# Melvin’s “magnetic universe”, the role of the magnetic tension and the
implications for gravitational collapse

Christos G. Tsagas1,2 and Panagiotis Mavrogiannis1
1Section of Astrophysics, Astronomy and Mechanics, Department of Physics
Aristotle University of Thessaloniki, Thessaloniki 54124, Greece
2Clare Hall, University of Cambridge, Herschel Road, Cambridge CB3 9AL, UK

###### Abstract

Electromagnetism appears to have the potential to alter the fate of
relativistic gravitational implosion. Coulomb forces, for example, can in
principle prevent the ultimate collapse of charged matter to a singularity.
Also, half a century ago, Mael Melvin noted that magnetic forcelines do not
collapse under their own gravity, no matter how strong the latter may be.
Instead, magnetic fields seemed capable of stabilising themselves against
gravitational self-contraction. Intrigued, Melvin wondered whether magnetism
could also support against the realistic collapse of charged matter. Here, we
look back into these issues by means of two complementary scenarios. The first
reconsiders Melvin’s highly idealised “magnetic universe” and reaches the same
conclusion, namely that magnetic fields do not self-gravitate. The second is a
realistic collapse scenario, like that of magnetised massive compact stars.
Looking for answers to Melvin’s question, we find that magnetic fields do show
a generic tendency to support against relativistic gravitational implosion.
The magnetic effect in question involves the curvature of the host space, it
grows stronger as the collapse progresses and the triggering agent is the
field’s tension. In other words, the reason is the elasticity of the field
lines and their inherent tendency to react against anything that distorts them
from equilibrium. In this case, the distorting agent is the ever increasing
curvature of the collapsing host space.

##  1 Introduction

Posing idealised physical questions allows one to push the laws of physics to
the extreme and thus expect to gain new insight into the way these laws work
[1]. This was what Mael Melvin did half a century ago, when he considered the
contraction and collapse of a cylindrical tube of source-free magnetic lines,
resting in an otherwise empty and static space, under their own gravity [2,
3]. He found that, irrespective of how strong the magnetic gravitational field
was, the forcelines never collapsed. Instead, the magnetic ($B$) field showed
an unexpected ability to stabilise itself against gravitational self-
contraction. Intrigued by his findings, Melvin wondered what they might mean
for realistic gravitational collapse. In particular, he posed the question as
to whether the magnetic ability to stabilise itself against self-contraction,
also reflected some underlying natural tendency of the $B$-field to resist the
collapse of charged matter as well [3]. Decades later, soon after the turn of
the millennium, independent studies suggested that magnetic fields might
actually have such an inherent tendency to resist relativistic gravitational
implosion [4, 5].

These are not the only occasions where the involvement of electromagnetism was
found to affect and eventually alter the traditionally expected outcome of
gravitational implosion. Studies of charged collapse, for example, have opened
the possibility that repulsive Coulomb forces could (in principle) lead to a
bounce and thus prevent the formation of singularities (e.g. see [6]-[9]).
With this in mind, we take an alternative look into Melvin’s work and its
potential implications by employing two complementary scenarios. The first
(see scenario (a) in § 3 below) reconsiders the gravitational self-contraction
and collapse of source-free magnetic forcelines resting in otherwise empty
space, thus reproducing Melvin’s highly idealised magnetic universe. Although
our approach is different, since we look at the convergence/divergence of the
(spacelike) magnetic forcelines by means of the Raychaudhuri equation, we
arrive at the same conclusion. We find that the magnetic field lines do not
converge under their own gravity no matter how strong the latter is. Moreover,
we identify the magnetic tension as the reason behind this unconventional
magnetic behaviour. The latter cancels out the contribution of the field’s
energy density to the local curvature and in so doing prevents the forcelines
from converging. In a sense, the magnetic tension ensures that $B$-fields
cannot “feel” their own gravitational pull.

Our second scenario goes beyond Melvin’s idealised model. There, we consider
the realistic gravitational collapse of highly conductive magnetised matter,
like that occurring in massive compact stars for example. In this effort, we
are also motivated by the question Melvin posed but left unanswered, namely
whether magnetism can support against the gravitational collapse of charged
media. We employ again the Raychaudhuri equation, though this time it is
applied to the timelike congruence of the matter particles (see scenario (b)
in § 3). The latter are tightly coupled to the magnetic forcelines, given the
high electrical conductivity of the collapsing fluid. Our study shows that
$B$-fields have a generic tendency to resist the collapse, which is purely
relativistic in nature. More specifically, the magnetic presence naturally
gives rise to supporting magneto-curvature stresses. These are triggered by
the magnetic tension, namely by the elasticity of field lines, and reflect
their inherent tendency to remain “straight” and to react against any agent
that distorts them from equilibrium. In this case, the line deformation is
caused by the curvature of the host space and, as a result, the supporting
tension stresses grow stronger as the gravitational contraction progresses and
the curvature distortion increases. Put another way, although the tension
stresses are triggered by the magnetic presence, their resisting effect is
driven by the increasing curvature of the collapsing host space. Therefore, in
principle at least, even a relatively weak $B$-field could lead to a
disproportionately strong magneto-curvature support during the advanced stages
of the collapse. In a sense, it appears as though the elastic properties of
the magnetic forcelines have been transferred into the fabric of the host
space, which seems to act like a spring under tension.

##  2 The Raychaudhuri equations

Theoretical studies of gravitational collapse typically employ the
Raychaudhuri equation, which has been at the centre of the singularity
theorems (e.g. see [10]-[12]). Although Raychaudhuri’s formula is usually
applied to the timelike worldlines of real (or hypothetical) observers, it can
be used (and it has been used) to study the “kinematics” of spacelike and null
curves as well (e.g. see [13, 14]). The magnetic forcelines, as seen by
observers moving along their own timelike paths, are examples of such
spacelike curves. Technically speaking, if $u_{a}$ (with $u_{a}u^{a}=-1$) is
the timelike 4-velocity of a family of observers, the magnetic field vector
($B_{a}$) seen by them satisfies the constraint $B_{a}u^{a}=0$. The observers’
4-velocity vector, together with the associated projection tensor
$h_{ab}=g_{ab}+u_{a}u_{b}$ (where $g_{ab}$ is the spacetime metric and
$h_{ab}u^{b}=0$) introduces an 1+3 decomposition of the spacetime into time
and 3-space. Similarly, if $n_{a}$ is a unit spacelike vector along the
direction of the magnetic lines (with $B_{a}=Bn_{a}$), the tensor
$\tilde{h}_{ab}=h_{ab}-n_{a}n_{b}$ projects onto the 2-dimensional surface
normal to $n_{a}$, namely $\tilde{h}_{ab}n^{b}=0$, thus achieving an 1+2
decomposition of the 3-space and leading to an overall 1+1+2 splitting of the
whole spacetime.111Following our 1+1+2 spacetime splitting, we will use
overdots to indicate time derivatives, namely differentiation along $u_{a}$
(i.e. ${}^{\cdot}=u^{a}\nabla_{a}$) and primes to denote differentiation along
$n_{a}$ (i.e. ${}^{\prime}=n^{a}{\rm D}_{a}$). Note that $\nabla_{a}$ is the
4-dimensional covariant derivative operator, while ${\rm
D}_{a}=h_{a}{}^{b}\nabla_{b}$ and $\tilde{\rm D}_{a}=\tilde{h}_{a}{}^{b}{\rm
D}_{b}$ are its 3-dimensional and 2-dimensional counterparts respectively (see
[15]-[17] for further details and references).

By construction, the $u_{a}$ and the $n_{a}$ fields are tangent to a family of
timelike worldlines and to a congruence of spacelike curves respectively.
Then, the variables $\Theta={\rm D}^{a}u_{a}$ and $\tilde{\Theta}=\tilde{\rm
D}^{a}n_{a}$ respectively define the 3-dimensional volume
expansion/contraction scalar and the 2-dimensional area scalar (spatial
divergence) of the aforementioned two groups of curves. Consequently, positive
values for $\Theta$ and $\tilde{\Theta}$ imply that the tangent curves move
apart (i.e. expansion), whereas in the opposite case we have contraction and
the curves are approaching each other. The kinematics of the
expansion/contraction are monitored by the associated Raychaudhuri equations.
In particular, the convergence or divergence of the timelike worldlines
tangent to the $u_{a}$-field follows from (e.g. see [15, 16])

$\dot{\Theta}=-{1\over
3}\,\Theta^{2}-R_{ab}u^{a}u^{b}-2\left(\sigma^{2}-\omega^{2}\right)+{\rm
D}^{a}\dot{u}_{a}+\dot{u}^{a}\dot{u}_{a}\,.$ (1)

On the other hand, the converegence/divergence of the spacelike curves tangent
to the $n_{a}$-field (i.e. of the magnetic forcelines) is governed by
[17]222Expression (1) monitors the convergence/divergence of the $u_{a}$-field
in time (i.e. along the $u_{a}$-field), while Eq. (2) does the same for the
$n_{a}$-congruence along a given spatial direction (i.e. along the spacelike
vector $n_{a}$).

$\tilde{\Theta}^{\prime}=-{1\over
2}\,\tilde{\Theta}^{2}-\mathcal{R}_{ab}n^{a}n^{b}-2\left(\tilde{\sigma}^{2}-\tilde{\omega}^{2}\right)+\tilde{\rm
D}^{a}n_{a}^{\prime}-n^{\prime\,a}n_{a}^{\prime}\,.$ (2)

Note that $R_{ab}$ and $\mathcal{R}_{ab}$ are respectively the 4-D (spacetime)
and the 3-D (spatial) Ricci tensors. Also, $\sigma^{2}={\rm D}_{\langle
b}u_{a\rangle}{\rm D}^{\langle b}u^{a\rangle}/2$ and
$\tilde{\sigma}^{2}=\tilde{\rm D}_{\langle b}n_{a\rangle}\tilde{\rm
D}^{\langle b}n^{a\rangle}/2$ are the (squared) magnitudes of the shear
tensors associated with the $u_{a}$ and the $n_{a}$ fields respectively, while
$\omega^{2}={\rm D}_{[b}u_{a]}{\rm D}^{[b}u^{a]}/2$ and
$\tilde{\omega}^{2}=\tilde{\rm D}_{[b}n_{a]}\tilde{\rm D}^{[b}n^{a]}/2$ are
the magnitudes of the corresponding vorticity tensors. Finally,
$\dot{u}_{a}=u^{b}\nabla_{b}u_{a}$ and $n_{a}^{\prime}=n^{b}{\rm D}_{b}n_{a}$
are the directional-derivative vectors ($\dot{u}_{a}$ is the 4-acceleration)
associated with the aforementioned two families of curves. When
$\dot{u}_{a}=0$ the observers’ worldlines are timelike geodesics, whereas
$n_{a}^{\prime}=0$ implies that the $n_{a}$-field is tangent to a congruence
of spacelike geodesics. The very close analogy between expressions (1) and (2)
is profound, with every term on the right-hand side of (2) having a direct
analogue in Eq. (1). In what follows, we will use the above two versions of
the Raychaudhuri equation to argue that:

  * •

(a) The magnetic lines do not self-gravitate, because they do not “feel” their
own gravity no matter how strong the latter may be (in agreement with Melvin).

