Electrovacuum solution

In general relativity, an electrovacuum solution (electrovacuum) is an exact solution of the Einstein field equation in which the only nongravitational mass-energy present is the field energy of an electromagnetic field, which must satisfy the (curved-spacetime) "source-free" Maxwell equations appropriate to the given geometry. For this reason, electrovacuums are sometimes called (source-free) Einstein-Maxwell solutions.
Mathematical definition.
In general relativity, the geometric setting for physical phenomena is a Lorentzian manifold, which is physically interpreted as a curved spacetime, and which is mathematically specified by defining a metric tensor formula_1 (or by defining a frame field). The Riemann curvature tensor formula_2
of this manifold and associated quantities such as the Einstein tensor formula_3, are mathematically well-defined. In general relativity, they can be interpreted as geometric manifestations (curvature and forces) of the gravitational field.
We also need to specify an electromagnetic field by defining an electromagnetic field tensor formula_4 on our Lorentzian manifold. These two tensors are required to satisfy two following conditions
The first Maxwell equation is satisfied automatically if we define the field tensor in terms of an electromagnetic potential vector formula_8. In terms of the dual covector (or potential one-form) and the electromagnetic two-form, we can do this by setting formula_9. Then we need only ensure that the divergences vanish (i.e. that the second Maxwell equation is satisfied for a "source-free" field) and that the electromagnetic stress-energy matches the Einstein tensor.
Invariants.
As in flat spacetime, the electromagnetic field tensor is antisymmetric, with only two algebraically independent scalar invariants,
Here, the star is the Hodge star.
Using these, we can classify the possible electromagnetic fields as follows:
Null electrovacuums are associated with electromagnetic radiation. An electromagnetic field which is not null is called non-null, and then we have a non-null electrovacuum.
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 formula_19 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 formula_20 direction and rotations about the formula_20 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 formula_20 axis; two further generators are the two "parabolic" Lorentz transformations aligned with the formula_20 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.
Eigenvalues.
The characteristic polynomial of the Einstein tensor of a "non-null" electrovacuum must have the form
Using Newton's identities, this condition can be re-expressed in terms of the traces of the powers of the Einstein tensor as
where
This necessary criterion can be useful for checking that a putative non-null electrovacuum solution is plausible, and is sometimes useful for finding non-null electrovacuum solutions.
The characteristic polynomial of a "null" electrovacuum "vanishes identically", even if the energy density is "nonzero". This possibility is a tensor analogue of the well known that a null vector always has vanishing length, even if it is not the zero vector. Thus, every null electrovacuum has one "quadruple eigenvalue", namely zero.
Rainich conditions.
In 1925, George Yuri Rainich presented purely mathematical conditions which are both necessary and sufficient for a Lorentzian manifold to admit an interpretation in general relativity as a "non-null" electrovacuum. These comprise three algebraic conditions and one differential condition. The conditions are sometimes useful for checking that a putative non-null electrovacuum really is what it claims, or even for finding such solutions.
Analogous necessary and sufficient conditions for a "null electrovacuum" have been found by Charles Torre.
Test fields.
Sometimes one can assume that the field energy of any electromagnetic field is so small that its gravitational effects can be neglected. Then, to obtain an approximate electrovacuum solution, we need only solve the Maxwell equations on a given vacuum solution. In this case, the electromagnetic field is often called a test field, in analogy with the term test particle (denoting a small object whose mass is too small to contribute appreciably to the ambient gravitational field).
Here, it is useful to know that any Killing vectors which may be present will (in the case of a vacuum solution) automatically satisfy the curved spacetime Maxwell equations.
Note that this procedure amounts to assuming that the electromagnetic field, but not the gravitational field, is "weak". Sometimes we can go even further; if the gravitational field is also considered "weak", we can independently solve the linearised Einstein field equations and the (flat spacetime) Maxwell equations on a Minkowksi vacuum background. Then the (weak) metric tensor gives the approximate geometry; the Minkowski background is unobservable by physical means, but mathematically much simpler to work with, whenever we can get away with such a sleight-of-hand.
Examples.
Noteworthy individual non-null electrovacuum solutions include:
Noteworthy individual null electrovacuum solutions include:
Some well known families of electrovacuums are:
Many pp-wave spacetimes admit an electromagnetic field tensor turning them into exact null electrovacuum solutions.