Why do clocks experience time dilation in special and general relativity?

[Galactic explosion]

I understand time dilation in both special and general relativity in terms of motion, ie. an object in a gravitational field will move slower than an object outside the field, as the spacetime causes the object in the field to take a longer path to the same destination. Same goes for two objects absent of gravity just traveling at a faster rate than the other object next to it, which will cause one of the objects to get to the destination faster, and therefore take less time.

The part of time dilation I don't understand, is the mechanical part of an actual clock. How do two physical clocks with an hour hand and a minute hand that started off with the exact same time, end up showing completely different intervals when one is subjected to extremely high speeds or warped gravitational fields? What exactly causes the hands of the watches to slow down? Do physicists mean this in a literal sense? I just can't wrap my mind around how one watch will change from the other just by one of them traveling near the speed of light. I understand how the time to get to a destination is faster, but the watch PHYSICALLY changes too? How does the watch know to change? The watch shouldn't care what's happening in reality, as it's programmed to a certain MECHANICAL time.

Apparently this has been proven by atomic clocks, measuring extremely small variations in terms of nanoseconds. But I still don't understand how the physical numbers on the clock will change.

 

[phinds]

What exactly causes the hands of the watches to slow down?

They DON'T slow down, they just take different paths through spacetime. If you drive a car at 60 mph from Boston to New York via a fairly straight path and a separate car from Boston to New York at 60 mph by way of Western New York, if they start off with identical odometer readings would you expect them to end up with identical odometer readings? Did either odometer "slow down" along the way?

 

[FactChecker]

In the physics of the moving clock reference frame, absolutely nothing is going on to change the speed of a clock. All the physics of the moving frame is identical to the physics of the "stationary" frame. It's a distortion of time and space that is very hard to detect. Only people in another reference frame can see a "problem" with the moving clocks. That is what made it so hard for people before Einstein to accept. Any non-accelerating reference frame measures the speed of light the same, no matter how fast that reference frame is moving compared to other reference frames. Einstein realized that the only explanation could be that clocks synchronized in one reference frame could not agree with clocks synchronized in another reference frames. That implied that the measurement of elapsed time and the concept of "simultaneous" depended on the reference frame. Clocks in a relatively moving reference frame appear to be moving slower than the stationary clocks. But people in the moving reference frame see nothing unusual about their clocks. Nothing about their clocks have changed -- it is their time and space that has changed.

 

[PeroK]

I understand time dilation in both special and general relativity in terms of motion, ie. an object in a gravitational field will move slower than an object outside the field, as the spacetime causes the object in the field to take a longer path to the same destination. Same goes for two objects absent of gravity just traveling at a faster rate than the other object next to it, which will cause one of the objects to get to the destination faster, and therefore take less time.

The part of time dilation I don't understand, is the mechanical part of an actual clock. How do two physical clocks with an hour hand and a minute hand that started off with the exact same time, end up showing completely different intervals when one is subjected to extremely high speeds or warped gravitational fields? What exactly causes the hands of the watches to slow down? Do physicists mean this in a literal sense? I just can't wrap my mind around how one watch will change from the other just by one of them traveling near the speed of light. I understand how the time to get to a destination is faster, but the watch PHYSICALLY changes too? How does the watch know to change? The watch shouldn't care what's happening in reality, as it's programmed to a certain MECHANICAL time.

Apparently this has been proven by atomic clocks, measuring extremely small variations in terms of nanoseconds. But I still don't understand how the physical numbers on the clock will change.

Unfortunately, your post is full of misconceptions about relativity. The two most important points to note are

1) All motion is relative. There is no such thing as an object moving at high speed; only moving at high speed relative to some other object or observer.

2) Time dilation is about time. Clocks work as normal as simply record the (proper) time that elapses. The clocks themselves are not physically or mechanically affected.

 

[Galactic explosion]

All of your guys' answers make sense. It's just that I don't think those are the answers I'm looking for. For example, I was watching the movie Interstellar a while ago. And the character Cooper gave a watch to his daughter that showed the exact same time as the watch on his wrist. He then traveled to another galaxy, experienced "gravitational time dilation" near a supermassive black hole, to which the time difference was in terms of decades. And when he came back, the time on his watch was stupidly off, relative to her watch. Is this what would literally happen in real life? Or did the movie just want to emphasise how time dilation actually works in a metaphorical sense, using watches?