  * •

(b) In an external gravitational field, the magnetic field develops naturally
curvature related stresses that resist gravitational contraction (in response
to Melvin).

In both cases, we will show that the agent responsible for such a unique and
unconventional behaviour is the magnetic tension.

##  3 The two scenarios

Scenario (a): For our purposes, the key difference between Eqs. (1) and (2) is
in their curvature terms. In particular, the former relation is affected by
the 4-Ricci curvature of the whole spacetime, while the latter contains the
3-Ricci tensor of the spatial hypersurfaces. These are

$\displaystyle R_{ab}={1\over 2}\,\kappa\left(\rho+3p\right)u_{a}u_{b}+{1\over
2}\,\kappa\left(\rho-p\right)h_{ab}+2\kappa q_{(a}u_{b)}+\kappa\pi_{ab}$ (3)

and

$\displaystyle\mathcal{R}_{ab}$ $\displaystyle=$ $\displaystyle{1\over
3}\,\mathcal{R}h_{ab}+{1\over 2}\,\kappa\pi_{ab}+E_{ab}-{1\over
3}\,\Theta\left(\sigma_{ab}+\omega_{ab}\right)+\sigma_{c\langle
a}\sigma^{c}{}_{b\rangle}-\omega_{c\langle a}\omega^{c}{}_{b\rangle}$ (4)
$\displaystyle+2\sigma_{c[a}\omega^{c}{}_{b]}\,,$

respectively (e.g. see [15, 18]). In the above $\kappa=8\pi G$ is the
gravitational constant, $\rho$ is the energy density of the (total) matter,
$p$ is its isotropic pressure $q_{a}$ the associated energy flux and
$\pi_{ab}$ the anisotropic pressure (i.e. the viscosity), with the last two
quantities vanishing in the case of a perfect fluid. Also,
$\mathcal{R}=2(\kappa\rho-\Theta^{2}/3+\sigma^{2}-\omega^{2})$ is the 3-Ricci
scalar (describing the mean curvature of the spatial sections) and $E_{ab}$ is
the electric Weyl tensor (associated with tidal forces and gravitational
waves). Note that $q_{a}$, $\pi_{ab}$ and $E_{ab}$ are spacelike, with the
last two also being symmetric and trace-free. A key (and also very well known)
difference between expressions (3) and (4) is that the former contains only
contributions from the local matter, while the latter carries additional
effects and in particular those of the long-range gravitational field, namely
of the Weyl curvature (see also § 4 below for additional related comments).

Magnetic fields correspond to an effective imperfect fluid with energy density
$\rho_{{}_{B}}=B^{2}/2$ (where $B^{2}=B_{a}B^{a}$), isotropic pressure
$p_{{}_{B}}=B^{2}/6$ and anisotropic pressure
$\pi_{ab}=\Pi_{ab}=(B^{2}/3)h_{ab}-B_{a}B_{b}$ (e.g. see [18] and references
therein). This tensor also carries the effects of the field’s tension, which
is manifested as negative pressure exerted along the direction of the magnetic
lines. Indeed, one can easily show that $\Pi_{ab}$ has a negative eigenvalue
(equal to $-2B^{2}/3$) along the direction of the $B$-field.

Let us now apply expressions (2) and (4) to Melvin’s “magnetic universe”. In
order to test the behaviour of the $B$-field under its own gravity, Melvin
considered a cylindrical tube of infinitely long, parallel (i.e. irrotational
and shear-free spacelike geodesics) sourceless forcelines, resting in an
otherwise empty and static space [2, 3]. In such an environment, where there
is nothing whatsoever to affect (or be affected) by the magnetic field, Eqs.
(2) and (4) reduce to

$\tilde{\Theta}^{\prime}=-{1\over
2}\,\tilde{\Theta}^{2}-\mathcal{R}_{ab}n^{a}n^{b}$ (5)

and

$\displaystyle\mathcal{R}_{ab}$ $\displaystyle=$ $\displaystyle{1\over
3}\,\kappa B^{2}h_{ab}+{1\over 2}\,\kappa\Pi_{ab}\,,$ (6)

respectively [17]. Note the term $\mathcal{R}_{ab}n^{a}n^{b}$ on the right-
hand side of (5), which measures the 3-curvature along the direction of the
$B$-field and essentially describes distortions in the shape of the magnetic
forcelines caused by the curvature of the host space. Contracting Eq. (6)
twice along $n_{a}$ gives $\mathcal{R}_{ab}n^{a}n^{b}=0$ irrespective of the
field’s strength, given that $h_{ab}n^{a}n^{b}=n_{a}n^{a}=1$ and
$\Pi_{ab}n^{a}n^{b}=-2B^{2}/3$ (see above). This null result is a direct
consequence of the magnetic tension, which cancels out exactly the
contribution of the magnetic energy density in Eq. (6) and in so doing
demonstrates the natural preference of the field lines to remain straight.

The vanishing of the curvature term on the right-hand side of (5) means that
the convergence/divergence of the forcelines (along their own direction)
proceeds independently of the field’s own gravity. In a sense, the magnetic
lines do not “feel” their own gravitational field, no matter how strong the
latter may be. As Melvin also claimed, the $B$-field does not self-
gravitate.333The absence of a magnetic contribution to the scalar
$\mathcal{R}_{ab}n^{a}n^{b}$ is independent of the presence of other matter
and of the strength of the $B$-field. If there are additional matter sources
around, the magnetic forcelines will generally “feel” the gravitational pull
of those sources, but never their own gravity (see scenario (b)). Then, the
convergence/divergence of the magnetic lines is fully determined by the
initial conditions. More specifically, when $\mathcal{R}_{ab}n^{a}n^{b}=0$,
Eq. (5) gives

$\tilde{\Theta}=\tilde{\Theta}(s)={2\tilde{\Theta}_{0}\over
2+\tilde{\Theta}_{0}s}\,,$ (7)

where $s$ is the proper length in the direction of the $B$-field.
Consequently, when $\tilde{\Theta}_{0}<0$, the forcelines converge to form
caustics within finite proper length (i.e. $\tilde{\Theta}\rightarrow-\infty$
as $s\rightarrow-2/\tilde{\Theta}_{0}$). In the opposite case (when
$\tilde{\Theta}_{0}>0$), the lines keep diverging. Most interestingly,
however, magnetic forcelines that happen to be at rest (i.e. with
$\tilde{\Theta}_{0}=0)$ will remain so indefinitely (see Fig. 1), unless an
external agent interferes (see scenario (b) next). The latter case profoundly
demonstrates the natural tendency of the field lines to remain straight/stable
and also reveals that the responsible agent is their tension, the negative
pressure of which cancels out the gravitational pull of the field’s own energy
density.

The pivotal role played by the magnetic tension in arriving at the above
results should be noted and emphasised. If there was no tension, namely if
$\Pi_{ab}n^{a}n^{b}\geq 0$, Eq. (6) would have led to
$\mathcal{R}_{ab}n^{a}n^{b}\geq\kappa B^{2}/3$. This in turn would have meant
$\tilde{\Theta}^{\prime}<0$, even when $\tilde{\Theta}=0$ initially (see
expression (5)). In such a case, the magnetic lines would have converged and
focused, under their own gravity alone, within finite proper length.

Figure 1: Magnetic lines lying parallel in empty 3-space along the unit vector
$\vec{n}$, with 2-D cross-sectional area $A$ (so that
$A^{\prime}/A=\tilde{\Theta}$). The forcelines will not self-gravitate and
converge (along their own direction), no matter how strong their gravitational
field is. Instead, they remain at rest, due to their tension properties,
provided $\tilde{\Theta}_{0}=0$ and no external agent intervenes.

In summary, our first scenario allowed for an alternative approach to Melvin’s
idealised magnetic universe, where the field lines form a cylindrical tube
resting in empty space [2, 3]. We have recovered Melvin’s conclusion on the
one hand, while on the other we have revealed the key role of the field’s
tension as the stability agent. Despite the idealised nature of the model, the
fact that the tension cancels out the magnetic energy input to
$\mathcal{R}_{ab}n^{a}n^{b}$ remains and holds irrespective of the presence of
matter, or of the magnetic strength. Next, we will use relativistic
magnetohydrodynamics (MHD) to consider the realistic collapse of a self-
gravitating magnetised fluid. In the process, we will also try to answer
Melvin’s question as to whether the $B$-field can support against the
relativistic implosion of charged matter [3].

Scenario (b): The magnetic tension reflects the elasticity of the field lines
and their tendency to react against any agent that distorts them from
equilibrium [19, 21]. In scenario (a), we considered the idealised situation
of magnetic forcelines self-gravitating in an otherwise empty space. There,
there was nothing to affect or to be affected by the magnetic presence. Here,
we will embed these lines into the gravitational field of a highly conductive
charged medium. In such an environment, which defines the ideal-MHD limit,
Ohm’s law guarantees that the electric field vanishes and the currents keep
its magnetic counterpart frozen into the fluid [19]-[21]. Then, Maxwell’s
formulae (Faraday’s law in particular) ensure that the magnetic strength is
enhanced/depleted by the compression/expansion of the host medium (see § 5.1
in [18] for the form of Maxwell’s equations at the MHD limit).

