is the relativistic torque, also second order antisymmetric tensor. The electromagnetic field tensor is another second order antisymmetric tensor field, with six components: three for the electric field and another three for the magnetic field. There is also the stress–energy tensor for the electromagnetic field, namely the electromagnetic stress–energy tensor. Metric The metric tensor allows one to define the inner product of two vectors, which in turn allows one to assign a magnitude to the vector. Given the four-dimensional nature of spacetime the Minkowski metric η has components (valid with suitably chosen coordinates) which can be arranged in a matrix: which is equal to its reciprocal, , in those frames. Throughout we use the signs as above, different authors use different conventions – see Minkowski metric alternative signs. The Poincaré group is the most general group of transformations which preserves the Minkowski metric: and this is the physical symmetry underlying special relativity. The metric can be used for raising and lowering indices on vectors and tensors. Invariants can be constructed using the metric, the inner product of a 4-vector T with another 4-vector S is: Invariant means that it takes the same value in all inertial frames, because it is a scalar (0 rank tensor), and so no appears in its trivial transformation. The magnitude of the 4-vector T is the positive square root of the inner product with itself: One can extend this idea to tensors of higher order, for a second order tensor we can form the invariants: similarly for higher order tensors. Invariant expressions, particularly inner products of 4-vectors with themselves, provide equations that are useful for calculations, because one doesn't need to perform Lorentz transformations to determine the invariants. Relativistic kinematics and invariance The coordinate differentials transform also contravariantly: so the squared length of the differential of the position four-vector dXμ constructed using is an invariant. Notice that when the line element dX2 is negative that is the differential of proper time, while when dX2 is positive, is differential of the proper distance. The 4-velocity Uμ has an invariant form: which means all velocity four-vectors have a magnitude of c. This is an expression of the fact that there is no such thing as being at coordinate rest in relativity: at the least, you are always moving forward through time. Differentiating the above equation by τ produces: So in special relativity, the acceleration four-vector and the