Constrained field theories on Kerr backgrounds
We analyze the constraints of gauge theories on Kerr and Kerr-de Sitter spacetimes, which contain one or more horizons. We find that the constraints are modified on such backgrounds through the presence of additional surface terms at the horizons. As a concrete example, we consider the Maxwell field and find that the Gauss law constraint involves surface corrections at the horizons. These surface contributions correspond to induced surface charges and currents on the horizons, which agree with those found within the membrane paradigm. The modification of the Gauss law constraint also influences the gauge fixing and Dirac brackets of the theory.
The horizons of black holes are a profound consequence of the General Theory of Relativity. Black holes present to the universe a closed surface of finite size, completely characterized by macroscopic parameters such as mass, charge and spin . Information about the internal structure of a black hole is unobservable from the outside due to the presence of the horizon, at least classically. The seminal discovery by Hawking  that a black hole radiates like a black body with a finite temperature, following Bekenstein’s suggestion that a black hole possesses an entropy proportional to the surface area of its horizon , implies the possibility that a black hole has associated with it a very large number of microscopic states. It is natural to think that these states are in some way related to the degrees of freedom of the horizon. This view has been strengthened in approaches that treat fields on black hole backgrounds as those of manifolds with boundaries. For gravity, this approach leads to a quantum description in which an infinite set of observables are localized on the boundary [4, 5, 6, 7].
There has been a resurgence of interest in studying the behaviour of quantum fields near black hole horizons, motivated by various paradoxes and puzzles related to the information problem [8, 9]. Based on the asymptotic symmetries of fields on the null boundaries of conformally compactified flat spacetimes [10, 11, 12, 13, 14, 15], there have been recent proposals for the existence of soft black hole hairs [16, 17, 18, 19, 20]. The significance of the horizon is highlighted in the membrane paradigm, where one replaces the black hole by a membrane with certain classical properties at the stretched horizon, i.e. a small distance outside the event horizon (an excellent overview is provided by the collection of articles in ). This is a sensible description from the perspective of an external stationary observer, who finds that particles cannot classically leave the interior of the black hole or reach the horizon from the outside in finite time. Thus the classical or semi-classical dynamics of fields on black hole backgrounds may be studied by considering the bulk and the horizon, and completely ignoring what happens in the interior of horizon.
Boundary conditions on the fields play a crucial role in all these investigations. In most of these papers, though not all of them, the fields (or their derivatives) are set to vanish on the horizon. For many field theories, this is a convenient way of ensuring that invariants constructed out of the stress energy tensor remain finite at the horizon. For the Kerr black hole spacetime, boundary conditions on the components of electric and magnetic fields relate the charge and surface currents at the horizon [22, 23, 24]. These conditions allow for the extraction of electromagnetic energy from Kerr black holes through a magnetic Penrose process . The boundary conditions for gauge fields are special in that we can ensure the finiteness of gauge invariant observables without necessarily imposing the finiteness on the components of gauge fields. In addition, assuming any particular values for gauge fields is not particularly meaningful, as they are defined always up to gauge transformations.
Gauge theories are characterized by the presence of redundant degrees of freedom, which leads to the presence of constraints. The formalism for studying the dynamics of constrained systems was discovered by Dirac  and independently by Bergmann et al. [27, 28], and has been applied to numerous theories of interest over the years [29, 30, 31]. While the formalism for constrained field theories set up by Dirac generalizes to curved backgrounds , the more general formulation in terms of shift and lapse variables was introduced by Arnowitt et al. . In particular, this formulation has been used to understand the initial value problem of fields theories , the behaviour of the fields near the horizons of stationary black hole spacetimes , and its quantization . Until recently, a noticeable absence in the literature involved the formulation of constrained theories on curved backgrounds with horizons. The modification of constraints due to spatial boundaries on flat backgrounds were investigated in [37, 38], while in [39, 40] the quantization of the Chern–Simons theory on a disk and the role of boundaries on the vacuum structure of the theory has been covered in detail. It is the boundary conditions on gauge fields at the horizon that concerns us in this paper. The point is that the value of a gauge field at a boundary can be changed by a gauge transformation. The only way to fix the boundary value of a gauge field is to restrict to gauge transformations which vanish at the boundary. However, there is no sensible reason to do that when the said boundary is not a physical singularity, so it is sufficient to keep the gauge transformations regular at the horizon. We will find that this seemingly innocuous condition leads to a modification of the system of constraints when a horizon is present.
