# Short question about length contraction

1. Aug 15, 2014

### HALON

The at rest distance between Betty the astronaut and a flag in open space is $1$ unit. If Betty approaches the flag at constant speed while laying out metre rulers, will she have laid out $1$ unit of at rest rulers when she reaches the flag? Or will the number of rulers laid out exceed $1$ due to length contraction?

2. Aug 15, 2014

### Staff: Mentor

You'll have to specify two more things before we can answer your question.

First, exactly what do you mean by "the rest distance between Betty and the flag"? Betty and the flag are moving relative to one another so the distance between them is constantly changing. I think what you mean is equivalent to: there are two flags, they are at rest relative to another, they are one unit apart according to an observer at rest relative to both flags, and moving Betty starts placing meter sticks as she passes one of the flags moving towards the other.

Second, when Betty places the meter sticks, are they at rest relative to her or relative to the flags after she places them?

3. Aug 15, 2014

### HALON

Yes, I see how it might get tricky. We could say the distance between Betty’s starting point and the flag was pre-determined by radar signal. The flag may be on a planet (but let’s forget gravity) then Betty accelerates from a previous position only until she reaches the starting point where she begins to lay out a ruler, waits until her motion carries her to the end of the ruler, then lays out another ruler and so on. Hmmm, I guess the “laying out” part involves acceleration which makes it complicated. I’m sorry I asked now!

4. Aug 15, 2014

### Simon Bridge

You can see how you have to be very careful with your descriptions in relativity...
Was it pre-determined by a radar signal from a radar range-finder that was stationary wrt the flag, or one that was moving?
This is the most important bit of information.

Do you want the meter sticks to end up stationary wrt the flag when Betty "lays them out"?

Say Betty runs a course that is proper length $D$ meters long, in Betty's frame that would be $D/\gamma$ meters long: length contraction.

Betty drops a marker every one of her meters, starting with one that hits the start line.
The marker hits and sticks to the course and Betty is a perfect shot.
Kinda like a relativistic fence-builder putting in fence posts.

That what you had in mind?

5. Aug 15, 2014

### HALON

Yes, it was stationary wrt the flag. Betty and an observer at the flag agreed beforehand on the distance.

To end up stationary would involve negative acceleration which would affect the number of markers counted. If she just brushed the flag in passing it might be easier to calculate.

I just thought of someone pacing out a distance, originally.

6. Aug 15, 2014

### Simon Bridge

Betty could brush the start and finish lines and time how long it takes for them to pass (remember, in Betty's frame, the start and finish are moving!) Betty knows v and has timed t so gets d=v/t ... and finds out that d measured this way is smaller than the d obtained by the radar.

... sure - so we imagine that Betty is "pacing out" a distance. The faster she goes, the fewer paces it takes to get from the start flag to the finish flag.
OK you'd normally lengthen your stride to go faster, but Betty is ultra-disciplined and always keeps the same pace length as measured by the ruler she is holding at the time.

That better?

Thing is - in each one of your descriptions, you are leaving something out... you kinda trail off.
Like you suggested touching the flag, but not what that was supposed to do; and pacing the distance but not what it was about the paces you wanted to know.
Try making a complete description.

7. Aug 15, 2014

### HALON

I'll try again

8. Aug 15, 2014

### HALON

Touching the flag is the signal for Betty to know when to stop counting paces.
This is the part I wasn't sure about. In a rotation, it is said, it takes more paces to complete a revolution, so this is an interesting difference.

I’ll try to formulate it. First, the gamma factor is
$\gamma=1/(1-(v/c)^2)^{1/2}$

I'll call the distance between two Einstein synchronized points (Betty’s flag and your flag) the rest frame distance, or $d$. So the travel time between her flag and your flag for the rest frame distance is $d=vt$.
In your frame, if $t$ is the time travelled by Betty to your flag, then in Betty’ frame $d_{Betty}=vt(1/γ)$.
To square this up with your distance we get $d_{Betty}=d_{Simon}(1/γ)$
From the previous relation we see Betty's distance is contracted relative to yours.

