By Jishnu Bhattacharyya, Mattia Colombo and Thomas Sotiriou.

Black holes are perhaps the most fascinating predictions of General Relativity (GR). Yet, their very existence (conventionally) hinges on Special Relativity (SR), or more precisely on local Lorentz symmetry. This symmetry is the local manifestation of the causal structure of GR and it dictates that the speed of light is finite and the maximal speed attainable. Accepting also that light gravitates, one can then intuitively arrive at the conclusion that black holes should exist — as John Michell already did in 1783!

One can reverse the argument: does accepting that black holes exist, as astronomical observations and the recent gravitational wave direct detections strongly suggest, imply that Lorentz symmetry is an exact symmetry of nature? In other words, is this ground breaking prediction of GR the ultimate vindication of SR?

Jishnu Bhattacharyya, Mattia Colombo and Thomas Sotiriou from the School of Mathematical Sciences, University of Nottingham.

These questions might seem ill-posed if one sees GR simply as a generalisation of SR to non-inertial observers. On the same footing, one might consider questioning Lorentz symmetry as a step backwards altogether. Yet, there is an alternative perspective. GR taught us that our theories should be expressible in a covariant language and that there is a dynamical metric that is responsible for the gravitational interaction. Universality of free fall implies that matter fields couple minimally to this metric and local flatness makes this picture perfectly compatible with the fact that the Standard Model of particle physics respects Lorentz symmetry. Now, keep all of these lessons and combine them with a lesson from field theory: symmetries of the action can be broken at the level of the solutions. Assume that there is a scalar field that couples to the metric, but does not couple to matter. Assume further that this scalar field has the following property: in every single solution it is always non-trivial and admits a timelike gradient. This means that every solution, including flat spacetime, comes together with a set of spacelike surfaces on which the scalar is constant — a preferred foliation. Matter does not know about this foliation, but gravity (e.g., metric perturbations) does.

This theory is so close to and yet so far from GR. It is just GR coupled to an, admittedly peculiar, scalar field, yet it is known to have a type of Einstein’s ‘spooky action at a distance’: an instantaneous mode that can transmit information infinitely fast along the leaves of the preferred foliation [1,2]. Consequently, causality in this theory is not determined by the metric but by the preferred foliation. Local Lorentz symmetry is absent in the gravity sector.

It is very tempting to conclude that black holes cannot exist in such a theory. Remarkably, this is not true! Conventional black hole horizons indeed fail to define a black hole here. However, a new type of horizon, dubbed the universal horizon, exists [1,3]. The leaves of the preferred foliation, which defines causality, normally extend to infinity. The universal horizon is a leaf that has barely detached itself from infinity and instead cloaks the singularity, much like a conventional black hole horizon. As a leaf of a spacelike foliation, it can only be traversed in one direction when travelling into the future. That is, nothing — even perturbations travelling infinitely fast — can exit from the region it cloaks without travelling backwards in time. Hence, the notion of a black hole survives!

Originally the universal horizon was found in static, spherical symmetric solutions [1,3]. Though it was later shown to exist in slowly rotating solutions [4,5] or lower-dimensional rapidly rotating solutions [6], its identification strongly hinged on stationarity and either spherical or circular symmetry and was rather ‘operational’. In our recent CQG paper [7] we present a rigorous analysis of the causal structure of theories with a preferred foliation. This allowed us to provide a global definition for the universal horizon that does not hinge on symmetries at all. Moreover, we were able to give a local characterisation of the universal horizon when stationary is assumed. This is particularly useful as it allows one to determine if a given solution actually contains a universal horizon or not. Recall that in GR event horizons are also Killing horizons for stationary solutions and this is the property we exploit to find them: we look of the locus of points where the relevant Killing vector becomes null and a generator of the horizon. Remarkably, universal horizons turn out to be located inside Killing horizons, where the Killing vector lies entirely on a leaf of the foliation and can thus be considered as a generator of that leaf. In our work we also touched upon the issue of causal development and evolution in such theories and we proved the conjecture [1] that universal horizons are Cauchy horizons.

We’ve taken a decisive step in understanding causality and black holes in absence of Lorentz symmetry, but clearly this is only the beginning. In the dawn of gravitational wave astronomy, black holes are about to become natural laboratories for fundamental physics. Testing Lorentz symmetry in novel ways is certainly something worth pursuing, given its prominent role in the foundations of the Standard Model. It has been argued that theories with a preferred foliation, suitably supplemented with some higher order corrections, exhibit improved behaviour in the ultraviolet [8]. Hence, they present themselves as good quantum gravity candidates. Even though this was not essential for our discussion, it certainly adds to the motivation for considering such theories.

References

[1] D. Blas and S. Sibiryakov, Horava gravity versus thermodynamics: The Black hole case, Phys. Rev. D 84, 124043 (2011), doi:10.1103/PhysRevD.84.124043 [arXiv:1110.2195 [hep-th]].

[2] J. Bhattacharyya, A. Coates, M. Colombo and T. P. Sotiriou, Evolution and
spherical collapse in Einstein-Æther theory and Hořava gravity, Phys. Rev. D 93, 064056 (2016) doi:10.1103/PhysRevD.93.064056 [arXiv:1512.04899 [gr-qc]].

[3] E. Barausse, T. Jacobson and T. P. Sotiriou, Black holes in Einstein-aether and Horava-Lifshitz gravity, Phys. Rev. D 83, 124043 (2011)
doi:10.1103/PhysRevD.83.124043 [arXiv:1104.2889 [gr-qc]].

[4] E. Barausse and T. P. Sotiriou, A no-go theorem for slowly rotating black holes in Hořava-Lifshitz gravity, Phys. Rev. Lett. 109, 181101 (2012). Erratum: Phys. Rev. Lett. 110, 039902 (2013). doi:10.1103/PhysRevLett.110.039902, 10.1103/PhysRevLett.109.181101 [arXiv:1207.6370 [gr-qc]].

[5] E. Barausse and T. P. Sotiriou, Slowly rotating black holes in Horava-Lifshitz gravity, Phys. Rev. D 87, 087504 (2013) doi:10.1103/PhysRevD.87.087504 [arXiv:1212.1334 [gr-qc]].

[6] E. Barausse, T. P. Sotiriou and I. Vega, Slowly rotating black holes in
Einstein-æther theory, Phys. Rev. D 93 , 044044 (2016)
doi:10.1103/PhysRevD.93.044044 [arXiv:1512.05894 [gr-qc]].

[7] J. Bhattacharyya, M. Colombo and T. P. Sotiriou, Causality and black holes in spacetimes with a preferred foliation, Class. Quant. Grav. 33, 235003 (2016) doi:10.1088/0264-9381/33/23/235003 [arXiv:1509.01558 [gr-qc]].

[8] P. Horava, Quantum Gravity at a Lifshitz Point, Phys. Rev. D 79 , 084008 (2009) doi:10.1103/PhysRevD.79.084008 [arXiv:0901.3775 [hep-th]].

Read the full article in Classical and Quantum Gravity:
Causality and black holes in spacetimes with a preferred foliation
Jishnu Bhattacharyya et al 2016 Class. Quantum Grav. 33 235003

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