The formulation of gauge theories on spherically symmetric backgrounds with horizons was considered in , where it was found that the constraints received contributions from terms localized at the horizon. In particular, this is true for the Gauss law constraint in electrodynamics, which now has an additional contribution from the horizon. This resulted in a vanishing charge for an observer situated at the horizon of a Reissner–Nordström black hole, while not affecting the usual charge observed by the oberver at infinity. In the present work we investigate the classical constraints of electrodynamics in Kerr spacetimes. We will find that like in the static spherically symmetric case, the Gauss law constraint in the stationary axisymmetric spacetime picks up a horizon term. While it is an expected result, we think it was worthwhile to check that it was not an artifact of spherical symmetry. Thus gauge transformations which do not vanish on the horizon are allowed for axisymmetric black hole spacetimes. We will also show that the horizon term is equivalent to a ‘surface’ charge density on the horizon which is not visible to outside observers, and that it induces a ‘surface’ current density on the horizon.
The organization of our paper is as follows. In Sect. 2, we set up our notations and conventions for the analysis of constraints on the Kerr background. In Sect. 3, we consider Maxwell’s theory and explicitly derive the surface contributions to the constraint on the horizon. Gauge fixing is considered in both the radiation gauge and axial gauge. Finally in Sect. 5, we discuss the physical consequences of our results. This involves a modification of the usual solution for the electromagnetic scalar potential known in the absence of boundaries. The calculations behind certain results used in the main body of the paper appear in three Appendices.
2 General algorithm
2.1 Kerr backgrounds
The time coordinate is measured along \(\chi _a\) and is constant on the hypersurface \(\Sigma \). In what follows we will consider the time evolution vector to be along \(\xi ^a\). With this choice, \(\alpha \omega ^a\) and \(\beta \) represent what are known as the shift and lapse of the time evolution vector. It is the lapse function \(\beta \) which vanishes at the horizons.
2.2 Hamiltonian formulation
In the following, we will demonstrate this by considering the Maxwell field.
3 The Maxwell field
3.1 The Dirac–Bergmann formalism
A remark regarding the Hamiltonian and its relation to time in this space is in order. Even though the integral defining the Hamiltonian in Eq. (3.19) is over \(\Sigma \) which is orthogonal to \(\chi \), and not to the timelike Killing vector field \(\xi \), the time evolution generated by this Hamiltonian provides the correct form of the Maxwell equations. This is shown in Appendix C.
3.2 Gauge fixing
4 Charges and currents
The induced charges and currents that we find on black hole horizons have been introduced before in the literature. It was noted in  that when an electric charge is lowered into a Schwarzschild black hole, the electric flux lines terminate on the horizon. This required the introduction of an induced surface charge density on the horizon, and the electric potential was calculated as the superposition of that due to the external charge and that due to the induced charge. This result was generalized to describe an induced surface current density on the horizon of a rotating black holes in an asymptotically flat spacetime in [22, 23]. The induced surface charges and currents can be described within the membrane paradigm as conditions on the electromagnetic fields on the membrane [21, 35] as well as through a surface action for the electromagnetic field on the membrane . The induced charges and currents on the horizon help describe the Blandford-Znajek mechanism , a magnetic Penrose process which provides a model for the source of pulsars, quasars and active galactic nuclei [49, 50]. Our result demonstrates that induced charges and currents on the horizon arise naturally as part of the general Gauss law constraint on black hole backgrounds. In the membrane paradigm, the induced charge density on the horizon appears as a consequence of boundary conditions . The vanishing electric flux at the horizons, following our treatment, could provide a means to investigate soft limits and their relation to gauge parameters at the horizon. In this regard, we note the proposal in , where soft hairs were defined as charges on the future horizon of the black hole, considered as a ‘holographic plate’, which are associated with non-vanishing large gauge transformations on the horizon. It will be interesting to investigate if such charges also result for the quantized electromagnetic field as a consequence of gauge parameters and constraints at the horizons.
In this paper we have considered the constrained dynamics of field theories on Kerr backgrounds with one or more horizons and have argued that the constraints of the theory will receive additional contributions from these horizons. We explicitly considered the example of the Maxwell field, and found that the Gauss law constraint must include contributions from the horizon(s). Such surface contributions will not arise on spacelike surfaces of the background, but they appear on horizons in part due to our inability to observe past the horizon, as well as the fact that gauge fields can in principle take on arbitrary values at the horizon provided gauge invariant quantities constructed from them remain finite. More precisely, the non-vanishing of gauge parameters and their derivatives at the horizons leads to a Gauss law constraint with surface contributions.
