Constant term in action term in general relativity?

Question

Question

So I recently pondered the following. Let's say I have an $2$ actions $S_1$ and $S_2$ which differ by a constant:

$$ S_1(\dot x_i, x_i) = S_2(\dot x_i, x_i) + \tilde c$$

Now their equations of motion will be identical in classical mechanics (without General Relativity) upon varying the coordinates $x_i \to x_i + \delta x_i$. Intuitively, I know this constant term will make a difference in general relativity. Is this hunch correct? What does the constant term $\tilde c$ look like in the form of Einstein Field Equations?

$$ G^{\mu \nu}+ \Lambda g^{\mu \nu}= \frac{8 \pi G}{c^4} T^{\mu \nu} $$

Or is there a better way to get the equations of motion in general relativity? Directly from the classical (without General Relativity) action?

Answer 1

1. An additive constant $\tilde{c}$ in the action functional cannot affect the Euler-Lagrange (EL) equations, i.e. in OP's case the EFE.

2. Such constant $\tilde{c}$ renders the action functional non-local unless we can write it as an integral over spacetime.

3. The cosmological constant term $\int\!d^4x~\sqrt{-g}\Lambda$ in the EH action is in contrast a local term that depends on the metric $g$, and affects the EFE.