Stressenergy tensor
Stress Energy Momentum Tensor[edit  edit source]
The stressenergy tensor for short of elements contains information about the stress, pressure, energy density, and momentum density of the matter in the spacetime. For rectilinear inertial frame coordinates

is a statment of energy conservation. The closest thing to such a statment for general relativity where globally rectilinear inertial frames don't exist when there is Riemannian spacetime curvature present is

which is a statment of energy conservation for local free fall frames for which the Christoffel symbols vanish reducing it to the expression just above.
For such a rectilinear inertial frame the elements of the stress energy tensor have the following interpretations
is the coordinate frame energy density.
is a flow of momentum per area in the direction or the pressure on a plane whose normal is in the direction.
is the component of momentum per area in the direction or describes a shearing from stresses.
is the volume density of the ith component of momentum flow.
Of Dust[edit  edit source]
Given that an observer local to and moving with an element of dust moving uniformly with the bits around it finds that its local energy density is then the stressenergy tensor according to a frame for which it moves at 4velocity is

Of an Ideal Fluid[edit  edit source]
Given that an observer local to and moving with an element of fluid finds that its local energy density is and that its local pressure is p, then the stressenergy tensor according to a frame for which it moves at 4velocity and the contravariant metric tensor is is

Of the Massless Electromagnetic Field[edit  edit source]
In Newtonian gravitation it is specifically the mass of a thing that gravitates. The discovery of the Higgs particle settles the question on whether the photon has any mass. Since it does not interact with the Higgs field, it does not. General relativity describes gravitation differently than Newtonian physics though. In General relativity/Einstein equations it is the macroscopic feild's stressenergy tensor that couples to the Einstein curvature tensor in the field equations, not the mass of the field's mediating particles. If a field of particles gives rise to a field and the stressenergy tensor of that field is then the fields particles are massless iff . The stressenergy tensor can still have gravitational effect, even though the field particles are massless because can be nonzero even when , and it is that is the source term in the field equations, not .
Given an electromagnetic field tensor of the stressenergy tensor for the electromagnetic field is

where in "local Cartesian coordinates" the electromagnetic field tensor is given by

And a test charge will experience a 4force from it of

In terms of E&B in Cartesian inertial frame coordinates for flat spacetime the stressenergy tensor of an electromagnetic field can be written out as


