It is a manifestly covariant equation, but at the same time its three-dimensional structure is evident.

Classical electromagnetic theory predicts that precisely the same microscopic eddy currents will be produced, but they will be due to an electric force.The laws and mathematical objects in classical electromagnetism can be written in a form which is manifestly covariant.Here, this is only done so for vacuum (or for the microscopic Maxwell equations, not using macroscopic descriptions of materials such as electric permittivity), and uses SI units.…

Tensor densities, especially integrands and variables of integration, may be allowed in manifestly covariant equations if they are clearly weighted by the appropriate power of the determinant of the metric.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.…

The equation above contains partial derivatives and is therefore not manifestly covariant.

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.An equation may be Lorentz covariant even if it is not manifestly covariant.…

These tensor equations are manifestly-covariant, meaning the equations can be seen to be covariant by the index positions.This short form of writing Maxwell's equations illustrates an idea shared amongst some physicists, namely that the laws of physics take on a simpler form when written using tensors.…

In general relativity, a manifestly covariant equation is one in which all expressions are tensors.The operations of addition, tensor multiplication, tensor contraction, raising and lowering indices, and covariant differentiation may appear in the equation.…

Secondly, it sheds light on the relationship between electricity and magnetism, showing that frame of reference determines if an observation follows electrostatic or magnetic laws.Third, it motivates a compact and convenient notation for the laws of electromagnetism, namely the "manifestly covariant" tensor form.Maxwell's equations, when they were first stated in their complete form in 1865, would turn out to be compatible with special relativity.…

Forbidden terms include but are not restricted to partial derivatives.Tensor densities, especially integrands and variables of integration, may be allowed in manifestly covariant equations if they are clearly weighted by the appropriate power of the determinant of the metric.Writing an equation in manifestly covariant form is useful because it guarantees general covariance upon quick inspection.…

A single two-body Dirac equation similar to the Breit equation can be derived from the TBDE.Unlike the Breit equation, it is manifestly covariant and free from the types of singularities that prevent a strictly nonperturbative treatment of the Breit equation.In applications of the TBDE to QED, the two particles interact by way of four-vector potentials derived from the field theoretic electromagnetic interactions between the two particles.…

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.An equation may be Lorentz covariant even if it is not manifestly covariant.…

In quantum field theory, and in the significant subfields of quantum electrodynamics and quantum chromodynamics, the two-body Dirac equations (TBDE) of constraint dynamics provide a three-dimensional yet manifestly covariant reformulation of the Bethe-Salpeter equation for two spin-1/2 particles.Such a reformulation is necessary since without it, as shown by Nakanishi, the Bethe-Salpeter equation possesses negative-norm solutions arising from the presence of an essentially relativistic degree of freedom, the relative time.…