four-vector

The four-velocity is the tangent four-vector of a world line.

The invariant of the velocity four-vector is c 2.

These four space-time coordinates form a four-vector with components.

One motivation for doing so is that the four-potential is a mathematical four-vector.

The displacement four-vector is defined to be an "arrow" linking two events:

Einstein rejected the Newtonian concept and identified t as the fourth coordinate of a space-time four-vector.

The invariant of any other four-vector is formed in the same way by taking the difference of the squares of its time and space components.

The energy-momentum four-vector is invariant.

They used Excel to reconstruct the invariant mass of a particle when given the four-vector of that particle's decay products.

So in relativity, the acceleration four-vector and the velocity four-vector are orthogonal.

The four-current is the contravariant four-vector which combines electric current density J and electric charge density ρ.

As proper time is an invariant, this guarantees that the proper-time-derivative of any four-vector is itself a four-vector.

In general relativity the elements of the acceleration four-vector are related to the elements of the four-velocity through a covariant derivative with respect to proper time.

Thus, because the components of all four-vectors transform in precisely the same way, it follows that Therefore the invariant length of any local four-vector is unaffected by general transformations.

Recognising this, we can turn the awkward looking law about composition of velocities into a simple statement about transforming the velocity four-vector of one particle from one frame to another.

In quantum field theory, a slash through a symbol, such as , is shorthand for γa, where a is a covariant four-vector, the γ are the gamma matrices, and the repeated index μ is summed over according to the Einstein notation.

Correspondingly, the simplest guess for a suitable equation of motion for test particles might seem to be where the dot signifies differentiation with respect to proper time, subscripts following the comma denote partial differentiation with respect to the indexed coordinate, and where is the velocity four-vector of the test particle.