 

[PeroK]

All of your guys' answers make sense. It's just that I don't think those are the answers I'm looking for. For example, I was watching the movie Interstellar a while ago. And the character Cooper gave a watch to his daughter that showed the exact same time as the watch on his wrist. He then traveled to another galaxy, experienced "gravitational time dilation" near a supermassive black hole, to which the time difference was in terms of decades. And when he came back, the time on his watch was stupidly off, relative to her watch. Is this what would literally happen in real life? Or did the movie just want to emphasise how time dilation actually works in a metaphorical sense, using watches?

You have a common sense idea that time is "absolute". It isn't. If two people start a given point and one sets out and makes a return journey, then when they meet again they will have experienced a different amount of time. If their watches are accurate, this difference will be shown on the watches.

In other words, there is no giant clock somewhere in the universe that keeps time for everyone.

If you want to understand this fully, you need to start to learn relativity.

 

[jartsa]

The part of time dilation I don't understand, is the mechanical part of an actual clock. How do two physical clocks with an hour hand and a minute hand that started off with the exact same time, end up showing completely different intervals when one is subjected to extremely high speeds or warped gravitational fields? What exactly causes the hands of the watches to slow down?

Well, if we have charged particles bouncing around in electric fields, then magnetism has an effect on how the charged particles move. And the magnetism is a frame dependent thing.

https://www.physicsforums.com/threa...lain-electromagnetism-with-relativity.932270/I am not claiming that this is the answer. It might perhaps be an answer in one special case.

 

[Ibix]

And when he came back, the time on his watch was stupidly off, relative to her watch. Is this what would literally happen in real life?

That's not time dilation (although a lot of pop sci sources will describe it as such), it's differential aging. Time dilation is a related phenomenon, but not the relevant one here.

In short, your wristwatch measures "distance" through spacetime in a sense similar to your car's odometer measuring distance through space. It's no particular surprise that two factory-fresh cars with zero mileage that meet up later might have different mileage at their second meeting - they took different routes to get there. Any answer you'd accept for why that might be can be applied to the two watches. They took routes of different "lengths" through spacetime.

It probably is surprising that your watch does that, but there you go. You might want to look up the "twin paradox" which is a simpler scenario with the same outcome.

 

[NoTe]

Nothing about their clocks have changed -- it is their time and space that has changed.

I would add: it's not the nature of their time and space that has changed, just the amounts. So it's not about 'slower time' or 'compressed space' or the like.

 

[phinds]

All of your guys' answers make sense. It's just that I don't think those are the answers I'm looking for. For example, I was watching the movie Interstellar a while ago. And the character Cooper gave a watch to his daughter that showed the exact same time as the watch on his wrist. He then traveled to another galaxy, experienced "gravitational time dilation" near a supermassive black hole, to which the time difference was in terms of decades. And when he came back, the time on his watch was stupidly off, relative to her watch. Is this what would literally happen in real life? Or did the movie just want to emphasise how time dilation actually works in a metaphorical sense, using watches?

Reread post #2

 

[Galactic explosion]

That's not time dilation (although a lot of pop sci sources will describe it as such), it's differential aging. Time dilation is a related phenomenon, but not the relevant one here.

In short, your wristwatch measures "distance" through spacetime in a sense similar to your car's odometer measuring distance through space. It's no particular surprisethat two factory-fresh cars with zero mileage that meet up later might have different mileage. Any answer you'd accept for why that might be can be applied to the two watches. They took routes of different "lengths" through spacetime.

It probably is surprising that your watch does that, but there you go. You might want to look up the "twin paradox" which is a simpler scenario with the same outcome.

That, and other answers like yours are starting to make sense in my head now. I see that each times as the both clocks tick, there will be variations in the hand movements relative to each other based on how fast one clock is traveling through space than the other...I think. Kind of like how when you're traveling 100 km/h in a car relative to another car traveling at 100 km/h in the same direction. Both cars relative to each other will appear to be stationary. So, time for both cars will not exist. But as one car starts to travel a little faster, say 1 km/h faster relative to the other one, then that car will appear to only be traveling at 1 km/h relative to the other car (not 101 km/h) while the other car will appear to be stationary, and vice versa, depending on your position of observation. And similarly, time for one of the cars will also appear to TICK faster or slower depending on the relative position. So I guess that can also be applied to two clocks relative to each other in a gravitational field or just motion through space. I really think I understand it now.

