# Length contraction and time dilation

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TL;DR Summary
Is there a simple mathematical demonstration which derives length contraction and time dilation from light speed being constant across multiple reference frames.
A long time ago, I was very impressed by a lecture on elementary special relativity which showed in simple math how the correct conclusion to the null results of the Michelson-Morley experiment were obtained by concluding a constant c but then adding a length contraction and time dilation. I don’t remember the math demo except that it was simple and used Pythagorean theorem. I think the triangle sides were different lengths that light traveled based on the reference frames. Can anyone reproduce such a demo? By Searching for Lorentz transformations, I haven’t found it on the web.

Have a look for "light clock", which I think is what you are talking about. A light pulse bounces back and forth between mirrors separated by distance ##l##, taking time ##2l/c## to bounce backwards and forwards. However, viewed in a frame where the mirrors are moving perpendicular to the line between them, the light follows a diagonal path, which Pythagoras' Theorem tells you is of length ##2\sqrt{(vt')^2+l^2}## which must be equal to ##ct'##. A bit of algebra will net you time dilation.

World interval is invariant under transformation between IFRs, i.e.
$$s^2=c^2t^2-x^2-y^2-z^2=c^2t'^2-x'^2-y'^2-z'^2$$
You can deduce some results from it.

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Bondi has a derivation of the Lorentz transform in "Relativity and Common Sense". It's not quite what you asked for - it would take more work to go from the Lorentz transform to time dilation, length contraction, and last but not least, the relativity of simultaneity, henceforth abbreviated as ROS. The time dilation part is the easiest, the length contraction can be derived from this, but it needs more work, in specific one needs to compensate for the ROS to achieve the correct value of length contraction. ROS is very important to understanding special relativity, but is often not addressed by people who find the concept confusing and in response simply ignore it. Unfortuantely, simply ignoring it leads to various misunderstandings, false paradoxes, and general confusion.

The math is very simple. - Bondi has an explanation at a much greater length, but I can sketch the result in this post.

Suppose we have a reference observer R, and a moving observer O. Each of them has a clock that they carry with them, and both clocks are set to zero at the instant R and O are co-located.

At time T, R sends out a radar pulse as measured by R's clock. The radar pulse will be received by O at time kT (as measured by O's clock), where k is some constant. I won't attempt to justify this in this short post, but simply use the fact.

The pulse that was transmitted at T will be received by O at time kT, and reflected back to R, arriving back at R at time ##k^2 T## according to R"s clock. This is because the observers are interchangable by the principle of relativity.

Given that the speed of light is constant and isotropic, we can conclude that in frame R, the radar pulse as received by O at time ##(T + k^2T)/2## according to R's frame of reference, and that at that in frame R at time ##(T + k^2T)/2##, the total light travel time there and back was ##(k^2T - T)##, implying that the distance in frame R was ##\frac{1}{2} (k^2 - 1) T##.

The ratio of distance/time gives the velocity v as a function of k, namely ##c (k^2-1)/(k^2+1)## Furthermore, it illustrates time dilation. In frame O, the pulse has a time coordinate kT and a space coordinate of 0. In frame R, the time coordinates measured byh the radar setup was ##\frac{1}{2} (k^2 + 1) T##.

If we set k=2 for a quick example, this means that in frame R, the time coordinate of the reception event was 5/2 = 2.5T, whine in frame O the time coordinate of the reception event was 2T.

Wikipedia has an article on Bondi's technique, known as k calculus (though it only involves algebra, not calculus) at https://en.wikipedia.org/w/index.php?title=Bondi_k-calculus&oldid=1007986605

I don't know if this video would suit your taste:

bhobba
I am reminded of my second favorite approach, which doesn't require much math, but does require one to be able to draw and interpret space-time diagrams, in particular the space-time diagrams of a light clock. This is "Relativity on Rotated Graph Paper". The preprint version is available at arxiv at https://arxiv.org/abs/1111.7254, there is a peer-reviewed published version that is paywalled also. There are some differences between the two versions due to peer feedback, but the published version is harder to obtain.

