Manifest covariance

Writing an equation in manifestly covariant form is useful because it guarantees general covariance upon quick inspection. If an equation is manifestly covariant, and if it reduces to a correct, corresponding equation in special relativity when evaluated instantaneously in a local inertial frame, then it is usually the correct generalization of the special relativistic equation in general relativity.

Example

An equation may be Lorentz covariant even if it is not manifestly covariant. Consider the electromagnetic field tensor

${\displaystyle F_{ab}={\mathrm{\partial }}_a{A}_b-{\mathrm{\partial }}_b{A}_a}$

where ${\displaystyle A_a}$ is the electromagnetic four-potential in the Lorenz gauge. The equation above contains partial derivatives and is therefore not manifestly covariant. Note that the partial derivatives may be written in terms of covariant derivatives and Christoffel symbols as

${\displaystyle {\mathrm{\partial }}_a{A}_b={\mathrm{\nabla }}_a{A}_b+{\mathrm{\Gamma }}_{ab}^c{A}_c}$

${\displaystyle {\mathrm{\partial }}_b{A}_a={\mathrm{\nabla }}_b{A}_a+{\mathrm{\Gamma }}_{ba}^c{A}_c}$

For a torsion-free metric assumed in general relativity, we may appeal to the symmetry of the Christoffel symbols

${\displaystyle {\mathrm{\Gamma }}_{ab}^c-{\mathrm{\Gamma }}_{ba}^c=0,}$

which allows the field tensor to be written in manifestly covariant form

${\displaystyle F_{ab}={\mathrm{\nabla }}_a{A}_b-{\mathrm{\nabla }}_b{A}_a.}$