Length Contraction causes Time Dilation?

[Max™]

I had a big epic discussion on another board after someone stated that "if I were to travel to Betelgeuse at a sufficient velocity I would reduce the distance between myself and Betelgeuse until it is say, 2 light years, which means I would only experience 2 years or so during my journey", this jumped off the page at me as a huge error.

He and others insisted that I must not know what I'm talking about if I don't agree with him on this, and completely ignored when I explained that this would violate the invariance of the spacetime interval.


I pointed out that yes, the distance would appear shorter from your frame, but you wouldn't claim the distance actually contracted and time dilation is a result of you having to cross 2 light years rather than 640~ light years to Betelgeuse.

I've never seen someone actually claim that it is a real physical effect, or that length contraction means the rest of the universe shrinks due to your motion, have I been terribly mistaken all this time about relativity?

 

[Passionflower]

I had a big epic discussion on another board after someone stated that "if I were to travel to Betelgeuse at a sufficient velocity I would reduce the distance between myself and Betelgeuse until it is say, 2 light years, which means I would only experience 2 years or so during my journey", this jumped off the page at me as a huge error.

Consider the points A and B in space.
At one point in time person X is standing at A not moving relative to B, another person Y is just over A moving with relativistic speed towards B and finally another person Z is right over A moving with relativistic speed away from B.

The measured the distance at that time for Y and Z is smaller than for X.

Relativity is almost like magic, if Z accelerates away from B in a certain way he actually can get closer and closer to B!

However in your example if the measured distance of the traveler is 2 ly and we assume he does not accelerate then he must take longer than 2ly to get there as no object can reach the speed of light.

I pointed out that yes, the distance would appear shorter from your frame, but you wouldn't claim the distance actually contracted and time dilation is a result of you having to cross 2 light years rather than 640~ light years to Betelgeuse.

Why not?

I've never seen someone actually claim that it is a real physical effect, or that length contraction means the rest of the universe shrinks due to your motion, have I been terribly mistaken all this time about relativity?

It is what is measured so it is pretty real.

 

[soothsayer]

I explained that this would violate the invariance of the spacetime interval.

No, do the calculation for time dilation and length contraction, the spaceship is traveling a much shorter distance in a much shorter period of time than an outside observer would see:
L'=L/$$\gamma$$
t'=t/$$\gamma$$
It's easy to see that L'/t' = L/t, I pretty sure the space-time interval doesn't change.

I pointed out that yes, the distance would appear shorter from your frame, but you wouldn't claim the distance actually contracted and time dilation is a result of you having to cross 2 light years rather than 640~ light years to Betelgeuse.

That's exactly what you would claim. This is the absolute truth in the reference frame of the ship traveling to Betelgeuse.

I've never seen someone actually claim that it is a real physical effect, or that length contraction means the rest of the universe shrinks due to your motion, have I been terribly mistaken all this time about relativity?

It is definitely a real physical effect, for example, GPS systems must make use of relativistic calculations in order to work properly (GR calculation, but it's the same principle.) Remember, no one reference frame is any better than any other.

 

[Max™]

I'm not saying that relativistic effects don't occur.

I'm saying there are two events and a worldline connecting them. Earth now, and Betelgeuse 640 years from now, and there is no way to reach Betelgeuse any earlier than that without moving faster than a beam of light.

If the distance was contracted to 2 light years, which meant it took you roughly 2 years to travel there, would you arrive after a beam of light that left Earth when you did?

If you turned around and came back, would you just wind up 4 years in the future after traveling a contracted 4 light year distance from here to Betelgeuse and back?


I'm pretty sure you'd get back to Earth sometime later than 1280 years from the time you left, though you would claim from your reference frame that you only moved 2 light years in around 2 years.

He was claiming you would have only covered 2 light years, and would have wound up at Betelgeuse 2 years from now, rather than 640+ years from now.

