Are Solutions to GR Field Equations Chaotic?

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Discussion Overview

The discussion revolves around the nature of solutions to the field equations of general relativity (GR), particularly in the context of gravitational collapse of stars. Participants explore whether these solutions can exhibit chaotic behavior and how this might differ from classical solutions like the Schwarzschild black hole. The scope includes theoretical implications and potential numerical approaches to solving the equations.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant suggests that while there are exact solutions for symmetrical cases in GR, chaotic solutions may also exist, particularly during the gravitational collapse of stars.
  • Another participant notes that chaotic solutions are present even in Newtonian gravity, implying that GR might similarly exhibit chaotic behavior.
  • A later reply questions whether a real star would collapse into a classical Schwarzschild black hole or if it could evolve into a different state entirely, highlighting the limitations of the classical model.
  • One participant argues that while a real star is unlikely to collapse into a perfect Schwarzschild black hole due to simplifications in the model, it is expected that the final object will still be some form of black hole.
  • There is a mention of the predictability of outcomes in Newtonian gravity despite its chaotic nature, suggesting that GR may share similar characteristics.

Areas of Agreement / Disagreement

Participants express differing views on the nature of gravitational collapse and the likelihood of chaotic solutions in GR. There is no consensus on whether real stars will collapse into classical black holes or exhibit chaotic behavior, indicating ongoing debate.

Contextual Notes

Participants acknowledge that classical models like the Schwarzschild solution involve simplifications that may not apply to real astrophysical scenarios. The discussion also highlights the complexity of initial conditions and their impact on the outcomes of gravitational collapse.

Juan Largo
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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?
 
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Even Newtonian gravity has chaotic solutions so you'd expect GR to have them too.
 
Simon Bridge said:
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.
 
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.
 

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