In fact, there are only three physically distinct possibilities. (If the magnitude of depends on the choice of units, and this tells us something about us rather than anything about the physical world.)
The possibility yields the Galilean transformations of Newtonian ("non-relativistic") mechanics:
(The common practice of calling theories with this transformation law "non-relativistic" is inappropriate, inasmuch as they too satisfy the principle of relativity.) In the remainder of this section we assume that
Suppose that object moves with speed relative to object and that this moves with speed relative to object If and move in the same direction, what is the speed of relative to ? In the previous section we found that
This allows us to write
Expressing in terms of and the respective velocities, we obtain
which implies that
We massage this into
divide by and end up with:
Thus, unless we don't get the speed of relative to by simply adding the speed of relative to to the speed of relative to .
Consider an infinitesimal segment of a spacetime path In it has the components in it has the components Using the Lorentz transformation in its general form,
it is readily shown that
We conclude that the expression
is invariant under this transformation. It is also invariant under rotations of the spatial axes (why?) and translations of the spacetime coordinate origin. This makes a 4-scalar.
What is the physical significance of ?
A clock that travels along is at rest in any frame in which lacks spatial components. In such a frame, Hence is the time it takes to travel along as measured by a clock that travels along is the proper time (or proper duration) of The proper time (or proper duration) of a finite spacetime path accordingly, is
If then there is a universal constant with the dimension of a velocity, and we can cast into the form
If we plug in then instead of the Galilean we have More intriguingly, if object moves with speed relative to and if moves with speed relative to then moves with the same speed relative to : The speed of light thus is an invariant speed: whatever travels with it in one inertial frame, travels with the same speed in every inertial frame.
we arrive at the same conclusion: if travels with relative to then it travels the distance in the time Therefore But then and this implies It follows that travels with the same speed relative to
An invariant speed also exists if but in this case it is infinite: whatever travels with infinite speed in one inertial frame — it takes no time to get from one place to another — does so in every inertial frame.
The existence of an invariant speed prevents objects from making U-turns in spacetime. If it obviously takes an infinite amount of energy to reach Since an infinite amount of energy isn't at our disposal, we cannot start vertically in a spacetime diagram and then make a U-turn (that is, we cannot reach, let alone "exceed", a horizontal slope. ("Exceeding" a horizontal slope here means changing from a positive to a negative slope, or from going forward to going backward in time.)
If it takes an infinite amount of energy to reach even the finite speed of light. Imagine you spent a finite amount of fuel accelerating from 0 to In the frame in which you are now at rest, your speed is not a whit closer to the speed of light. And this remains true no matter how many times you repeat the procedure. Thus no finite amount of energy can make you reach, let alone "exceed", a slope equal to ("Exceeding" a slope equal to means attaining a smaller slope. As we will see, if we were to travel faster than light in any one frame, then there would be frames in which we travel backward in time.)