Einstein Field Equations - Einstein–Maxwell Equations

Einstein–Maxwell Equations

If the energy-momentum tensor is that of an electromagnetic field in free space, i.e. if the electromagnetic stress–energy tensor

is used, then the Einstein field equations are called the Einstein–Maxwell equations (with cosmological constant Λ, taken to be zero in conventional relativity theory):

Additionally, the covariant Maxwell Equations are also applicable in free space:

where the semicolon represents a covariant derivative, and the brackets denote anti-symmetrization. The first equation asserts that the 4-divergence of the two-form F is zero, and the second that its exterior derivative is zero. From the latter, it follows by the Poincaré lemma that in a coordinate chart it is possible to introduce an electromagnetic field potential A_α such that

in which the comma denotes a partial derivative. This is often taken as equivalent to the covariant Maxwell equation from which it is derived. However, there are global solutions of the equation which may lack a globally defined potential.