We have derived from certain astrometric and cosmological observations how physical constants evolve in time (https://doi.org/10.3390/galaxies7020055). The theoretical basis used is rather unsatisfactory since we use the Einstein equations derived from Einstein-Hilbert action which treats speed of light c, gravitational constant G and cosmological constant Λ as constant. While many papers state the problem (e.g. https://arxiv.org/abs/gr-qc/0305099 and https://arxiv.org/abs/gr-qc/0007036), none to my knowledge provides a solution, possibly because it is too difficult. However, since we know how they should vary from our paper cited above, it should be possible to develop an action approximation that would yield the desired variations of physical constants. I wonder if someone competent in this field would be able address this problem and possibly collaborate.
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Varying G and Λ is described by introducing fields, beyond the metric, that can't be absorbed into a redefinition of the energy-momentum tensor, in a way that describes matter, dark or not. The first example was provided by scalar-tensor models, the most general example is provided by supergravity.
Varying c doesn't make sense, as a statement, because, in general relativity, c isn't a global invariant and relative velocity isn't well defined.
And any metric can be reduced to the Minkowski metric at any given point in spacetime.
The meaningful statement would be that the spacetime of our Universe isn't asymptotically flat; but it's known, since 1998, that the spacetime of our Universe isn't asymptotically flat, but de Sitter.
This problem is very difficult with mathematical viewpoint. We must obtain the solution of Einstein equations with right part, which includes the terms: 1) energy-momentum tensor with coefficient kappa = 8 pi G/c^2 ; 2) the term lambda g_alpha beta. Then it is necessary to obtain the solution of the 10 equations. Then you can postulate the structure of G and lambda (to formulate concrete variations of them) and choice the concrete kind of energy-impulse tensor. It can descrribe ideal liquid or physical vacuum: in this case density and pressure are connected as p = - rho c^2. It is necessary also to hive the mathematical description of variations. Then it is necessary to solve these equations.
Dear Stam Nicolis,
Stam: Varying G and Λ is described by introducing fields, beyond the metric, that can't be absorbed into a redefinition of the energy-momentum tensor, in a way that describes matter, dark or not. The first example was provided by scalar-tensor models, the most general example is provided by supergravity. Varying c doesn't make sense, as a statement, because, in general relativity, c isn't a global invariant and relative velocity isn't well defined.
With you contribution you didn't answer the question but only mentioned that there are problems by doing so. Mathematics does not give a comment if something is difficult or not. It gives an answer what is the solution. On the question how can we make that the equation y= ax2 + bx + c always has two values for x that yield y=0 then mathematics will come with the solution that if you invoke special mathematics with imaginary numbers, then you will find that this equation has the required properties.
So, the question that is asked here is not answered by stating what General Relativity assumes to be conserved. To answer the question how a flat force field of acceleration could be set equal to the curved field of gravity Einstein had to come with the concept of curved spacetime. Would he have stayed in flat euclidian space and flat Minkowski spacetime, then he wouldn't have had success with his equations.
Rμν-1/2Rgμν+Λgμν = 8πG/c4 Tμν
Gμν = 8πG/c4 Tμν
Is an equation where it is assumed that the values for G, gravity, and c, speed of light, are constants. Now the question is how do we need to convert these equations so that G, c, and Λ become a variable.
If we assume, contrary to Einstein, that space is flat and c is variable, then we again have Minkowski spacetime under the limiting case of constant c. However, we have to make c a tensor cμν.
Now, with your excellent knowledge of the GR mathematics you might be able to come with a good equation that is the correct pendant of the GR EFE with the assumption of constant space and variable c. There are many questions that can be asked. There are solutions that are more and that are less practical to use. If these solutions make sense to you from the view of general relativity is not important. If there is a practical solution for that in the future then it will have its tax.