# What can we accept in thought experiments in relativity?

Although title is more broad, and you are welcome to give examples, I will ask about why we accept certain things as acceptable in Einstein's thought experiments using a specific experiment:

Consider the famous mirror in a train example:

There is a train moving with velocity $$v$$ relative to earth. The train has height $$h$$ and there is a mirror in the ceiling and a light source at the bottom directly below mirror. Alice , inside train, measures the time it takes for light to go back and forth once , $$t_{\text{Alice}}$$. Bob, a person on ground, measures the time it takes for light to go back and forth once $$t_{\text{Bob}}$$. Relation between times is $$\gamma t_{\text{Alice}} = t_{\text{Bob}}$$ for $$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$.

Now a proof without using Lorentz transformations is as follows (based on that from David Morin's Classical Mechanics):

$$2h = ct_{\text{Alice}}$$ for obvious reasons, and $$c^2(\frac{t_{\text{Bob}}}{2})^2 = v^2(\frac{t_{\text{Bob}}}{2})^2 + h^2$$ based on fact that light will move piecewise straight until it reaches mirror and back and equation is based on Pythogoras theorem.

My question is really motivated from simultaneity being frame dependent, so that one person saying they saw something is a bit subjective : A person could say they saw 2 light sources light up at the same time, but other could say they light up at different times.

Then based on above, why we agree that both Alice and Bob saw light to hit the mirror in the first place? (i.e in general what constitutes a lie and what is just frame dependent truth in relativity?)

Note: I specifically omitted Lorentz transformations, since they can be accepted as axioms, which is fine, and equations guarantee that light hits the mirror, but Einstein probably wasn't basing his arguments directly to fit some formalism other than his own postulates.

• You are wrong in the interpretation of this. The full Lorentz transformations can only be arrived at after obtaining time dilation and length contraction. We do not postulate that the Lorentz transformation is correct. We postulate that the speed of light in vacuum is constant for all observers. Also, we are not asserting that Alice and Bob would say that the light hits the mirror at the same time; we only asserted that both of them see something, and then we try to relate them together. Commented Jan 5 at 9:57
• Also bear in mind that the word "see" used here is a very specialized interpretation of the word, which has nothing to do with light entering anyone's eyes! Commented Jan 5 at 12:42
• I am aware of meaning of "see". Also, postulating Lorentz transformation is equivalent to postulating const speed of light by the notion of invariant time. Commented Jan 5 at 18:01

Recommendation: the article Nothing but relativity by Palash B. Pal.

That article stands in the following approach to special relativity: how far can you push while using only the principle of relativity of inertial motion?

So that is an approach where the starting point is as follows: assume there is an equivalence class of coordinates systems such that all members are in inertial motion relative to each other. What is the most general form of a transformation that can be used to transform between any pair of coordinate systems?

(That approach has a long history: Palash B. Pal offers that his contribution is in achieving a more comprehensive and better flowing presentation than in prior treatments.)

In the article it is demonstrated that the principle of relativity of inertial motion alone is sufficient to narrow down to just two possibilities:

• Lorentz transformation
• Galilean transformation

And of course: the Galilean transformations are a limiting case; with infinite speed of causality the Lorentz transformations simplify down to Galilean transformation.

With the possibilities narrowed down to just two: it is then possible to have experimental result deciding which one to go with.

(Of course, what you then want is several mutually corroborating experimental results, some experiments probing propagation of light, other measuring motion of particles at relativistic velocity.)

More general consideration:
The purpose that thought experiments are useful for is to probe the implications of a theory of physics.
If the thought experiment leads to a self-contradiction then you know something is wrong.

In my opinion thought experiments are very much ill suited for the purpose of introducing some theory of physics, or for the purpose of motivating some theory of physics

• Awesome answer! Indeed Pal’s paper is an excellent resource.
– Dale
Commented Jan 5 at 13:28
• I think i got the answer to my question : the truth told, or what is the world at a snapshot of time is well defined through Lorentz transformation (buy algebra is messy ). Then the total history is the truth in a given frame. Commented Jan 5 at 18:06
• But other than that, above answer motivates why the mentioned principle is associated with Lorentz transformation (with support from experiment), so it is helpful. Commented Jan 5 at 18:08

Thought experiments are simply a way of thinking about physical principles. Whether a thought experiment is valid depends on whether it is self-consistent and leads to an outcome that is consistent with experimental results.

