Terms in the stress-energy tensor of a pair of masses coupled by a Hooke oscillator Think of the simplest system that can be constructed with masses and non-gravitational forces that keep the system bounded: Two masses bounded by a hook spring:
$$ T^{\alpha\beta}({\bf x},t) = \gamma_0 m_0 v_0^\alpha v_0^\beta + \gamma_1 m_1 v_1^\alpha v_1^\beta. $$
I'm missing a term that includes Hooke potential energy $k_{01}^2 (x_0^i - x_1^i)(x_0^i - x_1^i)$ (possibly with the positions at retarded times) where $i$ is a latin index summed on the spatial components only. I'm not sure how to express this in a manifestly covariant manner
 A: Since we expect to have some means that would transfer the energy (and momentum) from one mass to the other located in another place we must have some material entity that would contain the energy/momentum transfered (since energy-momentum conservation is a local law in GR). 
But that locality would mean that this medium must have some dynamics of its own since we could restrict our attention to the a small space not containing the point masses and concern ourselves with the local flow of energy-momentum in this small volume.
Of course, we could simply couple masses to some field and consider the field dynamics in the presence of two point sources.
But alternatively we could connect the two masses by a line and postulate that energy-momentum would flow only along this line (as it evolves in time, sweeping a 2d surface in a spacetime). So in fact, we would have two point masses connected by a string.
The action for such system would be a sum of the actions of point particles plus the action for the string itself:
$$
S=-m_1 \int_{\xi_1} ds -m_2 \int_{\xi_2}ds-S_\text{sting}[x^\mu(\sigma_0,\sigma_1),\ldots].
$$
Here, the sting action is calculated along the string worldsheet parametrized by functions $x^\mu(\sigma_0,\sigma_1)$ depending on two worldsheet coordinates $\sigma_a$, ($a=0,1$) and may also include additional degrees of freedom 'living' on the string (i.e. depending on $\sigma_a$). Minimizing this action would give us the equation of motion for the particles as well as equations for the evolution of the string.
If we want to reproduce Hooke's law in nonrelativistic limit (i.e. energy of a string is proportional to a square of separation) then we do need such additional degrees of freedom. But the simplest relativistic action for a string would be a Nambu–Goto action which does not have any additional degrees of freedom and is proportional to the area (2d volume) of worldsheet:
$$
S_\text{string}=-\lambda \int\sqrt{-\det \gamma_{ab}}\mathrm{d}^2\sigma,
$$
where $\lambda$ is the string tension and $\gamma_{ab}=g_{\mu\nu}\partial_ax^\mu\partial_bx^\nu$ is the metric induced on string worldsheet.
For classical dynamics of such particles/string system I would recommend looking into a textbook on string theory but here we can write  at least $T^{\mu\nu}$. The stress-energy tensor is obtained by variation of the action with respect to the metric $g_{\mu\nu}$. 
For point particle(s) we have
$$
T_{(i)}^{\mu\nu}= m_i \int_{\xi_i} u^\mu u^\nu \frac{\delta^4(x-x(s))}{\sqrt{-g}}ds.
$$
For (Nambu-Goto) string:
$$
T^{\mu\nu} = \lambda \int \mathrm{d}^2\sigma \sqrt{-\gamma} \gamma^{ab}\partial_a x^\mu\partial_bx^\nu \frac{\delta^4(x-x(\sigma))}{\sqrt{-g}}.
$$
