Final answer:
The RHS of the Einstein field equations cannot take the form Tμν plus some coefficient multiplied with g_μνT because it violates the principle of covariant conservation. In the specific example T_μν + g_μνT/4, the coefficient κ can be related to the gravitational constant G and the speed of light c by κ = 16πG/c⁴ in the Newtonian limit of general relativity.
Step-by-step explanation:
The reason the RHS of the Einstein field equations cannot take the form Tμν + some coefficient multiplied with g_μνT is because it violates the principle of covariant conservation. In general relativity, the RHS of the field equations represents the energy-momentum tensor Tμν, which describes the distribution of matter and energy in spacetime. The way Tμν is defined, it is already covariantly conserved, meaning its divergence is zero. If we were to add a coefficient multiplied by g_μνT to the RHS, its divergence would no longer be zero and covariant conservation would be violated.
Regarding the specific example T_μν + g_μνT/4, this is known as the perfect fluid form of the energy-momentum tensor, where T is the trace of the energy-momentum tensor. In the Newtonian limit of general relativity, where the effects of gravity are weak and velocities are much smaller than the speed of light, the coefficient κ can indeed be related to the gravitational constant G and the speed of light c by κ = 16πG/c⁴.