Other Energy-Momentum Tensors

 

 

Electromagnetic Energy Momentum Tensor

 

We can define an antisymmetric tensor called the Maxwell Electromagnetic Field tensor, which contains all of the information about the electric and magnetic field at a given point in spacetime. This antisymmetric tensor is defined in terms of the electric field  and the magnetic field  as follows:

 

(16.1)

 

We will be using  units in this section on the electromagnetic field. Defining the 4-vector current density as

 

(16.2)

 

 

we can write Maxwell's Equations in covariant tensor form. The standard 3-D form of the 4 Maxwell's Equations is as follows

 

(16.3)

 

(16.4)

 

(16.5)

 

(16.6)

 

These 4 equations are written succinctly as the following 2 tensor equations.

 

(16.7)

and

 

(16.8)

 

 

 

By a variational principle it is possible to get the energy-momentum tensor for the electromagnetic field shown below:

 

(16.9)  

 

 

This energy-momentum tensor, when combined with the Einstein Equations, gives rise to a set of differential equations called the Einstein-Maxwell Equations.

 

 

(16.10)  

where the units are such that not only is , but the gravitational constant  as well.

 

 

Energy-Momentum Tensor of a Scalar Field

 

In quantum field theory the use of a scalar fields is quite common to represent energy fields that are spread out everywhere in spacetime without any directional information built in (i.e. it's non-vector energy distribution). Scalar fields have been added to General Relativity so as to give a different type of gravitational field that is all pervasive through the universe forming source independent gravitational energy background. The Einstein Equations have been solved for energy-momentum tensors of scalar fields. The energy momentum tensor for a scalar field  can be found from a variational principle to have the following form:

 

(16.11) 

where  is the rest mass of the scalar field particle and  is the Planck Constant from quantum mechanics. Note that this tensor has a strong resemblance to the energy-momentum tensor for a perfect fluid. The pressure and the energy density act like scalar fields so this is not altogether surprising. When the scalar field particle not only has a mass but also has a charge, the scalar field is represented as a complex function. A similar energy-momentum tensor to that shown above can be used except that the product terms  and  in the tensor are replaced by complex-conjugate-type products  and , respectively.