9. Aug 15, 2014

### ghwellsjr

You had Betty accelerating from a previous position to the "starting" point.

I have a suggestion:

Why don't you make that position far removed from the starting point and why don't you have her remain at rest for awhile there so that she will have time to use radar to measure the distance between the "starting" point and the flag? Then have her accelerate rapidly to her final speed but reach it long before she gets to the "starting" point so that she will have time to use radar to again measure the distance between the "starting" point and the flag. She can also use radar to measure her final speed relative to the "starting" point (and the flag). Then she can measure how long it takes for her to traverse the distance from the "starting" point to the flag and confirm that the distance she traveled matches the distance she determined from radar?

I hope you have given up on the idea of her laying out rulers between the "starting" point and the flag because Special Relativity cannot address that issue as it would require an analysis of the structure of the rulers and how they deform as a result of extreme accelerations. And you would have to tell us exactly the process by which Betty determines when to lay out each next ruler. You might think that it would be obvious but it's not. For example, does she accelerate just one end of each ruler? Which end? How can she be sure that when the acceleration force propagates and dissipates through each ruler, the two ends of each ruler butt up against the adjacent rulers? Then you have to tell us what the final Proper Length of the rulers are after the vibrations dampen out. And that will be the answer to your question of how many rulers are laid out between the "starting" point and the flag.

To reiterate: you have to tell us that answer because Special Relativity can't.

10. Aug 15, 2014

### ghwellsjr

You keep adding and changing things:

Originally you had one flag and one "starting" point. Now you have two flags called Betty's flag and your flag. Who are you? And which of these new flags corresponds to the original flag? Why don't you just label them starting flag and ending flag or final flag? And now you're talking about different frames. You should define your scenario according to a single frame and show how Betty makes her measurements or does whatever you want her to do. Then you can transform to any other frame and see how the same scenario looks in that new frame but it won't change any observerations, any measurements, any calculations, any conclusions.

11. Aug 15, 2014

### HALON

It's a tough one, yes. Yet other authors have used the "pacing out" method on spinning disk's to conclude that more of them will fit per revolution than ought to be measured when the speed is close to zero. And that's supposed to be a special relativity problem.

But even if one walks there is acceleration, despite overall speed being uniform, because the feet are planted while the body's momentum does the rest.

12. Aug 15, 2014

### ghwellsjr

You're comparing apples and oranges. In your scenario, the rulers are momentarily accelerated from a rest state with Betty to a rest state with the "starting" point and the ending flag. In the spinning disk there is constant acceleration.

Also, there is neither any assurance that the Proper Lengths under both conditions before and after acceleration are the same (but you can stipulate that) nor that any particular way that Betty ejects that rulers results in them butting up against each other (but you can stipulate that). If you make both stipulations, then you have just answered your question and the proplem becomes of no interest because you have just glossed over the real issues that Special Relativity can't handle.

13. Aug 15, 2014

### HALON

Speaking of which..

Length contraction under conditions of constant acceleration in a rotation, compared to between inertial states, seems more intuitive. A rotating body and an inertial body can compare data in real time if their separation is constant; everything is asymmetric. It’s useful too, as evidenced by GPS for our devices.

Compare how messy the physics is for a twin to travel to some faraway planet and back, and having to account for simultaneity, and positive and negative acceleration and all the weird experiences in between. Even when two inertial bodies pass each other what they see is not “real” because everything is suspended in symmetry, so to speak. They can’t interact. You have to wait for the symmetry to break to compare data.

14. Aug 17, 2014

### ghwellsjr

It's not intuitive to me.

I think you are extrapolating the asymmetric Time Dilation between those two bodies and assuming that a similar trivial Length Contraction relationship also exists. The reason why it is trivial for Time Dilation is that we conceptualize clocks that have no dimensions and which maintain their ideal time keeping abilities under any acceleration. But you can't do the same thing with extended bodies specifically where you want to discuss Length Contraction. As I have said before, Special Relativity can't answer the question of how a body is deformed under acceleration and so you can't just set up a scenario by simply saying that you have two identical bodies, one inertial and the other rotating around it. You have to supply additional information on the changing shape of the rotating object.