But if I still got that wrong, then whatever. I guess it doesn't take a day to learn this, right? I'll keep learning. Thanks for all the answers!

 

[russ_watters]

The part of time dilation I don't understand, is the mechanical part of an actual clock. How do two physical clocks with an hour hand and a minute hand that started off with the exact same time, end up showing completely different intervals when one is subjected to extremely high speeds or warped gravitational fields? What exactly causes the hands of the watches to slow down? Do physicists mean this in a literal sense?

The clocks aren't changed, it's the amount of elapsed time itself that is different.

 

[FactChecker]

@Galactic explosion, @phinds post #2 is more profound and important than you may realize. If you are interested in Special Relativity, you should get familiar with Minkowski diagrams (aka spacetime diagrams). They let you use simple diagrams to see how moving objects are going through space-time. That tells you how the measurements of space and time are different for a moving object.

One description that I like is that everything is moving through spacetime at velocity c. At one extreme, a "stationary" object is only moving in time at velocity c and not in space at all. At the other extreme, light is only moving in space at velocity c and not in time at all. Anything in between is moving through both space and time at a total velocity c.

 

[NoTe]

One description that I like

...
Wow, that description may be good for some but confusing for others (me for one)! One thing I don't like about it, is the suggestion that there is a continuum from the first extreme to the second. This is not so. There are massive bodies moving at whatever velocity on the one hand, and light on the other hand. They are completely complementary, light is NOT some kind of limit case. Mathematically, the description is in order, though.

 

[FactChecker]

...
Wow, that description may be good for some but confusing for others (me for one)! One thing I don't like about it, is the suggestion that there is a continuum from the first extreme to the second. This is not so.

Maybe I am not understanding you. There is a continuum from one extreme to the other. If v_t and v_s are the "velocities" of an object through time and space, respectively, then c² = v_t² +v_s². An object can be at any non-negative combination of v_t and v_s that satisfies the equation.

There are massive bodies moving at whatever velocity on the one hand, and light on the other hand. They are completely complementary, light is NOT some kind of limit case.

Yes, light is a limit case. No object can exceed the speed of light. An object that is traveling through space at nearly the speed of light must be going through time at nearly 0 "speed" (to the "stationary" observer).

 

[NoTe]

Yes, light is a limit case. No object can exceed the speed of light.

I meant that it is no limit case because no object can even REACH the speed of light (let alone exceed it!)...

For the remaining, your post is touching the very heart of SRT, imho. From the view point of one observer, you could say that the spectrum of velocities is a continuum all the way to c, but I find this very tricky! Remember, that even for an object moving at 0.99c, the speed of light is still measured c (in its IRF). That's what I meant, when I said that the two worlds (of massive bodies and light) are completely complementary and can never 'touch'.

 

[phinds]

...
Wow, that description may be good for some but confusing for others (me for one)! One thing I don't like about it, is the suggestion that there is a continuum from the first extreme to the second. This is not so. There are massive bodies moving at whatever velocity on the one hand, and light on the other hand. They are completely complementary, light is NOT some kind of limit case. Mathematically, the description is in order, though.

It's a heuristic, not to be taken too literally.

 

[robphy]

One description that I like is that everything is moving through spacetime at velocity c. At one extreme, a "stationary" object is only moving in time at velocity c and not in space at all. At the other extreme, light is only moving in space at velocity c and not in time at all. Anything in between is moving through both space and time at a total velocity c.

I think this side discussion of this "moving through spacetime at velocity c"
is related to the idea of "how one takes the limit" from

• a massive particle with timelike 4-velocity with magnitude c and spatial-velocity with magnitude less than c
  to
  
• a massless particle with lightlike 4-momentum with magnitude 0 and spatial-velocity with magnitude c.

