Modern Physics/Changing Frames

So far, we have seen how length and duration both look different to a moving observer:

• lengths are contracted by a factor γ, when measured at the same time.
• durations are contracted by a factor γ, when measured in the same place.

We can extend these results to allow for measurements taking place at different times and places.

Lets consider two observers, O and O' such that O' is moving at velocity v along the x-axis with respect to O. We'll use primed variables for all the measurements O' makes.

We can assume for now both observers have the same origin and x-axis because we already know how to allow for oberservers being relatively rotated and displaced. We can put these complications back in later.

Now any length or duration can be written as the difference between two coordinates, for the two ends of the body or the start and end of the event, so it is sufficient to know how to change coordinates from one frame to the other.

We know how to do this in classical physics,

$\begin{array}{ccc}{x}^{\prime }& =& x-vt\\ t^{\prime }& =& t\end{array}$

we need to extend this to relativity.

Notice that the in classical physics the relationship is linear; the graphs of these equations are straight lines. This makes the maths much simpler, so we will try to find a linear relationship between the coordinates for relativity, i.e equations of the general form

$\begin{array}{ccc}{x}^{\prime }& =& mx+nt\\ t^{\prime }& =& px+qt\end{array}$

where m, n, p and q are all independent of the coordinates.

To begin with know that

• when t=0, x′ = γx (Lorentz contraction)
• when x=0, t′ = γt (Time dilation)

and that O' is travelling at velocity v. They measure their position to be at x′=0, but O measures it to be at vt so we must have

x′=0 when x-vt=0

The only relationship between x and x′ that satisfies these criteria is

$x^{\prime }=\gamma (x-vt)$

Both observers must measure the same speed for light,

$x=ct\Rightarrow x^{\prime }=c{t}^{\prime }$

or, substituting and rearranging,

$x=ct\Rightarrow t^{\prime }=\gamma \frac{c-v}{c}$

The only linear relationship between t and t′ that satisfies these criteria is

$t^{\prime }=\gamma (-\frac{vx}{c^2}+t)$

So the primed and unprimed coordinates are related by

$\begin{array}{ccc}{x}^{\prime }& =& \gamma (x-vt)\\ t^{\prime }& =& \gamma (-\frac{vx}{c^2}+t)\end{array}$

These equations are called the Lorentz transform.

They look simpler if we write them in terms of ct rather than t

$\begin{array}{ccc}{x}^{\prime }& =& \gamma (x-\frac{v}{c}ct)\\ c{t}^{\prime }& =& \gamma (-\frac{v}{c}x+ct)\end{array}$

Written this way they look much like the equations describing a rotation in three dimensions. In fact, once we allow for the different Pythagorean theorem, they are exactly like the equations for rotation.

• If observers are moving relative to each other, their coordinate systems are rotated in the (x,ct) plane.