An event has a position and a time . Relativity writes these four as one set, . The time is multiplied by so that all four have the same units.
In a Lorentz transformation to a frame moving along , and mix, while and are left as they are. The figure reads the same event with the axes tilted by different amounts.
Any set of four numbers that transforms by the same rule as is called a four-vector. itself is the four-position.
The squared length of an ordinary vector is , unchanged when the axes are rotated. For a four-vector the following quantity is used instead.
A Lorentz transformation leaves this value unchanged. It is the same combination as the spacetime interval.
The arrow in the figure is a four-vector for which this value is , read in five different frames. The values of and change from frame to frame, but the tip always lies on the same hyperbola and stays at .
Ordinary velocity is position differentiated by the time . But differs between frames, so does not transform by the four-vector rule.
So differentiate instead by the proper time of a clock moving with the body. comes out the same in every frame, so is a four-vector, called the four-velocity.
From , its components are . The squared length is , so at any speed the length is .
At , and the components are . Since , the length is the same as for a body at rest, .
The four-velocity multiplied by the mass is called the four-momentum. Its components are ; the time component times , , is called the energy , and the space component the momentum .
The length of the four-momentum is , independent of the frame. Rewriting gives
The tip in the figure stops at rapidity , and . At (), and , and every point lies on the same curve.
In a collision, the total energy and the total momentum are both unchanged. Written with four-momenta, this becomes a single equation: the sum of the four-momenta is the same before and after.
Changing frames mixes the components, but the difference between the sums before and after is a four-vector equal to , and it reads in every frame. If the conservation law holds in one frame, it holds in all of them.
In the collision in the figure, the sum of is both before and after, and the sum of is (in units with ). The masses of the two particles are and before and and after, so the sum of the masses is not conserved.