Are spherically symmetric and isotropic the same

[binbagsss]

If space-time is isotropic does this imply it is spherically symmetric?

why doesn't it need to be both isotropic and homogeneous?

 

[andrewkirk]

I think it depends on whether we are talking about perfect isotropy or homogeneity, or iso and hom 'on the large scale', which is the (vague) concept employed in the Cosmological Principle. I think that perfect isotropy and perfect homogeneity are equivalent, each implying the other - at least I can't think of any model that would have one and not the other. However that is not the case for large-scale. There are models that are large-scale homogeneous but not large-scale isotropic. I can't think of any models that are large-scale isotropic but not large-scale homogeneous.

Perhaps isotropy always entails homogeneity, whether perfect or large-scale, but homogeneity entails isotropy only if the homogeneity is perfect.

If a space is not homogeneous then we can establish a preferred direction, or a preferred proper subset of directions by the following:

If the homogeneity is not complete then there exist two points P and Q that have different characteristics, say different curvature scalar. Let S be the midpoint of the geodesic that connects the two, of length 2L. Then there is a preferred direction at S along the geodesic since the curvature at distance L in one direction is different from that in the opposing direction.

 

[PAllen]

A universe that is isotropic everywhere(the usual meaning), can be considered to be spherically symmetric about any point.

As pointed out previously, isotropy everywhere implies homogeneity. This is actually considered in exercise 27.1 in MTW.

 

[binbagsss]

As pointed out previously, isotropy everywhere implies homogeneity. This is actually considered in exercise 27.1 in MTW.

MTW?

 

[binbagsss]

As pointed out previously, isotropy everywhere implies homogeneity.

why then in sean carroll lecture notes on gr, page 218, does it say The usefulness of homogeneity and isotropy is that they imply that Σ must be a maximally symmetric space. (Think of isotropy as invariance under rotations, and homogeneity as invariance under translations. Then homogeneity and isotropy together imply that a space has its maximum possible number of Killing vectors.) as a pose to him just stating that isotropy is implying this?

Also, when deriving the Schwarzschild metric, in these notes, using spherical symmetry, the procedure is to use the definition that spherical symmetry is defined as : there exists a set of Killing vectors whose Lie brackets form the Lie algebra of SO(3),
as a pose to using the fact that we must have the maximum number of Killing vector fields (as he does use when deriving the FRW metric, based on the assumptions of isotropic and homogeneous space, but is never used in the derivation of the Schwarzschild metric).Thanks

 

[PAllen]

MTW?

"Gravitation" by Misner, Thorne, and Wheeler. So famous, that everyone just calls it MTW.

 

[PAllen]

why then in sean carroll lecture notes on gr, page 218, does it say The usefulness of homogeneity and isotropy is that they imply that Σ must be a maximally symmetric space. (Think of isotropy as invariance under rotations, and homogeneity as invariance under translations. Then homogeneity and isotropy together imply that a space has its maximum possible number of Killing vectors.) as a pose to him just stating that isotropy is implying this?

Why Carroll does not mention that isotropy implies homogeneity I don't know. I'm sure he knows. Perhaps he didn't see the pedagogical need to present the argument, and it is certainly simpler to assume both independently.

Also, when deriving the Schwarzschild metric, in these notes, using spherical symmetry, the procedure is to use the definition that spherical symmetry is defined as : there exists a set of Killing vectors whose Lie brackets form the Lie algebra of SO(3),
as a pose to using the fact that we must have the maximum number of Killing vector fields (as he does use when deriving the FRW metric, based on the assumptions of isotropic and homogeneous space, but is never used in the derivation of the Schwarzschild metric).Thanks

Spherical symmetry is isotropy around one point. I have stated explicitly I refer to 'isotropy everywhere'. Then, any inhomogeneity leads to the existence of some point that sees anisotropy.

 

[PAllen]

A last point worth mentioning is that homogeneity does not necessarily imply isotropy. A simple example is the flat cylinder 2 manifold. This is obviously homogeneous - no point can be distinguished from any other - but it is not isotropic. Geodesics in one direction are unbounded, while in another direction are closed. Thus it is nowhere isotropic, while being homogeneous.

 

[binbagsss]

Spherical symmetry is isotropy around one point. I have stated explicitly I refer to 'isotropy everywhere'. Then, any inhomogeneity leads to the existence of some point that sees anisotropy.

Apologies, so spherically symmetric is isotropic about the origin/centre of the sphere?Also, just looking at a single point, Isotropic about a single point does not imply homogenous about this point?

 

[PAllen]

Apologies, so spherically symmetric is isotropic about the origin/centre of the sphere?Also, just looking at a single point, Isotropic about a single point does not imply homogenous about this point?

Yes and yes. However, the center of spherical symmetry need not be part of the manifold, as in Schwarzschild geometry. And, in the literature, isotropy is normally used to mean isotropy everywhere, same for homogeneous.