Guest post – Jeff Wyss: The Relativistic Train April 30, 2008Posted by dorigo in Blogroll, mathematics, physics, science.
Jeff is a physics professor at the University of Cassino, and a long-time colleague and friend of mine. He worked in the SLD and CDF collaborations as a particle physicist, but later moved on to study radiation damage on silicon detectors for particle and astroparticle applications.
Besides admiring him for his wicked sense of humor, which he uses to make the workplace around him always a pleasant place, I have the highest esteem of Jeff as a professor, because he is quite skilled in explaining physics concepts in simple terms. He always looks for the most intuitive way to understand things, as you might appreciate in the contribution he offers below.
The following describes a very elegant and simple derivation of the relativistic formula for the addition of velocities, .
It is due to David Mermin. I fell in love with it and have been telling it for the past four years now to the students of my general physics course. The students are first year telecommunications and electrical engineering students. Before sitting in on my course all of them have heard about Einstein and most of them heard the expression “the velocity of light is constant”. I do not have the time to discuss special relativity in detail. My course is quite traditional. I discuss reference frames, inertial frames, Galilean transformations and covariance of Newton’s laws. I then point out that when describing mechanical waves the frame that is stationary respect to the medium is a special reference! In particular the wave motion can be made to disappear by moving respect to the medium with a velocity equal to that of the wave. It is clear at this point that the constancy of the velocity of light cannot be understood by assuming Newton’s laws and then modeling light as a mechanical wave in a medium (the ether). I then restate the constancy of the velocity of light and begin Mermin’s derivation.
The derivation uses:
- only one reference frame (no use of Lorentz transformations),
- simple kinematics (always good to brush up on),
- the constancy of the velocity of light (something that every telecommunications and electrical engineering student should know),
- the idea that some things are invariant; i.e. while many quantities are relative, observers will agree on some absolutes.
Consider a train of length L moving along the x-axis at a constant velocity v respect to an inertial frame of reference (the observer watching the events unfold). At the trailing end of the train a loaded gun is aimed in the forward direction and fired at time : the bullet and flash of light emerge and travel in the forward direction with different speeds: w the velocity of the bullet, c the velocity of light. A mirror at the front end of the train reflects the light back towards the advancing bullet. Let f be the fraction of the length of train that the reflected light travels before meeting up with the bullet. The constancy of light (Einstein’s dictum) tells us that the velocity of light in the forward direction is equal to the velocity of light in the backward direction; i.e. .
The space-time plot looks like this:
Let be the time for the light flash to reach the forward-going mirror and be the time the reflected light needs to return from the mirror and meet up with the forward-moving bullet. Simple kinematics allows us to label the space-time plot:
It is important to note that the expression for f we just obtained is valid if the velocity of light in the forward and backward direction are equal. Note:
- A classical pre-Einstein physicist would say this expression is valid only if the observer is stationary respect to the ether frame.
- On the other hand Einstein says that any inertial observer would use the same velocity of light; i.e. Einstein tells us that this expression is valid for any observer (generic inertial frame).
Following Einstein we consider a particular observer (frame), one that is moving along with the train. For this observer the velocity of the train is . For clarity let us use the symbol u to indicate the velocity of the bullet with respect to this observer; i.e. with respect to the train.
Suppose the train has 10 windows and the reflected light and the bullet meet up at the third window from the front (). It is important to realize that all observers will agree on the value of f. The fraction f is an invariant!
The constancy of the velocity of light allows us to impose the invariance of f the following way: