Electrovacuum Solution - Einstein Tensor

Einstein Tensor

The components of a tensor computed with respect to a frame field rather than the coordinate basis are often called physical components, because these are the components which can (in principle) be measured by an observer.

In the case of an electrovacuum solution, an adapted frame

can always be found in which the Einstein tensor has a particularly simple appearance. Here, the first vector is understood to be a timelike unit vector field; this is everywhere tangent to the world lines of the corresponding family of adapted observers, whose motion is "aligned" with the electromagnetic field. The last three are spacelike unit vector fields.

For a non-null electrovacuum, an adapted frame can be found in which the Einstein tensor takes the form

where is the energy density of the electromagnetic field, as measured by any adapted observer. From this expression, it is easy to see that the isotropy group of our non-null electrovacuum is generated by boosts in the direction and rotations about the axis. In other words, the isotropy group of any non-null electrovacuum is a two dimensional abelian Lie group isomorphic to SO(1,1) x SO(2).

For a null electrovacuum, an adapted frame can be found in which the Einstein tensor takes the form

From this it is easy to see that the isotropy group of our null electrovacuum includes rotations about the axis; two further generators are the two parabolic Lorentz transformations aligned with the direction given in the article on the Lorentz group. In other words, the isotropy group of any null electrovacuum is a three dimensional Lie group isomorphic to E(2), the isometry group of the euclidean plane.

The fact that these results are exactly the same in curved spacetimes as for electrodynamics in flat Minkowski spacetime is one expression of the equivalence principle.