Magnetic-line deformations are usually caused by electrically charged
particles, but in the context of general relativity spacetime curvature (i.e.
gravity in Einstein’s interpretation) can also act as a deforming agent (e.g.
see [22]-[24]). When this happens, the field’s tension gives rise to magneto-
curvature stresses that react against the cause of their distortion, namely
against gravity itself. To demonstrate all this, assume that the host
spacetime is neither static nor empty, but dynamical and filled with highly
conductive charged matter. In such an environment, the magnetic lines are no
longer infinite, but closed according to Gauss’s law (i.e. ${\rm
D}^{a}B_{a}=0$) and standard MHD. Then, the process of gravitational
contraction is monitored by the usual form of the Raychaudhuri equation (see
expression (1) here, as well as [11]). The latter is applied to the timelike
worldlines of the matter particles, instead of the spacelike magnetic
forcelines of scenario (a). Put another way, in scenario (b), it is the highly
conductive matter that collapses and in the process pulls the frozen-in
magnetic field along with it.

In the magnetic presence, the worldlines of the charged particles are no
longer timelike geodesics. This is a crucial difference between scenario (b)
and the typical (non-magnetised) models of gravitational collapse, where
matter “moves” along geodesic worldlines under the effect of gravity alone
(see [11, 12] and also footnote 6 here). Note that the non-geodesic nature of
timelike worldlines is guaranteed by the nonzero 4-acceleration vector in
Raychaudhuri’s formula ($\dot{u}_{a}$ – see Eq. (1) in § 2), which in turn
manifests the presence of non-gravitational forces. This is a crucial common
feature between this scenario and those claiming that Coulomb forces can
prevent the ultimate collapse of charged matter and the formation of
singularities (see [6]-[9]). In our case, however, it is not the Coulomb but
the magnetic Lorentz force that plays the pivotal role. At the ideal MHD
limit, the Lorentz force satisfies the momentum conservation law

$\displaystyle\left(\rho+p+B^{2}\right)\dot{u}_{a}=-{\rm D}_{a}p-{1\over
2}\,{\rm D}_{a}B^{2}+B^{b}{\rm D}_{b}B_{a}+\dot{u}^{b}B_{b}B_{a}\,,$ (8)

where the second and third terms on the right-hand side split the magnetic
Lorentz force into its pressure and tension parts respectively [15, 16]. To
proceed, let us suppose that both the matter and the magnetic energies have
nearly uniform distribution (i.e. assume that ${\rm D}_{a}\rho\simeq
0\simeq{\rm D}_{a}p\simeq{\rm D}_{a}B^{2}$). Then, to leading approximation,
we may bypass the effects from any inhomogeneities may exist in the fluid and
the magnetic energy densities.444Addressing the fully nonlinear case remains
an intractable task, even by numerical methods, since it requires solving the
complete set of the Einstein-Maxwell equations. Simplifying steps are
therefore necessary and the homogeneity assumption is a standard practise,
also adopted by the singularity theorems [11, 12]. On the other hand, we allow
for $B^{b}{\rm D}_{b}B_{a}\neq 0$ and therefore incorporate effects caused by
distortions in the shape of the magnetic forcelines. This enables us to focus
on the magnetic tension (see decomposition (8) above), which is at the centre
of this study.

Some clarifying notes are due here. Strictly speaking, hereafter, our analysis
refers to matter fields with spatially homogeneous energy-density
distribution. In practice, however, our results also apply to nearly
homonegeous environments, where the matter fields involved have been
sufficiently “randomised”. It should also be noted that spatial homogeneity is
not an ad hoc assumption of ours, but a rather common approximation. In fact,
exact spatial homogeneity is a standard feature/prerequisite in all typical
singularity theorems (e.g. see [11, 12] and also footnotes 4 and 6 here).
Recall that these theorems consider geodesics worldlines, a condition that no
longer holds when matter and pressure inhomogeneities are present. Moreover,
assuming a homogeneous (or nearly homogeneous) matter distribution does not
compromise the core of our argument, since gradients in the fluid and in the
magnetic density distribution are known to inhibit gravitational contraction,
even within Newtonian physics. Here, on the other hand, we focus upon the
purely general relativistic effects. On these grounds, the 3-divergence of Eq.
(8) combines with the 3-Ricci identities and with Maxwell’s formulae to give
[4, 5]

${\rm
D}^{a}\dot{u}_{a}=c^{2}_{a}\,\mathcal{R}_{ab}n^{a}n^{b}+2\left(\sigma^{2}_{{}_{B}}-\omega^{2}_{{}_{B}}\right)\,,$
(9)

where $c^{2}_{a}=B^{2}/(\rho+p+B^{2})$ is the Alfvén speed. Note that the
3-curvature term $\mathcal{R}_{an}n^{a}n^{b}$ is nonzero now, due to the
presence of matter, though still without any magnetic input. Also, the scalars
$\sigma^{2}_{{}_{B}}={\rm D}_{\langle b}B_{a\rangle}{\rm D}^{\langle
b}B^{a\rangle}/2(\rho+p+B^{2})$ and $\omega^{2}_{{}_{B}}={\rm
D}_{[b}B_{a]}{\rm D}^{[b}B^{a]}/2(\rho+p+B^{2})$ are the magnetic analogues of
the shear and the vorticity respectively.

All the terms seen on the right-hand side of Eq. (9) result from the magnetic
tension and reflect the “elastic” reaction of the field lines to distortions
triggered by spatial curvature, shear and rotation respectively (see also
expression (10) and related discussion below). The most important for our
purposes is the first term, which is a (purely relativistic) magneto-curvature
tension stress, since it contains contributions from the $B$-field and the
3-Ricci tensor (contracted twice along the direction of the field lines). More
specifically, the contraction $\mathcal{R}_{ab}n^{a}n^{b}$ encodes the
reaction of the field lines to their “bending” caused by the curvature of the
host 3-space.555The magneto-curvature term in Eq. (9) results from the vector
nature of the $B$-field and the geometrical interpretation of gravity that
Einstein’s theory advocates. These two combine to guarantee a purely
geometrical coupling between magnetism and spacetime curvature, which comes
into play through the Ricci identities and goes beyond the standard
interaction between matter and geometry seen in the Einstein equations. Among
others, this coupling has been known to affect the propagation of
electromagnetic radiation in curved spacetimes [22]-[24]. Substituting the
3-divergence ${\rm D}^{a}\dot{u}_{a}$ from Eq. (9) into Raychaudhuri’s formula
(see expression (1)), the latter reads

$\displaystyle\dot{\Theta}+{1\over
3}\,\Theta^{2}=-R_{ab}u^{a}u^{b}+c^{2}_{a}\,\mathcal{R}_{ab}n^{a}n^{b}-2\left(\sigma^{2}-\sigma^{2}_{{}_{B}}\right)+2\left(\omega^{2}-\omega^{2}_{{}_{B}}\right)+\dot{u}^{a}\dot{u}_{a}\,.$
(10)

with $R_{ab}$ and $\mathcal{R}_{ab}$ given by (3) and (4) respectively. When
the right-hand side of the above is negative, an initially contracting
congruence of worldlines will focus at a point (i.e.
$\Theta\rightarrow-\infty$) within finite proper time [11, 12]. In the
opposite case, however, caustic formation and the ultimate collapse to a
singularity will be averted. Therefore, negative terms on the right-hand side
of the above assist implosion, while positive ones act in the opposite way
[11, 12].666It is worth comparing Eq. (10) to the Raychaudhuri formula used in
standard singularity theorems. The latter consider the convergence and
focusing of irrotational, timelike geodesics, in which case we have [11, 12]
$\dot{\Theta}+{1\over 3}\,\Theta^{2}=-R_{ab}u^{a}u^{b}-2\sigma^{2}\,,$ (11)
without vorticity and 4-acceleration contributions (since
$\omega=0=\dot{u}_{a}$ by default). This means that only stresses that assist
the collapse have been included to the right-hand side of Eq. (11). Moreover,
the absence of any 4-acceleration terms in the above relation implies that all
the matter fields involved are homogeneously distributed. In this respect, the
possibility of avoiding the ultimate singularity by involving non-
gravitational forces (e.g. see [6]-[9] and scenario (b) here), does not
contradict the aforementioned theorems.

Given that $\dot{u}_{a}\dot{u}^{a}>0$ always, the 4-acceleration stress seen
at the end of Eq. (10) resists contraction at all times. The remaining
stresses act (in pairs) against each other in a way that manifests the
counterintuitive impact of the tension effects. Since, in all cases of
realistic stellar collapse, $\mathcal{R}_{ab}$ takes positive values along
every spatial direction (including that of the $B$-field), the magneto-
curvature stress ($c^{2}_{a}\,\mathcal{R}_{ab}n^{a}n^{b}$) supports against
the gravitational pull of the local matter, which itself is driven by the
4-Ricci curvature term ($R_{ab}u^{a}u^{b}$). Also, while kinematic-shear
distortions ($\sigma$) assist the collapse, the magnetic shear
($\sigma_{{}_{B}}$) inhibits contraction. The magnetic vorticity
($\omega_{{}_{B}}$), on the other hand, helps implosion, whereas vorticity
proper ($\omega$) does the opposite. In all three cases, the magnetic tension
ensures that the field lines react and try to counterbalance the effects of
curvature, shear and rotation, namely of the agents that caused their
deformation in the first place.

The fate of the magnetised collapse, as monitored by Eq. (10), depends on the
balance between the counteracting stresses seen on the right-hand side of that
expression. In extreme situations, like those occurring during the
gravitational implosion of a magnetised compact star or/and a black hole, it
sounds plausible to argue that the ultimate fate of the collapse will be
decided by the balance between the two curvature-related stresses on the
right-hand side of (10). After all, these are the only purely general
relativistic terms, since the rest will appear in flat Euclidean spaces as
well (with the covariant derivatives replaced by partial ones). It also makes
physical sense to claim that gravity (i.e. curvature in Einstein’s
interpretation) should dominate the final stages of the collapse. Then, to
leading order, Eq. (10) reduces to

$\dot{\Theta}+{1\over
3}\,\Theta^{2}=-R_{ab}u^{a}u^{b}+c^{2}_{a}\,\mathcal{R}_{ab}n^{a}n^{b}\,.$
(12)

In the above one can clearly see the competing effects of all the matter
fields driving the implosion on the one hand, and of the magnetic tension that
acts against contraction on the other. Recall that
$R_{ab}u^{a}u^{b}=\kappa(\rho+3p)/2>0$ for all types of conventional matter
and that the scalar $\mathcal{R}_{ab}n^{a}n^{b}$ takes positive values in all
realistic collapse scenarios. As we mentioned earlier, this magneto-curvature
tension stress also carries the reaction of the field lines to distortions
caused by the curvature of the host space.

