- Planck stars
Carlo Rovelli, Francesca Vidotto
To understand why this is a really radical proposal, let me first give you some context. When matter collapses to a black hole, its radius shrinks and its density increases. Quantum gravitational effects are expected to become strong when the curvature reaches the Planckian regime. The curvature is the inverse of a length square, so that means the curvature is the inverse of the Planck length square or smaller. At which radius the collapsing matter reaches this regime depends on the total mass: The higher the mass, the larger the radius.The radius at which the collapsing matter reaches the Planckian regime is larger than the Planck length if the mass is larger than the Planck mass. The radius is however always smaller than the horizon radius, so it doesn’t really matter exactly what happens because it’s not in causal contact with the exterior. The curvature at the horizon is weak as long as the total mass of the black hole is larger than the Planck mass. This is somewhat unintuitive, but the curvature at the black hole horizon goes with the inverse of the mass square, ie the higher the mass of the black hole, the smaller the curvature. Thus the often made remark that you wouldn’t notice crossing the black hole horizon - there’s nothing there and space-time can be almost flat if the black hole is large. In particular, you don’t expect any quantum gravitational effects at the horizon.
But the mass of the black hole decreases due to Hawking radiation. Keep in mind that Hawking radiation is not a quantum gravitational effect. It’s quantum fields in a classical gravitational background, a combination often referred to as ‘semi-classical’. If the mass of the black hole has shrunken to the Planck mass, the curvature reaches the Planckian regime and that’s when the semi-classical limit breaks down and quantum gravity becomes important. At that point also Hawking’s calculation breaks down and information can be released. However, the standard argument goes that by this time it’s already too late to get all the information out. Details are subtle but that’s a different story. Suffices to say that Rovelli and Vidotto want information release to be possible earlier, when the radius of the black hole is still much larger than the Planck length and its mass much above the Planck mass.
The only way to do this is to have strong quantum gravitational effects in a region where the curvature of the semi-classical metric is small, much below the Planck scale. In the paper they don’t explicitly say that this is what they do, but of course they have to. You see this most easily when you look at the metric they suggest, equation (14). The third term (containing α) is the correction term that supposedly has a quantum gravitational origin. The validity of the semi-classical limit means essentially that the third term should be smaller than the second as long as the second term is smaller than one. If you convert this into inequalities you find α < m, and that is explicitly the situation they do not consider. Instead α is supposed to start at m and then increase. They do not give any reason given as to why this should be so or what the meaning is of α or what the necessary source terms are for that.
At this point you are probably ready to throw the paper away. There is a reason one of the postulates of black hole complementarity is the validity of the semi-classical approximation near the horizon of a black hole with mass above the Planck mass. That’s because the curvature there is small and no quantum gravitational effects are at your disposal to screw up the semi-classical limit. However, allow me to exercise some good will. I think what Rovelli and Vidotto suggest may be possible if the Planckian-density core behind the horizon displays a very unusual behavior, though that’s a big “if”.
The behavior would have to be such that as the total mass is shrinking, the mass is taken from the center only, leaving behind an increasingly thinner shell of high density at a constant radius (or even an increasing one). This shell would eventually intersect the horizon of the black hole and could do so conceivably at a radius much above the Planck radius. This isn’t a priori in conflict with the semi-classical limit because there is a high density now and also a high curvature region.
However, the metric that is used by Rovelli and Vidotto does not describe such a scenario. (The metric inside a shell has to be flat while their metric is actually singular at the center.) Besides this, there exists no approach to quantum gravity that suggests such a hollow-core behavior. There doesn’t even exist a model that describes such a situation. I also strongly suspect that such a solution, even if it can be created by help of some quantum gravitational pressure (this is almost certainly possible), would be unstable under non-spherical perturbations and just recollapse to form a smaller Planckian-density core. Iterate and end at Planck scale radius as usual.
In summary, this is an ad-hoc proposal. It is not based on anything we know of quantum gravity. Neither is it a complete model. I am reasonably sure that the metric they use cannot describe the situation they want while still maintaining energy-conservation. They do not calculate the curvature that belongs to that metric to check whether their modification is consistent. Neither do they calculate the necessary source that presumably contains a quantum-gravitationally induced stress-energy. It is an interesting suggestion, but I do not think it is very plausible. Planck stars almost certainly do not exist.
Acknowledgements: Carlo Rovelli has been very patient explaining his idea by email, but as you can tell I’ve remained unconvinced...