Solving the Spherically Symmetric Einstein Equation

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Discussion Overview

The discussion centers on the possibility of rewriting the Einstein equation in a simpler form under the assumption of a spherically symmetric distribution of mass-energy, without requiring the distribution to be stationary. Participants explore the implications of this symmetry on the structure of the equations and seek resources for further learning.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant inquires whether the Einstein equation can be simplified for spherically symmetric mass-energy distributions, expressing a desire for a form that emphasizes derivatives.
  • Another participant provides links to resources that may assist in understanding the topic, including worksheets and course notes on General Relativity.
  • A third participant asserts that the Einstein equation can indeed be rewritten in a simpler form for spherically symmetric cases, noting that this scenario reduces the number of unknown functions in the metric from ten to two. They describe a common approach involving diagonal coordinates and the areal radius.
  • This participant references an Insights article that discusses the vacuum case of the Einstein field equations and suggests that the methods can be generalized to non-vacuum cases by incorporating appropriate stress-energy tensor components.

Areas of Agreement / Disagreement

Participants generally agree that the Einstein equation can be simplified under the assumption of spherical symmetry, but the specifics of how this is achieved and the implications for non-vacuum cases remain open for further exploration.

Contextual Notes

The discussion does not resolve the complexities involved in generalizing the solutions to non-vacuum cases or the specific forms of the stress-energy tensor that may be required.

Tomas Vencl
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TL;DR
When suppose only spherically symmetric distribution of mass-energy, how it simpify the Einstein equation ?
Can be Einstein equation rewrited into some simpler form, when suppose only spherically symmetric (but not necessarily stationary) distribution of mass-energy ?
If yes, is there some source to learn more about it ?
Thank you.
edit: by simpler form I mean something with rather expressed derivatives, than more compact form (if this makes sense)
 
Last edited:
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Tomas Vencl said:
Can be Einstein equation rewrited into some simpler form, when suppose only spherically symmetric (but not necessarily stationary) distribution of mass-energy ?

Yes. For the spherically symmetric case, there are only two unknown functions of the coordinates in the metric (a general metric with no symmetry has ten unknown functions of the coordinates, one for each of the independent metric components). The most common way of treating this case is to choose coordinates such that the metric is diagonal and the areal radius ##r## (given by ##r = \sqrt{A / 4 \pi}##, where ##A## is the radius of the 2-sphere containing the event in spacetime whose coordinates one is evaluating) is one of the coordinates. The general form of the metric with these assumptions, along with the significant components of the Einstein tensor, is given in this Insights article:

https://www.physicsforums.com/insights/short-proof-birkhoffs-theorem/

The article only considers the actual solution of the EFE for the vacuum case; but one can easily generalize what is done there to the non-vacuum case by simply putting the appropriate stress-energy tensor components (as functions of ##t## and ##r##) on the RHS of the equations.
 
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Thank you both, I will look at the links.
 

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