Spacetime and Geometry · Problem 9

Manifolds — Sean Carroll

Physics — General Relativity · energy-momentum-tensor-perfect-fluid

Problem

For a system of discrete point particles the energy-momentum tensor takes the form Tμν=∑apμ(a)pν(a)p0(a)δ(3)(x−x(a)),T_{\mu\nu} = \sum_a \frac{p^{(a)}_\mu p^{(a)}_\nu}{p^{0(a)}} \delta^{(3)}(\mathbf{x} - \mathbf{x}^{(a)}), where the index aa labels the different particles. Show that, for a dense collection of particles with isotropically distributed velocities, we can smooth over the individual particle worldlines to obtain the perfect-fluid energy-momentum tensor (1.114).

Related theory

Browse the problem library