# Planck Stars

Tl;dr: “A black hole is essentially a collapsing and bouncing star that appears frozen because it is seen in slow motion.”

## Problems of current black hole theory

In the standard theory of black hole formation, a massive star (greater than ~ 30 solar masses) collapses to a point or singularity. At this point the equations of general relativity become singular, nothing really works, and everything goes to infinity. At a certain radius outside this singular point is the event horizon - the region within which the escape velocity is greater than the speed of light, and is given for a non-spinning (Schwarzchild) black hole of mass $M$ by,

The existence of singular points in general relativity is problematic. At the moment we think that all singularities are hidden behind the event horizon and can have no causal impact on the wider universe. This is the Cosmic Censorship Conjecture; that all singularities in Nature exist behind the black hole event horizon. Should the conjecture be untrue, and naked singularities exist, this would throw up issues relating to causality, determinism and the break down of physical laws. This inspired Stephen Hawkins pronouncement that “Nature abhors a naked singularity”

A further problem comes from trying to mesh quantum mechanics (QM) with GR. A quantum theory of gravity is expected to quantize spacetime itself. As such, the smallest unit of space is the Planck length $l_p$, given by,

Now, over time black holes evaporate due to Hawking radiation and their radius $r_s$ (i.e. the event horizon) shrinks. The restriction of unitarity in QM means that information must be conserved in a system, whilst the No Hair Theorem implies that information must remain trapped inside the black hole. Consequently, at the end of the evaporation, when the radius is very small, the black hole would have to store a large quantity of information that is incompatible with the expected Planckian lengthscale.

## The Solution: Planck Stars?

A possible solution to both the $r=0$ singularities and the information storage problems was proposed by Rovelli & Vidotto, 2014. The concept is that during the final collapse of a large star, the core could compress into a small but finite, highly dense core, with a density of the order of the Planck density:

The key insight here is that the density can be of Planck order, before the lengthscale becomes Planckian. The size of a Planck star can be estimated as,

Therefore a stellar mass black hole would collapse to a Planck star with radius $\sim 10^{-10}$ cm, much larger than the Planckian lengthscale.

The collapse then does not result in a singularity but instead a quantum gravitational phase where the inwards gravitational forces are balanced by the outwards quantum pressure. A star in this phase is known as a Planck Star. Such a Plank Star would exist behind the event horizon of a black hole.

The question then becomes: can a Planck Star could exist for the typical lifetime of a black hole?. The answer proposed by Rovelli & Vidotto is very cool; as measured by an observer riding the collapse, the lifetime (proper time) is very short - essentially a bounce. However, to some distant observer the lifetime is very long owing to the extreme gravitational time dilation. That is, a distant observer sees the expansion of the Planck Star after the bounce in such slow motion that it appears stationary (of course, the observer doesn’t actually ‘see’ anything at all as the Planck Star is hidden behind the event horizon).

Therefore we have two radii: the event horizon radius which is shrinking due to Hawking radiation and the Plank star radius which after the initial collapse bounces back and begins to grow. Once the event horizon becomes less than the Planck horizon then the information that fell beneath the event horizon is then available to the universe and the unitarity of QM is preserved.

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