What is most intriguing is that, as the collapse proceeds, the resistance of
the magneto-curvature tension stress seen in Eq. (12) increases. This is
physically explained by the fact that the $B$-field is frozen into the
collapsing matter and by the elasticity of the magnetic forcelines. More
specifically, in realistic stellar collapse scenarios, the curvature of the
imploding 3-space increases along all directions, including that of the
$B$-field. This means that the scalar $\mathcal{R}_{ab}n^{a}n^{b}$ becomes
progressively more positive as the collapse progresses and also that the
curvature induced magnetic-line distortion increases too. As a result, the
resistance of the “elastic” magneto-curvature tension stress in Eq. (12) grows
stronger with the collapse. Therefore, although the aforementioned tension
stress is triggered by the magnetic presence, its effect is driven by the ever
increasing curvature of the imploding space. This makes it possible for
relatively weak magnetic fields (i.e. those with $c_{a}^{2}<1$) to trigger
disproportionately strong magneto-curvature tension stresses in strong-gravity
environments. In fact, as the 3-curvature diverges (i.e. as
$\mathcal{R}_{ab}n^{a}n^{b}\rightarrow\infty$) during the final stages of the
collapse, the resisting effect appears to become arbitrarily strong as well,
even in the presence of a relatively weak $B$-field. Intuitively speaking, it
appears as though the elasticity of the magnetic forcelines has been injected
into the fabric of the host space, which seems to backreact against its own
implosion. This entirely counterintuitive behaviour is what distinguishes the
magneto-curvature tension stress in Eq. (12) from other supporting stresses
(like those due to pressure gradients, or to the vorticity, for example) and
makes it unique.

With this in mind, it is worth going back to Melvin’s 1965 paper [3], where he
poses the following two questions: “As the matter falls inward and carries the
magnetic field with it, does this magnetic field inhibit the collapse? Does
the magnetic field give rise to a counterpressure of decisive importance to
the dynamics of collapse?”. Half a century later, our work points towards
positive answers to both of these questions. We found that magnetic fields
exhibit a generic tendency to resist gravitational contraction, by naturally
giving rise to (purely general relativistic) curvature-related supporting
stresses that grow stronger as the collapse progresses. Moreover, by
approaching the issue via a different route, we showed that the physical agent
behind this magnetic behaviour is not the field’s pressure but its tension,
which makes the magnetised space act like an elastic spring under tension.

##  4 Discussion

Gravitational collapse does not only lead to compact stars and black holes,
but ultimately it is also expected to cause the formation of spacetime
singularities. Nevertheless, although the latter appear inevitable when
dealing with geodesics, they could be averted when non-gravitational forces
(and therefore non-geodesics) are involved. Recall that repulsive Coulomb
forces can in principle prevent the ultimate collapse of charged matter to a
singularity [6]-[9]. Is it then conceivable that magnetism could be nature’s
way of preventing the catastrophic fate of a singularity? Following (12), it
seems plausible that worldline focusing could be averted if the (tension
induced) 3-Ricci stress outgrows its 4-Ricci counterpart (see Fig. 2), namely
when

$c_{a}^{2}\mathcal{R}_{ab}n^{a}n^{b}>R_{ab}u^{a}u^{b}\,.$ (13)

At this stage, it is still uncertain whether this (tentative)
condition/requirement is naturally fulfilled in physically realistic
situations. In principle, it depends on which of the two stresses grows faster
towards the singularity. With this in mind, it sounds promising that the
resisting magneto-curvature stress carries an additional direct geometrical
(i.e. gravitational) input. The latter comes from the Weyl field and more
specifically from the electric component of the Weyl tensor (i.e. from
$E_{ab}$ – see Eq. (4) and comments immediately after). In other words, the
magnetic lines also react to distortions caused by tidal forces and
gravitational waves, both of which are expected to grow arbitrarily strong as
the collapse reaches its final stages. Note that, as it is well known, the
Weyl field does not explicitly contribute to $R_{ab}$ (see expression (3)
earlier) and therefore to the 4-Ricci stress that drives the implosion in Eq.
(12). It is then conceivable that this “selective” contribution of the Weyl
field could prove significant, by “tilting” the balance in favour of the
supporting stress in condition (13).

Having said that, we should also emphasise that avoiding the focussing of the
non-geodesics worldlines of the charged particles, does not automatically
guarantee a non-singular spacetime. Indeed, the latter may still contain
timelike (as well as null) geodesics that are incomplete and singular (see
also footnote 6 here). In this respect, our study is in no conflict with the
Hawking-Penrose singularity theorems, at least no more than the work of
[6]-[9] is. Recall that, in the latter studies, the non-geodesic worldlines of
the charged particles were prevented from focusing by repulsive Coulomb
stresses.

Figure 2: Timelike worldlines of magnetised matter, with 4-velocity $u_{a}$
normal to the spatial 3-D hypersurface $S$ and an associated volume element
$V$ (so that $\dot{V}/V=\Theta$). When gravity dominates and
$R_{ab}u^{a}u^{b}>c^{2}_{a}\,\mathcal{R}_{ab}n^{a}n^{b}$, converging wordlines
focus to form caustics within finite time (left). On the other hand, if the
magneto-curvature tension stresses take over (with
$c^{2}_{a}\,\mathcal{R}_{ab}n^{a}n^{b}>R_{ab}u^{a}u^{b}$), the collapse could
proceed indefinitely and focusing may never happen (middle). There is also the
possibility of a bounce, provided $\dot{\Theta}$ in Eq. (12) turns positive
(right).

Even if $R_{ab}u^{a}u^{b}>c_{a}^{2}\mathcal{R}_{ab}n^{a}n^{b}$ and gravity
ultimately wins, $B$-fields still show a remarkable generic tendency to reduce
(or perhaps in some cases eliminate) the gravitational effects. The physical
reason responsible for this (purely general relativistic) effect is the
field’s tension. In particular, it is the magnetic tension that cancels out
the $B$-field’s energy contribution to the curvature of the 3-D space. This
leaves the twice contracted 3-curvature tensor, namely the scalar
$\mathcal{R}_{ab}n^{a}n^{b}$, free of any magnetic input and therefore ensures
that the field lines do not feel their own gravitational pull, no matter how
strong the latter may be. In the absence of any other sources, this ensures
that magnetic forcelines that were originally parallel will remain so, in
agreement with Melvin [2, 3]. During the realistic gravitational contraction
of charged matter, on the other hand, the scalar $\mathcal{R}_{ab}n^{a}n^{b}$
takes positive values, deforms the magnetic forcelines and gives rise to
curvature-related tension stresses that support against the contraction of the
magnetised medium. No other known source behaves this way. What is most
intriguing is that, as the collapse progresses and the curvature distortion
increases (seemingly unrestrained), the resistance of the magneto-curvature
tension stresses also grows (potentially unbounded as well), even when the
$B$-field remains weak relative to the matterial content. Metaphorically
speaking, one might say that the magnetised space reacts to the pull of
gravity and tries to bounce back, as if it has itself acquired the elastic
properties of the field lines.

Acknowledgments: The authors would like to thank Naresh Dadhich and Yuan Ha
for their comments. CGT was supported by the Hellenic Foundation for Research
and Innovation (H.F.R.I.), under the ‘First Call for H.F.R.I. Research
Projects to support Faculty members and Researchers and the procurement of
high-cost research equipment Grant’ (Project Number: 789). PM acknowledges
support by the Foundation for European Education and Culture.

## References

  * [1] K.P. Thorne, Black Holes and Time Warps: Einstein’s Outrageous Legacy (Picador, London and Basingstoke, 1994).
  * [2] M.A. Melvin, _Phys. Lett._ 8 (1964) 65.
  * [3] M.A. Melvin, _Phys. Rev. B_ 139 (1965) 225.
  * [4] C.G. Tsagas, _Phys. Rev. Lett._ 86 (2001) 5421
  * [5] C.G. Tsagas, _Class. Quantum Grav._ 23 (2006) 4323.
  * [6] I.D. Novikov, Sov. Astron. 10, 731 (1967).
  * [7] V. de la Cruz and W. Israel, Nuovo Cimento A 51, 744 (1967).
  * [8] A.K. Raychaudhuri, Ann. Inst. Henri Poincare 22, 229 (1975).
  * [9] A. Krasinski and K. Bolejko, Phys. Rev. D 73, 124033 (2006).
  * [10] S.W. Hawking and G.F.R. Ellis, The large scale structure of space-time (Cambridge University Press, Cambridge, 1973).
  * [11] R.M. Wald, General Relativity (University of Chicago Press, Chicago, 1984).
  * [12] E. Poisson, A Relativist’s Toolkit (Cambridge University Press, Cambridge, 2004).
  * [13] S. Kar and S. Sengupta, _Pramana_ 69 (2007) 49.
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  * [16] G.F.R. Ellis, R. Maartens and M.A.H. MacCallum, Relativistic Cosmology (Cambridge University Press, Cambridge, 2012).
  * [17] V. Giantsos and C.G. Tsagas, _Mon. Not. R. Astron. Soc._ 472 (2017) 869.
  * [18] J.D. Barrow, R. Maartens and C.G. Tsagas, _Phys. Rep._ 449 (2007) 131.
  * [19] E.N. Parker, Cosmical Magnetic Fields (Clarendon Press, Oxford, 1979).
  * [20] Y.B. Zeldovich, A.A. Ruzmaikin and D.D. Sokoloff, Magnetic fields in Astrophysics (Gordon and Breach, New York, 1983).
  * [21] L. Mestel, Stellar Magnetism (Oxford University Press, Oxford 1999).
  * [22] T.W. Noonan, _Astrophys. J._ 341 (1989) 786.
  * [23] F. De Felice and C. Clark, Relativity in Curved Manifolds (Cambridge University Press, Cambridge, 1990).
  * [24] C.G. Tsagas, _Class. Quantum Grav._ 22 (2005) 393.


