

##  3 Waves traveling in one direction

Instead of parametrizing the brane embedding on the Poincaré patch via
$z(t,x,y)$, one can find the equation of motion for $y(t,x,z)$. Let us take
the wave ansatz

$y(t,x,z)=p(x^{-},z)\qquad\textrm{where}\quad x^{-}\equiv t-x$ (22)

The equation of motion simplifies

$\partial_{z}p+(\partial_{z}p)^{3}-{z\over
3}\partial_{z}^{2}p+\kappa\left[1+(\partial_{z}p)^{2}\right]^{3\over 2}=0$
(23)

Note that the equation only contains $z$-derivatives. Let us first set
$\kappa=0$. Then the solution is given by

$p(x^{-},z)={z^{4}\over 4A(x^{-})^{3}}\,{}_{2}F_{1}\left({1\over 2},\,{2\over
3};\,{5\over 3};\,{z^{6}\over A(x^{-})^{6}}\right)+B(x^{-})$ (24)

with $A$ and $B$ are two arbitrary functions.

Let us shift $B$

$B_{\pm}(x^{-}):=\pm{\sqrt{\pi}\,\Gamma({5\over 3})\over 4\Gamma({7\over
6})}A(x^{-})+C(x^{-})$ (25)

Then the two solutions

$p_{-}={z^{4}\over 4A^{3}}\,{}_{2}F_{1}\left({1\over 2},\,{2\over 3};\,{5\over
3};\,{z^{6}\over A^{6}}\right)+B_{-}$ (26)

and

$p_{+}=-{z^{4}\over 4A^{3}}\,{}_{2}F_{1}\left({1\over 2},\,{2\over
3};\,{5\over 3};\,{z^{6}\over A^{6}}\right)+B_{+}$ (27)

connect smoothly. The membrane touches the boundary along two curves which are
parametrized by $A$ and $C$. The curves move with the speed of light on the
boundary and the membrane follows their motion as a rigid body. A snapshot of
such a solution on the Poincaré patch is shown in Figure 6.

At non-zero $\kappa$, the equation of motion (23) can be solved in terms of
Appell hypergeometric functions

$\displaystyle
p(x^{-},z)=-{z\kappa\over\sqrt{1-\kappa^{2}}}\,F_{1}\left({1\over 3};\,{1\over
2},\,{1\over 2};\,{4\over
3};\,{z^{3}\over(\kappa+1)A(x^{-})^{3}},\,{z^{3}\over(\kappa-1)A(x^{-})^{3}}\right)$
$\displaystyle+{z^{4}\over
4A(x^{-})^{3}\sqrt{(1-\kappa^{2})}}\,F_{1}\left({4\over 3};\,{1\over
2},\,{1\over 2};\,{7\over
3};\,{z^{3}\over(\kappa+1)A(x^{-})^{3}},\,{z^{3}\over(\kappa-1)A(x^{-})^{3}}\right)+B(x^{-})$

where $A$ and $B$ are again two arbitrary functions.

Figure 7:  Segmented string on the Poincaré patch of AdS3 [4]. The string
consists of two patches (arcs) which have constant normal vectors. The kink
between the two segments moves with the speed of light.

##  4 Attaching patches

Segmented strings in flat space are piecewise linear solutions. Kinks between
the segments move with the speed of light and their worldlines form a
rectangular lattice on the string worldsheet. This idea can be generalized to
AdS3 where the embedding is built from AdS2 patches (see [4, 5])222For recent
developments, the reader is referred to [8, 6, 7, 10].. The segmented string
construction provides an exact discretization of the non-linear string
equations of motion. In Figure 7 we show an example which consists of two
elementary patches (arcs). Any smooth string can be approximated by segmented
strings to arbitrary accuracy by increasing the number of segments.

In this section we would like to generalize these ideas for the case of two-
branes in AdS4. The membrane has one transverse direction and therefore in
certain cases its classical dynamics is similar to that of a string in AdS3.

For simplicity, we henceforth set $\kappa=0$. Let us consider two linear brane
patches (see section 2.1) with constant normal vectors
$N_{1},N_{2}\in{\mathbb{{R}}}^{2,3}$. At a given time, the two patches may
intersect along a curve. This curve is analogous to the kink in Figure 7. We
want the worldsheet of this curve to be a null surface in AdS4 so that it is a
shockwave on the membrane, propagating with the speed of light. In the case of
a string, this happens precisely when the normal vectors (which are elements
of ${\mathbb{{R}}}^{2,2}$) satisfy the compatibility condition [4]

$N_{1}\cdot N_{2}=1$ (28)

In this case the kink moves with the speed of light linearly in the usual
Poincaré coordinate system. We will see that the same condition applies for
the case of two membrane patches.

Figure 8:  Two compatible linear patches on the Poincaré patch. The vertical
black arc is a shockwave which travels with the speed of light. The
construction is a higher-dimensional analog of the segmented string in Figure
7.

