Question

So in flrw metric it's quite reasonable to take eigenvalue of the time-like component of the stress energy tensor and identify it with mass density.

Now, if someone argues the cosmological constant comes from the stress energy tensor. Wouldn't they also have to account for the change in mass density? In the Newtonian limit I'm pretty sure this is a change in mass density.

How do people who argue it doesn't matter where (stress energy or Einstein tensor) the cosmological constant comes from explain this discrepancy?

Answer 1

In both formulations of Einstein's field equations, with the cosmological constant included in the stress-energy tensor or as part of the geometry, it acts as vacuum energy with negative pressure affecting the universe's expansion. In Newtonian mechanics, mass density changes when considering energy, but cosmologists incorporate dark energy to assess the total energy density of the universe, beyond the scope of Newtonian physics.

Step-by-step explanation:

The cosmological constant, Ω, introduced by Einstein into his field equations of General Relativity, is often associated with the stress-energy tensor Tµ³, as both describe energy density components that influence the curvature of spacetime. When considering the time-like component of the stress-energy tensor, often associated with mass density in a Friedmann-Lemaître-Robertson-Walker (FLRW) metric, adding Ω can be viewed as introducing an effective energy density into the universe. Some argue that the placement of the cosmological constant—whether on the left-hand side of the field equations as a modification of the geometry described by the Einstein tensor or on the right-hand side as part of the stress-energy tensor—is not a fundamental concern, as both approaches result in the same equations of motion. Nevertheless, if Ω is incorporated into the stress-energy tensor, it corresponds to a uniform energy distribution throughout space, acting like a negative-pressure vacuum energy that drives acceleration of the universe's expansion.

In the Newtonian limit, considering the cosmological constant as a form of energy could alter calculations of mass density due to equivalence of mass and energy in Einstein's equation E = mc². However, cosmologists account for this by including dark energy, associated with the cosmological constant, when calculating the total energy density of the universe, which affects its overall geometry and dynamics. This calculation expands upon the Newtonian view by considering both the kinetic and potential energy components of mass-energy in the universe, leading to a more complete picture of cosmic evolution.