Are Solutions to GR Field Equations Chaotic?

[Juan Largo]

The field equations of general relativity are non-linear. There are exact analytic solutions to the equations for special symmetrical cases, e.g. Schwarzschild's solution for a black hole. But in general, wouldn't there be other chaotic solutions as well?

A chaotic system is analytically unpredictable; however, one or more so-called attractors constrain the system within defined regions of state space. These attractors in state space can be points, loops (periodic), or "strange" (non-periodic). I'm thinking that when a star undergoes gravitational collapse, it might not evolve into a classic, orderly, steady-state Schwarzschild black hole at all, but a chaotic system instead. And depending on the initial conditions of that star prior to the collapse, it could end up in any number of different chaotic states, depending on which attractor it encountered along the way.

Have there been attempts to solve the GR field equations either numerically or using analog computers, specifically for collapsing stars? Is there any literature on this subject?

 

[Simon Bridge]

Even Newtonian gravity has chaotic solutions so you'd expect GR to have them too.

 

[Juan Largo]

Even Newtonian gravity has chaotic solutions so you'd expect GR to have them too.

True, which is why the 3-body problem is analytically "unsolvable." But my real question is whether a real star would ever actually collapse into a classical Schwarzschild black hole, or become something else entirely.

 

[Simon Bridge]

The answer is no and yes.
It would be very surprising for a real star to collapse into a classical SBH as you say, since that model has a bunch of simplifications to it. However, we would expect the final object to be some form of black hole.

Notice that Newtonian gravity has predictable outcomes despite its chaotic nature - GR is the same.
You'd get black holes for the same reason you'd get big balls of gas. You won't get a BH exactly conforming to a simplified solution the same way that stars are not perfect spheres.

 

[pmacias]

I can confirm that the solutions to the GR field equations can indeed be chaotic. This is a well-known fact in the field of general relativity and has been extensively studied by researchers. The non-linearity of the equations allows for chaotic behavior, and as you mentioned, there are exact analytic solutions for special cases but for general cases, there can be a wide range of chaotic solutions.

The concept of attractors in state space is also relevant in this context. These attractors can indeed constrain the system to certain regions of state space, but they can also lead to chaotic behavior. In the case of gravitational collapse, it is possible that the system may not evolve into a classic Schwarzschild black hole, but instead, it may end up in a chaotic state depending on the initial conditions and the attractors encountered along the way.

There have been numerous attempts to solve the GR field equations numerically and using analog computers, particularly for collapsing stars. The use of numerical simulations has been a valuable tool in understanding the behavior of these systems. There is a vast amount of literature on this subject, and I would recommend exploring research papers and books on numerical relativity and gravitational collapse for further information.

In conclusion, the solutions to the GR field equations can indeed be chaotic, and there have been extensive efforts to study and understand this behavior. The use of numerical simulations has been a crucial tool in this research, and there is a wealth of literature available on this subject.