Time-Varying Cosmological Constant Implications in GR

[fhenryco]

Hi,

I want to be sure i understand correctly the implications of problem 2 in:

http://universeinproblems.com/index.php/Time-dependent_Cosmological_Constant

"when G is constant, Lambda is also a constant if and only if the ordinary energy-momentum tensor Tμν is also conserved "

As far as i know the ordinary energy momentum tensor covariant conservation is granted as soon as the corresponding fields :
1) have their own scalar action
2) are fully dynamical in this action e.g. their equations of motion are derived from them extremizing the action

so the case corresponding to a cosmological constant and matter fields exchanging energy (subject of next problem 3 ) implicitely imply that either the theory is not diffeomorphism invariant (1 not satisfied: which for me does not make any sense) or somehow the dynamics of the matter fields does not only follow from the action e.g. the fields are not completely dynamical in the usual sense: this is a bit less shocking for me ... but still very unusual

Do others confirm i understand correctly the implications of a time varying cosmological constant in GR ?

 

[Ambitwistor]

Hello,

You are correct in your understanding of the implications of a time-varying cosmological constant in general relativity (GR). In order for Lambda (the cosmological constant) to remain constant, the ordinary energy-momentum tensor Tμν must also be conserved. This means that the fields in the theory must have their own scalar action and be fully dynamical, as you mentioned.

However, as you pointed out, the case of a cosmological constant and matter fields exchanging energy (problem 3 in the link) raises some questions about the diffeomorphism invariance of the theory. This is because the matter fields would have to be non-dynamical or their dynamics would have to be determined by something other than the action, which goes against the principles of GR.

Some theories, such as scalar-tensor theories, do allow for a time-varying cosmological constant while still maintaining diffeomorphism invariance. In these theories, the scalar field is responsible for the dynamics of both the cosmological constant and matter fields. However, these theories are still not widely accepted and remain an active area of research.

In conclusion, your understanding of the implications of a time-varying cosmological constant in GR is correct. It does raise some questions about the diffeomorphism invariance of the theory, but there are some alternative theories that can accommodate a time-varying cosmological constant while maintaining this principle.