Why Is the Schwarzschild Solution Considered Spatially Symmetric?

[off-diagonal]

I've read Schwarzschild paper and I don't understand his conditions

"The solution is spatially symmetric with respect to the origin of the co-ordinate system in the sense that one finds again the same solution when x,y,z are subjected to an orthogonal transformation(rotation)"


Could anyone explain me about this??


Thank you

 

[lbrits]

He's assuming spherical or rotational symmetry. In other words, the solution/metric does not depend on which direction you're looking in (from the origin). This is a reasonable assumption for the metric around a star, since stars are to a high degree "round" =)

 

[tiny-tim]

"The solution is spatially symmetric with respect to the origin of the co-ordinate system in the sense that one finds again the same solution when x,y,z are subjected to an orthogonal transformation(rotation)"

Hi off-diagonal!

"orthogonal" means that it's a member of SO(3), the three-dimensional group of speical orthogonal transformations.

Basically, SO(3) means all rotations.

See http://en.wikipedia.org/wiki/Rotation_group for more details:

The rotation group is often denoted SO(3) for reasons explained below.