Energy-momentum tensor for a relativistic system of particles

• Frostman
In summary, when dealing with a system of non-interacting particles, the Lagrangian is the sum of the individual particle's Lagrangians and the energy-momentum tensor is the sum of the individual particle's energy-momentum tensors.

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?

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 $i$.

What is the energy-momentum tensor for a relativistic system of particles?

The energy-momentum tensor is a mathematical object used in Einstein's theory of general relativity to describe the distribution of energy and momentum in a system of particles. It is a 4x4 matrix that contains 10 independent components, representing the energy density, momentum density, and stress in different directions.

How is the energy-momentum tensor calculated?

The energy-momentum tensor is calculated by summing the individual contributions from each particle in the system. The energy-momentum tensor is a function of both the position and velocity of each particle, as well as the mass and energy of the particles.

What is the significance of the energy-momentum tensor in relativity?

The energy-momentum tensor plays a crucial role in Einstein's theory of general relativity. It is used to describe the curvature of spacetime, which is determined by the distribution of energy and momentum in a system. This tensor is also used to derive the equations of motion for particles and to understand the behavior of matter and energy in the presence of gravitational fields.

Can the energy-momentum tensor be used to describe non-relativistic systems?

While the energy-momentum tensor was originally developed for relativistic systems, it can also be used to describe non-relativistic systems. In this case, the tensor reduces to a simpler form with fewer components, as the effects of special relativity are not present. However, the energy-momentum tensor is still a useful tool for understanding the distribution of energy and momentum in any system of particles.

How does the energy-momentum tensor relate to conservation laws?

The energy-momentum tensor is closely related to the conservation laws of energy and momentum. In relativity, these laws are expressed as the conservation of the total energy-momentum, which is equivalent to the vanishing of the divergence of the energy-momentum tensor. This means that the total energy and momentum of a system are conserved as long as the energy-momentum tensor is symmetric and satisfies certain conditions.