# Evaporating Black-Holes, Wormholes, and Vacuum Polarisation: Must they Always Conserve Charge?

## Abstract

A careful examination of the fundamentals of electromagnetic theory shows that due to the underlying mathematical assumptions required for Stokes’ Theorem, global charge conservation cannot be guaranteed in topologically non-trivial spacetimes. However, in order to break the charge conservation mechanism we must also allow the electromagnetic excitation fields \(\mathbf {D}\), \(\mathbf {H}\) to possess a gauge freedom, just as the electromagnetic scalar and vector potentials \(\varphi \) and \(\mathbf {A}\) do. This has implications for the treatment of electromagnetism in spacetimes where black holes both form and then evaporate, as well as extending the possibilities for treating vacuum polarisation. Using this gauge freedom of \(\mathbf {D}\), \(\mathbf {H}\) we also propose an alternative to the accepted notion that a charge passing through a wormhole necessarily leads to an additional (effective) charge on the wormhole’s mouth.

## Keywords

Electromagnetism Topology Charge-conservation Constitutive relations Gauge freedom## 1 Introduction

It is not only a well established, but an extremely useful consequence of Maxwell’s equations, that charge is conserved [1]. However, this principle relies on some assumptions, in particular those about the topology of the underlying spacetime, which are required for Stokes’ Theorem to hold. Here we describe how to challenge the status of global charge conservation, whilst still keeping local charge conservation intact. We do this by investigating the interaction of electromagnetic theory and the spacetime it inhabits, and go on to discuss the potential consequences of such a scenario.

The relaxed assumptions about topology and gauge are not merely minor technical details, since many cosmological scenarios involve a non-trivial topology. Notably, black holes have a central singularity that is missing from the host spacetime [8, 9], and a forming and then fully evaporated black hole creates a non-trivial topology, which in concert with allowing a gauge freedom for now *non*-measureable \(\mathbf {D}\), \(\mathbf {H}\) fields, breaks the usual basis for charge conservation. We also consider more exotic scenarios, such as the existence of a universe containing a wormhole (see e.g. [10]), or a “biverse”—a universe consisting of two asymptotically flat regions connected by an Einstein–Rosen bridge. In particular we test the claim that charges passing through such constructions (wormholes) are usually considered to leave it charged [11, 12, 13].

Topological considerations and their influence on the conclusions of Maxwell’s theory are not new, but our less restrictive treatment of \(\mathbf {D}\), \(\mathbf {H}\) allows us a wider scope than in previous work. Misner and Wheeler, in [13] developed an ambitious programme of describing all of classical physics (i.e. electromagnetism and gravity) geometrically, i.e. *without* including charge at all. Non-trivial topologies, such as spaces with handles, were shown to support situations where charge could be interpreted as the non-zero flux of field lines, which never actually meet, over a closed surface containing the mouth of a wormhole. Baez and Muniain [14] show that certain wormhole geometries are simply connected, so that every closed 1-form is exact. In this case charge can then be *defined* as an appropriate integral of the electric field over a 2-surface. In another example, Diemer and Hadley’s investigation [15] has shown that it is possible, with careful consideration of orientations, to construct wormhole spacetimes containing topological magnetic monopoles or topological charges; and Marsh [16] has discussed monopoles and gauge field in electromagnetism with reference to topology and de Rham’s theorems.

It is important to note that our investigation here is entirely distinct from and prior to any cosmic censorship conjecture [17], the boundary conditions at a singularity, models for handling the event horizon [18], or other assumptions. Although an event horizon or other censorship arrangement can hide whatever topologically induced effects there might be, such issues are beyond the scope of our paper, which instead focuses on the fundamental issues—i.e. the prior and classical consequences of the violation of the prerequisites of Stokes’ Theorem in spacetimes of non-trivial topology.