From an older post https://www.physicsforums.com/threads/massless-photon.900960/page-3#post-5842652,

The bottom line... one should take care and be more explicit with how the limit is taken (what is being held fixed?).

My $0.02.

 

[robphy]

Now back to the OP...

The part of time dilation I don't understand, is the mechanical part of an actual clock. How do two physical clocks with an hour hand and a minute hand that started off with the exact same time, end up showing completely different intervals when one is subjected to extremely high speeds or warped gravitational fields? What exactly causes the hands of the watches to slow down?

[snip]

But I still don't understand how the physical numbers on the clock will change.

This might help.
It's an old visualization of mine that I recently uploaded to YouTube.

"Visualizing Proper Time in Special Relativity"

• Visualize the mechanism of a ticking light-clock drawn on a spacetime diagram.
• The spacetime paths of the light-signals are emphasized. (The worldlines of the mirrors are not drawn.)

The first is Time Dilation ( for v=(4/5)c ).
The second is the Clock Effect / Twin Paradox ( for there and back speeds of (4/5)c ).



More recently, I have used a slice of this spacetime diagram to perform visual calculations in special relativity on rotated graph paper.
Here is the Clock Effect / Twin Paradox ( for there and back speeds of (3/5)c ).

For more information, check out my PhysicsForums Insights:

• https://www.physicsforums.com/insights/spacetime-diagrams-light-clocks/
• https://www.physicsforums.com/insights/relativity-rotated-graph-paper/

 

[Grinkle]

I see many threads where I think the OP is asking for someone to make intuitive sense of time dilation and starting with the assumption that the best way to achieve this intuition is to discuss the mechanical behavior of clocks.

SR time dilation is a direct consequence of c being absolute. It is not at all intuitive that c is or should be absolute. If it were not an experimentally verified axiom, who would predict or believe it? (I mean that rhetorically but if there is any motivation to wonder if such might be the case I'd be very interested in knowing)

Its not surprising to me that other non-intuitive things spring from this astounding experimentally observed phenomena. Starting with trying to make time dilation intuitive is the wrong place to start from, imho. Start from the math that clearly shows how absolute c leads to time dilation, then ask oneself why an absolute c makes any intuitive sense - I don't have any answer for that, but transferring my "jaw drop" to the observation of absolute c helps me come to grips with time dilation much more easily.

 

[PAllen]

I see many threads where I think the OP is asking for someone to make intuitive sense of time dilation and starting with the assumption that the best way to achieve this intuition is to discuss the mechanical behavior of clocks.

SR time dilation is a direct consequence of c being absolute. It is not at all intuitive that c is or should be absolute. If it were not an experimentally verified axiom, who would predict or believe it? (I mean that rhetorically but if there is any motivation to wonder if such might be the case I'd be very interested in knowing)

Its not surprising to me that other non-intuitive things spring from this astounding experimentally observed phenomena. Starting with trying to make time dilation intuitive is the wrong place to start from, imho. Start from the math that clearly shows how absolute c leads to time dilation, then ask oneself why an absolute c makes any intuitive sense - I don't have any answer for that, but transferring my "jaw drop" to the observation of absolute c helps me come to grips with time dilation much more easily.

Have you seen our FAQ on this?

https://www.physicsforums.com/insights/speed-light-frames-reference/

Look particularly at the Pal reference which shows that the only ways to accommodate isotropy, homogeneity, and the equivalence of inertial frames is either with infinite invariant speed or finite invariant speed. Once you accept these are the only possibilities, any experiment in favor of SR establishes which is true for our universe.

 

[NoTe]

SR time dilation is a direct consequence of c being absolute. It is not at all intuitive that c is or should be absolute. If it were not an experimentally verified axiom, who would predict or believe it?

I agree: It would be nice if there were a theory from which it followed that c be absolute, instead of a theory that took this as a starting point...

 

[robphy]

Starting with trying to make time dilation intuitive is the wrong place to start from, imho. Start from the math that clearly shows how absolute c leads to time dilation...

In fact, that's how all of my diagrams above are constructed.

From the Relativity Principle and the Speed-of-Light Principle,
then you are led to time-dilation, length-contraction, relativity of simultaneity, headlight-effect, doppler effect, causal relations,...
all visible in the spacetime-diagrams of the light-clocks.
(If you look more carefully, you can also see the Lorentz transformation, the Bondi k-calculus, radar methods, and invariant intervals.)