There's also an insight article here on PF by the author, https://www.physicsforums.com/insights/relativity-rotated-graph-paper/. Both time dilation and length contraction are discusssed in the arxiv paper through the use of the diagrams of light clocks.

Drawing and interpreting space-time diagrams of a light clock is pretty much a prerequisite to using this technique. That's a rather modest bar, though I've found as modest as it is, it is frequently an obstacle.

vanhees71, robphy and Ibix
@pervect, I’m glad that you find my article helpful. One motivation for my approach is to visualize the size of timelike segments for the k-calculus.

vanhees71
@pervect, I’m glad that you find my article helpful. One motivation for my approach is to visualize the size of timelike segments for the k-calculus.
You're welcome - my first exposure back in the day to special relativity was the Bondi book, but I also like your approach. The Bondi approach will work for those who are comfortable with high school algebra, your approach avoids the need for even that much math, but requires drawing space-time diagrams. That shouldn't in theory be hard to do. On the practical side, I do suspect that not many casual readers wind up doing the work - I could be wrong.

Any approach at some point is going to run into the issue where the relativity of simultaneity must be addressed. However, explaining the realtivity of the simutaneity in a manner that's understood, and ideally also accepted, seems to be a very challenging task.

vanhees71
Have a look for "light clock", which I think is what you are talking about. A light pulse bounces back and forth between mirrors separated by distance ##l##, taking time ##2l/c## to bounce backwards and forwards. However, viewed in a frame where the mirrors are moving perpendicular to the line between them, the light follows a diagonal path, which Pythagoras' Theorem tells you is of length ##2\sqrt{(vt')^2+l^2}## which must be equal to ##ct'##. A bit of algebra will net you time dilation.
This is the "transverse" light clock, in which the light bounces back and forth in a path that is transverse to the direction of relative motion of the two frames.

If you also add a second light pulse and a second mirror, aligned parallel with the direction of relative motion, with the length in the clock's rest frame the same in both directions (so the two pulses, after being reflected off the mirrors, arrive back at the origin at the same event), you can then analyze this clock in the frame in which it is moving and recover length contraction as well as time dilation. (This configuration is similar to the one used in the Michelson-Morley experiment, except that we've left out measurement of interference fringes.)

bhobba and vanhees71
This is the "transverse" light clock, in which the light bounces back and forth in a path that is transverse to the direction of relative motion of the two frames.

If you also add a second light pulse and a second mirror, aligned parallel with the direction of relative motion, with the length in the clock's rest frame the same in both directions (so the two pulses, after being reflected off the mirrors, arrive back at the origin at the same event), you can then analyze this clock in the frame in which it is moving and recover length contraction as well as time dilation. (This configuration is similar to the one used in the Michelson-Morley experiment, except that we've left out measurement of interference fringes.)
Something like this:

PeterDonis, Motore and FactChecker
This is the "transverse" light clock, in which the light bounces back and forth in a path that is transverse to the direction of relative motion of the two frames.

If you also add a second light pulse and a second mirror, aligned parallel with the direction of relative motion, with the length in the clock's rest frame the same in both directions (so the two pulses, after being reflected off the mirrors, arrive back at the origin at the same event), you can then analyze this clock in the frame in which it is moving and recover length contraction as well as time dilation. (This configuration is similar to the one used in the Michelson-Morley experiment, except that we've left out measurement of interference fringes.)

This second clock is the "longitudinal light clock".
The usual story is that to make longitudinal light clock
agree with the time-dilation observed in the "transverse light clock",
the longitudinal light clock must be length contracted.
(Note: No contraction in the transverse direction using the "nails on a meterstick" argument

You can find this in, e.g., Griffith's Introduction to Electrodynamics 4e, Ch 12.

