Coordinate conditions in cosmology

[TrickyDicky]

The FRW cosmology is a solution of the FRW that can be foliated into 3D isotropic and homogeneous slices.
This foliation is implemented mathematically first by the use of a not generally covariant coordinate condition https://en.wikipedia.org/wiki/Coordinate_conditions#Synchronous_coordinates , namely the use of synchronous coordinates https://en.wikipedia.org/wiki/Synchronous_coordinates, with a synchronous time coordinate and drdt crossterms excluded .This coordinate condition is necessary to define a scale factor for the spacelike part and maximally symmetric(isotropic and homogeneous) spatial slices.

This is explained in any cosmology or GR's cosmology section textbook.

The question that I think has a simple enough answer is, from the above, isn't it obvious mathematically that the isotropicity and homogeneity properties of those 3D slices are not in general (not in general since in certain specific cases like for instance in flat spacetime or in de Sitter spacetime they are indeed geometric features of the 4-manifold) geometric features for the 4D spacetime given that they come imposed by the coordinate condition.

 

[Chalnoth]

Sort of. The point is that it's not always possible to produce a set of coordinates where the universe appears homogeneous and isotropic. The FRW space-time is special in that this choice could be made. One way of thinking of it is that the space-time has a built-in symmetry, and that symmetry makes the equations turn out simpler if the coordinate system exploits that symmetry.

 

[PeterDonis]

This coordinate condition is necessary to define a scale factor for the spacelike part

No, it isn't. The scale factor can be defined in a coordinate-independent manner.

and maximally symmetric(isotropic and homogeneous) spatial slices.

No. The presence of slices with those properties is not determined by what coordinates you choose. See below.

isn't it obvious mathematically that the isotropicity and homogeneity properties of those 3D slices are not in general (not in general since in certain specific cases like for instance in flat spacetime or in de Sitter spacetime they are indeed geometric features of the 4-manifold) geometric features for the 4D spacetime given that they come imposed by the coordinate condition.

The presence of a set of spacelike slices that are homogeneous and isotropic is a geometric feature of the 4D spacetime, independent of the coordinates. The presence of such a set of spacelike slices is, as Chalnoth says, what makes it possible to find coordinates with the properties of FRW coordinates. But that doesn't mean the coordinates determine the geometric properties.

 

[PeterDonis]

Note, btw, that we had a very similar discussion in a recent thread:

https://www.physicsforums.com/threads/implications-of-the-tensorial-form-of-the-efe.822294/

Please, let's not repeat what has already been discussed in that thread. PF rules are that once the moderators have determined that a thread topic is closed, it should not be reopened.

 

[Nugatory]

isn't it obvious mathematically that the isotropicity and homogeneity properties of those 3D slices are not in general (not in general since in certain specific cases like for instance in flat spacetime or in de Sitter spacetime they are indeed geometric features of the 4-manifold) geometric features for the 4D spacetime given that they come imposed by the coordinate condition

Not only is not obvious, it's not even true. The only thing the coordinates do is make it easier for us to identify the slices which will be isotropic and homogeneous - but these slices have that property no matter what coordinates we use to label events in the spacetime.

As PeterDonis has pointed out, this is an attempt to reopen a locked thread. Stop it.