# Causal structure of general relativistic spacetimes

E. M. Howard

###### Abstract

We present some of the recent results and open questions on the causality
problem in General Relativity. The concept of singularity is intimately
connected with future trapped surface and inner event horizon formation. We
offer a brief overview of the Hawking-Penrose singularity theorems Hawking
and discuss a few open problems concerning the future Cauchy development
(domain of dependence) geroch70 , break-down criteria and energy conditions
for the horizon stability. A key question is whether causality violating
regions, generating a Cauchy horizon are allowed.

We raise several questions concerning the invisibility and stability of closed
trapped surfaces from future null infinity and derive the imprisonment
conditions. We provide an up-to-date perspective of the causal boundaries and
spacelike conformal boundary extensions for time oriented Lorentzian manifolds
and more exotic settings.

###### Keywords:

causality, general relativity, causal structure, cosmic censorship, Cauchy
hypersurface, singularity theorems, global hyperbolicity

###### :

04.20.Gz, 04.20.Dw, 04.20.Gz, 04.20.Cv

##  1 Introduction

In this short paper I would like to present a brief overview of some open
questions in General Relativity with dramatic consequences for causality
theory, aiming to a deeper understanding of the (global) causal structure of
the space-time. Special attention is accorded to the problem of global
hyperbolicity, as the most important condition on causality and its relation
to time functions. seifert77  It is important to note that General Relativity
has been infiltrated with exotic geometries involving closed timelike curves
Thorne or other possible causal violations. The definition and understanding
of the nature of causality plays a fundamental role in the construction of
physical theories. The number of open issues concerning the causal behaviour
of the space-time is fairly large. Let us recall that the generic conformal
invariants that concern causality have been ordered in a ”causal ladder” with
a few variations regarding the number of properties included in the
hierarchy.minguzzi06c  (See Figure 1)

Roughly speaking, we define causality as the relation between two events
corelated in a regular pattern or between a cause an an effect. All physical
theories assume causation as an inherent fundamental assumption. In
relativity, an event can influence another event only if there is a causal
(timelike or null) curve connecting the two space-time points. We may consider
3 different levels of comprehension of the causal structure of the space-time:
the first one is an abstract-formal stage, with origins in Special Relativity
and assigning a light cone to every single event in space-time. The second
stage has a topological nature and considers the local differential behaviour
of geodesics on a Lorentzian manifold. The third aspect assumes a cosmological
level of our understanding of causality and incorporates classical global
problems in General Relativity, as the initial value problem, space-time
boundaries or the singularity theorems.

##  2 Remarks

Is there a current rigorous definition of the causal structure of the space-
time? A generic acceptable definition should contain all the information about
the Lorentzian manifold, causal properties and time orientability, a universal
assumption always lying behind any physical theory. The main question about
causality and its relation to time is here translated into an issue of
assuring a perfect consistency of the causal ladder of space-time.

The most important step in the causal hierarchy is the global hyperbolicity
property, located at the top of the ladder. The concept is central to General
Relativity and generates several open questions in other context areas:
initial value problem, singularity theorems, geodesic inextendability,
imprisonment or causal boundaries. Geroch has shown that if a space-time is
globally hyperbolic, it admits at least one submanifold that is intersected
once by an inextendible timelike geodesic. Briefly, the space-time is globally
hyperbolic if it admits a Cauchy hypersurface. Any such a topological surface
would be acausal as it is crossed only once by an inextendible timelike curve.
In this sense, an achronal causal curve is a lightlike geodesic. One of the
other recent results is also the statement that ”chronological spacetimes
without lightlike lines lines are stable causal” (Minguzzi). The physical
meaning of this theorem is the fact that if the space-time admits causality
violations, then either the chronology is violated or the space-time is
singular.

The standard definition of global hyperbolicity assumes two separate
conditions:

1\. strong causality (no ”almost closed causal curves”)Lobo  which was later
weakened by Bernal, Sanchez (2000)bernal06b  by replacing it with causality
(no closed causal curves exist), by introducing a new concept of ”causal
simplicity”.

2\. compactness of the space-time (”no naked singularities” condition),
strictly derived from the Weak Cosmic Censorship Conjecture.

If we consider causal boundaries over the space-time, other new open questions
appear. It has been proven that a globally hyperbolic space-time always admits
a causal boundary containing a timelike subset of causal curves. seifert71
Again, a causal gradient is implicitly asssumed in order to hold the
consistency of the causal theory.

Figure 1: Causal hierarchy of space-time

If an unknown factor that defines some sort of continuous increasing function
that applies to any future directed causal curve, is applied, the consistency
of the causal hierarchy of the spacetime is saved. The implicit or explicit
assumption of such a smooth time function containing a pre-ordering timelike
gradient sorkin96  has already been introduced by Hawking in his work ”The
existence of cosmic time functions”.hawking68  The cosmic time is defined as a
global function that increases along every future directed timelike or null
curve. The existence of such a function requires causal stability as a
fundamental condition (no closed timelike or null curves in any Lorentz metric
that is sufficiently near the space-time metric). Hawking proves the
equivalence bewtween two fundamental features of the space-time: stable
causality and global time.minguzzi08b

Is it necessary to make all these tacit assumptions about a global time
function, often associated with a cosmological flow? Can we speak about a
global time? Are there any constraints in the local physical laws that would
tell us anything about a global time? The concept itself of a global time is
taken for granted. The time function generates a total pre-ordering gradient
on the space-time manifold. It has been proven (Minguzzi) that under
physically reasonable conditions, the absence of a global time function would
imply a singular space-time.

##  3 Conclusions

The main problem is that we don’t have enough information about the evolution
of the space-time manifold from physically reasonable initial conditions and
if this evolution would generate naked singularities. There are no clear
results that prove the existence of global hyperbolicity. If the space-time is
non-globally hyperbolic, the initial Cauchy data can’t provide enough
information from past time-like infinity to future infinity, to completely
determine the current state of the universe.

If global hyperbolicity doesn’t hold, information coming from spatial future
null infinity should be taken into account. This concept could work if we
modify the notion of causal precedence and disconnect it from the ”hidden”
timelike gradient self-contained in the causality definition. From a
deterministic point of view, the principle of causality states that a physical
event described by various variables is fully determined at a given time by a
previous event in the causal chain.

In the new definition, the cause still precedes the effect. However, we
separate this expression from a ”temporal” point of view. In this case, the
cause wouldn’t necessarily precede the effect from past to future infinity.
Old philosophical definitions of ”final causality” could work here.
Superluminal causal propagator in scalar quantum field would easily secure
this concept. A simple illustration of a non-causal theory is the elliptic
Klein-Gordon scalar field, where information can propagate along a closed
curve in the space-time, any event along this curve being able to influence
itself. Another example could be the scalar field theory with a non-canonical
kinetic term (or k-essence), allowing superluminal propagation of information,
without threatening causality.

We basically want to stress here the fundamental role of chronology,
orientability and Cauchy problem in the standard definition of causality. We
are looking for a generalized definition of causality that wouldn’t involve
any chronology requirement in its expression (assuming a well posed Cauchy
problem).

The assumption is a satisfactory mechanism that helps to climb the causal
ladder, without violating Hawking’s chronology protection conjecture Hawk2 .
The ”well behaved” Cauchy initial data would indeed contain information from
spatial null infinity and it would lead to a ”stable” state of the Universe,
without contradicting any regular physical assumptions (without violating
either chronology or causality). Whether we assume the existence of a global
time function or a global hyperbolic space-time, stable causality, as the
strongest constraint in the causal ladder would be in this way saved.

## References

  * (1) S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Spacetime, Cambridge University Press, London (1973).
  * (2) S. W. Hawking, “Chronology Protection Conjecture,” Phys. Rev. D 56, 4745-4755 (1992).
  * (3) K. S. Thorne, “Closed Timelike Curves,” General Relativity and Gravitation 1992, Gleiser, J.L., et al. eds, Institute of Physics, Bristol (1992).
  * (4) F. Lobo and P. Crawford, “Time, closed timelike curves and causality,” NATO Sci. Ser. II 95, 289 (2003) [arXiv:gr-qc/0206078].
  * (5) Sorkin, R. D. and Woolgar, E.: A causal order for spacetimes with $C^{0}$ Lorentzian metrics: proof of compactness of the space of causal curves.  Class. Quantum Grav. 13, 1971–1993 (1996)
  * (6) Seifert, H.: The causal boundary of space-times.  Gen. Relativ. Gravit. 1, 247–259 (1971)
  * (7) Seifert, H. J.: Smoothing and extending cosmic time functions.  Gen. Relativ. Gravit. 8, 815–831 (1977)
  * (8) Minguzzi, E. and Sánchez, M.: _The causal hierarchy of spacetimes_ , Zurich: Eur. Math. Soc. Publ. House, vol. H. Baum, D. Alekseevsky (eds.), Recent developments in pseudo-Riemannian geometry, of _ESI Lect. Math. Phys._ , pages 299–358 (2008).
  * (9) Minguzzi, E.: $K$-causality coincides with stable causality.  Commun. Math. Phys. 290, 239–248 (2009)
  * (10) Malament, D. B.: Causal theories of time and the conventionality of simultaneity.  Noûs 11, 293–300 (1977)
  * (11) Geroch, R.: Domain of dependence.  J. Math. Phys. 11, 437–449 (1970)
  * (12) Hawking, S. W.: The existence of cosmic time functions.  Proc. Roy. Soc. London, series A 308, 433–435 (1968)
  * (13) Bernal, A. N. and Sánchez, M.: Globally hyperbolic spacetimes can be defined as ‘causal’ instead of ‘strongly causal’.  Class. Quantum Grav. 24, 745–749 (2007)


††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.††thanks: These authors contributed
equally to this work.