In my opinion, the diagram presents the result... (in a physics-first sort of way: "it's the ticking of a clock")
and then, since the construction encodes the above features,
encourages (for those interested) further development of the underlying formulas.
It seems to me, the other "math-first" (formula-first) approach has been a stumbling block for those trying to understand relativity.
(Abstract arguments about symmetries and reliance on algebraic derivations don't work for the average student.)

In my opinion,
the "math" is not really the "algebra" of Lorentz transformations or other coordinate transformations...
or "derivations" of time-dilation formula or length contraction formula...
rather
it's really the "geometry of spacetime".
The goal should be to understand the geometrical relationships among spacetime events.

 

[Janus]

That, and other answers like yours are starting to make sense in my head now. I see that each times as the both clocks tick, there will be variations in the hand movements relative to each other based on how fast one clock is traveling through space than the other...I think. Kind of like how when you're traveling 100 km/h in a car relative to another car traveling at 100 km/h in the same direction. Both cars relative to each other will appear to be stationary. So, time for both cars will not exist. But as one car starts to travel a little faster, say 1 km/h faster relative to the other one, then that car will appear to only be traveling at 1 km/h relative to the other car (not 101 km/h) while the other car will appear to be stationary, and vice versa, depending on your position of observation. And similarly, time for one of the cars will also appear to TICK faster or slower depending on the relative position. So I guess that can also be applied to two clocks relative to each other in a gravitational field or just motion through space. I really think I understand it now.

But if I still got that wrong, then whatever. I guess it doesn't take a day to learn this, right? I'll keep learning. Thanks for all the answers!

Let's take this example and tweak it a bit to bring it more in line with Relativity. Instead of both cars traveling in the same direction at different speeds, you have two cars traveling at the same speed but is slightly different directions. They both start off from the same point at 100 kph. But in different directions from each other. As each car travels along, it checks its progress against that of the other car, but each car judges this progress as being measured in the direction it itself is driving.
If we label the cars as A and B,this is how things look from A's perspective:

A makes faster progress in the direction that it s driving than B is making in that same direction.
B sees things like this:

B sees itself as making more progress than A in the direction that B is traveling.
This is the equivalent of time dilation due to relative motion in Special Relativity. In SR, the two clocks moving relative to each other measure each other as making less "progress through time" than they are, or in other words, aging slower.

Now consider what happens if B, after having traveled for some distance, alters its direction of travel, but not its speed, so that it is heading back towards the path of A.
This is what happens according to A

B ends up "behind" A. Even if A turns to travel in the same direction as A from this point on, it will always be behind A. With SR, this is the equivalent of a clock traveling away a some high speed and then returning at that speed and showing less time upon returning than the clock that "stayed home". In A's view this happens because B made slower progress during the entire trip.
B sees things a bit differently. Up until it makes it turn, A is "behind" it. But once it turns, A goes from behind to "in front". In the following animation this is shown by having the direction in which B is traveling at an given time as being in the vertical direction. After the turn, A goes from being below B to being above it. After that A still make slower progress than B in the vertical direction, but B never makes up all the distance before it reaches A's path. Again if after this point B turns to follow A it finds itself behind A.

This illustrate the difference that was alluded to in an earlier post between "time dilation" and "accumulated time difference". Time dilation is the difference in the rate of progress B measures for A at any given point of the trip, which during the two legs has B measuring A's clock as running slower. Accumulated time difference is the what we see at the end of the trip and is the result of the accumulation of the the time dilation during the legs, plus the "jump forward" that A appears to make as seen by B when B makes its turn.
In the animation this looks like a instantaneous jump. in reality it would take some none zero time. If we zoom in on A and B and slow down the turn, this what it would look like to B

The difference between the Car example and Relativity between clocks, is that with the car example, we are dealing with differences in direction that only involve two spatial dimensions. Relativity deals with space-time, and involve "direction changes" in both the spatial and time dimensions. The cars travel on different paths along in a two dimensional plane, while with SR, clocks traveling at different speeds take different paths through 4 dimensional space-time.

