Constraints on metric coefficients in General Relativity (GR)

[oldman]

My understanding is that in GR the way in which the components of the metric tensor can vary from location to location in spacetime is determined or constrained by a number of factors: (1) the choice of coordinates (a mathematical constraint); (2) the distribution of mass/energy (a physical constraint that is the heart of GR); and (3) the requirement that special relativity rule locally (a logical constraint also at the heart of GR). In specific situations there are, fourthly, additional constraints required to match observations (observational constraints). And there must be other constraints I am unaware of; e.g. perhaps the components of the metric tensor must be single-valued functions of whatever coordinates are chosen?

I find it difficult to appreciate just which features of a metric used as a tool for a particular purpose --- of the finished product, as it were --- are associated with what constraint.

I need help in the case of the Robertson-Walker (RW) metric of cosmology used for the accepted LCDM model. My understanding is that this metric is chosen subject to the additional constraints of: (5) the Copernican Principle, to match to a postulated homogeneous, isotropic universe; (6) a need to be able to define a universal time coordinate (a mathematical constraint?); (7) a need to account for the redshift (an observational constraint); (8) the simplifications introduced by the now-accepted Euclidean character of space sections (a recent and welcome constraint?); and finally, (9) the assumed and observed ubiquitous rule of the known laws of physics throughout spacetime, together with their incorporated constants, in particular c.

I would like to better understand just which features of the RW metric are uniquely and fully determined by the constraints I have listed above ... and by others I may have missed.

In particular, is the choice of a scale factor to describe change uniquely mandated by these constraints? I suspect not, but would like to know for sure.

 

[pervect]

I'm rather fond of http://www.eftaylor.com/pub/chapter2.pdf

see for instance figure 1

Reproducing the shape of an overturned rowboat (top) by driving nails around
its perimeter, then stretching strings between each nail and every nearby nail (middle).
The shape of the rowboat can be reconstructed (bottom) using only the lengths of string
segments—the distances between nails. To increase the precision of reproduction,
increase the number of nails, the number of string segments, the table of distances.

The metric is just the "table of distances" expressed as a formula. It depends on one's choice of the placement of "nails", which are the coordinates assigned to events.

It's convenient to assign constant coordinates to observers that see the universe as isotropic since such observers exist, that's the choice underlying the RW metrics.

 

[oldman]

First, thanks for the reference. I've done a lot of sailing, so I'm not as fond of upturned boats as you may be, but I do like this analogy. I only wish I could express myself as clearly as Taylor and Wheeler do. Faint hope!

Second, I was perhaps trying to clumsily do for the RW metric what they do elegantly for the Schwarzschild metric:

(p.2-21) ...we need not accept (this metric) uncritically. Here we check off the ways in which it makes sense...

What I wanted to ask is whether the choice of the RW metric with its conventional scale factor makes the only inevitable sense in the early stages of developing a cosmological model, given the factors that constrain its formulation, or not.

When you say:

It's convenient to assign constant coordinates to observers that see the universe as isotropic (co-moving observers?) since such observers exist, that's the choice underlying the RW metrics.

Are you implying that this alone fully determines the form of the metric? Surely not.