If you want to make claims that it is intuitive, please actually set up a precise scenario with all the required details and then show how each body measures the Length Contraction of the other body.

What evidence is there for Length Contraction by GPS in our devices? Are you thinking about Time Dilation?

Your rotating body scenario is messy because you are trying to explain Length Contraction with it. The twin scenario is very well understood and precisely defined under Special Relativity and nothing is weird about it because we always choose to ignore the Length Contraction of the traveling twin. I agree that if we wanted to include the Length Contraction of the traveling twin as a result of his accelerations, then Special Relativity would be inadequate but I have never seen an example of where anyone asked or anyone addressed that issue. Can you?

No, what they actually see and measure is very real. Why would you say something like that? And they can interact. Why would you say they can't? They can certainly compare data under a symmetrical situation. Why do you think otherwise?

Maybe before you try to understand accelerating bodies, you should learn how Special Relativity deals with inertial bodies. It's really very simple. Einstein said so. And he should know.

Last edited: Aug 17, 2014
15. Aug 17, 2014

### HALON

Actually, yes, that’s what I was thinking- time dilation, or $γ=1/\sqrt{1-v^2/c^2}$. I am aware of the difficulty in finding direct evidence for length contraction. Computer modelling of the length contraction of particles is just that, a model. (There may be other evidence that I’m not aware of, or can't understand). In any case I just imagine $γ$ acts as a kind of magnification factor when we apply $e=γmc^2$ to the rotating body. Because the total energy of the body’s system is invariant we need to multiply the previous energy mass conversion result by $1/γ$ to bring it back to terms. So it’s cancelled out. Length contraction is just the cancelling out procedure for time dilation. I struggle with the idea physical bodies deform due to length contraction; I don’t get it. It’s like saying a body viewed through binoculars must deform to fit into our normal view.

Yes, I was thinking of time dilation. Length contraction just balances the books.

I concede I was thinking of time, not length. To me, length contraction is not as real as time dilation because we can’t measure it directly. So what I really meant was that conceptually, time dilation in a rotation can be measured directly as it happens, continually. I guess the lengths of received wavelengths sent by the other body change, implying they contract in the rotating frame, but that’s not the kind of length that is meant.

Sure, it’s a short interaction only. They can interact for an instant, and whoosh! They are separated. Bob can’t remark to Alice “you seem so slow today” and wait for Alice to say “so do you”. In that sense each other’s apparent slowing is not real. How can they compare data in real time? As Bob and Alice approach each other their signals are blue shifted, then there is an instant of interaction...followed by red shifted signal exchanges. But if Alice is rotating around Bob, Alice can tell Bob “you really are quick today” while Bob can confidently say “that’s because you are so slow compared to me” and carry on continual interactions. Alice's body length won’t change; she can lie next to a measuring rod parallel to the direction of rotation. We can only assume Alice's length contracts due to time dilation.

You may be amused to know I have heard of Alfred Einstein and the Lorentz contraption. Something to do with pressing against the ether. Lorentz got a Nobel prize after he gave Alfred an equation, so Alfred stuck to the same terminology out of respect for the man who thought objects got squashed against the gas.

16. Aug 18, 2014

### ghwellsjr

What about the Michelson Morley Experiment where Length Contraction was devised as an explanation for the null result?

It's very easy to show that Length Contraction does not cancel out or balance out Time Dilation if you recall that Length Contraction applies only along the direction of motion whereas Time Dilation applies no matter the direction of motion. For example, consider a light clock where the light reflects back and forth perpendicular to the direction of motion and Length Contraction does not play a part in what is happening whereas if you rotate the light clock 90 degrees along the direction of motion it does.

Of course Bob can remark to Alice and wait for a response back from Alice. Why are you saying this?

Whatever is apparent to each of them is real. I don't understand why you say this.

If they are separated, then they will have to wait and compare data later. Why is that a problem?

True. You have been talking about Doppler not Time Dilation. Do you know the difference?

But they cannot do it in real time. They have to wait for the signals to propagate between them just like in the non-rotation case. Why do you see a significant difference between the two cases? I don't understand why you think one case is more real than the other.

