Stellar Collapse and Time Travel?

This post summarizes work done in collaboration with Prof. Jayashree Balakrishna (Harris Stowe State U.) and Dr. Christine Corbett Moran (Caltech). Our paper just appeared in the cosmology section of the Frontiers Journal.
 

Supernova explosion (artist: Mehau Kulyk)

Stellar Collapse. In the simplest stellar collapse model of classical General Relativity (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. Infinitesimally small particles would be then infinitely large in size, and could never be localized within the stellar horizon.  The smaller the mass of the constituent particles, the more significant the quantum mechanical effects become, breaking down the classical approximation.

Quantum Effects

When we include quantum effects,  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. 

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.

Classical paths with all initial velocities

We model stellar collapse using a path integral formulation.  In the special case of the dust ball collapse the paths can be computed analytically for all initial velocities (towards and away from the center of the star).  We derive analytic solutions to all classical paths (space-like, time-like, and light-like) in Schwarzschild (Table I and II in our paper) 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 (1) from an initial point to a final point directly, (2) reach its highest and fall back down or (3) escape. Thus some particles that initially move away from the star can return and contribute to the collapse. 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).  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.

Back in Time?
 In the dust-collapse model, the classical paths that turn back in time are space-like. This means that they are outside the light-cone, i.e., not in the realm of paths considered possible. Further understanding of these low probability paths that are traditionally ignored might lead to new physics. We conjecture the classical space-like paths and paths around them play an important role in the quantum mechanical collapse in the same way passing through the potential barrier is important in tunneling. Classically, it can never happen, and yet the tunneling probability cannot be ignored in quantum mechanics. It may be that time travel could be achieved by some kind of tunneling from time-like paths to the space-like paths that turn back in time.

Why include the space-like paths?  There is no theory of quantum gravity that works, but the theories that do exist and are accepted in the literature are causal. In our case, the propagator does not vanish outside the light-cone, and so we include all paths in our integration. The propagation on and around the space-like paths is acausal, i.e., backwards in time in some Lorentz frames. Whether or not this proves to be admissible in the ultimate theory of quantum gravity is beyond the purpose of this exploration since no such theory exists today.
 

Approximations
 We assume that the radius is the only spatial degree of freedom that is not frozen. Such restrictive approximations make the problem tractable analytically.

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 compute closed form solutions to the propagator in (1) the WKB approximation where an expansion is performed around the classical paths and (2) in the Schrödinger approximation. We compare the resulting wavefunctions and 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, while our approximation converged. Space-like classical paths give the dominant contribution in the construction of the WKB approximation outside the light-cone.

 

Black holes or boson stars?

Even with LIGO's detection, it is still unclear whether black holes exist or not. We have no proof of a singularity. Theories suggest that black holes can be mimicked by boson stars or that the black hole itself could be a collection of Bose-Einstein condensates (e.g., Dvali and Gomez).  However, to obtain black holes as massive as the ones just seen by LIGO, the scalar particles would have to be very light. Quantum effects would be important in this situation, and the gravitational waves might look different due both the size and nature of the star.

Classical boson stars are believed to form from dark matter particles (fundamental spin zero particles) that Bose-condense creating a macroscopic quantum object. The size of this object depends on the wavelength of the constituent particle and can either fit in your pocket or be larger than a galaxy. Since no fundamental spin zero particles have been discovered other than the perhaps the Higgs boson, it is difficult to know if boson stars exist. 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). If such stars exist in the mass-range that LIGO sees,  we might soon see a pair of co-orbiting boson stars. Additionally, detectors beyond LISA may see gravitational waves from perturbed supermassive boson stars.

Feeding on dark matter?

Macroscopic dark matter particles?

Dark matter particles come close to the attributes of classical dust. Unlike virialized dust halos or stars, the super-fluid dark matter particles would have negligible momentum and thus accrete easily. If somehow black holes can feed on dark matter particles,  the paucity of super-massive black holes immediately above a certain mass M (e.g., we don't see intermediate mass black holes) could be linked to the presence of 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. 

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 against men and believe in equal rights for everyone. Our collaboration just happens to be 100% female.