# Supplementary for:
“Identification of exciton complexes in a charge-tuneable Janus
W${}_{\mathrm{Se}}^{\mathrm{S}}$ monolayer”

Matthew S. G. Feuer  Cavendish Laboratory, University of Cambridge, 19 J. J.
Thomson Ave., Cambridge, CB3 0HE, UK  Alejandro R.-P. Montblanch  Cavendish
Laboratory, University of Cambridge, 19 J. J. Thomson Ave., Cambridge, CB3
0HE, UK  Mohammed Sayyad  Materials Science and Engineering, School for
Engineering of Matter, Transport and Energy, Arizona State University, Tempe,
Arizona, 85287, USA  Carola M. Purser  Cavendish Laboratory, University of
Cambridge, 19 J. J. Thomson Ave., Cambridge, CB3 0HE, UK  Cambridge Graphene
Centre, University of Cambridge, 9 J. J. Thomson Ave., Cambridge, CB3 0FA, UK
Ying Qin  Materials Science and Engineering, School for Engineering of Matter,
Transport and Energy, Arizona State University, Tempe, Arizona, 85287, USA
Evgeny M. Alexeev  Cambridge Graphene Centre, University of Cambridge, 9 J. J.
Thomson Ave., Cambridge, CB3 0FA, UK  Cavendish Laboratory, University of
Cambridge, 19 J. J. Thomson Ave., Cambridge, CB3 0HE, UK  Alisson R. Cadore
Cambridge Graphene Centre, University of Cambridge, 9 J. J. Thomson Ave.,
Cambridge, CB3 0FA, UK  Barbara L. T. Rosa  Cambridge Graphene Centre,
University of Cambridge, 9 J. J. Thomson Ave., Cambridge, CB3 0FA, UK  James
Kerfoot  Cambridge Graphene Centre, University of Cambridge, 9 J. J. Thomson
Ave., Cambridge, CB3 0FA, UK  Elaheh Mostaani  Cambridge Graphene Centre,
University of Cambridge, 9 J. J. Thomson Ave., Cambridge, CB3 0FA, UK
Radosław Kalęba  Cavendish Laboratory, University of Cambridge, 19 J. J.
Thomson Ave., Cambridge, CB3 0HE, UK  Pranvera Kolari  Materials Science and
Engineering, School for Engineering of Matter, Transport and Energy, Arizona
State University, Tempe, Arizona, 85287, USA  Jan Kopaczek  Materials Science
and Engineering, School for Engineering of Matter, Transport and Energy,
Arizona State University, Tempe, Arizona, 85287, USA  Kenji Watanabe  National
Institute for Materials Science, Tsukuba, Ibaraki 305-0034, Japan  Takashi
Taniguchi  National Institute for Materials Science, Tsukuba, Ibaraki
305-0034, Japan  Andrea C. Ferrari  Cambridge Graphene Centre, University of
Cambridge, 9 J. J. Thomson Ave., Cambridge, CB3 0FA, UK  Dhiren M. Kara
Cavendish Laboratory, University of Cambridge, 19 J. J. Thomson Ave.,
Cambridge, CB3 0HE, UK  Sefaattin Tongay<EMAIL_ADDRESS>Materials
Science and Engineering, School for Engineering of Matter, Transport and
Energy, Arizona State University, Tempe, Arizona, 85287, USA  Mete Atatüre
<EMAIL_ADDRESS>Cavendish Laboratory, University of Cambridge, 19 J. J.
Thomson Ave., Cambridge, CB3 0HE, UK

###### Contents

  1. Supplementary Note S1 Device characterisation
  2. Supplementary Note S2 Gate-dependent measurements in device 2
  3. Supplementary Note S3 Spectral mapping and homogeneity
  4. Supplementary Note S4 Power-dependent measurements
  5. Supplementary Note S5 Band structure calculations
  6. Supplementary Note S6 Magnetic-field dependent measurements

##  Supplementary Note S1 Device characterisation

Supplementary Figure S1 shows the characterisation of the layer thicknesses by
atomic force microscope (AFM) topography (Bruker Icon). Supplementary Fig. S1a
and b show the height profiles for the bottom and top multilayer hexagonal
boron nitride (ML-hBN), respectively. Step height analysis was performed in
Gwyddion, by obtaining the average height across an appropriate line cut
(black circles) and fitting the ‘smooth bent step’ function to extract the
layer thickness from the step height [1]. This gives the bottom and top ML-hBN
thickness as $27.2(2)$ nm and $12.1(3)$ nm respectively, as shown by the red
lines in Supplementary Figs. S1b and S1c. Supplementary Fig. S1c shows the
height profile for the few layer graphene (FLG), which gives the FLG thickness
as $6.3(2)$ nm.

Supplementary Figure S1: AFM height profiles of the device. (a) Height profile
for the bottom ML-hBN, obtained as the average height (black circles) across a
line cut. The red line is a fit to a ‘smooth bent step’, which gives the step
height $h$. (b) Height profile for the top ML-hBN, obtained as the average
height (black circles) across a line cut. The red line is a fit to a ‘smooth
bent step’, which gives the step height $h$. (c) Height profile for the FLG,
obtained as the average height (black circles) across a line cut. The red line
is a fit to a ‘smooth bent step’, which gives the step height $h$.
Supplementary Figure S2: Raman spectra from the device, acquired at room
temperature using 2.33 eV optical excitation. (a) Raman spectra from an un-
converted 1L-WSe2 (black curve) and fully converted Janus
1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ (red curve) region of the device. The
labelled peaks are the 1L-WSe2 $E^{\prime}$ \+ $A^{\prime}_{1}$ and the
1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ first-order Raman modes: $E^{1}$,
$A^{1}_{1}$, $E^{2}$ and $A^{2}_{1}$. (b) Raman spectra from the FLG (black
curve) and ML-hBN (red curve). The labelled peaks are the G and 2D Raman modes
from the FLG and the $E_{2g}$ Raman mode from the ML-hBN.

We perform room-temperature Raman spectroscopy in a commercial Horiba LabRam
Evolution system, using 2.33 eV optical excitation. Supplementary Figure S2a
shows the Raman spectra from an un-converted WSe2 monolayer (1L-WSe2) region
of the device as the black curve and fully converted Janus
W${}_{\mathrm{Se}}^{\mathrm{S}}$ monolayer
(1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$) region as the red curve. The peak at
$\sim 253$ cm-1 from 1L-WSe2 is assigned to the convoluted $E^{\prime}$ \+
$A^{\prime}_{1}$ modes, which are degenerate in 1L-WSe2 [2, 3]. The first-
order Raman peaks from 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ are assigned as the
$E^{1}$ mode at $\sim 205$ cm-1, the $A^{1}_{1}$ mode at $\sim 283$ cm-1, the
$E^{2}$ mode at $\sim 331$ cm-1 and the $A^{2}_{1}$ mode at $\sim 421$ cm-1
[4]. The other Raman peaks present arise from higher-order Raman modes, as
discussed in ref. [4]. The distinct Raman modes between 1L-WSe2 and Janus
1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ distinguishes the un-converted 1L-WSe2 and
fully converted 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ regions. Furthermore, the
1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ Raman spectra also confirms we have
ordered Janus 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ rather than a disordered
ternary alloy, which would show the representative Raman peaks of 1L-WSe2 and
1L-WS2 [4, 5, 6].

Supplementary Figure S2b shows the Raman spectra from the FLG (black curve)
and ML-hBN (red curve). The peak at $\sim 1366$ cm-1 from ML-hBN is assigned
to the $E_{2g}$ mode [7, 8]. The G peak from the FLG is seen at $\sim 1582$
cm-1 and the 2D band at $\sim 2720$ cm-1 [9].

Supplementary Figure S3: Comparison of spectra from 1L-WSe2 and
1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$, at room temperature and 4 K. (a) Room
temperature PL spectrum from an un-converted ML-hBN-encapsulated 1L-WSe2
region of the device. (b) Room temperature PL spectra from Janus
1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$, comparing the ML-hBN-encapsulated
1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ device (red curve) with unencapsulated
1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ (blue curve). Spectra are normalised to
the same peak height. (c) 4 K PL (red curve, left axis) and RC (black curve,
right axis) spectra from a location in the 1L-WSe2 region of the device. The
arrows mark the energy of the intra- and inter-valley trion and the dashed
line shows the energy of the neutral exciton, X0. (d) 4 K PL (red curve, left
axis) and RC (black curve, right axis) from a location in the
encapsulated1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ region of the device. The
dashed line marks the energy of the X0. All PL spectra were acquired with 2.33
eV excitation.  Supplementary Figure S4: Charge dependence of the reflectance
contrast for a second 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ device. (a) Optical
image of the second device. The 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ is
outlined in red, the top and bottom ML-hBN in blue and the top and bottom FLG
gates in black. The bulk TMD, bottom FLG and top FLG gates are electrically
contacted by gold. (b) The derivative of RC with varying electron doping
density $n$ (left axis) and applied bottom gate voltage (right axis) at 4 K.

The photoluminescence (PL) spectra from the 1L-WSe2 and
1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ regions of the device are compared in
Supplementary Figure S3. The left-hand panels (a and b) compare the room-
temperature PL spectra, with the spectrum from a ML-hBN-encapsulated un-
converted 1L-WSe2 region of the device in Supplementary Fig. S3a.
Supplementary Fig. S3b compares the spectrum from an unencapsulated
1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ sample (blue curve), which is directly on
a Si/SiO2 substrate and converted from exfoliated 1L-WSe2, to the ML-hBN-
encapsulated 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ (red curve) region of the
device. The emission peak intensity is spectrally shifted from $\sim$1.675 eV
in 1L-WSe2 to $\sim$1.8 eV in 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$. Peak widths
are estimated from a Gaussian fit to the high energy side of the spectra,
which gives $\sim$50 meV for the unencapsulated
1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$, $\sim$42 meV for the encapsulated 1L-WSe2
and $\sim$35 meV for the encapsulated 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$. The
width of the unencapsulated 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ emission is
similar to the best reported for unencapsulated
1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ at room temperature [10]. Spectral
narrowing is seen in the room temperature 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$
emission upon encapsulation, to below the width of the emission from the
encapsulated 1L-WSe2 region of the device. The highest peak at $\sim 1.8$ eV
in the encapsulated 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ spectrum likely arises
from the recombination of the neutral exciton, as identified in the
temperature dependence of ref. [10].