Alice's body length won't change from what? Do you realize that you can place a dozen measuring rods made of different materials side by side and them subject them to extreme identical acceleration and they may all end up different lengths? There's just no way to address this issue with Special Relativity.

The difference between Lorentz's explanation is that he thought it was necessary to understand Length Contraction in terms of an absolute stationary ether through which we on the surface of the earth are not stationary while Einstein realized that the same equations would work just as well if you assumed that we were stationary in that ether (in effect). That's why he called his theory simple.

17. Aug 18, 2014

### HALON

Length contraction was devised as an explanation, yes, but what experiment directly shows a “yardstick” to Lorentz contract? Only the number of ticks in a clock contract which, of course, is what is predicted by the length contraction hypothesis. So it is indirect evidence. I'm not saying it's not enough evidence to satisfy me. But it makes me wonder if length itself is a kind of abstraction.

What I meant by “cancelling out” was in the sense of “normalization” so that proper time and proper lengths remain invariant in the moving frame. When you say time dilation applies no matter the direction of motion you can imagine a body’s proper time extending infinitely, like an expanding balloon without boundary. To impose a boundary we must first specify a value. Then we can make the balloon’s surface contract parallel (every which way) to the time expansion because it is stretched by whatever value of time volume we specified. But we still can’t measure this contraction directly. It’s existence is implied by using clocks, not yardsticks.

I was thinking of the case where Alice moves away from Bob. Communications are affected by feedback delay.

It's only a minor point. Alice might have died, but Bob may still be receiving transmissions and think she’s alive. The delay causes phantoms.

This is my understanding. Between inertial frames, a clock moving toward Bob will seem to run fast right up until the point it reaches Bob. Then for an instant, the clock looks like it's running slow. Now, if the clock moves away it will seem to run even slower than the rate it runs according to time dilation. So the Doppler Effect is kind of independent of time dilation. However, the situation is simpler for a rotation. Here, what is called the Transverse Doppler Effect results in the perception agreeing with actual time rates, for both inertial and non-inertial observer, if the separation is constant. It's only when Alice leaves her orbit of Bob that the mismatch between signals and actual clock time appears.

I know that. But any evident length contraction due to material deformation would not be due to the Lorentz contraction.
Both cases are objectively real, it’s just that the currency of feedback between negligibly separated bodies in the rotation example makes it seem more relevant. In both cases there is a delay between propagation and reception. Yet in the rotation example the delay is minimal if Bob and Alice are close enough, which is as real as “real time” gets.

It's a very simple theory. I wish I'd thought of it.

18. Aug 18, 2014

### ghwellsjr

Length has a precise definition that is consistent with all measurements and all evidence. But it is also dependent on the frame of reference which I guess is why you think of it as an abstraction.

Proper Time and Proper Length are defined in the rest frame of the object, not a frame in which it is moving. I can't make sense of the rest of your paragraph except for the last sentence. Nowadays, length is not just implied by using a clock, it is defined by using a clock.

You should quit thinking of Length Contraction as something that happens to a body as a result of its acceleration but rather simply as a result of applying the Lorentz Transformation to the coordinates of a body when you change from one Inertial Reference Frame to another IRF moving with respect to the first one. It's simply a coordinate effect. As such, it doesn't have to be measured. It's like defining two temperature scales and being concerned about finding some experiment to prove that the formula for converting from one scale to the other is experimentally verifiable.

The transmissions wouldn't deceive him. If she says shes still alive at a particular age, then he will think she is still alive at that age. If she dies at some age and the transmissions stop, then later when he notices the lack of transmissions he can deduce the age at which she died. I don't know why you are concerned about these issues.

Also, when you say "Alice might have died" you are implying that there is an absolute time for which such a statement would be true. Even in Bob's rest frame, where there is a precise definition of simultaneity, it is understood that Bob cannot determine the coordinates of distant events until the evidence of them propagate to him at the speed of light according to his rest frame.

