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).