In Newtonian mechanics, several different vector quantities appear: position, velocity, acceleration, momentum, force. One extends the idea of the position vector to the four-dimensional world by appending ct to it as a fourth component. This extension has several technical advantages in the theory of relativity, the most important of which, for our present purpose, is that the length of the four-vector is an invariant quantity, whereas the ordinary three-dimensional length is not invariant, but depends on the state of motion of the observer. This process of generalizing vectors from three to four dimensions can be carried out not only for the position vector but for all of the vectors of mechanics, and some unexpected and interesting partnerships turn up. Next in interest to the joining together of space and time is the union of momentum and energy.
Momentum, because it is a three-dimensional vector, may be expressed in component form:
Since space has acquired a fourth dimension—time—momentum needs a fourth component, or time component. Remarkably, energy (or, more exactly, energy divided by the speed of light) proves to be the time component of momentum:
This merger of two of the key concepts of mechanics into a single four-dimensional entity was achieved by Hermann Minkowski in 1908, and later dubbed momenergy by John Wheeler. Soon other mergers were recognized, such as electric charge with electric current, and electric field with magnetic field, so that before long all of classical physics became imbedded in the new spacetime.
The four-dimensional momentum-energy (or momenergy) vector has a length, whose square is defined in much the same way as the square of the space-time interval. It is
px2 + py2 + pz2 – pt2 .
This combination, as a little algebra reveals, is exactly equal to the simple constant – m2c2, the same for all observers. Although both the energy and momentum of a particle depend on the state of motion of the observer, here is a certain invariant combination about which all observers agree.
Although spacetime vectors connecting pairs of events may be either “spacelike” or “timelike,” the momentum-energy vector of a material particle is always timelike, as indicated by the minus sign on the right of the equation
px2 + py2 + pz2 – pt2 = – m2c2 .
For a massless particle, always standing apart as a special case, the momentum-energy vector has zero length, although momentum and energy are of course not separately zero . Such a vector is called a null vector. A briefer and often useful way to write this equation, valid for particles with or without mass, is
E2 = p2c2 + m2c4 .
This expresses the total energy of a particle in terms of its momentum and its mass.
In general, what does the fourth component of any vector mean? It scarcely seems an adequate answer to say it is a component pointing in the time direction. We can appreciate the meaning of north, south, east, west, up, and down. But which direction is the time direction? Unfortunately humans don’t seem able to visualize four dimensions, yet that is just what the theory of relativity is asking us to do. The best one can do is try and extrapolate one’s pattern of thought from two dimensions to three and on to four, to form analogies, and to make use of what has already been learned about space and time. By way of analogy, we might think of a two-dimensional worm living out its life in a plane. The worm would know that space has two dimensions. If a more learned worm tried to explain to him the meaning of the third dimension, he might become irritated and exclaim, “But what can you mean by a third dimension It is nothing you can point to; it is not anything we can experience.” We three-dimensional creatures must likewise be content to accept the fourth dimension as something at best vaguely visualizable, yet vitally important in deepening our view of the world.
Relativity has revealed two important new aspects of energy. First, there exists a new kind of energy, the energy of mass. Second, energy is linked to momentum in the same way that time is linked to space. Just as space and time become mixed in relating the measurements of two different observers, so do momentum and energy become inextricably mixed, and the laws of conservation of momentum and of energy combine into a single more general conservation law. All this because the ether was rejected. How beautiful and how unexpected are the consequences of Einstein s two simple postulates.
1 The squared interval is sometimes written (ct)2 – x2 + y2 + z2, in which the time between a pair of events is emphasized rather than the distance between them.
2 How can the square of a real number be negative? It can’t. It is actually only the squared quantity, positive or negative, that enters into the equations of special relativity. It is called an interval-squared because it has the dimension of length-squared.