Let us first apply an $SO(2,3)$ transformation such that the unit normal
vectors are rotated into a frame in which

$\displaystyle N_{1}$ $\displaystyle=$ $\displaystyle(0,\,0,\,0,\,0,\,1)^{T}$
$\displaystyle N_{2}$ $\displaystyle=$
$\displaystyle(a,\,0,\,0,\,a+\epsilon,\,\sqrt{1-2a\epsilon-\epsilon^{2}})^{T}$

The intersection locus is two-dimensional in spacetime. Points on it satisfy

$X\cdot N_{1,2}=0\qquad X^{2}=-1$ (29)

This is solved by

$X=(\sigma_{1},\frac{\sqrt{a^{2}\left(\sigma_{2}^{2}+1\right)+\epsilon(2a+\epsilon)\left(1-\sigma_{1}^{2}+\sigma_{2}^{2}\right)}}{a+\epsilon},\sigma_{2},\frac{a\sigma_{1}}{a+\epsilon},0)$

where $\sigma_{1,2}$ parametrizes the worldsheet of the shockwave. The induced
metric on it is given by

$\det g=-{\epsilon(2a+\epsilon)\over
a^{2}(1+\sigma_{2}^{2})+\epsilon(1-\sigma_{1}^{2}+\sigma_{2}^{2})(2a+\epsilon)}$

Clearly, the shockwave defined by $X$ moves everywhere with the speed of light
iff $\epsilon=0$. (Here we have assumed that $a\neq 0$. If $a=0$ then the two
normal vectors are not compatible.) Note that $\epsilon=0$ is equivalent to
$N_{1}\cdot N_{2}=1$. This formula is invariant under $SO(2,3)$ and thus it is
valid for branes in general positions. We will call adjacent brane patches
that satisfy this constraint compatible333The compatibility condition
generalizes to higher-dimensional codimension one branes as well..

The membrane analog of Figure 7 is depicted in Figure 8.

Figure 9:  Three compatible linear patches with normal vectors $A$, $B$, and
$C$. The lines represent the shockwaves.

##  5 Trivalent vertex on the boundary

Once there are shockwaves on the brane, we need to understand what happens
when they collide. For simplicity, let us consider linear patches again with
$\kappa=0$. Is the trivalent vertex of shockwaves in Figure 9 possible? We
assume that the patches are pairwise compatible, i.e.

$A\cdot B=A\cdot C=B\cdot C=1$ (30)

If the triple intersection point is an ordinary point in spacetime, then by
zooming in on its vicinity, we can scale out the AdS radius. We then end up
with three planes intersecting along lines which themselves move with the
speed of light. Therefore the intersection point must move superluminally
which is impossible.

The only possibility is that the vertex moves on the boundary of AdS, see
Figure 10. Boundary points $X\in{\mathbb{{R}}}^{2,3}$ are characterized by
$X^{2}=0$ with the identification under rescaling $X\sim\lambda X$. After an
$SO(2,3)$ transformation the normal vectors can be written as

$\displaystyle A$ $\displaystyle=$ $\displaystyle(0,\,0,\,0,\,0,\,1)^{T}$
$\displaystyle B$ $\displaystyle=$ $\displaystyle(0,\,a,\,-a,\,0,\,1)^{T}$
$\displaystyle C$ $\displaystyle=$ $\displaystyle(b,\,c,\,-c,\,b,\,1)^{T}$

where $a,b,c$ are parameters. The triple intersection locus is a one-
dimensional subspace. Points on it satisfy

$X\cdot A=0\qquad X\cdot B=0\qquad X\cdot C=0\qquad X^{2}=0$ (31)

Figure 10:  Three compatible linear patches on the Poincaré patch. The
boundary of AdS is the plane on the top of the bounding box. The vertex (red
dot) moves on the AdS boundary with the speed of light. The three semi-circles
emanating from the vertex are the three shockwaves on the membrane.

Points on the boundary can be parametrized as

$X=\left({\frac{1}{2}}(-1+t^{2}-x^{2}-y^{2}),\,t,\,x,\,y,\,{\frac{1}{2}}(1+t^{2}-x^{2}-y^{2})\right)^{T}$

Eqn. (31) can be solved by setting $x(t)=-t$ and $y(t)=-1$. Hence, the vertex
moves in the $-x$ direction with the speed of light.

By adding extra vertices and additional compatible membrane patches one could
construct more complex membranes. The projection of the shockwaves on the
$z=0$ boundary is then a trivalent graph (with linear edges). In order to
compute the global time-evolution, one would need to understand what happens
when two trivalent vertices collide.

Figure 11:  Left: Schematic picture: a snapshot of the membrane which consists
of three linear patches. The normal vectors are denoted by $A,B,C$. The two
parallel shockwaves are moving towards each other. Right: A boost of the
system on the left in the vertical (parallel) direction produces crossing
shockwaves. The vertex where the shockwaves cross moves superluminally.

