Electrovacuum Solution - Mathematical Definition

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 (or by defining a frame field). The curvature tensor of this manifold and associated quantities such as the Einstein tensor, are well-defined even in the absence of any physical theory, but in general relativity they acquire a physical interpretation as geometric manifestations of the gravitational field.

We also need to specify an electromagnetic field by defining an electromagnetic field tensor on our Lorentzian manifold. These two tensors are required to satisfy two following conditions

1. The electromagnetic field tensor must satisfy the source-free curved spacetime Maxwell field equations and
2. The Einstein tensor must match the electromagnetic stress-energy tensor, .

The first Maxwell equation is satisfied automatically if we define the field tensor in terms of an electromagnetic potential vector . In terms of the dual covector (or potential one-form) and the electromagnetic two-form, we can do this by setting . 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.