In Sect. 2 we summarise the features of electromagnetism relevant to our analysis. In Sect. 3 we investigate under what circumstances charge conservation no longer holds, and its consequences for the electromagnetic excitation field \({{\mathbf {\mathsf{{H}}}}}\), which is the differential form version of the traditional \(\mathbf {D}\), \(\mathbf {H}\). Next, in Sect. 4 we describe further consequences, such as how a description of bound and free charges necessarily supplants a standard approach using constitutive relations based on \({{\mathbf {\mathsf{{H}}}}}\). Then, in Sect. 5 we see that topological considerations mean that \({{\mathbf {\mathsf{{H}}}}}\) can be defined in a way that has implications for the measured charge of wormholes. Lastly, after some discussions in Sect. 6, we summarise our results in Sect. 7.

## 2 Electromagnetism

### 2.1 Basics

*m*is the mass of the electron and length \(\ell \) is a small parameter.

However, fixing a constitutive relation where \({{\mathbf {\mathsf{{H}}}}}\) has a straightforward relationship to \({{\mathbf {\mathsf{{F}}}}}\), such as those given above, is of itself sufficent to enforce charge conservation. In contrast, we consider more general constitutive models, and so can investigate wider possibilities.

### 2.2 Conservation of Charge

*both*due to Stokes’ theorem:

- 1.
**Topology condition:**The first proof assumes that \(\mathcal {U}\) is the boundary of a topologically trivial bounded region of spacetime, i.e. \(\mathcal {U}={\partial }\mathcal {N}\), \(\mathcal {N}\subset \mathcal {M}\), within which \({{\mathbf {\mathsf{{J}}}}}\) is defined. A*topologically trivial*space is one that can be shrunk to a point i.e. it is topologically equivalent to a 4-dimensional ball. Then one hasthe last equality arising from (4), which we call the “topology condition”.$$\begin{aligned} \int _\mathcal {U}{{\mathbf {\mathsf{{J}}}}}= \int _{{\partial } \mathcal {N}} {{\mathbf {\mathsf{{J}}}}}= \int _\mathcal {N}{d} {{\mathbf {\mathsf{{J}}}}}= 0 , \end{aligned}$$(10) - 2.
**Gauge-free condition:**The second proof arises from integrating (3) over \(\mathcal {U}\), and presumes that \({{\mathbf {\mathsf{{H}}}}}\) is a well-defined 2-form field. We have thatwhere the last equality, which we call the “gauge-free condition”, results solely from the fact that \(\mathcal {U}\) is closed (i.e. \({\partial } \mathcal {U}=\emptyset \)), but does$$\begin{aligned} \int _\mathcal {U}{{\mathbf {\mathsf{{J}}}}}= \int _\mathcal {U}{d} {{\mathbf {\mathsf{{H}}}}}= \int _{{\partial } \mathcal {U}} {{\mathbf {\mathsf{{H}}}}}= 0 , \end{aligned}$$(11)*not*require that \(\mathcal {U}\) is itself the boundary of a compact 4-volume.^{1}

### 2.3 Non-conservation of Charge

To break the topology condition (10), it is sufficient that either there is no compact spacetime region \(\mathcal {N}\) such that \(\mathcal {U}={\partial } \mathcal {N}\), or that there are events in \(\mathcal {N}\) where \({{\mathbf {\mathsf{{J}}}}}\) is undefined. A test scenario is represented in Fig. 2, where a black hole forms in an initially unremarkable spacetime, i.e. one that contains spatial hypersurfaces that are topologically trivial. On formation this introduces a singularity, but then later as the black hole evaporates, the singularity also vanishes. The evaporation step also removes the event horizon, thus exposing any effects of the singularity—e.g. in charge conservation—to the rest of the universe. In this case the singularity, which exists for a period of time before evaporating [28] by means of Hawking radiation [29, 30], must either be removed from spacetime, meaning that \(\mathcal {N}\) is no longer topologically trivial, or alternatively that \({{\mathbf {\mathsf{{J}}}}}\) is not defined in all of \(\mathcal {N}\).

