This article considers four-vectors in the context of special relativity. Although the concept of four-vectors also extends to general relativity, some of the results stated in this article require modification in general relativity.
where = 0, 1, 2, 3, labels the spacetime dimensions and where c is the speed of light. The definition ensures that all the coordinates have the same units (of distance). These coordinates are the components of the position four-vector for the event. The displacement four-vector is defined to be an "arrow" linking two events:
(Note that the position vector is the displacement vector when one of the two events is the origin of the coordinate system. Position vectors are relatively trivial; the general theory of four-vectors is concerned with displacement vectors.)
where η is the Minkowski metric. Sometimes this inner product is called the Minkowski inner product. It is not a true inner product in the mathematical sense because it is not positive definite. Note: some authors define η with the opposite sign:
The inner product is often expressed as the effect of the dual vector of one vector on the other:
Here the s are the components of the dual vector of in the dual basis and called the covariant coordinates of , while the original components are called the contravariant coordinates. Lower and upper indices indicate always covariant and contravariant coordinates, respectively.
The relation between the covariant and contravariant coordinates is:
Four-vectors may be classified as either spacelike, timelike or null. Spacelike, timelike, and null vectors are ones whose inner product with themselves is greater than, less than, and equal to zero respectively.
In special relativity (but not general relativity), the derivative of a four-vector with respect to a scalar (invariant) is itself a four-vector.
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. It is agreed that time derivatives are taken with respect to the proper time (τ). As proper time is an invariant, this guarantees that the proper-time-derivative of any four-vector is itself a four-vector. It is then important to find a relation between this proper-time-derivative and another time derivative (using the time of an inertial reference frame). This relation is provided by the time transformation in the Lorentz transformations and is:
for i = 1, 2, 3. Notice that
The four-acceleration is given by:
Since the magnitude of is a constant, the four acceleration is (pseudo-)orthogonal to the four velocity, i.e. the Minkowski inner product of the four-acceleration and the four-velocity is zero:
which is true for all world lines.
The four-momentum for a massive particle is given by:
The four-force is defined by:
For a particle of constant mass, this is equivalent to
The power and elegance of the four-vector formalism may be demonstrated by seeing that known relations between energy and matter are embedded into it.
with f as above. Noticing that and expanding this out we get
for some constant S. When the particle is at rest (u = 0), we take its kinetic energy to be zero (K = 0). This gives
Thus, we interpret the total energy E of the particle as composed of its kinetic energy K and its rest energy m c2. Thus, we have
Using the relation , we can write the four-momentum as
Taking the inner product of the four-momentum with itself in two different ways, we obtain the relation
This last relation is useful in many areas of physics.
Examples of four-vectors in electromagnetism include the four-current defined by
formed from the current density j and charge density ρ, and the electromagnetic four-potential defined by
formed from the vector potential a and the scalar potential .
A plane electromagnetic wave can be described by the four-frequency defined as
where is the frequency of the wave and n is a unit vector in the travel direction of the wave. Notice that
so that the four-frequency is always a null vector.