Cameron R D Bunney

My paper C-metric in a (nut)shell in collaboration with Robert B Mann has just been published in the journal Classical and Quantum Gravity and I’d like to share a few thoughts and results in the paper.

In the paper, we investigate a spacetime known as the C-metric in 2+1 dimensions, a solution to Einstein’s equations with a negative cosmological constant. The 3+1 C-metric was discovered as early as 1917; however, the 2+1 C-metric was only discovered in 2011. Physically, a C-metric describes an accelerated black hole, either being pushed by a strut or pulled by a string of stress energy. This string is a defect in the spacetime due a conical deficit.

Due to this acceleration, the physics of C-metrics is very rich and the thermodynamics of the 3+1 solution was only resolved in 2016 in a wonderful paper published in Physical Review Letters.

Working within Israel’s thin-shell formalism, we consider an interior solution modelled by the 2+1 C-metric and match it by a thin shell to an exterior BTZ spacetime. We can imagine this scenario as follows: a black hole is being pulled by a string of stress energy that is in turn attached to a thin shell. Exterior to the shell, there is no string. We found an unexpectedly large variety of behaviours exhibited by this construction.

In the thin-shell formalism, you ensure that the induced metric on the shell is continuous and then interpret the jump in extrinsic curvatures across the shell as a shell of stress energy. For example, one may consider matching two Schwarzschild solutions with different masses. If the exterior mass is larger than the interior, then the shell stress energy is positive, providing the “extra mass” required to generate the exterior solution. However, if the exterior mass is small than the interior, then the shell stress energy must be negative to “eat up” some of the energy. By considering the stress energy of the shell, we can investigate what sort of energy conditions (strong, weak, null, dominant) the matter satisfies.

Between the positive and negative stress energies, there is a critical mass at which the stress energy is zero, interpreted as no shell. In the case of matching two Schwarzschild solutions, this occurs when the interior mass and exterior masses are equal: the interior and exterior spacetimes have the same geometry.

Our studies of thin shells surrounding 2+1 C-metric were originally intended to investigate the matter composing the shells. However, in the style of Sir Arthur Conan Doyle, we had our own “curious incident of the dog in the night-time”. In this Sherlock Holmes story, the curious incident is that a dog did not bark and certain conclusions are drawn from this.

In our investigations, we had a “curious incident of the shell in the night-time”, in which the shell stress energy vanished. Unlike matching two Schwarzschild solutions, our interior and exterior spacetimes are different: the interior solution contains a black hole accelerated due to a string and the exterior solution contains no string.

We must interpret our result as a new solution to Einstein’s equations describing an accelerating black hole pushed or pulled by a finite-length string.

This is the key difference between our solution and the usual C-metric—in the C-metric, the string extends from the black hole out to infinity.

A natural question is: what happens at the end of the string? We investigated this, finding a point particle sitting at the end of the string. Because the string originates from a conical deficit, the point particle’s (positive or negative) mass counteracts this deficit, allowing us to smoothly match the C-metric to a regular BTZ spacetime.

In total, the new solutions to Einstein’s field equations fall into three categories: an accelerated black hole pulled by a finite-length string with a point particle at the other end, an accelerated black hole pushed by a finite-length strut with a point particle at the other end, and an accelerated black hole pushed from one side by a finite-length strut and pulled from the other by a finite-length string, each with a point particle at the other end.