*single*constitutive relation for the Maxwell vacuum

It is worth noting that these two apparently distinct cases allowing for non-conservation of charge are related by topological considerations—the choice of spacetime with a line or point removed, the non-existence of a well-defined \({{\mathbf {\mathsf{{H}}}}}\), and the breaking of global charge conservation are all related to the deRham cohomology of the spacetime manifold.^{2}

## 3 Singularity

In this section we construct an orientable manifold \(\mathcal {M}\) on which charge is *not* globally conserved, even though (locally) \(d{{\mathbf {\mathsf{{J}}}}}=0\) everywhere on \(\mathcal {M}\). We start by assuming a flat spacetime with a Minkowski metric, except with the significant modification that a single event \(\mathbf {0}\) has been removed; i.e. \(\mathcal {M}={\mathbb R}^4\backslash \mathbf {0}\). This spacetime \(\mathcal {M}\) is sufficient to demonstrate our mathematical and physical arguments for charge conservation failure—but without introducing any of the additional complications of (e.g.) the Schwarzschild black-hole metric. As already noted in our Introduction, the discussion here is entirely separate from and prior to any assumptions about cosmic censorship, or any imagined model of the singularity behaviour.

### 3.1 Charge Conservation

*t*,

*x*,

*y*,

*z*) be the usual Cartesian coordinate system with \(\mathbf {0}=(0,0,0,0)\) and let \((t,r,\theta ,\phi )\) be the corresponding spherical coordinates.

^{3}Set \({\mathbb R}^{+}={\left\{ r\in {\mathbb R}|r\ge 0\right\} }\). Let us construct the smooth 3-form current density \({{\mathbf {\mathsf{{J}}}}}\) defined throughout \(\mathcal {M}\) as

*f*(and also of the function

*h*below), and it is replaced by

*r*/

*t*when the function is used to define fields. We then have

^{4}We note that as expressed in the Cartesian coordinates of (17), \({{\mathbf {\mathsf{{J}}}}}^{+}\) is well defined at the spatial origin for \(t>0\). In spherical polar coordinates we have

*not*an event in \(\mathcal {M}\), and the \({Q}\)’s appearance at \(t=0\) does not induce \({d}J\ne 0\) at some event in \(\mathcal {M}\). We see that the total charge is zero for the constant-time hypersurfaces

^{5}with \(t<0\), but for the constant-time hypersurfaces with \(t>0\) the charge is \({Q}\).

Similarly, over a region such as that shown in Figs. 1 and 2 we have that \(\int _\mathcal {U}{{\mathbf {\mathsf{{J}}}}}\ne 0\). Charge is therefore not conserved in \(\mathcal {M}\), despite the fact that \({d}{{\mathbf {\mathsf{{J}}}}}= 0\) everywhere in \(\mathcal {M}\).

*must*be the case since if it were not, then we could apply (11) to establish \({Q}=0\). Nevertheless we

*can*find two fields \({{\mathbf {\mathsf{{H}}}}}^{+}\) and \({{\mathbf {\mathsf{{H}}}}}^{-}\) with intersecting domains \(\mathcal {M}^{+}\) and \(\mathcal {M}^{-}\) such that on the intersection, \({{\mathbf {\mathsf{{H}}}}}^{+}= {{\mathbf {\mathsf{{H}}}}}^{-} + {d}\psi \), i.e they differ by just a gauge, as per (12). Let

The new gauge freedom for \({{\mathbf {\mathsf{{H}}}}}\) suggests the possibility of further generalisations to the vacuum constitutive relations. These could now go beyond rather prescriptive vaccum models such as e.g. the Euler-Heisenberg or Bopp–Podolsky ones in (6) and (7), whose Lagrangian formulations insist on a unique \({{\mathbf {\mathsf{{H}}}}}\).

