The relativity calculation leading to the ratio of times can be reached with less discussion if we postulate a few things rather than explain them. Doing this math correctly could get you 20 to 30 points on a test in physics. Once you are satisfied with the calculation you will probably be curious about the content that was cut out. Appendices will soon be added at the end of the blog to address this.
Postulate 1
Even though we have clocks when you got two different speeds, there is a starting point and stopping point for the experiment.
Postulate 2
A right triangle provides the foundation for the calculation. The sides of the triangle are distances from the story used for the calculation.
Postulate 3
The speed of light is independent of frame of reference and this forces two clocks to record different changes in time.
Postulate 4
We draw the triangle so that a hypotenuse connects a longer vertical side and a short horizontal. The vertical line has time relative to the train. The other two lines move in the horizontal direction because they show movement relative to the ground underneath the train.
Definitions
- c- speed of light
- v- velocity of the train
- d- a distance traveled between the start-and-stop point of the experiment
- we use a,o,h as labels for the sides of the right triangle, adjacent, opposite and hypotenuse respectively.
We use the Pythagorean Theorem with the sides of the right triangle:
Appendix A – Perspective
Trains move over the ground, and we introduced the train is moving at velocity ‘v’ to the west. However for the purposes of the calculation it may actually help to imagine the train standing still and the ground moving under the train at a velocity of v the east.
That’s what it actually looks like if you are on the train because to you it seems like the train isn’t moving. That helps explain why the line of the triangle corresponding to the photon of light light moves in a vertical line when moving relative to the train.
To the extent that the ground moves, light on the train moving relative to the ground has to move in the horizontal direction.
Appendix B – Starting and Stopping Points
This might be the most challenging part of the discussion. Previously, anytime we talked about starting and stopping points, we used time. Now we have time moving differently and something else has to govern the starting and stopping points. The person devising the train story might go to elaborate lengths with details to make their starting point and stopping points exist.
Appendix C
Two ideas are shown below. It seems possible that one of them explains the other but we have to decide which came first. Said another way, one has to be the postulate, then the other can be something that we get from the postulate.
- Nothing to have a speed greater than the speed of light.
- The speed of light (and other things in the same category) is invariant (does not change) to frame of reference.
The argument was made, that if the speed of light cannot change, then two things traveling different distances between a start-and-stop can only do so by having different times.
Rob Sterling 