##  6 Crossing shockwaves

In the previous section we discussed that a hypothetical vertex connecting
shockwaves between linear membrane patches necessarily moves faster than the
speed of light if it is located in the bulk of AdS. In order to understand
whether it is possible to construct a brane configuration with such a vertex,
let us first look at the case of segmented strings. Let us assume that the
string consists of three segments, with two kinks moving towards each other.
Let us denote the normal vectors by $A,B,C\in{\mathbb{{R}}}^{2,2}$, where $B$
corresponds to the middle segment. Compatibility requires $A\cdot B=B\cdot
C=1$. This ensures that the kinks move on null geodesics. When they collide,
$B$ will jump according to the reflection formula [4, 5]

$B^{\prime}=-B+4{A+C\over(A+C)^{2}}$ (32)

It is easy to check that $A\cdot D=D\cdot C=1$ after the collision.

Let us now add a transverse dimension to the string (and to the target space).
The configuration is depicted in Figure 11 on the left-hand side. (The figure
is schematic because the shockwaves are drawn as straight lines even though
lie in AdS.) After the shockwaves pass through each other, the middle patch
will have a new normal vector, again computed by (32) where it is now
understood that the vectors all lie in ${\mathbb{{R}}}^{2,3}$. If the membrane
is boosted in the vertical direction which is parallel to the shockwaves, then
the result will be the one on the right-hand side. Now on the same time-slice
all four patches can be seen, including the one corresponding to $B^{\prime}$.

Figure 12:  Degree four vertex on a timeslice of the Poincaré patch. It lies
at the intersection of four hemispheres (the four linear membrane patches
which are connected by the reflection formula as explained in the main text).

This idea allows us to construct brane configurations with crossing
shockwaves. Adjacent linear patches must be compatible and the four patches
should be connected by the reflection formula (32). An example on the Poincaré
patch of AdS is shown in Figure 12 (with $\kappa=0$). The hemispheres of the
four linear patches are shown in full. They touch the AdS boundary along their
equators (black circles). White arrow indicates the location of the
superluminal vertex where the shockwaves cross. The membrane solution consists
of the four patches (blue, orange, purple and gray) around it.

More complex initial conditions for membranes can be constructed by adding
further shockwaves to the setup. In order for the shockwaves to move with the
speed of light, adjacent linear patches must be compatible and patches around
four-valence vertices must be connected by the reflection formula.

Figure 13:  Schematic picture of the collision of three shockwaves on the
membrane. Left: Before collision the membrane consists of seven linear
patches. Right: After collision, a non-linear patch forms in the middle
(indicated by a gray disk). The boundary of this patch (black circle) lies on
the intersection of the membrane and the future lightcone of the collision
event. Normal vectors outside the lightcone are dictated by the reflection
formula.

##  7 Collision of three shockwaves

In the previous section we discussed how piecewise linear membrane
configurations can be built in AdS. A natural question arises: what happens
when three shockwaves collide and a patch vanishes?

The situation before collision is shown in Figure 13 on the left-hand side.
The three shockwaves on the membrane are moving towards each other and the
size of $D$ decreases (with a slight abuse of notation the same letter will be
used for patches and their normal vectors). Preliminary numerical simulations
show that after collision the membrane will not be linear any longer. This is
seen on the right-hand side of the figure. There are several new linear
patches on the brane: $U,V,W$ and three smaller ones $F,G,H$. These are all
computed by the reflection formula from three adjacent patches around a
vertex. For instance, $U$ can be computed from $A,Z$ and $Y$. In the middle of
the membrane a non-linear patch is formed. This is indicated by a gray disk in
the figure. The boundary of this patch (indicated by a black circle) lies on
the future lightcone of the collision event. Inside the gray region the
membrane ‘knows’ about the collision and interactions deform its shape.

##  8 Discussion

In this paper we discussed membrane embeddings in AdS4 which satisfy the
equations of motion of a Nambu-Goto membrane (coupled to a background three-
form). Various membrane solutions can be obtained by gluing linear patches.
Since the brane is a codimension one object in this target space, patches can
be characterized by their constant normal vectors. The curve where adjacent
patches intersect is a shockwave on the membrane. It has to propagate with the
speed of light, otherwise the brane will bend and lose its piecewise linear
nature. This imposes a compatibility condition on normal vectors of adjacent
patches. We have also discussed cases with crossing shockwaves where the
intersection point was either on the boundary of AdS (trivalent vertices) or
in the bulk (valence-four vertices). Although an underlying three-dimensional
discrete integrable theory is not present here, some of these solutions can be
described by a two-dimensional discrete integrable theory. Although we have
not discussed how to glue quadric patches, it also seems possible. For a
similar construction in higher dimension, see [11].

Acknowledgments

The author is supported by the STFC Ernest Rutherford grant ST/P004334/1.

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