## 4 Polarisation of the Vacuum

We now stay with the same scenario as in the previous section, but instead apply the bound current version of Maxwell theory as given by (15), interpreting \({{{\mathbf {\mathsf{{J}}}}}_{{\mathrm{b}}}}\) as representing the polarisation of the vacuum. It is known from quantum field theory that vacuum polarization occurs naturally for intense fields, with the first order correction to the excitation 2-form given by (6). Indeed, the strong magnetic fields associated with magnetars are known to induce non-trivial dielectric properties on vacuum [31]. An alternative model for the polarization of the vacuum is given by the Bopp–Podolski theory of electromagnetism, as outlined in (7). However in these cases the bound currents \({{{\mathbf {\mathsf{{J}}}}}_{{\mathrm{b}}}}^{\!\!\!\!{\mathrm{EH}}}=d{{\mathbf {\mathsf{{H}}}}}_{{\mathrm{EH}}}-{d}{\star }{{\mathbf {\mathsf{{F}}}}}\) and \({{{\mathbf {\mathsf{{J}}}}}_{{\mathrm{b}}}}^{\!\!\!\!{\mathrm{BP}}}=d{{\mathbf {\mathsf{{H}}}}}_{{\mathrm{BP}}}-{d}{\star }{{\mathbf {\mathsf{{F}}}}}\) correspond to a well defined excitation 2-form \({{\mathbf {\mathsf{{H}}}}}\) and therefore must conserve charge, regardless of topology. Nevertheless we are still free to consider more general versions of \({{{\mathbf {\mathsf{{J}}}}}_{{\mathrm{b}}}}\) which are not exact and contain more than just those corrections.

Since \({\star } {{\mathbf {\mathsf{{F}}}}}\) is well defined, and \({d}{\star } {{\mathbf {\mathsf{{F}}}}}={{{\mathbf {\mathsf{{J}}}}}_{{\mathrm{f}}}}+{{{\mathbf {\mathsf{{J}}}}}_{{\mathrm{b}}}}\), one can use the argument (11), replacing \({{\mathbf {\mathsf{{H}}}}}\) with \({\star } {{\mathbf {\mathsf{{F}}}}}\), to conclude that \({{{\mathbf {\mathsf{{J}}}}}_{{\mathrm{f}}}}+{{{\mathbf {\mathsf{{J}}}}}_{{\mathrm{b}}}}\) is globally conserved. We now examine whether it is necessary for \({{{\mathbf {\mathsf{{J}}}}}_{{\mathrm{f}}}}\) and \({{{\mathbf {\mathsf{{J}}}}}_{{\mathrm{b}}}}\) to be globally conserved *independently*. If \({{\mathbf {\mathsf{{H}}}}}\) is well defined then from (3) (which now becomes \(d{{\mathbf {\mathsf{{H}}}}}={{{\mathbf {\mathsf{{J}}}}}_{{\mathrm{f}}}}\)) \({d}{{{\mathbf {\mathsf{{J}}}}}_{{\mathrm{f}}}}=0\) and hence \({d}{{{\mathbf {\mathsf{{J}}}}}_{{\mathrm{b}}}}=0\). Under these circumstances, we find that \({{{\mathbf {\mathsf{{J}}}}}_{{\mathrm{f}}}}\) by itself is globally conserved, and likewise for \({{{\mathbf {\mathsf{{J}}}}}_{{\mathrm{b}}}}\) by itself.

In the following, we demand only that \({{{\mathbf {\mathsf{{J}}}}}_{{\mathrm{f}}}}\) and hence \({{{\mathbf {\mathsf{{J}}}}}_{{\mathrm{b}}}}\) are closed: \(d{{{\mathbf {\mathsf{{J}}}}}_{{\mathrm{f}}}}= d{{{\mathbf {\mathsf{{J}}}}}_{{\mathrm{b}}}}= 0\), and do not insist that they are globally conserved independently. This requires us to abandon the concept of a global macroscopic well-defined \({{\mathbf {\mathsf{{H}}}}}\), and to express the constitutive relations for our spacetime using the microscopic bound current, \({{{\mathbf {\mathsf{{J}}}}}_{{\mathrm{b}}}}\). Unlike \({{\mathbf {\mathsf{{F}}}}}\), the excitation \({{\mathbf {\mathsf{{H}}}}}\) cannot be measured directly using either the Lorentz force equation or the Aharonov–Bohm effect.

