Other EnergyMomentum 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 4vector current density as
(16.2) 


we can write Maxwell's Equations in covariant tensor form. The standard 3D 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 energymomentum tensor for the electromagnetic field shown below:
(16.9)
This energymomentum tensor, when combined with the Einstein Equations, gives rise to a set of differential equations called the EinsteinMaxwell Equations.
(16.10)
where the units are such that not only is , but the gravitational constant as well.
EnergyMomentum 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 nonvector 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 energymomentum 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 energymomentum 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 energymomentum tensor to that shown above can be used except that the product terms and in the tensor are replaced by complexconjugatetype products and , respectively.