Directly visualizing this 4 dimensional movement is a bit hard to say the least, which is why most people initially struggle with understanding what "causes" one clock to tick off less time than another.

 

[PAllen]

In fact, that's how all of my diagrams above are constructed.

From the Relativity Principle and the Speed-of-Light Principle,
then you are led to time-dilation, length-contraction, relativity of simultaneity, headlight-effect, doppler effect, causal relations,...
all visible in the spacetime-diagrams of the light-clocks.
(If you look more carefully, you can also see the Lorentz transformation, the Bondi k-calculus, radar methods, and invariant intervals.)

In my opinion, the diagram presents the result... (in a physics-first sort of way: "it's the ticking of a clock")
and then, since the construction encodes the above features,
encourages (for those interested) further development of the underlying formulas.
It seems to me, the other "math-first" (formula-first) approach has been a stumbling block for those trying to understand relativity.
(Abstract arguments about symmetries and reliance on algebraic derivations don't work for the average student.)

In my opinion,
the "math" is not really the "algebra" of Lorentz transformations or other coordinate transformations...
or "derivations" of time-dilation formula or length contraction formula...
rather
it's really the "geometry of spacetime".
The goal should be to understand the geometrical relationships among spacetime events.

I like all this, but the question was why is light speed invariant, not what or how you make deductions from it. The FAQ addresses this. After dealing with just assuming it, and understanding that well, understanding there are are only two distinguishable choices is about as close as you can come to why.

 

[weirdoguy]

I agree: It would be nice if there were a theory from which it followed that c be absolute, instead of a theory that took this as a starting point...

You can derive the existence of invariant speed by assuming principle of relativity, and homogeneity and isotropy of spacetime. It's harder mathematically, so it's not presented that often, but it exists. But still, you have to assume something.

 

[PAllen]

You can derive the existence of invariant speed by assuming principle of equivalnce, and homogeneity and isotropy of spacetime. It's harder mathematically, so it's not presented that often, but it exists. But still, you have to assume something.

Not the principle of equivalence, but the principle of relativity. The former has to do with gravity. And what you derive this way is that the invariant speed can be either infinite or finite, with Galilean relativity being the infinite case.

 

[weirdoguy]

Yep, corrected, thanks

 

[SiennaTheGr8]

Maybe I am not understanding you. There is a continuum from one extreme to the other. If v_t and v_s are the "velocities" of an object through time and space, respectively, then c² = v_t² +v_s². An object can be at any non-negative combination of v_t and v_s that satisfies the equation.
Yes, light is a limit case. No object can exceed the speed of light. An object that is traveling through space at nearly the speed of light must be going through time at nearly 0 "speed" (to the "stationary" observer).

I've never understood this "speed through time" business. It's always sounded like pop-sci nonsense to me, but maybe I'm missing something. Usually I've seen people suggest that there's an inverse relationship between "speed through time" and "speed through space," so that they must always sum to $c$, which would look something like this in your notation:

$c = v_t + v_s$

(I know you didn't say this). What's that about? Anyone know?

Now, the equation you wrote was:

$c^2 = v_t^2 + v_s^2$

What is $v_t$ here? The squared four-velocity magnitude looks similar, but there's a minus sign:

$c^2 = (\gamma c)^2 - (\gamma \vec{v})^2 = \left( c \, \dfrac{d t}{d \tau} \right)^2 - \left( \dfrac{d \vec{r} }{d \tau} \right)^2$.

So is the "speed through time" supposed to be $dt / d\tau$? And is this just supposed to be a rough analogy to the equation I wrote, or is there some other relativistic equation I'm not thinking of?

 

[robphy]

I like all this, but the question was why is light speed invariant, not what or how you make deductions from it. The FAQ addresses this. After dealing with just assuming it, and understanding that well, understanding there are are only two distinguishable choices is about as close as you can come to why.

EDIT:
Actually, the question I was addressing [and quoted above in #23 ] was

Starting with trying to make time dilation intuitive is the wrong place to start from, imho. Start from the math that clearly shows how absolute c leads to time dilation...

..which is not the same as why c is invariant.
end EDITYes, I agree [about the two distinguishable choices].
My point is that: the typical beginning student will NOT appreciate that feature... it's too abstract.
The less-typical but enthusiastic and more-mathematically equipped student may appreciate that feature.