 

[Dale]

If the distance was contracted to 2 light years, which meant it took you roughly 2 years to travel there, would you arrive after a beam of light that left Earth when you did?.

Yes.

He was claiming you would have only covered 2 light years, and would have wound up at Betelgeuse 2 years from now, rather than 640+ years from now.

In which reference frame?

 

[DaveC426913]

I pointed out that yes, the distance would appear shorter from your frame, but you wouldn't claim the distance actually contracted...

The distance is indeed shorter. Your frame of reference is not somehow fake just because you are traveling wrt the stars involved.

You could look at it as if you are stationary and the stars (and the rest of the universe) are hurtling toward you.

 

[Passionflower]

One thing Max is that it also depends on how you measure it.

Using light signals you will absolutely get a contracted distance. However if there was a road and you would measure the distance by calculating the number of full rotations a rolling wheel made then you would not see a contracted distance.

 

[DaveC426913]

Using light signals you will absolutely get a contracted distance. However if there was a road and you would measure the distance by calculating the number of full rotations a rolling wheel made then you would not see a contracted distance.

How would these two be reconciled if you performed both measurements at the same time?

 

[Max™]

In which reference frame?

In a frame stationary with respect to Betelgeuse or Earth, or with respect to a cosmic yardstick stretched between them?


I know that the effect happens, but like I said, you can't get to Betelgeuse in 2 actual years, so you can't claim that the 2 years you observed was a proper time, or that the 2 light year distance was a proper distance.

 

[darkhorror]

I'm not saying that relativistic effects don't occur.

I'm saying there are two events and a worldline connecting them. Earth now, and Betelgeuse 640 years from now, and there is no way to reach Betelgeuse any earlier than that without moving faster than a beam of light.

From the Earth it would take longer than 640 years because in Earth's frame of reference the distance is 640 light years, and the ship is unable to move faster than the speed of light.

But in the ship's frame of reference Betelgeuse is NOT 640 light years away, and if it's moving fast enough it could be much much less than 640 light years. So the ship will still be moving slower than the speed of light but the distance is much less also so they can get there in much less than 640 years.

If the distance was contracted to 2 light years, which meant it took you roughly 2 years to travel there, would you arrive after a beam of light that left Earth when you did?

Yes the beam of light would arrive before you get there. Because to you you see the beam of light moving at C in your frame of reference. If the distance is 2 light years in your frame of reference, then in that frame of reference light will take 2 years to reach Betelgeuse.

A link that can help you understand.
http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html

If you turned around and came back, would you just wind up 4 years in the future after traveling a contracted 4 light year distance from here to Betelgeuse and back?

I'm pretty sure you'd get back to Earth sometime later than 1280 years from the time you left, though you would claim from your reference frame that you only moved 2 light years in around 2 years.

He was claiming you would have only covered 2 light years, and would have wound up at Betelgeuse 2 years from now, rather than 640+ years from now.

Nether is right or wrong, In the Earth's frame of reference lots of time will have passed, in your frame of reference not much time at all will have passed. You should look up the "Twin paradox"

 

[DaveC426913]

In a frame stationary with respect to Betelgeuse or Earth, or with respect to a cosmic yardstick stretched between them?


I know that the effect happens, but like I said, you can't get to Betelgeuse in 2 actual years, so you can't claim that the 2 years you observed was a proper time, or that the 2 light year distance was a proper distance.

The occupants of the spaceship can get to Betelguese in an arbitrarily short time. In principle, they could get there in five minutes if their spaceship could accelerate and decelerate rapidly enough. Betelguese would become just a mile or two away (and very, very flat).

 

[darkhorror]

In a frame stationary with respect to Betelgeuse or Earth, or with respect to a cosmic yardstick stretched between them?


I know that the effect happens, but like I said, you can't get to Betelgeuse in 2 actual years, so you can't claim that the 2 years you observed was a proper time, or that the 2 light year distance was a proper distance.