As @PeterDonis notes, this is essentially the Michelson-Morley apparatus, as described in special relativity to explain the null result, which was unexpected from a pre-SpecialRelativistic viewpoint.
( https://en.wikipedia.org/wiki/Michelson–Morley_experiment ).
The apparatus was a clever way to measure the expected time-difference in the round trips since high-precision clocks weren't available then.

You can see the geometry involved in the MM-apparatus (the transverse and longitudinal light clocks)
in a [seemingly uncommon] spacetime diagram of the MM-apparatus.

pre-SR (without length-contraction): TX and TY are distinct reception events

SR (with length-contraction): TX and TY are coincident reception events

In addition to this txy-spacetime diagram,
you may wish to view
the xy-plane (for the transverse light clock) and
the tx-plane (for the longitudinal light clock)
using https://www.geogebra.org/m/XFXzXGTq

For an older animation,
based on an old paper of mine
(VPT) "Visualizing proper-time in Special Relativity" https://arxiv.org/abs/physics/0505134
[which would inspire the "Relativity on Rotated Graph Paper" (RRGP) approach mentioned by @pervect]
look at

In the above VPT-paper, the MM-apparatus is generalized into a "circular light clock"
whose ticks provide a visualization of proper-time...
which I use to physically argue what special relativity numerically predicts
from applying Einstein's postulates with light-clocks,
which can further be analyzed with textbook formulas... and various ways of interpreting the result.
I feel this argument and geometric construction is a "physics-first" approach,
which can be followed up by the typical textbook algebraic approach.

Time dilation (with length-contraction and relativity-of-simultaneity) [has sound!] :

and the Clock-Effect/Twin-Paradox [has sound!] :

for more details, go to my site: http://visualrelativity.com/LIGHTCONE/LightClock/

By the way,
in addition to my https://www.physicsforums.com/insights/relativity-rotated-graph-paper/ that @pervect mentioned,
there is one on the Bondi k-calculus
https://www.physicsforums.com/insights/relativity-using-bondi-k-calculus/

From these light-clock approaches, I try to regard
"length contraction" as a [supplementary] effect (and not a [primary] cause).
What I regard as more primary is the "causal diamond" between ticks,
because that is what is Lorentz-invariant and is more connected to "causality",
which I think is primary in relativity.
Then, from the Rotated Graph Paper approach, all of (1+1)-Minkowski geometry can be extracted.
(I would think (3+1)-Minkowski geometry should also be extractable from the full causal diamond.)

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Ibix
I have a question. Since relativity says that length contraction and time dilation are based solely on an observer of different relative position and velocity observing the thing that is moving, whereby the object did not really shrink nor change in its rate of flow of time according to an observer moving at the same velocity vector with the moving object, then how was Lorentz justified in stating that the westward arm of Michelson's interferometer physically shrank as the cause of the near-null result of the 0.02 fringe gap, since the light beams were all traveling at the same velocity vector towards the same receiver in line with them. Since the receiver saw ALL the light beams coming at it in the SAME direction, how could it be in a position to differentiate its observation of each light beam's relative velocity to begin with, such that the Lorentz Transform would be required at all? The only difference in the beams was their initial velocity vectors...so how could the end receiver even perceive that at all in ONE reference frame if it didn't "see" the initial start to begin with?

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how was Lorentz justified in stating that the westward arm of Michelson's interferometer physically shrank
Remember that Lorentz proposed this before relativity was developed. He was still thinking in terms of Newtonian physics plus an ether with mechanical properties. Ultimately this developed into Lorentz Ether Theory, which says that there is one inertial frame that is somehow "real" and objects moving in this frame do "really" time dilate and length contract. The way they do it is such that you can't tell which frame is the "real" one. So on a philosophical level, it does not respect the principle of relativity, but operationally it does. It's basically an alternative interpretation of special relativity. It's very much a minority view (maybe even non-existent in physics professional circles these days) but it's not wrong. It's just unpopular - disfavoured by Occam's Razor (which frame is the undetectable "real" one is an additional assumption) and hard to reconcile with some of the weirder and wonderfuller spacetimes general relativity allows.

