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Does general relativity violate determinism inside charged black holes?

30 Jan 2018
Artist’s impression of a black hole
Artist’s impression of a black hole

Under certain extreme conditions Einstein’s general theory of relativity seems to violate determinism, according to an international team of physicists. The group has shown that in a universe expanding under the influence of the cosmological constant, black holes generated by the collapse of highly charged stars should contain a region where physical conditions are not fixed by the stars’ initial state. At odds with a 40-year old idea known as cosmic censorship, the researchers say that signs of this indeterminism might show up in detections of gravitational waves.

Newton’s mechanics allow us in principle to calculate the exact state of a physical system at any point in the future, provided that we know its initial state perfectly. So too with general relativity: a precise knowledge of space’s geometry and its rate of change in the present enables us in theory to predict exactly how space-time will evolve. As such, Einstein’s theory is considered by most physicists to be entirely deterministic.

Charged black holes, however, challenge this deterministic picture. The “Reissner-Nordström” solution of general relativity describes a black hole created when a star that is electrically charged and spherical collapses in on itself under the force of gravity. Hidden from view inside such a black hole’s event horizon lies a second boundary known as the Cauchy horizon, beyond which space-time is smooth but indeterminate. In other words, the future can no longer be predicted.

Strong cosmic censorship

An idea put forward by British physicist Roger Penrose in the 1970s had appeared to forbid such non-deterministic behaviour. His “strong cosmic censorship” conjecture states that there is some mechanism within general relativity – a censor – that prohibits the appearance of Cauchy horizons. In the case of a charged black hole, he calculated that even the slightest perturbation in the initial conditions of the imploding star destroys the Cauchy horizon and yields a singularity in its place. At this point of infinite space-time curvature the relativistic field equations break down and determinism, or its absence, ceases to become an issue.

But in the new work, described in Physical Review Letters, Lisbon University’s Vitor Cardoso and colleagues find that there should be some circumstances when the singularity imposed by cosmic censorship does not form. They considered the net effect of two opposing influences on the Cauchy horizon – the amplification of any tiny perturbation by the immense gravity of a black hole on the one hand, and the damping effect of the black hole’s external environment on the other. Specifically, they worked out what would happen for a highly charged star collapsing in a universe whose expansion is being accelerated by a cosmological constant – as ours appears to be.

Quasinormal modes

To do so, Cardoso and team studied damped oscillations known as quasinormal modes. They have shown that when the collapsing star has enough charge, damping wins out over amplification and the oscillations die away quickly. As Cardoso explains, the charge and cosmological constant essentially provide repulsive forces that counteract the pull of gravity and so diminish its amplifying effects. The upshot, he and his colleagues say, is that the Cauchy horizon is damaged but not completely destroyed. As such, they conclude, there is indeed a region within the black hole where the relativistic field equations work but where determinism breaks down.

Group member João Costa says that this would be a more fundamental breakdown of determinism than that inherent to quantum phenomena. As he points out, although we can’t predict the outcome of any particular quantum measurement we can still work out the probability distribution for an ensemble of measurements. But, he says, beyond the Cauchy horizon such overall predictability would be impossible.

Falling Schrödinger’s cat

“Thinking about Schrödinger’s cat, we know we can assign probabilities to the cat being alive and dead,” says Cardoso. “But if the cat were to fall inside the Cauchy horizon we could not even compute these probabilities.” Although, he adds, “for the cat that is probably irrelevant, since it would be dead anyway.”

As Cardoso and colleagues point out in their paper, charged black holes are not expected to exist in nature. But they say that a close analogy between charge and angular momentum means they expect “very similar results” for neutral, rotating black holes. “Given the non-zero cosmological constant and the existence of rapidly rotating black holes in our universe,” they write, “these results cannot be taken lightly.”

Gary Horowitz of the University of California, Santa Barbara, who was not involved in the work, says the research provides “the best evidence I know for a violation of strong cosmic censorship in a theory of gravity and electromagnetism.” The next step, he adds, would be to fully model gravitational and charge effects – the present analysis relying on simplified massless scalar fields.

Unavoidable presence

In fact, in a paper recently posted to the arXiv server, Shahar Hod of the Ruppin Academic Center and the Hadassah Academic College in Israel claims to have done something similar. Hod has found that charged fields close to Reissner-Nordström black holes decay slowly enough to guarantee unstable Cauchy horizons. Given “the unavoidable presence” of these fields – since the collapsing stars themselves are charged – he concludes that such black holes must respect strong cosmic censorship.

Cardoso’s colleague Aron Jansen describes Hod’s counter-proposal as “an intriguing possible resolution” to the dispute but says more work needs to be done. One complicating factor, he says, is that actual charged matter would consist of fermions not the scalar particles investigated by Hod.

It is possible, adds Cardoso, that the issue could be settled by observing gravitational waves. He says that the idea of indeterminism would be bolstered by the existence of black holes that either have lots of charge or spin very quickly. Alternatively, he speculates, the fading of gravitational wave signals – due to the presence of a black hole’s event horizon – might be modulated by the presence of a Cauchy horizon. He points out, however, that such a signal might also mean dark energy cannot be explained in terms of the cosmological constant.

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