Yes, in the rotational scenario, the Transverse Doppler equals the Time Dilation but only for the one particular IRF in which the non-rotating body is at rest. In other IRF's they are not equal. I fail to see why you are placing so much emphasis on one particular scenario. Special Relativity handles all scenarios where we can ignore the effects of gravity equally well.

If you are trying to show a relativistic effect, then you can't relegate it to the realm of negligibility or the effect will go away. What you are talking about are examples such as where we ignore a finite acceleration interval when the total scenario is hundreds of times longer or where we ignore the finite size of a spaceship when the distance it is traveling is millions of times greater.

19. Aug 19, 2014

### HALON

OK

The rest frame is also a uniform moving frame if there is no acceleration.

Your observation about time not being dependent on direction (and the fact length is defined by time) leads to a geometric description. Namely, a body’s proper time extends in all directions, like a bubble. That's the part that didn't make sense to you, but I guess it's just a heuristic and nothing more. I blame all those introductory textbooks that inculcated me with the concept of relativistic mass, which is just mass multiplied by the time dilation factor; it's how the bubble idea arose in my mind.

OK

OK

OK

OK

OK

20. Aug 20, 2014

### ghwellsjr

The term "rest frame" has no meaning by itself. It's a shorthand way of saying a frame in which a particular object is at rest so you must always include the full expression, "the rest frame of the object".

You can always transform the coordinates of the rest frame of an object to the coordinates of another frame which is moving with respect to the rest frame but those are two different frames so that leaves your statement in limbo land. I don't know what you mean or why you would say that.

I think you have come to a wrong conclusion. Are you thinking that time stops for a photon, so that the time of each tick of a clock (so to speak) propagates outward in all directions at the speed of light like an expanding soap bubble? Is that what you're thinking?

Last edited: Aug 20, 2014
21. Aug 21, 2014

### HALON

Yes I think I understand your point. I just meant the rest frame of the object with respect to the coordinates of another object.

My next answer is longer than I thought it would be.

I'll put it in terms of the Doppler effect. When the distance between the signal emitter and the receiver widens, more wavelengths will fill the wider distance which causes the receiver to perceive them as a lower frequency as the signal is "stretched" over a longer distance. And the converse is true when the distance shortens. But in the very different circumstance of one body rotating about another body, the radial distance between them is constant. Now the central body receives a lower frequency, while the rotating body receives a higher frequency. This is called the transverse Doppler effect. So rather than the signal filling a longer or shorter distance, in this situation the signals fill a larger or smaller time volume. I don't know how else to picture it. (You can draw it quite simply using each clock "tick" as the radius of the time bubble, and the Lorentz transformation is the same). Anyway, Einstein said a body's acceleration is equivalent to a gravitational field- which is associated with mass. So if you use $e=γmc^2$ you can describe the bubble's increase in relativistic mass and come to an analogous conclusion regarding a body's relative increase in mass. Relative and relativistic mean different things. To illustrate the idea, see this image from Wikipedia

http://en.wikipedia.org/wiki/Gravitational_redshift#mediaviewer/File:Gravitational_red-shifting2.png

A body with greater relative mass receives a higher frequency of wavelengths, just as a body with greater relativistic mass does. You can re-imagine the larger sphere representing the rotating body compared to the central smaller sphere. But if you do so, the sphere's shown are now filled with relative time, not relative mass. (Or you can say each is composed of different relativistic mass, but not necessarily of different relative masses.).

Einstein introduced the equivalence principle. But equivalence does not mean identical. As I'm trying to convey, relativistic mass is not identical to relative mass. But as long as this distinction is clear, then it's OK to say a body's relativistic mass increases while its relative length contracts, although it's kind of using mixed terminology. Maybe that's why the term relativistic mass is avoided nowadays. In my early readings it was difficult to understand why writers would say the relativistic mass would increase, while elsewhere they would say it's length contracted. The distinction between relativistic and relative were not clear in the textbooks I read.

22. Aug 21, 2014

### Simon Bridge

I don't think that means anything - you can only have motion wrt a rest frame ... which is the frame in which some observer is at rest.

