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

Composition of velocitiesEdit

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 timeEdit

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 speedEdit

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.)