the electromagnetic four-current . The electromagnetic field at any point in spacetime is described by the antisymmetric (0,2)-rank electromagnetic field tensor The Lagrangian To obtain the dynamics for this field, we try and construct a scalar from the field. In the vacuum, we have We can use gauge field theory to get the interaction term, and this gives us The equations To obtain the field equations, the electromagnetic tensor in the Lagrangian density needs to be replaced by its definition in terms of the 4-potential A, and it's this potential which enters the Euler-Lagrange equations. The EM field F is not varied in the EL equations. Therefore, Evaluating the derivative of the Lagrangian density with respect to the field components and the derivatives of the field components obtains Maxwell's equations in vacuum. The source equations (Gauss' law for electricity and the Maxwell-Ampère law) are while the other two (Gauss' law for magnetism and Faraday's law) are obtained from the fact that F is the 4-curl of A, or, in other words, from the fact that the Bianchi identity holds for the electromagnetic field tensor. where the comma indicates a partial derivative. Gravitation After Newtonian gravitation was found to be inconsistent with special relativity, Albert Einstein formulated a new theory of gravitation called general relativity. This treats gravitation as a geometric phenomenon ('curved spacetime') caused by masses and represents the gravitational field mathematically by a tensor field called the metric tensor. The Einstein field equations describe how this curvature is produced. Newtonian gravitation is now superseded by Einstein's theory of general relativity, in which gravitation is thought of as being due to a curved spacetime, caused by masses. The Einstein field equations, describe how this curvature is produced by matter and radiation, where Gab is the Einstein tensor, written in terms of the Ricci tensor Rab and Ricci scalar , is the stress–energy tensor and is a constant. In the absence of matter and radiation (including sources) the 'vacuum field equations, can be derived by varying the Einstein–Hilbert action, with respect to the metric, where g is the determinant of the metric tensor gab''. Solutions of the vacuum field equations are called vacuum solutions. An alternative interpretation, due to Arthur Eddington, is that is fundamental, is merely one aspect of , and is forced by the choice of units. Further examples Further examples of Lorentz-covariant classical field theories are Klein-Gordon theory