A conserved energy for axially symmetric Newman–Penrose–Maxwell scalars on Kerr black holes

New paper by Nishanth Gudapati: “A conserved energy for axially symmetric Newman–Penrose–Maxwell scalars on Kerr black holes.” What follows is a brief summary of the findings. A PDF of the paper can be found here.

A link to the journal can be found here. 

Energy Extraction or Lack Thereof ?

Einstein’s general theory of relativity was a great revelation of 20th century science that provides a scientific basis for notions that exist in the realm of science fiction like worm holes, time travel, the big bang etc. Mathematically, Einstein’s equations for general relativity can be concisely represented as Rμν = 0. Despite their simple appearance, in practice they are a system of highly complex nonlinear equations. A solution of Einstein’s equations is a continuum of space and time. Arguably, black holes are the most exotic predictions of Einstein’s general theory of relativity that provide the motivation for seemingly fictional scenarios. For example, the popular movie ‘Interstellar’ has several instances where the story line is closely linked to surreal phenomena exhibited by black holes. In order for such stories to be realistic, we need to rigorously establish that black holes are stable and correspond to Einstein’s equations.

Following the discovery of Einstein’s general relativity, there was a general ex- pectation that there should be a solution of Einsteins equations that represents massive, rotating bodies. As such, attempts were made by Hermann Weyl, J. Ernst, A. Papapetrou and others to find such a solution. In 1963, when Roy Kerr arrived at a solution there was a wide consensus that one had arrived at the right solution. Indeed, it was soon conjectured that the Kerr family 1 of black hole space- times are unique, stable and the final state of all black holes found in the universe, thus underlining their importance.

A peculiar property of Kerr black holes is that one is always surrounded by a so called ergo-region. In the ergo-region, an object cannot remain stationary and is forced to move along with the rotating black hole. A surprising feature of the ergo-region can be explained using Roger Penrose’s gedanken experiment. Suppose we throw an object into a Kerr black hole in such a way that it splits into two pieces (as shown in the figure), where one piece enters the black hole and the other exits the ergo-region. The piece that exits can have higher energy than the original object. This is not a violation of the conservation of energy because the piece that goes into the black hole carries ‘negative’ energy.

This phenomenon can also be seen for waves propagating on Kerr black holes. Consider a wave propagating in a plucked string, it carries with it a potential energy and a kinetic energy, which add up to give a positive total energy. On the other hand, for a wave propagating on a Kerr black hole, it is not necessary that it has positive-definite energy. By analogy with Penrose’s gedanken experiment, this results in a phenomenon where it appears as though the black hole is emitting energy. This phenomenon is referred to as the Penrose process or superradiance or energy extraction.

Figure 1. Penrose process or energy extraction for a rotating Kerr black hole

This feature of the ergo-region has important implications for the mathematical problem of the stability of Kerr black hole spacetimes. The aforementioned model of waves propagating on a Kerr black hole is analogous to the disturbances in Einstein’s equations around a Kerr black hole spacetime. As it can be intuitively imagined, the stability of a system is closely linked to the decay of waves, which in turn is associated to ‘loss’. The fact that a Kerr black hole effectively ‘emits’ energy counteracts the decay of the disturbances.

In case of linear waves which are symmetric around the axis of a Kerr black hole, the problem of the ergo-region disappears on account of the fact that Kerr black holes are also axially symmetric. However, the problem reappears for axially symmetric electromagnetic waves propagating on Kerr black holes. Indeed, one can arrange for the axially symmetric electromagnetic waves such that their energy is locally negative in Kerr black hole spacetimes.

The question whether the axially symmetric electromagnetic waves admit a positive-definite total energy, somewhat analogous to linear waves, has been one of those perplexing ‘borderline’ open problems for decades.

In this work, we have been able to prove that this question can be answered affirmatively. In axial symmetry, the problem of the ergo-region does not go away immediately, but we start with the usual energy density for electromagnetic waves, which has indefinite sign, and make a series of transformations in such a way that the final total energy is positive-definite. Importantly, these transformations, which are motivated by the identities constructed by D. C. Robinson in 1974 for black hole uniqueness theorems, do not affect the dynamics of the electromagnetic waves (‘canonical transformations’).

The fact that the electromagnetic waves carry a positive-definite total energy for Kerr black holes is a strong indication that the (total) energy extraction or superradiance may not be possible in this case.