Note that PF policy is that we don't discuss Lorentz Ether Theory outside it's historical context. Its predictions are identical to SR and fighting about which theory is "really real" gets old, fast.

The only difference in the beams was their initial velocity vectors...so how could the end receiver even perceive that at all if it didn't "see" the initial start to begin with?
It isn't just light that is affected by the length contraction, it's the whole apparatus. So Lorentz (Fitzgerald first, in fact) was proposing that the apparatus was actually shorter in one direction, so that the shorter length exactly canceled the extra flight time of light for a fixed distance. If the original ether theory had been correct the extra flight time along one arm would not have been canceled and light along one arm would have returned later than light traveling along the other arm - this slight difference in return time is what Michelson and Morley were looking for.

bhobba and mitochan
I have spent several hours drafting this reply...and I shouldn't have to. I feel like I'm in a court room every time I ask a question on here lol. That's pretty much why I never post on here...it makes it very difficult to learn with such restrictions...I'm at loss as to how to exactly ask questions on here. Yes, I just reviewed the terms and I had no idea that Lorentz was given a specific mention in that regard. How can I word my questions in such a way that I can actually learn things without being corrected (albeit very politely so thank you for that) when I ask about them?

I still don't have an answer - at least for the second question. Citing relativity, which says that any object moving relative to an observer does measurably shrink according to any outside observer, but in stark contradiction does NOT shrink in its own reference frame, how is it possible for BOTH to be true in the SAME space? I have no idea how else to ask this question, since I am trying to learn and reconcile just EXACTLY how an object can physically shrink AND yet maintain its original dimensions in the same space. I don't see how space itself can contract and...yet not. Moreover, if the contraction of length cannot ever be observed...then how can it be measured and proven? How do you measure something you cannot detect to begin with? Is the transform equation all there is, without observation against which it may be tested?

I guess the crux of it is this: according to the scientific method, if something can't be observed, then how can it be tested? (I do hope that yet another rule that I may not have noticed or yet committed to memory arises here.)

Hello.

then how was Lorentz justified in stating that the westward arm of Michelson's interferometer physically shrank as the cause of the near-null result of the 0.02 fringe gap, since the light beams were all traveling at the same velocity vector towards the same receiver in line with them.
As you deduce no shrink takes place on the arms of interferometer that stays still in our IFR.
Lorentz hypothesized that light speed along the arm change in go and return. In order to meet his hypothesis and the result of no interferometry he has to conclude that the arm shrinks. But as you know the hypothesis was wrong. Einstein clearly stated that light speed for any direction in any IFR is constant.

This is at the heart of the necessity to change our fundamental concepts of space and time we are used to from everyday life, where the "Newtonian approximation" is by far accurate enough. If it comes to motions of objects relative to you as an observer which comes close to the speed of light in a vacuum you have to rethink the very definitions of measuring space and time intervals, i.e., distances between and duration of "events".

That's, because according to the observed fundamental laws of Nature, in this case all electromagnetic phenomena as summarized brillantly by Maxwell in his famous equations, we have to use the special-relativistic space-time model rather than the Newtonian, as has been finally formulated by Einstein (1905) and mathematically analyzed and made as simple as possible by Minkowski (1908).

Einstein in 1905 figured out that there are just two very fundamental conclusions from Maxwell's equation you have to use to come to this better space-time model: (a) as in Newtonian mechanics there exists an inertial frame of reference (and thus infinitely many), where Newton's 1st Law holds and (b) the velocity of light as measured in any inertial reference frame is independent of the velocity of the light source.

The latter postulate is incompatible with Newton's space-time description and thus you have to redefine how space and time are described in terms of measurable quantities, i.e., what's measured with "rods and clocks".

bhobba
Citing relativity, which says that any object moving relative to an observer does measurably shrink according to any outside observer, but in stark contradiction does NOT shrink in its own reference frame, how is it possible for BOTH to be true in the SAME space?
That is the relativity of simultaneity at work. The length of an object is the distance between where one end is and where the other end is at the same time. Two observers using different notions of “at the same time” are measuring different things when they speak of “the” length of the object.