You can talk about how the rest frame of an object is moving wrt the rest frame of another object - in special relativity that would just involve stating the relative velocity of the two frames. But in that case, "frame" has the same meaning as "rest frame". You've seen this in introductory lessons which talk about the S frame and the S' frame. S' is the rest frame of observer O' and S is the rest frame for observer O. It is usually set up so that S' moves at speed v in the +x direction in frame S while S moves at (the same) speed v in the -x direction in the S' frame.

... um... when the receiver moves away from the emitter, the wave peaks arrive further apart in time - which results in the reduced frequency.

Or do you really mean that the farther apart receiver and emmitter, the longer the wavelength?
Even when they are stationary with respect to each other?

Note: gravity is subject to GR, your questions are in SR. Please try to avoid mixing the models up.
You also need to be more careful with your descriptions ... i.e.
Time has only one dimension so it cannot have a volume. So what is it that you are calling a "time volume"?

The old "relativistic mass" is better understood as part of kinetic energy. "Rest mass" is just "mass" or "invarient mass". You are correct that it is easy to get confused when you use the term "relativistic mass" - so just don't use the term.

I'll leave ghwellsjr to decide if you have made the mistake suspected - it sounds like that to me.

23. Aug 21, 2014

### HALON

I understand that.

You could look it at that way too. When a train blows its whistle while moving away, the sound waves must cover the extra distance before reaching the observer. Doesn’t a moving light source follow the same principle? We have red-shifts to indicate stars are moving away.

Not quite. If emitter and receiver are not moving with wrt each other then the wavelength is what it is in the rest frame of the observer. You need to introduce acceleration in an orbit for asymmetry to arise.
This is the fascinating part. They are not exactly stationary; one body is rotating about the central observer. It was pointed out to me that the central observer will receive red-shifted signals from the rotating emitter, while the rotating emitter must therefore receive blue-shifted signals from the central emitter. Again, it was pointed out to me this is called the transverse Doppler effect. Yet the signals are not covering any extra or lesser distance, they are only covering dilated time and contracted time, if that makes sense.

Time is a scalar; distance is a vector. So yes, it is alien to speak of time as having volume. But in a strictly confined context you can use it to describe a calculation. I’ll try be careful...and I will claim in advance there is nothing new about it, or contradictory. Here goes:

It strikes me that in the rotation case we have GR and SR rolled into one. There is constant acceleration of the angular velocity- that’s the GR part- and there's time dilation and length contraction- which is the SR part. After a few steps in calculation, we find the diagonal path (DP) traced by a photon in a moving light clock relative to a "stationary frame" may be given as $1h/(c^2-v^2)^{1/2}$, where $h$ is the separation between the mirrors. It turns out the result is always longer than $1$ for the light clock that is moving (when viewed from a stationary frame). Now, imagine the DP of a photon in a light clock that is momentarily co-moving with the rotating frame for some very short distance at uniform speed (i.e. as part of a polygon fitted around the circle). Then the wider DP for the co-mover compared to the vertical path traced by a photon in a relatively “stationary” light clock (which always equals $1h/c$) must correspond to a higher frequency of received wavelengths if sent from the central frame. So the wider DP exactly equals the time dilation factor, $1/(1-v^2/c^2)^{1/2}$, if you substitute $1$ for $h$ in the first equation. This surely isn’t new.

When I think of dilation I think of something that "dilates" like the pupil in the eye. The higher received wavelengths (the Transverse Doppler effect) “fit” into the dilated dimension of the rotating frame. Because the dimension is dilated, more wavelengths have to be accommodated for each “tick” of the clock that approaches synchronization with the earlier mentioned co-moving light-clock. Therefore the wavelengths travel longer in the dimension of time (from the "stationary" view) just like they travel longer in the dimension of distance between widening inertial frames. We can view it as a relativistic time expansion so the properties of mass expand relativistically too, which in no way should be confused with absolute quantities of mass.

There are interesting papers pro and against the use of “relativistic mass”. Some properties depend on motion relative to the observer (TR Sandin, In Defense of Relativistic Mass, American Journal of Physics, 1991) I found Sandin’s argument persuasive. Rest mass is rest mass, which does not change just because its relativistic mass changes. Anyway, that sounds like another thread starter. But I think it comes down to being careful with the term and its correct context, rather than total avoidance.