In my opinion,
this letter to the editor
http://aapt.scitation.org/doi/10.1119/1.17728
"Lapses in Relativistic Pedagogy" by Mermin*
makes some good points

...Lorentz transformation doesn't belong in a first exposure to special relativity. Indispensable as it is later on, its very conciseness and power serve to obscure the subtle interconnnectedness of spatial and temporal measurements that makes the whole business work. Only a loonie would start with real orthogonal matrices to explain rotations to somebody who had never heard of them before, but that's how we often teach relativity. You learn from the beginning how to operate machinery that gives you the right answer but you acquire little insight into what you're doing with it.

(* Mermin has given one of these derivations-without-explicitly-using-the-speed-of-light-postulate http://aapt.scitation.org/doi/abs/10.1119/1.13917 )

 

[FactChecker]

I agree: It would be nice if there were a theory from which it followed that c be absolute, instead of a theory that took this as a starting point...

I don't know if you saw @PAllen 's post one minute before yours. He refers to this paper https://arxiv.org/pdf/physics/0302045.pdf. It makes some very natural and intuitive assumptions (although the following math may not be so intuitive) and shows that there are only two mathematical possibilities: 1) old Galilean relativity and 2) Einsteinian relativity for some constant maximum velocity, a. When experiments showed that the speed of light was constant, it followed that Einsteinian relativity with a = c was correct.

 

[Galactic explosion]

Let's take this example and tweak it a bit to bring it more in line with Relativity. Instead of both cars traveling in the same direction at different speeds, you have two cars traveling at the same speed but is slightly different directions. They both start off from the same point at 100 kph. But in different directions from each other. As each car travels along, it checks its progress against that of the other car, but each car judges this progress as being measured in the direction it itself is driving.
If we label the cars as A and B,this is how things look from A's perspective:
View attachment 217916
A makes faster progress in the direction that it s driving than B is making in that same direction.
B sees things like this:
View attachment 217917
B sees itself as making more progress than A in the direction that B is traveling.
This is the equivalent of time dilation due to relative motion in Special Relativity. In SR, the two clocks moving relative to each other measure each other as making less "progress through time" than they are, or in other words, aging slower.

Now consider what happens if B, after having traveled for some distance, alters its direction of travel, but not its speed, so that it is heading back towards the path of A.
This is what happens according to A
View attachment 217918
B ends up "behind" A. Even if A turns to travel in the same direction as A from this point on, it will always be behind A. With SR, this is the equivalent of a clock traveling away a some high speed and then returning at that speed and showing less time upon returning than the clock that "stayed home". In A's view this happens because B made slower progress during the entire trip.
B sees things a bit differently. Up until it makes it turn, A is "behind" it. But once it turns, A goes from behind to "in front". In the following animation this is shown by having the direction in which B is traveling at an given time as being in the vertical direction. After the turn, A goes from being below B to being above it. After that A still make slower progress than B in the vertical direction, but B never makes up all the distance before it reaches A's path. Again if after this point B turns to follow A it finds itself behind A.
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This illustrate the difference that was alluded to in an earlier post between "time dilation" and "accumulated time difference". Time dilation is the difference in the rate of progress B measures for A at any given point of the trip, which during the two legs has B measuring A's clock as running slower. Accumulated time difference is the what we see at the end of the trip and is the result of the accumulation of the the time dilation during the legs, plus the "jump forward" that A appears to make as seen by B when B makes its turn.
In the animation this looks like a instantaneous jump. in reality it would take some none zero time. If we zoom in on A and B and slow down the turn, this what it would look like to B
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The difference between the Car example and Relativity between clocks, is that with the car example, we are dealing with differences in direction that only involve two spatial dimensions. Relativity deals with space-time, and involve "direction changes" in both the spatial and time dimensions. The cars travel on different paths along in a two dimensional plane, while with SR, clocks traveling at different speeds take different paths through 4 dimensional space-time.

Directly visualizing this 4 dimensional movement is a bit hard to say the least, which is why most people initially struggle with understanding what "causes" one clock to tick off less time than another.

That was beautifully explained! Thank you!