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The Schwarzschild solution is a static spherically symmetric metric. But I wanted to know that how would the space-time interval look in a Non-Static case. I tried to work it out and got $$ds²= Bdt² - Cdtdr - Adr² - r²dΩ²$$ Where $A,B,C$ are functions if $r$ and $t$. Being an amateur I am not sure whether this is correct.

Qmechanic
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Manvendra Somvanshi
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2 Answers2

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While Qmechanic's answer is not incorrect, I think it is misleading and misses the point of the question.

First, there is the well-known "Vaidya metric", which is a simple generalization of Schwarzschild to achieve a non-static spherically symmetric spacetime. Of course, in agreement with Birkhoff's theorem, this is also a non-empty (non-vacuum) spacetime — but the OP never restricted to vacuum solutions.

Second, the question strikes me as being more immediately about the general form of a spherically symmetric metric. There's a nice treatment given by Schutz in his chapter 10.1, which is also mostly reproduced here, as well as chapter 23.2 and box 23.3 of MTW. The answer is that yes, a spherically symmetric metric can generally be put in the form given by the OP's equation. But it's also worth pointing out that the $dt\, dr$ term can be eliminated by redefining $t$ by a linear combination of $t$ and $r$ involving $B$ and $C$ (as MTW explain).

Mike
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  • If I may ask, how is the Schwarzschild solution static inside the horizon? If I understand this correctly, static means that the metric coefficients don't depend on time, but the radial coordinate is time inside the horizon and the metric does depend on this coordinate inside. – safesphere Jan 13 '19 at 17:48
  • PS. I see that Wki states (under Examples) that only exterior Schwarzschild solution is static. However, this whole solution is derived based on the assumption of being static. Thus the interior part of the solution must not be valid. No? https://en.wikipedia.org/wiki/Static_spacetime – safesphere Jan 13 '19 at 17:55
  • You might want to ask these questions as a separate question for fuller responses, but I'll try to answer. First, "static" does not mean that "the metric coefficients don't depend on time" in any particular coordinate system; rather it means that there exist coordinates in which the metric coefficients don't depend on time. To approach it from another direction, suppose you had a metric with coefficients that don't depend on time. You know that a smooth change of coordinates doesn't change anything physically, but you can probably see how to add time dependence to those coordinates. – Mike Jan 14 '19 at 17:12
  • Second, you are correct that the region inside the Schwarzschild horizon is not static — meaning that it's impossible to find coordinates where the metric coefficient is time-independent. In fact, in the Kerr metric there's even a region that's outside the horizon but is not static. – Mike Jan 14 '19 at 17:15
  • Third, it is true that the Schwarzschild solution is usually derived using the assumption that it is static, but it is also true that the result you get with that approach is a metric that makes sense (is nonsingular) everywhere except $r=0$ and $r=2M$, and you can just check to make sure that even for $r \leq 2M$ the metric does solve the vacuum Einstein equations. There's no reason it has to work that way; it just does. So the inside of the horizon is a valid solution. – Mike Jan 14 '19 at 17:19
  • Thanks @Mike this helps. The logic of a non static solution being valid under the initial static assumptions seems debatable. The Schwarzschild solution is often described as two very different universes joined at the horizon. So the validity of describing the inner non static universe by a solution for the outer static universe is questionable at least. Thanks again for your insight! I'll refer to textbooks for more information. – safesphere Jan 14 '19 at 18:00
  • Well, the fact that you can change coordinates to get a continuous vacuum solution of Einstein's equations everywhere means that it's a valid solution — even though those new coordinates won't be manifestly static outside the horizon. I've actually never heard of Schwarzschild itself being described as two separate universes, but I have heard the Kruskal-Szekeres extension described in that way. That's considered a different spacetime, but doesn't invalidate Schwarzschild. – Mike Jan 14 '19 at 18:28
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It is actually a consequence of Birkhoff's theorem that a spherically symmetric vacuum solution to EFE is necessarily static. See also this related Phys.SE post.

Qmechanic
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