The thought experiment involving a light clock can be considered in a more abstract way without any observers at all. You can simply consider light leaving the origin in some reference frame, traveling directly up the y axis, striking a horizontal mirror and returning back to the origin. In another frame moving inertially with respect to the first at 90 degrees to the y axis, the light follows a diagonal path, and therefore must take longer to complete the round trip. From that you can work out the formula for time dilation.

The more abstract version of the thought experiment contains a number of implicit assumptions. For example, it assumes that the vertices distance from the origin to the mirror is the same in both frames. It assumes that there is no significant delay involved in either frame when the light changes direction upon encountering the mirror. You can of course question all such assumptions, but ultimately what counts is whether they lead to experimentally verifiable conclusions.

Then based on above, why we agree that both Alice and Bob saw light to hit the mirror in the first place? (i.e in general what constitutes a lie and what is just frame dependent truth in relativity?)

There is an assumption involved here that is often (or even usually?) left unstated because it seems so extremely manifestly self-evident. That assumption is that, even though time and space are each frame dependent, localized events are objectively absolute, independent of frame. Changing your reference frame may change when and where any specific event occurred, but it cannot change whether the event occurred.

The Big Bang happened in the reference frame you are using, and changing to another reference frame cannot change that objective fact.

The Earth formed in the reference frame you are using, and changing to another reference frame cannot change that.

You viewed this answer in the reference frame you are using, and changing to another reference frame cannot change that.

Likewise, the light hit the mirror in the thought experiment, and changing your choice of reference frame cannot change that. Changing your reference frame can change the time and location of the event of the light hitting the mirror, but cannot change that the light did hit the mirror.

This principle is an axiom that is so commonly implicit that I'm not sure it even has an official name.

• Perfect! This is exactly answer of my question. Commented Jan 6 at 20:31
• In terms of Lorentz transformations, clearly transformation is continuous (and so is its inverse), thus , it seems relevant to say I think, locally we can focus on one part of event and look at how it did transform than the global picture.E.g there are 2 light bulbs and in frame $S$ they light up at the same time but in $S'$ they didn't. Now the term 'one part of event' is really if we choose a locality around a light bulb that doesn't contain the other, then that part of system transforms in such a way that in both frames light turns on, but not both of them at the same time. Commented Jan 6 at 20:35

In this type of thought experiment, Alice and Bob are only made human observers in order to make the thought experiment more "relatable". There is an assumption (often unstated) that Alice and Bob are perfect objective observers and neither make mistakes nor lie about their observations. Conceptually, you could replace Alice and Bob with very precise and accurate photodetectors and timing devices.

First of all, you should examine the train experiment more carefully. Read the version in Einstein's book Relativity. The train experiment can be translated mechanically into the thought experiment in the 1905 paper. There is no difference.

The anomaly is that Einstein says point M and M' "fallt zwar...zusammen." The English translation has "naturally coincides."

The problem with this term is that it is not defined in the argument. Einstein does use the Euclidean definition of the coincidence of points (which is: ________ [you fill that in]). But there is no definition of a "natural" coincidence of points.

Thus, this term has no place in the argument. Because that is so, we cannot move beyond it to establish the relativity of simultaneity, or special or general relativity, or the standard model.

We can't remove the "naturally" either, since that leaves us with one Cartesian coordinate system, when we are supposed to assume two--a contradiction.

So this term is the pea under the mattress.

Restatements of the train experiment always unconsciously attempt to correct this error--the authors know there is a problem, but are unaware of it. The most glaring example is the 1920 Italian translation of the book. Compare the discussion of points M and M'. The attempted correction is right before your eyes. Do you see it?

So, in the first place, make sure you do actually have a thought experiment and not, as in the case of the relativity of simultaneity, a sleight of hand.

• As it’s currently written, your answer is unclear. Please edit to add additional details that will help others understand how this addresses the question asked. You can find more information on how to write good answers in the help center.
– Community Bot
Commented Feb 4 at 8:23
• This was a truly pointless objection the first time you made it and it remains pointless this time too. Under a linear transform, the midpoint of a segment naturally transforms to the midpoint of the transform of the segment. It is entirely correct, well understood, and completely non-objectionable. This is not a sleight of hand nor a pea under the mattress. There may be variations in translation, but the math and science are clear
– Dale
Commented Feb 4 at 19:14