Energy Tensor & Field Equation | Einstein Theory

[shubham agn]

Hello!

The Einstein field equation relates the curvature of space-time to the energy tensor of mass-energy. This is fine. These field equations are derived by varying the Hilbert action. Now the Hilbert action is an integral of scalar curvature (R) over volume. So, when we vary this action, we must get the energy tensor of the field. How then do we naively take this to be the energy tensor of mass-energy and claim to have derived the field equation?

Thank you!

 

[ShayanJ]

Varying the EH action alone, gives you the field equations in the absence of sources. But when you vary the action $S_{tot}=S_{EH}+S_{m}$ ,where $S_{m}$ is the action for any matter or non-gravitational field, there will be an RHS term resulting from the variation of $S_m$ w.r.t. metric which we define to be the SEM tensor of those fields or particles. This definition of SEM tensor turns out to be compatible with other areas of physics.

 

[PWiz]

As Shyan has already stated, one needs to account for the Lagrangian of matter fields while minimizing the Einstein-Hilbert action to obtain a useful equation. The variation of this Lagrangian with respect to the metric is by definition proportional to the stress-energy tensor associated with the matter affecting spacetime:
$T_{μν} = \frac {-2 \delta L}{\delta g^{μν}} + g_{μν} L$ (L= lagrangian describing matter fields)
So when we're deriving the EFE, what we're actually doing is setting the action $S$ equal to $\int ( \frac{R}{2α} + L) \sqrt{-g} d^4 x$ and varying the entire integrand with respect to the metric tensor, where $α$ is a constant which is set equal to $8πG$ so that GR reduces to Newtonian gravity $∇^2 Φ = 4πGρ$ when velocities are much lower than $1$ and gravitational curvature is small enough for it to be considered as a perturbation of flat Minkowski space.