# Motivation for tensor theory of gravity

In class we were shown that $$\rho = \frac{dm}{dV}$$ has the transformation properties of the 00 component of a rank 2 tensor. So we'd like to turn the classical Poisson equation for gravity into a Tensor Equation. This is fine, however I've also been told countless times to shy away from relativistic mass, so my intuition was to use the proper mass (rest mass). This would lead me to a vector theory of gravity though. Is there a better way to think about this? In terms of some kind of momentum density perhaps? If so, how can this be motivated classically?

• The RHS of your equation isn't a tensor its a single number. (It is the the $00$ component of a tensor, but I don't see how that information could be obtained without knowing the other components?) Oct 15, 2019 at 17:19
• Sorry, that's what I meant. I edited the question. You know from special relativity how Tensors transform, and how volumes and relativistic mass transform. You then observe these are the same Oct 15, 2019 at 17:33

Your intuition is correct in my opinion. The relativistic equivalent of $$\rho$$ would be $$j^{\mu} = (\rho c, j^i)$$ which leads to a vector equation. However you should not think of masses being the source of the gravitational field but more about energy in general. This leads to the stress-energy-tensor. $$\begin{equation} T^{\mu \nu} = \rho \gamma^2 c^2\left( \begin{array}{rrrr} 1 & v^1/c & v^2/c & v^3/c \\ v^1/c & & & \\ v^2/c & & v^i v^j /c^2 & \\ v^3/c & & & \\ \end{array} \right) = \rho u^\mu u^\nu \end{equation}$$ But notice that this is a special case, in which the pressure and electromagnetic energy (and other sources) were neglected. Also in Einstein's field-equations the energy and momentum of the gravitational field are considered as well (but they are are not in the stress-energy-tensor of matter).
Here $$\rho$$ is a lorentz scalar but $$T^{00}$$ can be interpreted as relativistic energy-mass-density.