Lorentz Transformation in Different Reference Frames

Question

TO ALL: Thanks to the help of many dedicated forum members, I have learned that this problem can be explained by understanding Minkowski diagrams. Here is a consolidated list of helpful links: https://www.khanacademy.org/science/physics/special-relativity/minkowski-spacetime-2016-01-18T22:56:14.718Z/v/starting-to-set-up-a-newtonian-space-time-diagram https://www.physics.byu.edu/faculty/allred/222%2011/minkowski%2011.pdf

How can the speed of light remain constant in reference frames that experience time differently?

Maybe some context would be helpful, so let’s do a quick thought experiment: Imagine that you are standing on the platform of a railway station. A train approaches you at 99% the speed of light. It’s headlight flashes when the locomotive is directly in front of you*. At that instant, what do you see?

• Note: The fact that the headlight is directly in front of the observer when it flashes is crucial to this version of the thought experiment. The light source is not moving away from the observer in any direction, thus simplifying things by eliminating the relativistic doppler effect.

Although an overall analysis of this thought experiment would be greatly appreciated in the comments, there is a specific question that I am trying to answer: Is the absolute, (as opposed to observed) speed of light truly constant? Let’s take two perspectives to examine this scenario from. An engineer in the locomotive would observe that the photons emitted from the headlight are traveling at the speed of light, C, relative to him. He is also experiencing time at the same rate, or is in the same “time bubble” so to speak, as the light source because they are both moving at the same velocity. On the other hand, at what speed would the photons appear to be traveling from the perspective of the (relatively) stationary observer on the platform? This person is in a different time bubble than the engineer and the light source, since he is traveling at a different velocity. If the observed speed of light is constant, then the person on the platform would observe that the photons are traveling at C, relative to him. However, wouldn’t this mean that light was traveling at two different absolute speeds? In fact, it seems as though the absolute velocity of the photons would be greater for the observer on the platform than for the engineer, since the person on the platform observes the photons traveling away from the moving train at the same velocity that the engineer observers them when moving along with the train.

Essentially, how can time dilation explain why the observed speed of light remains constant in a scenario where the observer experiences time at a different rate than the light source which is being observed?

Answer 1

(From a comment by the OP)

To my understanding, Lorentz transformations adjust all variables in order to maintain a constant ratio of Distance/Time. However, how can these variables be adjusted proportionally if the observer is not experiencing the same Lorentz transformations as the light source?

First, observers don't experience Lorentz transformations.

The Lorentz transformation is a coordinate transformation that relates the spacetime coordinates in one inertial reference frame (IRF) to the spacetime coordinates of another, relatively moving IRF.

To be clear, there is no preferred IRF from which we 'Lorentz transform' from. For an inertial observer, all other relatively moving IRFs that are moving while she is at rest.

Second, if an entity has speed $c$ in one IRF, that same entity will have speed $c$ in all IRFs; this is easy to show using the Lorentz transformations.

Third, the Lorentz transformation preserves the spacetime interval, a kind of 'distance' through spacetime. Two events, A and B have an invariant interval $\Delta s^2_{AB}$; all inertial observers find the same interval for events A and B.

Fourth, in special relativity (SR), there is a distinction between observing (recording the spacetime coordinates of an event) and seeing (taking a photograph).

In your post, you write:

It’s headlight flashes when the locomotive is directly in front of you*. At that instant, what do you see?

You observe (record the coordinates with your rods and synchronized clocks at rest) that the headlight has flashed at that instant and at that position but you don't see the light until later because it must first propagate (at speed $c$) the distance between the headlight and your eyes (or camera).

If event A is the headlight flashing and event B is the arrival of the light at your eyes (camera), you will find that the interval $\Delta s^2_{AB}$ is null (light-like).

However, other observers will disagree on the spatial distance $\Delta x_{AB}$ and the temporal distance $c\Delta t_{AB}$ but they all agree that $\Delta x_{AB}^2 = (c\Delta t_{AB})^2$ since the interval is null.

Answer 2

They both measure the same speed, which is independent of the speed of the source. The two observers not only see that the other observer's clock run slower, they also see length contraction and clock desynchronization. Suppose the guy on the train synchronize clocks with the guy on the platform at the time the pass each other and the light is emitted. Both observers measure the speed of light by dividing the distance to the next station to the time it takes the light to reach the station. The observer in the train will see that the distance is smaller. He will also see that the clock at the next station is out of synchrony with the one at the original station. If, instead of a clock at the next station, the clock at the emitting station were used, then the observer on the train will see that the person will stop the clock after the light, according to him, has reached the second station.

Answer 3

I think you are getting confused about the simultaneity of events.

The person on the platform observes the train passing and the headlight as being turned on simultaneously.

From the point of view of the train driver the two events do not happen simultaneously.

Both see the light travelling at c.