Energy-momentum tensor for a relativistic system of particles

[Frostman]

Homework Statement

Construct the energy-momentum tensor for a relativistic system of non-interacting particles and show explicitly that it is conserved.

Relevant Equations

$T^{\alpha\beta}=\frac{\partial L}{\partial \varphi/_\alpha}\varphi/^\beta-g^{\alpha\beta}L$
$T^{\alpha\beta}/_\alpha=0$

I think it is quite simple as an exercise, following the two relevant equations, but at the beginning I find myself stuck in going to identify the lagrangian for a relativistic system of non-interacting particles.
For a free relativistic particle I know that lagrangian is:
$$L=-\frac{m_0}{\gamma}$$
But for a system of non-interacting particles I can use this one?
$$L=\sum_i-\frac{m_{0i}}{\gamma}$$
But when I step to energy-momentum tensor I don't have any covariant formalism in this lagrangian. Somebody can help me?

 

[Nate810]

A:No, the Lagrangian for a system of non-interacting particles is not just the sum of the Lagrangians for each individual particle. The Lagrangian for a system of non-interacting particles is simply the sum of the individual particle's Lagrangians.$$L = \sum_i L_i$$where$$L_i = -\frac{m_{0i}}{\gamma_i}$$The total energy-momentum tensor for such a system is then given by$$T^{\mu \nu} = \sum_i T_i^{\mu \nu},$$where $T_i^{\mu \nu}$ is the energy-momentum tensor of particle $i$.