What is the stress-energy tensor in this case?

Question

In case of static point mass we have the following stress energy tensor

$$T_{00}=m\delta^3(\vec{r})$$

And other components are zero.

What are the components of this tensor in case of moving point mass with angular momentum?

Answer 1

The energy-momentum tensor for a moving point particle with mass $m$ following trajectory $\mathbf{x}(\tau)$ on some background metric $g_{\mu\nu}$ is given by

$$ T^{\mu\nu} = m \int_{-\infty}^\infty u^\mu(\tau) u^\nu(\tau) \frac{\delta^4(\mathbf{x}-\mathbf{x}(\tau))}{\sqrt{-g}} d\tau, $$

where $u^\mu(\tau)$ is the 4-velocity of the particle and $g$ is the determinant of $g_{\mu\nu}$.

If the particle has an intrinsic spin dipole moment, this gets augmented by

$$ T^{\mu\nu} = \int_{-\infty}^\infty u^{(\mu}(\tau) S^{\nu)\alpha}(\tau) \nabla_\alpha\frac{\delta^4(\mathbf{x}-\mathbf{x}(\tau))}{\sqrt{-g}} d\tau, $$

where $S^{\mu\nu}(\tau)$ is the spin-tensor corresponding to the particle's spin.

Answer 2

A Stress-Energy-Tensor for a point particle with no electromagnetic nor gravitational field present is just the momentum current:

$$T^{\mu\nu}=P^{\mu}J^{\nu}$$

If you used the definition of the 4-current using delta functions, you get your answer.