24. Aug 21, 2014

### Simon Bridge

How do you know if it is the light source or the light observer who is moving?

Oh you intended your statements to be read in the context of a non-inertial reference frame. I'll leave that to others to avoid hitting you with too many voices at once.

Distance is a scalar also - it is the magnitude of the displacement vector. I'm being pedantic because I suspect that your habit of being imprecise is the cause of much confusion between yourself and those trying to talk to you.

You'll also notice that, in relativity, time is another length.
It is common to use the invariant interval to define a metric as a way to deal with geometry in space-time.

Fair enough - I'll let someone else wade through that ;)
Still don't see what you mean by the volume of time.

No wonder you are having trouble - time is 1D and the pupil of an eye is 2D, so there is no useful simile to be had there either.

Try to avoid poetic language in your descriptions if you want to be understood.
Perhaps you can demonstrate how the wave fits into the dilated time by a space-time diagram or by reference to the metric or something more standard like that?

Avoid it all together. You are having enough problems with imprecise and ambiguous terms already.

Meantime I'll watch how others answer you some more ;)

25. Aug 22, 2014

### HALON

I don’t have the skills to arrive at the equivalence principle using space-time diagrams. Maybe somebody can explain that. This is how I get there:

Time is 1D, but a clock face is 2D (and potentially even 3D). From the outset, this is a purely idealized imaginary clock (old fashioned circular type) but what it’s made of is irrelevant.

The double diagonal path of a photon in the Michelson-Morley experiment is $2h/(c^2-v^2)^{1/2}$. For simplicity, let the separation between the mirrors $h$ and speed of light $c$ be equal to $1$. Next, plug in any value for $v$ which is a proportion of $c$ and calculate. Whatever you found can equal the big hand’s length, the radius $r$ of the imaginary clock. So the double diagonal path = $r$

Here is an exercise. You found the radius already. Now find the circumference and area of the clock face when $v=0$. Now divide the circumference by the area. What does it equal? Next divide the area by the circumference. What does it equal? (Hint: circumference=$2πr$ and area=$πr^2$)

Now do the same exercise for any value of $v$, like 0.6, 0.8 or whatever. Do you notice anything familiar? Compare the results with the Lorentz factor and length contraction factor!

The imaginary clock face’s relativistic mass has increased by $γ$. Because energy is conserved in a system, the big hand’s absolute perimeter speed is the same as what it was when $v=0$. So naturally it will take longer to complete a revolution of the dial, because some of the energy has been diverted into the dimension of time. It will take $γ$ times longer to complete a revolution.

This is what is meant by time dilation and I credit Brian Greene, author the Elegant Universe for helping me think this way.

When $v$ increases, the double diagonal path increases in length. Imagine a stream of photons bouncing between the mirrors. As the path traced by the stream lengthens, then whoever travels with the mirrors (i.e as co-moving temporary observers) will see the same frequency of light for each cycle of reflection. But a perceived light sent from an inertial central frame will increase in frequency by $γ$. That’s because some of the wavelengths have been diverted into the dimension of time. More of them are crammed into this dimension, due to the asymmetry of time between accelerating and inertial bodies. So they are perceived as being more frequent.

Now, this imaginary clock’s hand rotates about the centre. Thus the hand is constantly accelerating. As the clock-face’s relativistic mass increases, the frequency of centrally emitted light as perceived in the clock-face’s frame increases by an equivalent amount to the case of acceleration caused by gravity. That’s called the equivalence principle.

And that’s how, in my bumbling way, I see GR and SR morphing into one.

Avoid relativistic mass? I didn’t invent the term. Relativistic mass helps me conceive the equivalence principle directly and simply. I agree it is used imprecisely and ambiguously by others. Some authors employ relativistic mass to explain why an object can’t go faster than light. They say it takes more and more energy to accelerate a larger and larger relativistic mass, implying that relativistic mass is something that gets more physically massive. That’s just wrong.