Can a 4-Potential Approximate Gravitational Field in Weak-Field Limit?

[raopeng]

I just read some basic concepts on General Relativity, and this idea pops up: I know we should use variations of metrics for gravitational field in the Lagrangian. But considering the resemblance of gravitational field(weak-field) to electromagnetic field, can we construct a 4-potential similar to that of the electromagnetic field, say $A_{G} = (ψ(gravitational potential),0, 0, 0)$. So the Action for the effect of gravitational field would be: $\int -\frac{m}{c} A_{G}dx^{i}$. Would that be a good approximation for weak field?

 

[haushofer]

No. What you wrote down is identically zero. You should consider the weak field limit directly for the point particle action. So, take this action, write the metric as a
perturbation of e.g. Minkowski spacetime, and perform a Taylor expansion on the square root. Taking static gravity and slowly moving particles should then on its turn give Newtonian gravity, as in e.g. Eqn 2.10 of arxiv:1206.5176.

In the weak field limit GR becomes Fierz-Pauli theory, massless spin 2. A vector potential would mean that gravity is represented by spin-1. In the Newtonian approx. gravity is effectively reduced to a Gallilean scalar field, which could be interpreted as spin 0 (but then with spin defined wrt to the Galilei group).

 

[raopeng]

Eh I was trying to make a parallel comparison to $S_{mf}$, so the integral is actually $\int -\frac{m}{c}ψ d(ct)$ and when v is relatively small the whole expression does degrade into the classical Lagrangian. But it is a very imprudent thought as I only start to scratch the surface of Relativity Theory. Thank you for your reply that I know where my problem is.