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11 April 2003

An Amazing Result

by George Hrabovsky, President of MAST

News from MAST

Hello everyone. Well, none of the proposals sent to the NSF have been accepted. This has been a very good learning experience, and we hope to do much better next time.

Everyone Has Their Own Coordinate System

Recall from last time that we have established the spacetime diagram.

[Graphics:Images/index_1.gif]

We have also established that every observer is a coordinate system. Since we all share the same spacetime (if this were not true then things would be really weird), we should be able to represent the same events on the spacetime diagrams of different observers.

So far we have not said anything particularly startling. Here is something interesting; since we can observe the same event on different spacetime diagrams, we should be able to use that event to place one spacetime diagram into another by drawing the relative coordinate axes.

Suppose we have an observer located at the origin of a coordinate system, ÷~, and we have a different observer located at the origin of a second coordinate system Overscript[÷~, ~]. The squigly line above the scripted O is the tilde symbol, and we shall call the second coordinate system the tilde system. We assume that the tilde system is moving with some velocity v with respect to the first coordinate system. For the first coordinate system we have t and x as normal, but where do we place Overscript[t, ~] and Overscript[x, ~]?

The time axis is the locus of constant Overscript[x, ~] = 0. Since the tilde system is moving at some velocity v the axis will not be vertical, it will be slanted in the x direction.

[Graphics:Images/index_9.gif]

This is the worldline for the tilde system.
    Let's look at an event, ÷, located at (t, x) in the ÷~ system and (Overscript[t, ~], Overscript[x, ~]) in the Overscript[÷~, ~] system.

[Graphics:Images/index_15.gif]

In order for ÷~ to illuminate ÷ at some time t it must shine a light beam from some point t - x.

[Graphics:Images/index_20.gif]

This light beam crosses the Overscript[÷~, ~] worldline at Overscript[t, ~] - Overscript[x, ~]. This light beam can then be reflected back, arriving at the ÷~ system at the point t + x.

[Graphics:Images/index_25.gif]

The reflected light beam crosses the Overscript[÷~, ~] worldline at Overscript[t, ~] + Overscript[x, ~].

If we assume that the light beam leaves at time t _ 1 and is reflected back at t _ 2 we can fix the coordinates of our event,

(t, x) = (1/2 (t _ 1 + t _ 2), 1/2 (t _ 1 - t _ 2)) .

We can make this more useful by stating that the initial time occurs at some unit T, t _ 1 = T, the event itself occurs at some factor of the initial time later, t = k T, and t _ 2 = k^2 T. We now have

(t, x) = (1/2 (k^2 + 1), 1/2 (k^2 - 1)) .

    Recall from elementary kinematics,

v = x/t

    = (1/2 (k^2 - 1))/(1/2 (k^2 + 1))

    = (k^2 - 1)/(k^2 + 1) .

We can solve this for k in terms of v,

k = (1 + v)/(1 - v)^(1/2) .

We can see that Overscript[t, ~] - Overscript[x, ~] will occur a factor, k, later than t - x, so

Overscript[t, ~] - Overscript[x, ~] = k (t - x),

and similarly,

t + x = k (Overscript[t, ~] + Overscript[x, ~]) .

If we solve this system of equations we get,

Overscript[t, ~] = (t - v x)/(1 - v^2)^(1/2)

and

Overscript[x, ~] = (x - v t)/(1 - v^2)^(1/2) .

These are the famous Lorentz transformations, and they tell us how to look at one coordinate system from another.

From this we see that, Overscript[t, ~] = 0 if and only if

0 = (t - v x)/(1 - v^2)^(1/2)

    = t - v x

t = v x .

Thus we have the ordered pair (x, v x) for the tilde time coordinate.
    Similarly, Overscript[x, ~] = 0 if and only if,

0 = (x - v t)/(1 - v^2)^(1/2)

     = x - v t .

x = v t

t = x/v .

This gives us the ordered pair (x, x/v) for the spatial coordinate.

We see that the spacetime diagram now looks something like this,

[Graphics:Images/index_57.gif]

We will explore some ideas about how lengths can be represented on spacetime diagrams next time.

Answer to the Theory Challenge from Last Time

The task is to think about how an oscillator behaves if we introduce a force to keep it oscillating. In the absence of damping there is no point to having such a force. If such a force, F(t), is present, we would expect the equation of motion for the damped oscillator to be written

m d^2 x/d t^2 + b d x/d t + k x = F(t) .

Theory Challenge

What happens when the applied forcing term is a sinusoidally oscillating force?

Books That I Like

Bernard F. Schutz (1990), A First Course in General Relativity, Cambridge University Press. This has a nice introduction to spacetime diagrams and derives how to view other coordinate systems.

Ray D'Inverno (1992), Introducing Einstein's Relativity, Oxford University Press. This is my favorite introduction to relativity. This introduces the k-factor in a more formal way than I have here.

Converted by Mathematica  (April 11, 2003)