Stellar Collapse in Semi-classical Gravity

Core-collapse simulation image (Philipp Mösta)

Supernova explosion (artist: Mehau Kulyk)

In the simplest stellar collapse model of classical General Relativity developed by Oppenheimer & Snyder 1939, the collapsing star is idealized as a uniform ball of dust that contracts under the pull of gravity. The dust particles that make up the star are assumed to be classical and thus infinitely small, infinitely light, and interact only gravitationally with other matter. This approximation is not perfect since infinitesimally small particles would be infinitely large in size, and could never be localized within the stellar horizon.  

The inclusion of quantum effects is unavoidable both when determining whether a black hole will form and when understanding the final stages of the stellar collapse. When we include quantum effects (arXiv:1501.04250, submitted to Physical Review D),  some of the assumptions above are lifted. A particle on the surface of the star is no longer localized, but is instead represented by its wavefunction. Every particle now has a finite probability of escaping the gravitational pull of the star. This allows for the possibility that some configurations will not collapse to black holes, but will instead disperse or even form stable new configurations. The smaller the mass of the constituent particles, the more significant the quantum mechanical effects become.

Classical paths with all initial velocities

To simulate the stellar collapse using a path integral formulation, we have to integrate over all possible paths towards and away from the center of the star. This includes classical paths with all initial velocities. In the special case of the dust ball collapse these paths can be computed analytically.  We first derive analytic solutions to all classical paths (space-like, time-like, and light-like) in Schwarzschild (Table I and II in arXiv:1501.04250) and Kruskal coordinates (see Table V and VI).

The evolution of the wavefunction

In Schwarzschild coordinates, we can only study the collapse outside r=2M. The motion of the particle on the surface of the star is analogous to the vertical motion of a ball moving under gravity, which can go from an initial point to a final point directly or reach its highest and then come down or escape. Each of these paths is unique taking a different amount of time to complete.

In Kruskal coordinates,  we can model the behavior of a particle on the surface of the collapsing star up to the physical singularity at r=0.  We find that classical time-like paths are unique. A path between an initial and final point can be either direct or indirect (turns back in space). Thus some particles that initially move away from the star can return and contribute to the collapse. Space-like paths can turn back in time, but cannot turn back in space. They are also no longer unique when the final point lies inside r=2M. Classically, no information can exit the black hole. However, by integrating around the classical paths one might be able to extract information from inside the horizon. We only compute the paths in the Kruskal case, and leave the computation of the wavefunction and full exploration of the quantum collapse to future work.

The minisuperspace approximation

Models that apply quantization procedures to general relativity operate in superspace, where a 4-geometry space is represented as a trajectory within space-like 3-geometries. A minisuperspace is a symmetry reduced space where these trajectories are limited to a finite number of parameters describing the constant t slices. We assume that the radius is the only spatial degree of freedom that is not frozen. Such restrictive approximations make the problem tractable analytically. Most physicists reading this have likely worked in minisuperspace unknowingly.

The wavefunction of a particle on the surface of the collapsing star

WKB & Schrödinger comparison

t=5 M, multiple masses

The initial wavefunction is taken to be a Gaussian centered far away from r=2M.  In Schwarzschild coordinates, we then compute closed form solutions to the propagator in the WKB approximation where an expansion is performed around the classical paths and in the Schrödinger approximation, and compare the resulting wavefunctions. We find that the two solutions converge towards each other at intermediate times. They are out of step at early and late times with the Schrödinger solution being more exact at early times, and the WKB approximation being more accurate at late times. For lower particle mass, the star resists collapse longer and the probability that it disperses increases. Further work is needed (including the addition of higher order corrections to the Schrödinger solution) to determine the mass limit at which stellar configurations no longer collapse.

M=0, dotted (WKB), solid (Schrödinger)

In the limit when the mass of the star is zero, the WKB approximation converges to the Schrödinger solution at all times. This checks that the WKB approximation is accurate. The Wheeler-DeWitt equation converges to Schrödinger equation in this limit making the Schrödinger solution the exact solution of the free particle problem (Redmount and Suen 1992). Note that Redmount and Suen did not find a good agreement between their WKB approximation and the exact Schrödinger solution due to numerical issues.

 

Black holes, black stars or boson stars?

The dust ball collapse is a testbed for relativistic corrections to the Schrödinger equation. Recent work by Dvali and Gomez argue that black holes are a collection of Bose-Einstein condensates. Quantum effects are very important in this situation, and could add potentially new semi-classical corrections that can be investigated.

Other authors argue that there are no black holes and that nature has black stars or boson stars instead. Neither have an event horizon. 

Classical boson stars are formed from dark matter particles that are spin zero bosons and Bose-condense creating a macroscopic quantum object that depending on the particle size can either fit in your pocket or be the size of a galaxy. Since no spin zero particles have been discovered other than the perhaps the Higgs boson, it is difficult to predict which theory is correct. Supermassive boson stars could lie in the centers of galaxies. Upcoming instruments sensitive to light may detect supermassive boson stars through lensing (e.g., see Boson stars as Gravitational lenses and Method for detecting a boson star at Sgr A* through gravitational lensing). In the future, it may also be possible to detect gravitational waves from a pair of co-orbiting boson stars. Additionally, detectors beyond LISA may see gravitational waves from perturbed supermassive boson stars, which would have quasinormal modes that damp slower than if the object had been a black hole.

Macroscopic dark matter particles?

Feeding on dark matter?

Dark matter particles come close to the attributes of classical dust. If somehow black holes can feed on dark matter particles,  the paucity of super-massive black holes immediately above a certain mass M could be linked to the presence of an ultra-light particle halos that black holes heavier than M can feed on. Black holes lighter than M would be unable to capture such dark matter particles in the same way mini black holes produced at the LHC cannot be grown on atomic matter and pose no danger to the Earth. As soon as a black hole reaches mass M, it starts feeding on dark matter and grows rapidly, perhaps swallowing the entire halo. It is important to note that unlike virialized dust halos or stars, the super-fluid dark matter particles would have negligible momentum and thus accrete easily.

Why should we care about black holes?

Supermassive black hole. Artist conception.

In order to understand the universe, we have to understand black holes. They are the most permanent objects in our universe. All life revolves around them. Every galaxy has a black hole at its center. In the early universe, black holes acted as the seeds around which material collected, and eventually galaxies as complex as our own formed.  Black holes are also believed to be the only objects that will be left in the very far future as our universe continues to expand growing cold and empty. They do evaporate, but the evaporation of black holes that are stellar mass or higher happens on timescales longer than the age of the universe and thus is not observable. To truly understand black holes we have to understand their quantum nature, which is not only important in understanding the final stages of stellar collapse, but also in understanding whether they will form at all. The lighter the constituent particles, the more important the quantum mechanical effects become.

This post is based on arXiv:1501.04250, which is work done in collaboration with Prof. Jayashree Balakrishna (Harris Stowe State U.) and Dr. Christine Corbett-Moran (Caltech), and has been submitted to Physical Review D.

This is the first article I have written where all the authors are women. It is also coincidentally my most technical article to date. Note that my co-authors and I do not discriminate and believe in equal rights for everyone. Our collaboration just happens to be 100% female this time.