The PL (red curve) and reflection contrast (RC) spectra (black curve) at 4 K
are shown in the right-hand panels (c and d) of Supplementary Figure S3, for
both the 1L-WSe2 region of the device in Supplementary Fig. S3c and the
1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ region in Supplementary Fig. S3d. The PL
spectra narrow considerably from room temperature and multiple resolvable
peaks can be seen in both regions. In the 1L-WSe2 region, the PL spectrum is
consistent with previous reports [11] and we can identify the the neutral
exciton, X0, at 1.741 eV (width 5 meV); the inter-valley negatively charged
trion, X${}^{-}_{\textrm{inter}}$, at 1.720 eV (width 6 meV); and the intra-
valley negatively charged trion, X${}^{-}_{\textrm{intra}}$, at 1.710 eV
(width 9 meV). The 1L-WSe2 RC spectrum shows the neutral exciton feature at
1.741 eV, coincidental in energy with the X0 PL peak. On moving to the
1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ region, the spectral range of the emission
shifts to above $\sim$ 1.77 eV but the spectral widths remain similar. As
discussed in the main text, the highest energy peak is identified as arising
from 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ X0 and the RC signal is coincidental
with this.

The RC signal from 1L-WSe2 uses a different light source (Thorlabs M730L5)
compared to the RC from 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ (Thorlabs M660L4),
due to the different spectral range of the transitions.

##  Supplementary Note S2 Gate-dependent measurements in device 2

Supplementary Figure S5: Integrated Raman and PL maps from the device. (a) -
(c) Integrated Raman maps, acquired at room temperature and using 2.33 eV
optical excitation. The integration range is over Raman shifts 250 to 300 cm-1
in a, 253 to 255 cm-1 (1L-WSe2 $E^{\prime}$ \+ $A^{\prime}_{1}$ Raman mode) in
b and 283 to 285 cm-1 (1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ $A^{1}_{1}$ Raman
mode) in c. (d) - (f) Integrated PL maps, acquired at 4 K and using 2.33 eV
excitation. The integration range is over the spectral band 1.63 to 1.91 eV in
d, 1.63 to 1.75 eV (1L-WSe2 spectral band) in e and 1.77 to 1.91 eV
(1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ spectral band) in f. All maps are over
the same region of the device as highlighted by the white box in Fig. 1b of
the main text. Each panel is normalised to the maximum intensity within the
panel. The red outline corresponds to the 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$,
the blue outline to the top ML-hBN and the black outline to the FLG.

We have measured similar results to those discussed in the main text on a
second sample (device 2). Supplementary Figure S4a shows the optical image of
device 2. This second device consists of a converted region of
1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ (outlined in red), attached to bulk
unconverted WSe2. The 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ is encapsulated by
61 nm of bottom ML-hBN and 43 nm of top ML-hBN (both outlined in blue). A
bottom FLG gate is under the bottom ML-hBN and top FLG gate above the top ML-
hBN (both outlined in black). Electrical contacts to the bulk TMD, top and
bottom FLG gates are provided by gold leads. The device is fabricated using
the same polydimethylsiloxane (PDMS) transfer technique as for device 1 (see
Methods) and the Janus conversion uses the same SEAR in-situ process [5, 10].
Layer thicknesses are confirmed by AFM topography.

Supplementary Figure S4b shows the RC derivative as we tune the doping
density, $n$, for device 2. The doping density is tuned here by applying a
voltage between the TMD and bottom FLG gate and $n$ is calculated as described
in Methods, using the 61 nm bottom ML-hBN thickness. We set the band edge
$n=0$ to be where the neutral exciton RC signal vanishes and find the
intrinsic doping to be $n_{i}=0.7\times 10^{12}\textrm{ cm}^{-2}$. In the
n-doped regime we see the same transitions and behaviour as for device 1 (see
Fig. 3 of the main text), namely the neutral exciton X0, the inter- and intra-
valley negative trions (X${}^{-}_{\textrm{inter}}$ and
X${}^{-}_{\textrm{intra}}$, respectively) and the X-′ transition. Device 2 has
a lower level of intrinsic doping than device 1; however, as we change to
positive voltage and attempt to enter the p-doped regime we observe that the
neutral exciton persists and thus we are unable to reach the p-doped regime.

We find that the neutral exciton transition X0 is at 1.907 eV at this location
on device 2, which is a 17 meV shift from the average across device 1 and
possibly arises from a difference in strain on the device. We see the
X${}^{-}_{\textrm{inter}}$ transition at 1.875 eV and
X${}^{-}_{\textrm{intra}}$ at 1.869 eV, which gives binding energies relative
to X0 of 32 meV and 38 meV, respectively, agreeing with those measured on
device 1. This gives an exchange splitting of 6 meV, agreeing with both our
calculation (see Supplementary Note S5) and measurement on device 1. As in
device 1, we see the X-′ transition, which emerges at 1.854 eV and red shifts
with increased doping. The doping densities where the various transitions
appear in device 2 are also similar to those in device 1. Specifically, the
cross-over between negative trions and X-′ occurs around $(2-3)\times
10^{12}\textrm{cm}^{-2}$ in both devices.

##  Supplementary Note S3 Spectral mapping and homogeneity

Supplementary Figure S6: Raman and PL spectra across paths in the Raman and PL
maps. (a) Raman map of the device from Fig. 1c of the main text, acquired at
room temperature and using 2.33 eV optical excitation. (b) PL map of the
device from Fig. 1d of the main text, acquired at 4 K and using 2.33 eV
excitation. (c) Raman spectra along the red path in a, from the un-converted
1L-WSe2 region (point A), through the partially converted region (point B) to
the fully converted 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ region (point C). The
spectra are vertically offset for clarity. (d) PL spectra along the red path
in b, from the un-converted 1L-WSe2 region (point A), through the partially
converted region (point B) to the fully converted
1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ region (point C). The spectra are
vertically offset for clarity.

Supplementary Figure S5 presents integrated spectral maps over the device, for
both Raman and PL spectroscopy, which are used to calculate the relative
intensity maps in Figs. 1c and d of the main text. Supplementary Figs. S5a -
S5c show the room-temperature integrated Raman maps of the characteristic
Raman modes from 1L-WSe2 and 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$.
Supplementary Fig. S5a shows the integrated Raman intensity over the spectral
range (250 to 300 cm-1) covering both the 1L-WSe2 and
1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ characteristic modes. Supplementary Fig.
S5b shows the integrated intensity from the 1L-WSe2 $E^{\prime}$ \+
$A^{\prime}_{1}$ Raman mode (253 to 255 cm-1) and Supplementary Fig. S5c shows
the integrated intensity from the 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$
$A^{1}_{1}$ Raman mode (283 to 285 cm-1). This shows that the flake has a
central region of fully converted 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$, as
shown by the yellow region in the relative intensity map of Fig. 1c from the
main text.

Supplementary Figs. S5d - S5f show the PL maps, acquired at 4 K, of the
integrated spectral bands associated with 1L-WSe2 and
1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$. Supplementary Fig. S5d shows the
integrated PL intensity over the spectral band (1.63 to 1.91 eV) covering both
the 1L-WSe2 and 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ emission. Supplementary
Figs. S5e and S5f show the integrated intensity from the 1L-WSe2 (1.63 to 1.75
eV) and 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ (1.77 to 1.91 eV) spectral bands,
respectively. The integrated PL maps show a central region of converted
1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$, consistent with the integrated Raman maps
and shown by the yellow region in the relative intensity map of Fig. 1d from
the main text.

Supplementary Figure S6 shows the variation in Raman and PL spectra across
paths of varying Janus conversion. Supplementary Figure S6c and S6d plot the
Raman and PL spectra along the paths indicated in Supplementary Figs. S6a and
S6b, respectively. Note that these paths were taken on the same region as the
one shown in Figs. 1c and 1d from the main text. The paths are along the red
lines from the un-converted 1L-WSe2 region (point A), through the partially
converted region (point B), to the fully converted
1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ region (point C). The partially converted
regions show the distinctive Raman modes from both 1L-WSe2 and
1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$, along with PL emission in both spectral
bands. The un-converted and fully converted regions show the Raman modes
corresponding to either 1L-WSe2 or 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$, and PL
emission from a single spectral band.

Supplementary Figure S7: PL and RC spectra from
1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ across different locations. (a) - (i) PL
(red curve, left axis) and RC (black curve, right axis) spectra at different
locations in the 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ region of the device.
Each panel shows the PL and corresponding RC spectra at a single location. All
spectra were acquired at 4 K, and the PL spectra with 2.33 eV excitation.
Panel a appears as Fig. 2b of the main text.

Supplementary Fig. S7 presents the variation in PL and RC spectra at nine
locations across the sample in the 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ region
of the device. Across the sample, we see changes in the relative intensity of
the different PL peaks and width of the redder emission tail. However, the
general structure of the higher energy peaks persists across different
locations, and specifically the peaks labelled in Fig. 2a of the main text (1,
2, 3 and X0) are identifiable. Across the device, the RC feature is
coincidental with the X0 peak in PL but with contrast dependent on location.
As discussed in the main text, calculating statistics across the device gives
the average X0 PL peak energy as 1.890(1) eV and FWHM as 8.4(4) eV. A similar
variation in PL spectra is also seen in the best WSe2 samples [11, 12] and
likely arises from a variation in strain and fabrication inhomogeneity.

##  Supplementary Note S4 Power-dependent measurements

Supplementary Figure S8: Excitation-power PL dependence of the
1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ emission, acquired at 4 K and using 2.33
eV excitation. (a) Power dependence of the PL spectra, normalised by
integration time and excitation power. The excitation power is varied between
15 nW (grey curve) to 56 $\mu$W (black curve). Spectra are vertically offset
for clarity. (b) Same as in a, but showing only the neutral exciton X0. (c)
Excitation power dependence of the PL intensity of the labelled peaks 1, 2 and
X0 from a and b. These are the resolvable peaks at high power. The data is
plotted on a double logarithmic scale and the solid line is a fit to
$I~{}\propto~{}P^{n}$, with peak intensity $I$, power $P$ and power law
scaling $n$. All three peaks give linear power law scaling. (d) Same as in c
but for the unlabelled peaks in a as indicated by coloured arrows, which
appear at low excitation power. The solid line is a fit to
$I~{}\propto~{}P^{n}/(P^{n}+P_{\textrm{sat}}^{n})$, with saturation power
$P_{\textrm{sat}}$. The dashed line shows a linear power scaling $n=1$. All
these peaks show an initially linear power dependence, before their intensity
saturates.