- (i)
\(h(\xi )\) is defined from (21), i.e. \(h(\xi ) = \frac{1}{\xi ^3}\int _0^\xi f({\hat{\xi }}){\hat{\xi }}^2d{\hat{\xi }}\).

- (ii)
\(f: {\mathbb R}^+\rightarrow {\mathbb R}^+\) is again a bump function satisfying \(f(\xi ) \ge 0\) for \(0< \xi < 1/2\), \(f(\xi ) = 0\) for \(\xi \ge 1/2\), and \(f^{(n)}=0\) for \(n \ge 1\).

- (iii)
\({{\mathrm{hv}}}:{\mathbb R}\rightarrow {\mathbb R}\) is the Heaviside step function.

- (iv)
\({\chi }:{\mathbb R}^{+}\rightarrow {\mathbb R}\) is a bump function with \({{\chi }(\xi )}=1\) for \(0\le \chi <\tfrac{2}{3}\) and \({{\chi }(\xi )}=0\) for \(\xi >\tfrac{5}{6}\).

*local*sense, not globally. For our example, they are respectively given by

In writing (24) the distinction between free and bound current densities, as arise in the subsequent calculations, is introduced artificially. So whilst our introduced example is certainly artificial, it can be taken to be representative and illustrative of a scenario in which the 2-form field \({{\mathbf {\mathsf{{H}}}}}\) is no longer globally defined; but that the sum of free and bound charge densities in vacuum is globally conserved, while the two types of charge are not collectively globally conserved, and exist independently in disjoint regions of space. This echoes the previous section, but here we use the non-exactness of \({{\mathbf {\mathsf{{J}}}}}\) rather than the gauge freedom for \({{\mathbf {\mathsf{{H}}}}}\); thus suggesting generalisations to the charge and polarization properties of the vacuum constitutive relations.

## 5 Wormhole

*not*break conservation of global charge, but instead address the issue of whether a wormhole necessarily gains the charge of any matter passing through it. One simple way of describing this standard viewpoint is to note that the usual process of drawing field lines for a charge, as it moves, forbids them from swapping their end-points from one place to another. This means that a positive charge passing through a wormhole “drags” its field lines behind it like a tail, and the resulting collection of field lines re-entering the wormhole looks like a negative charge, and then as they exit the other side they look like a positive charge; as depicted in Fig. 6.

*t*, we have

*q*located within the sphere \(\mathcal {S}_{{\mathrm{I}}}\) passes though the throat of the wormhole to \(\mathcal {M}_{{\mathrm{I}}{\mathrm{I}}}\), an observer in \(\mathcal {M}_{{\mathrm{I}}}\) who has merely integrated \({{\mathbf {\mathsf{{H}}}}}\) over \(\mathcal {S}^{{\mathrm{I}}}\) to establish the conserved quantity \({Q}_t^{{\mathrm{I}}}\), no longer sees

*q*in their part of the universe. They rather say that after the charge has passed though the throat, the wormhole has

*gained*charge

*q*[11, 12, 13].

There may still be aspects of this standard “charged wormhole” view that worry some. Of course, if a charge enters a box, the charge will still be in the box whenever we subsequently look inside, and we can reasonably say that the box has acquired that charge. However, in the current case, after passing to universe \(\mathcal {M}_{{\mathrm{I}}{\mathrm{I}}}\), the charge *q* might subsequently move *arbitrarily far* from the throat.^{6} At such a distance, some might consider it unreasonable to have the steady-state field of the charge still influenced by some prehistoric transit from \(\mathcal {M}_{{\mathrm{I}}}\). Nevertheless, the standard viewpoint insists that an observer in \(\mathcal {M}_{{\mathrm{I}}}\) still sees that the wormhole has acquired, *and retained*, charge *q*.

