Introduction To Special Relativity - Clock Delays and Rod Contractions: More On Lorentz Transformations

Clock Delays and Rod Contractions: More On Lorentz Transformations

Another consequence of the invariance of the spacetime interval is that clocks will appear to go slower on objects that are moving relative to the observer. This is very similar to how the 2D projection of a line rotated into the third-dimension appears to get shorter. Length is not conserved simply because we are ignoring one of the dimensions. Let us return to the example of John and Bill.

John observes the length of Bill's spacetime interval as:

whereas Bill doesn't think he has traveled in space, so writes:

The spacetime interval, s2, is invariant. It has the same value for all observers, no matter who measures it or how they are moving in a straight line. This means that Bill's spacetime interval equals John's observation of Bill's spacetime interval so:

and

hence

.

So, if John sees a clock that is at rest in Bill's frame record one second, John will find that his own clock measures between these same ticks an interval t, called coordinate time, which is greater than one second. It is said that clocks in motion slow down, relative to those on observers at rest. This is known as "relativistic time dilation of a moving clock". The time that is measured in the rest frame of the clock (in Bill's frame) is called the proper time of the clock.

In special relativity, therefore, changes in reference frame affect time also. Time is no longer absolute. There is no universally correct clock; time runs at different rates for different observers.

Similarly it can be shown that John will also observe measuring rods at rest on Bill's planet to be shorter in the direction of motion than his own measuring rods. This is a prediction known as "relativistic length contraction of a moving rod". If the length of a rod at rest on Bill's planet is X, then we call this quantity the proper length of the rod. The length x of that same rod as measured on John's planet, is called coordinate length, and given by

.

These two equations can be combined to obtain the general form of the Lorentz transformation in one spatial dimension:

or equivalently:

where the Lorentz factor is given by

The above formulas for clock delays and length contractions are special cases of the general transformation.

Alternatively, these equations for time dilation and length contraction (here obtained from the invariance of the spacetime interval), can be obtained directly from the Lorentz transformation by setting X = 0 for time dilation, meaning that the clock is at rest in Bill's frame, or by setting t = 0 for length contraction, meaning that John must measure the distances to the end points of the moving rod at the same time.

A consequence of the Lorentz transformations is the modified velocity-addition formula: