This Quantum World/Appendix/Relativity/Composition theorem and proper time

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Composition of velocities[edit]

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

and 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 .

Proper time[edit]

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

An invariant speed[edit]

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.

Starting from

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.)