However, since our biverse scenario has a non-trivial topology, we can again consider \({{\mathbf {\mathsf{{H}}}}}\) to be undefined in an absolute sense. Having decided that \({{\mathbf {\mathsf{{H}}}}}\) is not defined, one is free to consider how to replace it. We consider here a simple extension to Maxwell’s equations in which the charge the wormhole gains depends on the distance from the charge to the throat, and is *no longer* affected by whether or not the charge made a one-way transit through that throat in the past.

*q*, and define \({{Q}^{{\mathrm{I}}}({r_p})}\) as a function of the radial position, \({r_p}\) of the charge. Thus we can set

*and*eqn. (40).

Another attractive feature of our proposed modification occurs in relation to a wormhole connecting two distinct regions (\(\mathcal {A}\) and \(\mathcal {B}\), say) in the same universe. In this topology, a charge *q* can circulate multiple (*n*, say) times by entering at \(\mathcal {A}\) and exiting at \(\mathcal {B}\). Standard Maxwell theory then predicts that \(\mathcal {A}\) has a charge of *n**q*, and \(\mathcal {B}\) a charge of \(-n{q}\), which can become arbitrarily large. The modification to Maxwell’s theory of (38) avoids this problem, as integrating around \(\mathcal {A}\) will yield a charge that does not exceed *q*.

## 6 Discussion

There are mechanisms for charge conservation that exist independently of the topology or gauge-free conditions that we have discussed above. One of the most notable is a consequence of Noether’s theorem for a *U*(1) gauge invariant Lagrangian, which enforces local charge conservation \({d} {{\mathbf {\mathsf{{J}}}}}= 0\). For example, if \(\Lambda [{{\mathbf {\mathsf{{A}}}}},\alpha ] \in \Gamma \Lambda ^4 \mathcal {M}\) is invariant under substitutions \(\alpha \rightarrow e^{\imath \phi } \alpha \) and \({{\mathbf {\mathsf{{A}}}}} \rightarrow {{\mathbf {\mathsf{{A}}}}} + \imath {d}\phi \), then the 3-form \(\partial \Lambda /\partial {{\mathbf {\mathsf{{A}}}}}\) is locally conserved, i.e. \({d}( \partial \Lambda /\partial {{\mathbf {\mathsf{{A}}}}} ) = 0\). Since the variations are purely local, this makes no statement about the *global* conservation of charge in non trivial spacetimes. It should also be noted that most Lagrangian formulations of electromagnetism implicitly assume a model for \({{\mathbf {\mathsf{{H}}}}}\). For example the Maxwell vacuum where \(\Lambda \) contains the term \(\Lambda ^{\mathrm{EM}} = \frac{1}{2} {d}{{\mathbf {\mathsf{{A}}}}} \wedge \star {d}{{\mathbf {\mathsf{{A}}}}}\), or a model of a simple non-dispersive “antediluvian” medium^{7} where \(\Lambda ^{\mathrm{EM}} = \frac{1}{2} {d}{{\mathbf {\mathsf{{A}}}}} \wedge \star {{\mathbf {\mathsf{{Z}}}}}({d}{{\mathbf {\mathsf{{A}}}}})\) and \({{\mathbf {\mathsf{{Z}}}}}\) is a constitutive tensor [32]. It would also be interesting to attempt to construct Lagrangians which do not imply a well defined excitation 2-form.

We might also broaden our examination of conservation laws beyond just charge to those of energy and momentum, by looking at the divergence-free nature of the stress-energy tensor \({\mathbf{T}}\). In our discussion of Sect. 3, the total energy of the current and electromagnetic field must be zero before the singularity, i.e. on a hypersuface in \(\mathcal {M}^{-}\). Likewise, although we did not define the energy, the existence of fields after the singularity, implies that the total energy would be non zero. However, just as in the case of charge conservation, this lack of global energy conservation is not inconsistent with the local energy conservation law \({d}({\varvec{\tau }_{K}}) = 0\), obtained from the energy 3-form \(\varvec{\tau }_{K} = \star {\mathbf{T}}({K},-))\), where *K* is a timelike Killing field. Of course, in the general relativistic case of an evaporating black hole there are challenges about defining the total energy, but one should not be surprised if an appropriate measure of total energy were also not conserved.

