Proving the Existence of Event Horizon in Homogeneous Spacetimes

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SUMMARY

The discussion centers on the existence of event horizons in homogeneous spacetimes, specifically within the context of Schwarzschild spacetime. It is established that if the singularity from a point mass is removed, the event horizon may also disappear. The participants suggest that proving the absence of an event horizon can be achieved by analyzing the metric of stars with homogeneous mass distributions, particularly focusing on the behavior of timelike coordinates. References to Schutz's textbook and MTW's "Gravitation" provide foundational material for further exploration of this topic.

PREREQUISITES
  • Understanding of Schwarzschild spacetime
  • Familiarity with null geodesics and their implications for event horizons
  • Knowledge of metrics in general relativity, particularly for homogeneous mass distributions
  • Ability to interpret and analyze mathematical expressions related to spacetime geometry
NEXT STEPS
  • Study the metric of the interior of stars with homogeneous mass distribution as detailed in Schutz's textbook
  • Explore the concept of null geodesics and their role in determining event horizons
  • Review MTW's "Gravitation" for insights on spacetime metrics and event horizons
  • Investigate previous discussions on Physics Forum regarding static stars and their metrics
USEFUL FOR

The discussion is beneficial for theoretical physicists, astrophysicists, and students of general relativity who are exploring the properties of event horizons and spacetime metrics in homogeneous conditions.

hellfire
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Consider a Schwarzschild spacetime. If the singularity due to the point mass is removed (e.g. with an homogeneous matter distribution), does the event horizon disappear? If yes (I assume this is the case), how can be proven that there exists no event horizon if there is no singularity? May be it is enough to show that in the metric for the interior of stars (with homogeneous mass distribution) there is no change of the timelike coordinate from t to r at any r, as in case of the vacuum Schwarzschild solution for r = 2GM (Schutz p. 290)? I was not able to find an expression for the metric of the interior of stars to check this.
 
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There's an analysis of the metric of a star with a constant density rho online

http://www.pma.caltech.edu/Courses/ph136/yr2002/chap25/0225.1.pdf

It's very detailed and long, being a chapter from a textbook, you'll probably have to print out & study it to get very far.

The same results are also in MTW's gravitation. This topic came up late in the thread "The Mass of a Body" that you'll find here on Physics Forum, though I'm not sure if there will be anything useful in that discussion for your question.

I think another poster, DW, once posted some formulas about static stars as well, you might try searching the physics forum for that, too.

As far as the event horizon goes, I think the best test is to consider whether or not light can escape from an object. If there are null geodesics that connect the interior of the star to the exterior, it doesn't have an event horizon. [I haven't double-checked this definition though.]

[add]
I should clarify, there should always be ingoing null geodesics, but if there is a horizon, there won't be outgoing ones.
 
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Thanks, pervect, I will take a look. Your proposal to consider the outgoing geodesics to prove whether an event horizon exists or not seams to be the best one, but I can imagine it may be difficult to do the calculations. What do you think about this possibility:

hellfire said:
May be it is enough to show that in the metric for the interior of stars (with homogeneous mass distribution) there is no change of the timelike coordinate from t to r at any r, as in case of the vacuum Schwarzschild solution for r = 2GM (Schutz p. 290)?
 
Last edited:

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