There’s a similar “how is it possible?” question with time dilation: how is it possible that A’s clock is both faster (using the frame in which A is at rest) and slower (using the frame in which B is at rest) than B’s clock? Again, the apparent paradox goes away when you allow for relativity of simultaneity.

vanhees71
Since the receiver saw ALL the light beams coming at it in the SAME direction, how could it be in a position to differentiate its observation of each light beam's relative velocity to begin with, such that the Lorentz Transform would be required at all?
Not for answering your question but for showing extraordinary of relativity, say you switch on flashlight. Light goes everywhere to make a sphere light sphere expanding with speed constant c. You observe it sphere. Your brother driving a car also observe it a sphere shape.
How is it possible? In order to make it possible time and space change its nature.

But how can this phenomenon actually be observed as both cases in order to be put to the test of experiment?

...and the definition of clock synchronization within clocks distributed in space for any inertial frame, i.e., if you choose to synchronize the clocks, all at rest relative to each other within one inertial frame, such that the speed of light signals (in a vacuum!) are measured to be ##c## within this inertial frame, and you do this for two different inertial frames, then necessarily the clocks of one frame, being synchronized in this frame, are not synchronized with the clocks, which are synchronized in the other inertial frame.

This implies the said "relativity of simultaneity", i.e., if you observe two spatially separated events in one inertial frame as simultaneous, i.e., the clocks synchronized in this frame and being at the locations of these two events show the same time, are not simultaneous when observed in another inertial frame moving against the first one, because the two clocks, synchronized in the 2nd frame, at the locations of the two events are not synchronized with the clocks in the 1st frame and thus the events are observed to be at different times from the point of view of an observer at rest wrt. the 2nd inertial frame.

It is indeed pretty difficult to get used to this, and there are many attempts in the literature to visualize this. Two frequently used also in this forum are Minkowski diagrams and the Bondi description, aka the use of "rotated graph paper" of your member @robphy . Which one is most simple to follow, you have to figure out yourself. For me the most simple way is not to use the spacetime diagrams of different kinds but simply the mathematical descriptions of Minkowski space as a pseudo-Euclidean affine manifold ;-).

In my relativity FAQ I use Minkowski diagrams together with the algebraic description:

https://itp.uni-frankfurt.de/~hees/pf-faq/srt.pdf

robphy
it makes it very difficult to learn with such restrictions...

But still it makes you learn, instead of wandering in wrong directions and wasting your time

How can I word my questions in such a way that I can actually learn things without being corrected (albeit very politely so thank you for that) when I ask about them?
I think that, as long as you don't try to start a fight about Lorentz Ether Theory being superior to vanilla SR, you're fine. The point is that everybody and all textbooks (as far as I'm aware) regard LET as a historical footnote, so whatever your personal preference you need to conversant with the usual form of SR. So it's better to learn that.
Citing relativity, which says that any object moving relative to an observer does measurably shrink according to any outside observer, but in stark contradiction does NOT shrink in its own reference frame, how is it possible for BOTH to be true in the SAME space?
"Space" is a 3d slice through 4d spacetime. Two reference frames in relative motion do not use the same slice - so you've hit on the key point. They do not share a space.

The point is that the thing you think of as "an object" is a 3d slice through a 4d structure. A Euclidean analogy is that the full structure is a cylinder, but you only see a slice through it which is a circle - except if the slice is at an angle to the cylinder you don't see a circle, you see an ellipse. That's analogous to length contraction. It's not that the object is simultaneously a circle and an ellipse. It's a cylinder. But at any given height, depending on how you measure it, its cross-section can be a circle or an ellipse and two people may make different measurements without contradicting each other. Minkowski spacetime is a little different (the cross-section shortens in the direction the cylinder tilts (its direction of motion) rather than lengthens), but essentially that's length contraction.

