Do the Math, Time

The Theory of Relativity should really be called the Theory of Invariants since scientists look for things that remain constant in different moving reference frames, then use these constants to do calculations.

One invariant is the speed of light, it is the same in all reference frames.

Another invariant connects space and time.

A fundamental concept in relativity is the idea of an event, and event is something like the flash of a camera flashbulb, an event happens at a place, x, and a time, t.

In one frame event 1, will occur at x1, t1 in another, moving frame of reference, the event will occur at x'1 and t'1.

The time between two events 1 and 2 is then t2-t1 = Dt

and the space between two events is x2 - x1 = Dx

An invariant can be made by combining these differences:

Dx^2 - Dt^2 = a constant in all frames

so

Dx^2 - Dt^2 = Dx'^2 - Dt'^2

In a primed frame in which a watch remains strapped to my wrist and the watch is at both event 1 and event 2 then the distance in space between the two events is Dx' = 0 and this last equation becomes

Dx^2 - Dt^2 = -Dt'^2

or

Dt'^2/Dt^2 + Dx^2/Dt^2 = 1

which is seen to be the equation form a circular arc x^2 + y^2 = 1

where x and y are played by Dx/Dt the speed of the moving clock

and Dt'/Dt the rate at which it runs.

If we measure the rate of a moving clock relative to our stationary clock it runs slow.

The slowness is determined by a term called gamma = 1/(1-v^2/c^2)^0.5

The moving clock runs at a rate which is slower by 1/gamma times the stationary clock.

For example when a clock moves at v = 0.87c then the clock rate is 0.5 times the rate of the stationary clock.

 Scientific Explorations with Paul Doherty © 2005 8 May 2005