A Gauss law constraint with horizon corrections implies some novel consequences on our understanding of the electromagnetic field, and more generally of gauge field theories, on black hole spacetimes. Let us briefly note the new results in this paper and how they relate with known results in the literature. Firstly, for gauge fields on spacetimes with horizons, existing literature sets gauge parameters or their derivatives to vanish on the horizon exactly as on spatial boundaries. Then the Gauss law constraint involves no surface contribution. A recent alternative approach introduces, at spatial boundaries, additional dynamical fields whose transformations cancel the surface contributions resulting from gauge transformations [51, 52]. What we find, by not fixing the gauge parameters to vanish at a horizon, is that we can choose more general boundary conditions which allow not only horizon terms in the Gauss law constraint but also gauge fixing conditions which involve additional contributions from the horizon. As we discuss below, these terms could help explore the physical degrees of freedom at the Killing horizons.
A second novel result we have found is that the horizon contribution to the Gauss law leads to a vanishing electric flux across the horizon. Thus the horizon term can be interpreted as an induced ‘surface’ charge density at the horizon which in turn induces a surface current at the horizon. Previous work based on the membrane paradigm had postulated the existence of such a surface charge and surface current [23, 47]. We found that the same charge and current appears on the horizon as a consequence of the modification of Gauss law brought about by our choice to allow gauge parameters to not vanish on the horizon.
While some of the results in this paper are similar to those previously presented by some of us for spherically symmetric black hole spacetimes , the class of Kerr spacetimes considered in this paper lead to some interesting new results. The spatial hypersurfaces on which we have integrated the Hamiltonian density cannot be taken to be orthogonal to the timelike Killing vector field, but are orthogonal to a timelike vector field which coincides with the Killing field only at the horizons. Thus it is not a priori obvious that the unconstrained Hamiltonian would generate time evolution, but we have shown in Appendix C that this Hamiltonian does indeed produce Maxwell equations modulo first class constraints. Another key difference involves the implications of the Gauss law constraint on currents at Killing horizons. As we mentioned in the previous section, the surface terms in the Gauss law constraint can be directly associated with surface charges and currents postulated previously in the literature. A third difference is that there are many more gauge choices compatible with the symmetries of the Kerr-type background than for the spherically symmetric case. This also implies that surface terms in the gauge fixed theory are more general than those on spherically symmetric spacetimes, which we will now describe.
Another implication of the Gauss law constraint involves the charges and currents on Kerr spacetimes. We noted in Sect. 4 that the horizon correction in the Gauss law can be identified with the induced surface charge on the horizon of a black hole. This term was considered previously in the literature through boundary conditions on the normal component of the electric field. In addition, Maxwell’s equations further imply an induced surface current as a consequence of the induced surface charge, which is related to components of the magnetic field parallel to the horizon. Thus corrections to the Gauss law constraint resulting from Killing horizons of the background lead to a natural identification of an induced surface charge and induced surface current in Eqs. (4.5) and (4.10) respectively. The induced surface charge in particular implies the vanishing of electric flux lines on the horizon. Non-vanishing gauge parameters are associated with soft charges at null infinity on asymptotically flat spacetimes . It will be interesting to consider if the classically vanishing electric flux at Killing horizons, which arises as a consequence of non-vanishing gauge parameters following our treatment, is also related to a description of soft hairs on Killing horizons in the quantized theory .
Finally, we note that the BRST formalism provides an interesting and powerful means to investigate quantized fields in the Hamiltonian framework. Following our analysis in this paper and in , it can be argued that the BRST charge operator will involve the additional surface terms contained in the constraints. Thus the physical states defined by the cohomology of the BRST charge will have to satisfy non-trivial conditions on the horizon. Within the BRST formalism, we can expect that the surface corrections in the BRST charge and gauge-fixing fermion will affect the path integral and resulting quantum theory. The exploration of these issues lie outside the scope of the present work. We have recently considered some of these questions in the case of spherically symmetric spacetimes elsewhere . We leave the investigation on axisymmetric spacetimes and physical states at the horizon for future work.
- 2.S. W. Hawking, Particle Creation by Black Holes. Commun. Math. Phys. 43, 199 (1975). https://doi.org/10.1007/BF02345020 [Erratum: Commun. Math. Phys. 46, 206 (1976)]
- 5.A. P. Balachandran, L. Chandar, A. Momen, Edge states in canonical gravity. Talk delivered at the Montreal-Rochester-Syracuse-Toronto (MRST) Conference, 1995. arXiv:gr-qc/9506006 (unpublished)
- 29.P.A.M. Dirac, Lectures on Quantum Mechanics (Yeshiva University, New York, 1964)Google Scholar
- 30.A.J. Hanson, T. Regge, C. Teitelboim, Constrained Hamiltonian Systems (Accademia Nazionale dei Lincei, Rome, 1976)Google Scholar
- 53.K. Fernandes, A. Lahiri, Hamiltonian BRST formalism for gauge fields on black hole spacetimes. S.N.Bose Centre. arXiv:1809.05729 [gr-qc] (preprint)
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