Four-vector

[2]: ch1 Four-vectors describe, for instance, position xμ in spacetime modeled as Minkowski space, a particle's four-momentum pμ, the amplitude of the electromagnetic four-potential Aμ(x) at a point x in spacetime, and the elements of the subspace spanned by the gamma matrices inside the Dirac algebra.

The action of a Lorentz transformation on a general contravariant four-vector X (like the examples above), regarded as a column vector with Cartesian coordinates with respect to an inertial frame in the entries, is given by

(matrix multiplication) where the components of the primed object refer to the new frame.

For an example of a well-behaved four-component object in special relativity that is not a four-vector, see bispinor.

Similar remarks apply to objects with fewer or more components that are well-behaved under Lorentz transformations.

Here the standard convention is that Latin indices take values for spatial components, so that i = 1, 2, 3, and Greek indices take values for space and time components, so α = 0, 1, 2, 3, used with the summation convention.

The split between the time component and the spatial components is a useful one to make when determining contractions of one four vector with other tensor quantities, such as for calculating Lorentz invariants in inner products (examples are given below), or raising and lowering indices.

in which the matrix Λ has components Λμν in row μ and column ν, and the matrix (Λ−1)T has components Λμν in row μ and column ν.

All four-vectors transform in the same way, and this can be generalized to four-dimensional relativistic tensors; see special relativity.

For two frames rotated by a fixed angle θ about an axis defined by the unit vector:

Contrary to the case for pure rotations, the spacelike and timelike components are mixed together under boosts.

The dot product of the basis vectors is the Minkowski metric, as opposed to the Kronecker delta as in Euclidean space.

in which case ημν above is the entry in row μ and column ν of the Minkowski metric as a square matrix.

Following are two common choices for the metric tensor in the standard basis (essentially Cartesian coordinates).

Nevertheless, this type of expression is exploited in relativistic calculations on a par with conservation laws, since the magnitudes of components can be determined without explicitly performing any Lorentz transformations.

A particular example is with energy and momentum in the energy-momentum relation derived from the four-momentum vector (see also below).

Some authors define η with the opposite sign, in which case we have the (−+++) metric signature.

If r is a function of coordinate time t in the same frame, i.e. r = r(t), this corresponds to a sequence of events as t varies.

The definition R0 = ct ensures that all the coordinates have the same dimension (of length) and units (in the SI, meters).

When considering physical phenomena, differential equations arise naturally; however, when considering space and time derivatives of functions, it is unclear which reference frame these derivatives are taken with respect to.

This relation is provided by taking the above differential invariant spacetime interval, then dividing by (cdt)2 to obtain:

Note the basis vectors are placed in front of the components, to prevent confusion between taking the derivative of the basis vector, or simply indicating the partial derivative is a component of this four-vector.

Since the magnitude of U is a constant, the four acceleration is orthogonal to the four velocity, i.e. the Minkowski inner product of the four-acceleration and the four-velocity is zero:

The geometric meaning of four-acceleration is the curvature vector of the world line in Minkowski space.

The four-force acting on a particle is defined analogously to the 3-force as the time derivative of 3-momentum in Newton's second law:

where s is the entropy per baryon, and T the absolute temperature, in the local rest frame of the fluid.

The wave four-vector has coherent derived unit of reciprocal meters in the SI.

Note this is consistent with the above case; for photons with a 3-wavevector of modulus ω / c , in the direction of wave propagation defined by the unit vector

The Feynman slash notation is a shorthand for a four-vector A contracted with the gamma matrices:

appear, in which the energy E and momentum components (px, py, pz) are replaced by their respective operators.