Lorentz would say that an object moving in the ether frame actually does shorten, but observers moving in the ether frame have simultaneity problems and don't actually measure the length but some combination of length and velocity times their timing failure. It ends up with the same answer, but it's messier.
I don't see how space itself can contract and...yet not.
Space doesn't do anything. Different frames mean different slices of spacetime by the word "space", and comparing them directly is a mistake.
Moreover, if the contraction of length cannot ever be observed...then how can it be measured and proven? How do you measure something you cannot detect to begin with? Is the transform equation all there is, without observation against which it may be tested?
The Michelson-Morley experiment is blind to length contraction. That doesn't mean all experiments are blind to it. The classic one is the arrival of cosmic ray muons at the Earth's surface. In the Earth's rest frame the muons survive because their clocks are time dilated, but in the muons' rest frame they make it because the atmosphere is length contracted (again, this is really just two different explanations of the same phenomenon - they aren't contradictory).
But how can this phenomenon actually be observed as both cases in order to be put to the test of experiment?
Muons, as I mentioned. Purcell's explanation of the behaviour of a current carrying wire requires length contraction, so an ordinary electromagnet is a test of length contraction. In principle, you could accelerate a pole to a significant fraction of light speed and actually do the pole-and-barn experiment (probably using photodetectors instead of actual barn doors), but the energy requirements are absurd.

jbriggs444, vanhees71 and robphy
I do love the visual of the barn and the stick. Have we done that in experiment yet, as in using lasers pointed perpendicular to the path of a moving object to represent the barn doors?

I do love the visual of the barn and the stick. Have we done that in experiment yet, as in using lasers pointed perpendicular to the path of a moving object to represent the barn doors?
the energy requirements are absurd.

What would the requirements be?

Ibix
...or the total power output of a large power station (##\sim 10^9\mathrm{W}##) for most of a year.

It may help to note that I'm a blithering idiot compared to those guys. That said, let's say I build a simple EM rail gun and accelerate a tiny steel rod one cm long out to whatever distance I choose. To measure the moving length versus the static length (in whatever rest frame I choose) using a series of, say, twenty lasers separated by precisely one cm intervals, each pointed perpendicular to the path of motion, I don't see how very much power would be required other than the power to run the lasers, the rail gun, and the hardware. Now, of course, in my mind, at any velocity where contraction does not occur, the lasers would be sequentially interrupted in pairs, whereas if there is a measurable contraction, then only one laser would be interrupted at at time in sequence. Is this not viable? Thanks.

Is this not viable?
It's viable in principle, yes. Viable in practice is another matter. Have you done the math to calculate what velocity you would need to accelerate the rod to to show measurable length contraction?

Nope, but convention would hold that it would have to be a significant fraction of the speed of light LOL, something you really couldn't achieve with the mass of a steel rod in the first place, no matter how tiny you can make it. Good point. However, if that is indeed the case, then how was the LT applied to the M-M experiment in the first place, since the Earth does not move at a relativistic speed? (I hope I asked the right question. ;) )

if that is indeed the case, then how was the LT applied to the M-M experiment in the first place, since the Earth does not move at a relativistic speed?
Because the M-M experiment was measuring interference fringes of light, which is a much more sensitive measurement than the one you proposed, and which can therefore distinguish much smaller effects. M-M had calculated the size of the effect they expected to see, and it was more than large enough to be distinguished by their apparatus. That's why its absence was so surprising: because the effect they expected to see wasn't just at the borderline of detection, it was something they expected to see loud and clear in their data.

As @PeterDonis notes, this is essentially the Michelson-Morley apparatus, as described in special relativity to explain the null result, which was unexpected from a pre-SpecialRelativistic viewpoint.
( https://en.wikipedia.org/wiki/Michelson–Morley_experiment ).
The apparatus was a clever way to measure the expected time-difference in the round trips since high-precision clocks weren't available then.
I bolded my earlier comment.