For momentum, if \(\mathcal {M}\) possesses a spacelike Killing vector *K*, then *K* is locally conserved, i.e. \({d}({\varvec{\tau }_{K}}) = 0\), but again this has said nothing about the *global* conservation of momentum. We see from (18) that the construction of \({{\mathbf {\mathsf{{J}}}}}\) that it is spherically symmetric and hence will not change the total momentum. However, this was a *choice* and non-spherically symmetric currents can easily be obtained by introducing a Lorentz boost. Of course, when considering the total momentum, i.e. that of the electromagnetic field plus that of the response of the medium, one encounters the thorny issue of the Abraham–Minkowski controversy [33, 34] and choice of Poynting vector [35]. From the perspective here, the question of which momentum is most appropriate would be further complicated by the non existence of the excitation 2-form.

As a final remark, our results presented here raise the possibility of developing a way to prescribe dynamic equations for the electromagnetic field \({{\mathbf {\mathsf{{F}}}}}\) without introducing or referring to an excitation field \({{\mathbf {\mathsf{{H}}}}}\) at all. One possibility is to combine Maxwell’s equation (3) directly with the constitutive relations, thus eliminating the need for \({{\mathbf {\mathsf{{H}}}}}\) [36].

## 7 Conclusion

In this paper we have clarified physical issues regarding electromagnetism on spacetimes with a non-trivial topology—either missing points, as can be introduced by the singularity at the heart of a black hole, or the presence of wormhole-like bridges between universes, or between two locations in the same universe.

We have unambiguously shown that such cases have significant implications for charge conservation—i.e. that it need not be conserved; and the role of (or need for) the electromagnetic excitation field \({{\mathbf {\mathsf{{H}}}}}\) (i.e. the Maxwell \(\mathbf {D}\), \(\mathbf {H}\) vector fields)—i.e. that it is not always globally unique, and thus has a subordinate or even optional status as compared to the more fundamental \({{\mathbf {\mathsf{{F}}}}}\) comprising the Maxwell \(\mathbf {E}\), \(\mathbf {B}\) vector fields. All of these considerations are purely electromagnetic, and are prior to any considerations about the physics of singularities, such as cosmic censorship hypotheses. Similar statements can be made about the global versus local conservation of leptonic and baryonic charges.

Although our results show that Maxwell’s equations need not conserve charge on topologically non-trivial spaces, neither do they guarantee that they will not (or cannot). But they do insist that charge conservation is not a fundamental property, and can only be maintained with additional assumptions. Further, wormhole mouths do not—or need not—be considered to accumulate a charge that is the sum of all charge that passes through; it is possible to construct a self-consistent electromagnetic solution where the wormhole only temporarily accommodates a passing charge.

## Footnotes

- 1.
The fact that \({{\mathbf {\mathsf{{H}}}}}\) is well-defined has been used in invoking \(\int _\mathcal {U}{d} {{\mathbf {\mathsf{{H}}}}}= \int _{{\partial } \mathcal {U}} H\) in (11). Compare integrating \({d}\theta \) around the unit circle \(\mathcal {C}\) to obtain the fallacious result \(\oint _\mathcal {C}{d} \theta = \oint _{{\partial } \mathcal {C}}\theta = 0\), since \({\partial } \mathcal {C}= \emptyset \). The problem is that \(\theta \) is not well defined (and continuous) on all of \(\mathcal {C}\). By defining two submanifolds \(\mathcal {U}_1\) and \(\mathcal {U}_2\) such that \(\mathcal {U}_1 \cup \mathcal {U}_2 = \mathcal {C}\) with respective coordinate patches \(\theta \) and \(\theta +2\pi \), then a careful integration around \(\mathcal {C}\) yields the correct answer of \(2\pi \).

