Saturday 17 March 2018

Exercise 1.03 Events ABC

The question was; Three events, A, B, C, are seen by observer O to occur in the order A B C. Another observer, O', sees the events to occur in the order C B A. Is it possible that a third observer O'' sees the events in the order A C B? Support your conclusion by drawing a spacetime diagram.

I couldn't see how to do this without a space time diagram, moreover to make life easier I created excel charts for them so that I could easily compare temporal orders.

I started with a diagram like this

It shows the x, t coordinates that B must have to get the sequence ABC in O and CBA in O'. B must be inside the yellow parallelogram. The boost is 0.6.

Next we have a boost of  0.16.
For the temporal order ABC in O and ACB in O'', B must be in the green triangle. (Actually it is a an open sided parallelogram with the lower side at t = ta.)

To fulfil both the conditions, B must be in the overlap of the yellow parallelogram and the green triangle, which is a small triangle, shown here:
By this time I had moved A to (0,0) to make life easier.

It then became possible to draw a fateful triangle for various boosts. Here are a few samples.

To see all the workings read the pdf here.

Thursday 15 March 2018

Exercise 1.02 Three torus space

The question was

Imagine that space (not spacetime) is actually a finite box, or in more sophisticated terms, a three-torus, of size L. By this we mean that there is a coordinate system xμ = (t,x,y,z) such that every point with coordinates (t,x,y,z) is identified with every point with coordinates (t,x+L,y,z), (t,x,y+L,z), and (t,x,y,z+L). Note that the time coordinate is the same. Now consider two observers; observer A is at rest in this coordinate system (constant spatial coordinates), while observer B moves in the x-direction with constant velocity v. A and B begin at the same event, and while A remains still, B moves once around the universe and comes back to intersect the world line of A without having to accelerate (since the universe is periodic). What are the relative proper times experienced in this interval by A and B? Is this consistent with your understanding of Lorentz invariance?

Read my answer here.