Spherically symmetric spacetime

[maxverywell]

I know from classical physics that, for example, an electric field is spherically symmetric if its magnitude depends only on the distance $r$ to the origin (and not on the angles $\phi$, $\theta$) and it's in radially inward or outward direction.

But, what does it mean when spacetime is spherically symmetric? Does it mean that the metric depends only on $r$ and not on the angles?

 

[WannabeNewton]

It means that the isometry group $G$ of the space-time $(M,g_{ab})$ has a subgroup $H\subseteq G$ such that $H\cong SO(3)$ and such that the orbits of the group action associated with $H$ are topological 2-spheres. You should think of spherical symmetry in this way and not in the way you tried to characterize it because that is a coordinate dependent characterization (and is false by the way the metric doesn't only depend on $r$ in the coordinate basis - the Schwarzschild metric also depends on $\theta$ in the coordinate basis) whereas spherical symmetry of the space-time is a geometric property independent of coordinates. The isometries are related to one-parameter families of local diffeomorphisms that generate killing vector fields related to rotational symmetry so very loosely put, a spherically symmetric space-time is "invariant under rotations".

EDIT: See this introduction: http://www.physto.se/~ingemar/sfar.pdf

 

[Popper]

I know from classical physics that, for example, an electric field is spherically symmetric if its magnitude depends only on the distance $r$ to the origin (and not on the angles $\phi$, $\theta$) and it's in radially inward or outward direction.

But, what does it mean when spacetime is spherically symmetric? Does it mean that the metric depends only on $r$ and not on the angles?

You're using a 3D object and asking what it's like in 4D. Since the spatial part of an object does not apply to time then I don't see how any meaning can be given to this situation.

 

[maxverywell]

Here http://en.wikipedia.org/wiki/Deriving_the_Schwarzschild_solution it says:

A spherically symmetric spacetime is one in which all metric components are unchanged under any rotation-reversal $\theta \to -\theta$ or $\phi \to -\phi$

Why is that true and how it is related to what WannabeNewton wrote?

 

[WannabeNewton]

Ugh, I wish people would stop reading that wiki article. It is sacrilegious to describe a geometric property of space-time using meaningless coordinates so please don't take wiki's "definition" as an actual definition. It is merely a consequence of the definition I wrote above. As for why, see here (starting with page 171 of the PDF): http://arxiv.org/pdf/gr-qc/9712019v1.pdf

 

[maxverywell]

Thanks WannabeNewton, you helped a lot!

 

[WannabeNewton]

Thanks WannabeNewton, you helped a lot!

Anytime mate! Feel free to ask any further questions you might have after reading the PDF. I just want to stress again, geometry > coordinates Cheers!

 

[Bill_K]

Ugh, I wish people would stop reading that wiki article. It is sacrilegious to describe a geometric property of space-time using meaningless coordinates so please don't take wiki's "definition" as an actual definition. It is merely a consequence of the definition I wrote above.

Well, if you feel that way, the logical thing to do would be to edit the Wikipedia page and replace their definition with yours.

 

[WannabeNewton]

Well, if you feel that way, the logical thing to do would be to edit the Wikipedia page and replace their definition with yours.

Apparently there is already a wiki article already that has the coordinate independent definition: http://en.wikipedia.org/wiki/Spherically_symmetric_spacetime but I have no idea how to link this to the other one or edit wiki articles n' stuff It seems safer to just learn the definitions from a textbook rather than from wiki (the above article cites Wald for example, who gives the exact same definition in his text when deriving the Schwarzschild metric).