# Minkowski diagram

This is on Wikiversity instead of Wikipedia because it contains material that is either unsourced or original. The effort is purely pedagogical, not intended to develop any new theories regarding special relativity. While the Minkowski diagram is probably not a suitable topic for a beginning student in special relativity, the ideas developed here might someday lead to a reassessment of that view. The nice thing about writing for Wikiversity is that we don't need to worry about such issues. This website suggests that these ideas are old (no surprise to me): http://hubpages.com/education/Minkowski-Diagram
Figure 1 explains why superluminal communication violates causality. At time, t=0, Alice will send Bob a message he will receive before she sends it. The train is equipped with a hypothetical instantaneous signal delivery device. Two photons at the center C are sent to the front F and rear R to verify that the message is received the instant it is sent. Alice waits till the front reaches her to send the message that Bob had received a short while ago when the rear of the train passed him. The consequences of a prompt reply (from Bob to Alice) using a similar device is sufficient to convince most that superluminal communication is impossible.
In Figure 2, extra diagonal elements have been added. One is drawn to look like a train, and the other is drawn to mark time an looks like the red 7-segment display of a digital clock. It is convenient if the train is one light-year long (as measured in the the rest frame of Alice and Bob). A time travelSer who "dropped" into an empty seat on the diagonal train might witness the passengers celebrating day when the magic telephone sent the instantaneous message from the front to the back of the train (in their frame it is a simultaneous celebration, but it might not seem so due to look-back effects). A time traveler who "dropped" onto the red digital clock line would be at the center of the train where a one-year clock was marking the time between when two pulses of light were emitted at the center of the train. The endpoints of this digital clock line mark when the pulses were emitted and when they return. Since the train is one light-year long in the reference frame of Alice and Bob, the digital time clock must mark more than one year because the "proper" length of the train is longer than one light-year.
Figure 3 shows a similar diagram with the A train traveling to right, and after rearranging the lines we get Figure 4. Is this related to the Minkowski diagram of Figure 5?

## Trigonometry

From Figure 3, taking the train's length in to be 1 light-year (in the rest frame of Alice), apply the law of sines to the two triangles at the bottom of the diagram. Then apply the sine sum angle rule to the identities shown in that figure: ${\frac {\sin \delta }{0.5}}={\frac {\sin \epsilon }{b}}$, and, ${\frac {\sin \eta }{0.5}}={\frac {\sin \epsilon }{b}}.$
Since tanφ=b/a is the velocity v (if c=1), we have:
$a={\frac {1}{\sqrt {2}}}{\frac {\cos \phi }{\cos \phi +\sin \phi }}={\frac {1}{\sqrt {2}}}{\frac {1}{1+v}}$
$b={\frac {1}{\sqrt {2}}}{\frac {\cos \phi }{\cos \phi -\sin \phi }}={\frac {1}{\sqrt {2}}}{\frac {1}{1-v}}$

## Instructions for creating you own Minkowski diagram

The figure shows how to create a diagram with the train traveling in the negative-x direction.