Kinds of relativity

A simple way to look at the world is to assume that space and time are absolute: the locations, the distances, the durations, speeds, and so forth as measured by one person are the same for everyone. That is, if my automobile speedometer shows 50 mph (80 kph), then the police with a laser gun at the side of the road will show the same speed.

For many purposes of everyday life, that works just fine. But for those who think about it more or those who perform experiments, that breaks down. Galileo Galilei was the first express a principle of relativity in his 1632 work Dialogue Concerning the Two Chief World Systems using the example of a ship travelling at constant velocity on a smooth sea: any observer doing experiments below the deck would not be able to tell whether the ship was moving or stationary. He still accepted absolute time, however.

We can call Galilean relativity “spatial relativity” since it applies only to space. Since we have seen the symmetry between space and time, we could develop a similar “temporal relativity” in which time is relative but space is not. This may seem odd at first but it is as consistent (and limited) as spatial relativity. For reference, here are the transformations for spatial and temporal relativity, given two reference frames, S and S’, with an event having space and time displacements r and t (r’ and t’) respectively, with S’ moving at constant velocity v relative to S, then:

r’ = r – vt and t’ = t for spatial (Galilean) relativity, and

r’ = r and t’ = t – r/v for temporal relativity.

Both of these relativities are nonconvertible: knowing the spatial displacement tells us nothing about the temporal displacement and vice versa. Other relativities are convertible; these have finite conversion factors between space and time. The transformation for finite relativity was given here and here:

r´ = (1 – v/c) r and = (1 – v/c) t for finite relativity.

What is that conversion factor between space and time? In everyday life we may use a typical travel speed to tell others, for example, that “it’s two hours from Baltimore to Philadelphia,” which assumes an average speed of 50 mph (80 kph) and so is equivalent to 100 mi (160 km). However, a typical speed is relative to a particular time and place, and perhaps a particular driver or opinion of typical traffic conditions. What is the conversion between time and space for everyone, for all times and places, for the whole physical world?

Einstein was the first to answer the question by combining the principle of relativity with the speed of light as absolute. This led to his derivation of the Lorentz transformation which in addition to the finite speed of light includes the property that the speed of light is the same for all observers traveling with constant speed:

= (1 – v/c) γr and= (1 – v/c) γt,

where the Lorentz factor γ2 = 1 / (1 – (v/c)2). This could be called “isoluminal relativity” because the conversion factor between time and space is the constant speed of light.

Because the Lorentz factor is not a real number if v > c, we either have to assume that this never happens or we have the alternate situation described in Lorentz with 3D time, which we can express in a symmetric way as:

r’ = (1 – c/v) gr and t’ = (1 – c/v) gt,

where g2 = 1 / (1 – (c/v)2).

Note that γ2 + g2 = 1. Also note that γ is real only if v < c and g is real only if v > c. The latter is called superluminal motion if c is the speed of light. It is controversial whether such speeds exist (in contrast to subluminal motion). But if c is just a typical speed used to relate space and time in a transportation mode, it is not an absolute and actual speeds may easily be larger or smaller.

As v and c diverge, the Lorentz transformations lead to those of finite relativity. This implies that speeds greater than the conversion speed also lead to an alternate transformation in general:

r’ = (1 – c/v) r and t’ = (1 – c/v) t.

In conclusion, there are several kinds of relativity principles: spatial relativity (in which time is absolute), temporal relativity (in which space is absolute), finite relativity (in which a finite conversion factor relates time and space), and isoluminal relativity (in which the conversion between time and space is the absolute speed of light).