- 2.
The

*k*-th deRham cohomology \({{\mathbf {\mathsf{{H}}}}}^k_{{\mathrm{dR}}}(\hat{\mathcal {M}})\) of the manifold \(\hat{\mathcal {M}}\) is defined to be the equivalence class of closed \(k-\)forms modulo the exact forms. In the topologically trivial case all the \({{\mathbf {\mathsf{{H}}}}}^k_{{\mathrm{dR}}}(\hat{\mathcal {M}})=0\), with \(k>0\), and hence all closed forms are exact. In the language here, this implies that since \({{\mathbf {\mathsf{{J}}}}}\) is closed, \({d}{{\mathbf {\mathsf{{J}}}}}=0\) there must exist a 2-form \({{\mathbf {\mathsf{{H}}}}}\in \Gamma \Lambda ^2\hat{\mathcal {M}}\) such that \({d}{{\mathbf {\mathsf{{H}}}}}={{\mathbf {\mathsf{{J}}}}}\). In general \({{\mathbf {\mathsf{{H}}}}}\) is not unique but it is globally defined. In the case of an evaporating black hole, the deRham cohomology \({{\mathbf {\mathsf{{H}}}}}^3_{{\mathrm{dR}}}(\hat{\mathcal {M}})={\mathbb R}\). Therefore even though \({{\mathbf {\mathsf{{J}}}}}\) is closed, it is not exact, i.e. there is no \({{\mathbf {\mathsf{{H}}}}}\in \Gamma \Lambda ^2 \hat{\mathcal {M}}\) such that \({d}{{\mathbf {\mathsf{{H}}}}}={{\mathbf {\mathsf{{J}}}}}\), and thus \({{\mathbf {\mathsf{{J}}}}}\) need not be not globally conserved. A similar analysis is connected with magnetic monopoles. If we remove a world-line from spacetime, then the \({{\mathbf {\mathsf{{H}}}}}^2_{{\mathrm{dR}}}(\hat{\mathcal {M}})={\mathbb R}\). This implies that there need not exist an electromagnetic potential \({{\mathbf {\mathsf{{A}}}}}\), where \({d}{{\mathbf {\mathsf{{A}}}}}= {{\mathbf {\mathsf{{F}}}}}\). Hence \(\int _{S^2} {{\mathbf {\mathsf{{F}}}}}\ne 0\) where \(S^2\) is a sphere at a moment in time enclosing the “defect”. - 3.
Let \(\mathcal {M}\) have signature \((-,+,+,+)\) and orientation \({\star } 1={dt}\wedge {dx}\wedge {dy}\wedge {dz}\).

- 4.
One may think of our proposed \({{\mathbf {\mathsf{{J}}}}}\) as an application of deRham’s second theorem. Since the unit 3-sphere about the origin is a 3-cycle which is

*not*a boundary, deRham’s second theorem states that for any real valued*Q*there always exists a 3-form \(\varvec{\omega }\) on \(\mathcal {M}\) such that \(\int _{S^3} \varvec{\omega } = {Q}\). Here our \(\varvec{\omega }\) is \({{\mathbf {\mathsf{{J}}}}}\), which is chosen so that after the initial “impulse” at \(\mathbf {0}\), it subsequently respects causality. - 5.
In fact any Cauchy hypersurface with \(t<0\) suffices here.

- 6.
Ignoring dynamical constraints it could even move to \(\infty _{{\mathrm{I}}{\mathrm{I}}}\) and therefore pass out of universe \(\mathcal {M}_{{\mathrm{I}}{\mathrm{I}}}\), changing the overall charge of the biverse!

- 7.
A a non-dispersive medium would not produce rainbows.

## Notes

### Acknowledgements

The JG and PK are grateful for the support provided by STFC (the Cockcroft Institute ST/G008248/1 and ST/P002056/1) and EPSRC (the Alpha-X Project EP/N028694/1). PK would like to acknowledge the hospitality of Imperial College London. The authors would like to thank the anonymous referees for their useful suggestions.

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