Supplementary Figure S8 shows the excitation power dependence of the PL
emission from 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$, over the full spectral
range in Supplementary Fig. S8a and over the spectral range of the neutral
exciton in Supplementary Fig. S8b. The spectra remain of a similar shape as
the excitation power is changed from 15 nW to 50 $\mu$W (corresponding to 3
Wcm-2 to 104 Wcm-2), except for the appearance of low energy peaks at low
powers. The excitation power dependence of the integrated intensity of the
resolvable peaks (labelled 1, 2 and X0) in the high power spectrum are shown
in Supplementary Fig. S8c. A power law fit shows that these three peaks follow
a linear power law scaling, consistent with assignment of these peaks as
arising from free excitonic species [11]. Specifically, the linear power
dependence of the X0 peak provides evidence of it arising from the
recombination of the neutral exciton, as discussed in the main text. We see no
evidence of a blueshift to these spectral peaks with increasing excitation
power, up to the maximum accessible power of $\sim 50~{}\mu$W (104 Wcm-2). The
power dependence of the intensity of the marked low energy peaks (labelled by
coloured arrows) are shown in Supplementary Fig. S8d. In contrast to the
higher energy peaks, we see that their intensity follows a linear power
scaling at low powers and saturates at powers in the range 50 to 500 nW (10 to
100 Wcm-2). As noted in the main text, this points to the presence of
localised defects displaying quantum light emission [13, 14, 15].

##  Supplementary Note S5 Band structure calculations

Supplementary Figure S9: DFT calculation of the band structure for 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$. (a) Energy dispersion plotted along the $\Gamma-M-K-\Gamma$ path. The calculation uses the PBE functional, including spin-orbit coupling. The blue and red arrows indicate the spin of the bands at the K point (spin up and down respectively) and the black arrow shows the energy gap. The Fermi energy is set to zero. (b) The exchange energy for the inter-valley negative trion, between the excess electron (grey circle) and the paired electron-hole pair (black and white circle). $J^{\textrm{ee}}_{\textrm{ex}}$ and $J^{\textrm{eh}}_{\textrm{ex}}$ are the electron-electron and electron-hole exchange energies, respectively. $\sigma$ indicates the paired electron-hole pair. Spin up bands are shown in blue and spin down in red. Complex | Binding energy (meV)
---|---
Negative trion (X-) | 32.2
Positive trion (X+) | 32.0
Biexciton (XX0) | 21.0
Quinton (XX-) | 54.2
Supplementary Table S1: Binding energy of the excitonic complexes compared to
the neutral exciton, for free-standing 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$.
Supplementary Figure S10: Magnetic field dependence of the PL for
1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$, acquired at 4 K and using 2.33 eV
excitation. (a) PL spectra as a function of magnetic field, under right-
circularly ($\sigma^{+}$) polarised excitation and detection (co-polarised).
(b) Same as for a but with right-circularly ($\sigma^{+}$) polarised
excitation and left-circularly ($\sigma^{-}$) polarised detection (cross-
polarised). (c) PL spectra at 8 T, under $\sigma^{+}$ excitation and with both
$\sigma^{+}$ (black curve) and $\sigma^{-}$ (red curve) detection. The inset
shows only the neutral exciton X0. (d) Energy splitting $\Delta E$ as a
function of magnetic field $B$ between the $\sigma^{+}$ and $\sigma^{-}$
detected PL for the labelled peaks in c. The solid lines are linear fittings,
with the g factor found from the gradient as $\Delta E=-g\mu_{B}B$.

We perform density functional theory (DFT) to understand the band structure
and observed excitonic transitions of 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$.
Supplementary Figure S9a shows the band structure along the $\Gamma-
M-K-\Gamma$ path. This is calculated using the DFT-PBE functional, including
spin-orbit coupling with a k-grid of $24\times 24\times 1$. The DFT
calculation shows that, similar to conventional tungsten-based TMDs (1L-WSe2
and 1L-WS2) [16, 17, 18], 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ is direct
bandgap at the K points with spin ordering such that the upper valence band is
opposite in spin to the lower spin-split conduction band, with 37 meV spin
splitting.

The effective masses obtained from the DFT calculation are used in the Mott-
Wannier model and quantum Monte Carlo (QMC) [19, 11] to calculate the expected
binding energies of the excitonic charge complexes in free-standing
1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$, as given in Supplementary Table S1.
Experimental differences arise from the difference in dielectric environment
caused by the ML-hBN encapsulation [20].

The fine structure of the negative trion arises from the exchange interaction
between the excess electron and the paired electron-hole, which causes an
energy increase for the intervalley relative to intravalley negative trion
[21, 22, 23]. Supplementary Figure S9b shows the electron-electron
($J^{\textrm{ee}}_{\textrm{ex}}$) and electron-hole
($J^{\textrm{eh}}_{\textrm{ex}}$) exchange energies for the intervalley trion.

The magnitude of the exchange energies are [24, 21]:

$\displaystyle J^{\textrm{ee}}_{\textrm{ex}}$ $\displaystyle\approx
g^{\textrm{ee}}V(\textrm{{K}})|\langle
u_{\textbf{K}^{\prime}c\downarrow}|u_{\textbf{K}c\downarrow}\rangle|^{2}$
$\displaystyle J^{\textrm{eh}}_{\textrm{ex}}$ $\displaystyle\approx
g^{\textrm{eh}}V(\textrm{{K}})|\langle
u_{\textbf{K}^{\prime}v\downarrow}|u_{\textbf{K}c\downarrow}\rangle|^{2}$

where $u$ are the periodic Bloch wave factors at the $K$ and $K^{\prime}$
points in the Brillouin zone. The indices $c\downarrow$ and $v\downarrow$
denote the spin down conduction band and spin down valence band, respectively,
and correspond to the bands shows in Supplementary Fig. S9b. $g^{\textrm{ee}}$
and $g^{\textrm{eh}}$ are the electron-electron and electron-hole contact pair
distribution function (PDF), respectively, which correspond to the probability
of the finding the particles in the same position. $V(\textrm{{K}})$ is the
unscreened form of the 2d Coulomb interaction.

We calculate the PDFs for 1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ by using the
quantum Monte Carlo method [19] and find $g^{\textrm{ee}}$ to be much smaller
than $g^{\textrm{eh}}$, which is due to the Coulomb repulsion between two
electrons. Hence, we neglect $g^{\textrm{ee}}$ and
$J^{\textrm{eh}}_{\textrm{ex}}$ dominates over
$J^{\textrm{ee}}_{\textrm{ex}}$. Using DFT-PBE, we calculate $|\langle
u_{\textbf{K}^{\prime}v\downarrow}|u_{\textbf{K}c\downarrow}\rangle|^{2}=0.1$
and therefore we calculate the exchange energy splitting of the negative trion
to be $J_{\textrm{ex}}\sim 6$ meV, which agrees well with our observation of
an energy difference of 7 meV between X${}^{-}_{\textrm{inter}}$ and
X${}^{-}_{\textrm{intra}}$ on device 1 (as discussed in the main text).

##  Supplementary Note S6 Magnetic-field dependent measurements

We perform a magnetic field dependence of the
1L-W${}_{\mathrm{Se}}^{\mathrm{S}}$ PL emission, with circularly polarised
excitation and detection. Supplementary Figs. S10a and b show the PL spectra
as a function of magnetic field, under right circularly polarised excitation
and for co- and cross-polarised detection, respectively. As we apply the out-
of-plane magnetic field, we see the various PL peaks shift in energy.
Supplementary Fig. S10c shows the co- and cross-polarised (black and red
curves, respectively) PL spectra at an applied field of 8 T, with the inset
showing the energy splitting for the X0 peak. The energy splitting $\Delta E$
between the opposite circularly polarisations with magnetic field is shown in
Supplementary Fig. S10d, for the peaks labelled 1, 2 and X0. A linear fitting
to the energy splitting gives the g factor as $\Delta E=-g\mu_{B}B$ for each
of the peaks: we find a g factor 4.14(6) for the neutral exciton X0 (as shown
in Fig. 4 of the main text), a g factor 5.51(7) for peak 1 and 6.7(1) for peak
2. As discussed in the main text, a g factor $\sim$4 is expected for bright
excitons in conventional TMDs. The X0 g factor from PL agrees with that
extracted from RC, 4.5(2), and is as expected for the bright neutral exciton.
A higher g factor is also seen for lower energy states in 1L-WSe2 [25, 26, 27]
and, similarly to in 1L-WSe2, these peaks likely arise from various excitonic
complexes. Full attribution of the different transitions apparent in PL
requires further work.

Supplementary Figure S11: Gate voltage dependence of the g factor of the
different excitonic species. (a) Voltage dependence of the g factor for the
X-′ peak, between -22 to -5 V. (b) Voltage dependence of the g factor for the
X${}^{-}_{\textrm{inter}}$ (black points) and X${}^{-}_{\textrm{intra}}$ (red
points) peaks, between 8 to 14 V. (c) Voltage dependence of the g factor for
the X0 peak, between 17 to 21 V. All g factors are extracted from a linear
fitting to the energy splitting in RC between $\sigma^{+}$ and $\sigma^{-}$
detection polarisation with magnetic field. Error bars are given by the
fitting error from the linear fitting.

We find that the g factors, as measured in RC, of the various charged
complexes shown in Fig. 4 of the main text have a gate-voltage dependence.
Supplementary Figure S11 shows this gate dependence for the X-′ peak in panel
a, X${}^{-}_{\textrm{inter}}$ and X${}^{-}_{\textrm{intra}}$ in panel b (black
and red data points, respectively) and X0 in panel c. At each gate voltage,
the g factor is extracted from a linear fitting to the magnetic field
dependence of the energy splitting of the peak. We see that for X0 the g
factor remains constant at $\sim$4.5 over the entire range of voltages. The g
factor of the X-′ peak increases as the doping level increases, from $\sim$3
when the peak first appears, towards $\sim$4 V at -20 V. The trion peaks show
a larger gate voltage dependence, as discussed in the main text: the inter-
and intra-valley trion g factors appear anti-correlated and change from
$\sim$3 to $\sim$13\. Reported values of the g factors of the negatively
charged trions in 1L-WSe2 vary between similar values [28, 29, 30, 31] and
this variation is attributed to the doping dependence of the trion g factor in
ref. [32]. A similar doping dependence of the negatively charged trion in
1L-MoS2 has also been measured and is attributed to many-body interactions
with the Fermi sea of electrons [33].

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