And yet the original transform for length contraction was invented and applied, what was it, maybe two years after the 1887 experiment had been conducted, and thus about eight years after the first, since Fitzgerald had tried to get one of his pupils, W. Preston, to publish its earliest form in his 1890 book? Where were the prior observations, and what were they specifically, that proved the contraction? They had what they would consdier a hypothesis, but it really couldn't have been that since it was contrived without observation of any particular phenomenon...only an interpretation of experimental data. They should've at least tried to observe this even after the scientific method was not completely satisfied. Where are those observations recorded? I would really rather not have to hang my hat of research on a formula that was derived a priori. All I'm trying to do is find observations that led to it. I know I sound like a broken record, but I love science, and even more I love history. Thanks.

the original transform for length contraction was invented and applied, what was it, maybe two years after the 1887 experiment had been conducted
Yes.

Where were the prior observations, and what were they specifically, that proved the contraction?
What? The M-M experiment itself, and the totally unexpected null result, doesn't count as a prior observation?

I don't think you have thought this through very well. When you have an observation that doesn't match the current theoretical models, what else can you do but come up with new models that could account for it? That's exactly what the contraction was: a new, hypothesized theoretical model to account for the unexpected null result of the M-M experiment. What's wrong with that? That's how science progresses.

They had what they would consdier a hypothesis, but it really couldn't have been that since it was contrived without observation of any particular phenomenon...only an interpretation of experimental data.
I don't know where you are getting this weird rule for what can count as a "hypothesis". Certainly not from "the scientific method" or the history of science.

They should've at least tried to observe this even after the scientific method was not completely satisfied.
Observe what? Length contraction? If the M-M result itself doesn't count as such an observation, how would they do that? You have already admitted that your proposed experiment with a steel rod and lasers is not practical, now, in 2021. How are you expecting scientists in the 1880s to directly observe length contraction?

All I'm trying to do is find observations that led to it.
And it seems to me that you have such an observation staring you in the face--the M-M experiment itself--but your weird rules about how you think science is supposed to be conducted, which have no valid basis that I can see, are preventing you from seeing it.

bhobba and weirdoguy
And yet the original transform for length contraction was invented and applied, what was it, maybe two years after the 1887 experiment had been conducted, and thus about eight years after the first, since Fitzgerald had tried to get one of his pupils, W. Preston, to publish its earliest form in his 1890 book? Where were the prior observations, and what were they specifically, that proved the contraction? They had what they would consdier a hypothesis, but it really couldn't have been that since it was contrived without observation of any particular phenomenon...only an interpretation of experimental data. They should've at least tried to observe this even after the scientific method was not completely satisfied. Where are those observations recorded? I would really rather not have to hang my hat of research on a formula that was derived a priori. All I'm trying to do is find observations that led to it. I know I sound like a broken record, but I love science, and even more I love history. Thanks.
Well, of course, an important point about mathematics and physics is that the history of how theorems and theories came about is irrelevant to their validity. You can always start with a clean sheet of paper and develop the theory afresh.

In terms of space and time, for example, you can start with a few basic assumptions about homogeneity and isotrophy and show that there are only two possibilities:

1) Newtonian space and time, withe the Galilean transformation.

2) SR spacetime, with the Lorentz transformation.

To decide between the two you need an experiment: any experiment that distinguishes between the two options will do. One option is that you build a particle accelerator and give particles enough energy so that if 1) were the case they would be traveling at hundreds of times the speed of light; and, if 2) were the case they would only be getting close to the speed light.

In any case, 1) and 2) imply very different theories of energy and momentum. All of modern physics at high energies uses the SR theory of energy and momentum. It is of no practical relevance how the theory came about or whether length contraction is practically testable. The theory of SR rests on its wider predictions at the heart of particle physics, general relativity and quantum field theory.

vanhees71, PeterDonis and bhobba