# Spherically symmetric metric in isotropic coordinates ?

1. Apr 2, 2008

### smallphi

spherically symmetric metric in isotropic coordinates ???

The most general spherically symmetric metric can be written as:

$$ds^2 = - dt^2 + X^2(r,t) dr^2 + R^2(r,t) (d\theta ^2 + sin^2(\theta) d\phi^2)$$

Is there an isotropic form of that metric which shows more explicitly there is no difference looking along x or y or z direction from the center. It must look something like

$$ds^2 = - A^2(r,t) dt^2 + B^2(r,t) (dx^2 + dy^2 + dz^2 )$$
where $$r = \sqrt{x^2 + y^2 + z^2}$$

I know the above is spherically symmetric, the question is wether it's the most general form? Any references I should check?

As an example the isotropic form of a general FRW metric is:

$$ds^2 = - dt^2 + \frac{a(t)^2}{(1+kr^2/4)^2} (dx^2 + dy^2 + dz^2 )$$

Last edited: Apr 2, 2008
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