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La Guinee
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Hi. Birkhoff's Theorem says that the Schwarzschild metric is the unique spherically symmetric vacuum solution. Isn't the Robertson-Walker metric spherically symmetric?
May? The FRW solution contains only matter there is no vacuum in this solution.atyy said:The Schwarzschild solution is a vacuum solution. The Robertson-Walker metric may include matter.
MeJennifer said:Beware, it might be similar to asking a catholic priest too many questions about hell, he may answer it is the place whe you go if you keep asking those questions.
La Guinee said:Sorry, I still don't understand. You have the Robertson Walker metric. From this you can construct the Ricci tensor. Now set the components of the Ricci tensor equal to 0. This gives you a condition on the scale factor, namely that it's linear in time. What's wrong with this?
atyy said:Is there any way to change coordinates to get it into the Schwarzschild form?
La Guinee said:Why can't the Robertson Walker metric be a vacuum solution? The Ricci tensor for the Robertson Walker metric can be easily calculated. Choosing a linear scale factor with appropriate coefficients satisfies R_mu nu = 0 (or am I doing something wrong)?
La Guinee said:Thank you for the replies. I don't see how the FRW can reduce to the schwarzschild metric with a suitable coordinate change because the FRW is time dependent.
Careful. Massless FRW reduces to a patch of flat Minkowski spacetime in unusual coordinates. Minkowski spacetime is isotropic.
La Guinee said:Consider the following FRW metric:
ds2 = -dt2 + 9t2 [ dr2 / (1+9r2) + r2 dOmega2 ]
This satisfies einsteins equations in vacuum. So are you're saying under suitable coordinate change this reduces to Minkowski? This seems weird because one is time dependent and the other isn't.
Birkhoff's Theorem states that the Schwarzschild metric is the unique spherically symmetric, vacuum solution of Einstein's field equations in general relativity. In simple terms, it is a mathematical theorem that proves the uniqueness of the Schwarzschild metric in describing the gravitational field outside of a spherically symmetric, non-rotating mass.
The Schwarzschild metric is a solution to Einstein's field equations in general relativity that describes the spacetime geometry outside of a spherically symmetric, non-rotating mass. It is a fundamental solution in understanding the gravitational effects of massive objects, such as planets and stars.
Birkhoff's Theorem was first proposed by physicist George Birkhoff in 1923. He used mathematical techniques to prove the uniqueness of the Schwarzschild metric as a solution to Einstein's field equations. This theorem has since been verified and accepted by the scientific community.
Birkhoff's Theorem has significant implications in understanding the gravitational field of massive objects in our universe. It allows us to accurately describe and predict the behavior of objects, such as planets and stars, that have a spherically symmetric distribution of mass.
While Birkhoff's Theorem holds true for spherically symmetric, non-rotating masses, it does not apply to more complex systems such as rotating masses or systems with multiple masses. In these cases, the Schwarzschild metric is no longer unique and other solutions must be used to describe the gravitational field.