Sure you can, in the spaceship that's what the distance and time are going to be. Earth is just another random frame of reference. That frame of reference is just as good as the ships. If you threw in another frame of reference it would take the ship a different amount of time and have to travel a different distance.

 

[Max™]

If the ship is measuring using a beam of light, then you can resolve it if you know that you're moving slower (at a different angle) through time, and that your measurements of distances are skewed by that.

The absolute minimum proper spacetime distance between Earth and betelgeuse is 640 light years and 640 years elapsed at the speed of light.

An observer on Earth (ignoring the motion of Earth/Betelgeuse/time dilation from gravity) would move 640 years through time and nothing through space while a beam of light made the journey. If you moved to Betelgeuse at anything slower than the speed of light you would cross 640 light years and something more than 640 years in a reference frame at rest with respect to the origin/destination.

If there was a cosmic yardstick set out where you could measure your distance with a rolling wheel, you could determine that you are moving slowly through time compared to when you left/when you arrived/the inertial frame of the yardstick.

You'd observe that a measurement taken with a beam of light traversing a certain distance would produce a contracted distance, but if you understood relativity you'd know you were timing the beam of light from a dilated frame, because you had to accelerate, which breaks the symmetry between all frames, doesn't it?


Naturally it isn't incorrect to say you measure/observe the universe contracting while you remain unchanged if you assume you're in an inertial frame. How can you make the assumption that you are actually in an inertial frame if you know you accelerated/will need to decelerate at the end of the journey?

 

[Dale]

In a frame stationary with respect to Betelgeuse or Earth, or with respect to a cosmic yardstick stretched between them?

In that frame the trip does indeed take >640 years.

I know that the effect happens, but like I said, you can't get to Betelgeuse in 2 actual years, so you can't claim that the 2 years you observed was a proper time, or that the 2 light year distance was a proper distance.

Why not? What makes the Earth frame better than the ship frame? The whole point of relativity is that there is no frame whose time can claim to be "actual years". The ship frame is just as good a frame as the Earth frame and in the ships frame it takes 2 years. The 2 years in the ship frame have just as much claim to be "actual years" as the 640 in the Earth frame.

 

[Dale]

The absolute minimum proper spacetime distance between Earth and betelgeuse is 640 light years and 640 years elapsed at the speed of light.

No, the proper distance is a maximum, not a minimum. In all other frames the distance is less than 640 ly.

Naturally it isn't incorrect to say you measure/observe the universe contracting while you remain unchanged if you assume you're in an inertial frame. How can you make the assumption that you are actually in an inertial frame if you know you accelerated/will need to decelerate at the end of the journey?

You are correct that the ships rest frame is a non-inertial frame, however you can still analyze the ship's motion from the inertial frame where it is at rest during the outbound leg. That frame is just as valid as Earth's frame, and in that frame the distance is <640 ly.

 

[DaveC426913]

Look at it form the point of view of the spaceship sitting stationary in space. Let's pretend it's not Sol and Betelguese, let's pretend it's Hotrod A and Hotrod B. Both have been ejected from some massive galactic collapse and are whizzing towards the spaceship at .999995c.

Hotrod A is first in front and Hotrod B is eating its dust a mere 2 light years behind it. Also, as they approach us, we can see that, due to their extreme relativistic passage, both Hotrod A and Hotrod B are highly compressed along their direction of travel.

Now, I take our spaceship's turbo shuttle and decide to intercept Hot Rod A. I accelerate my shuttle to match its speed. When I move into orbit around Hotrod A, I take a measurement on the distance to Hotrod B. I measure the distance as 640 light years.

Which frame of reference has the "real" distance between A and B?

 

[Max™]

I understand that part, but not the bit where you experience time dilation because you measure a shorter distance.


I've always seen it the other way, that your time dilation causes you to measure a shorter distance.


Why are the frames of Earth or Betelgeuse better than the ship frame? The ship frame has undergone acceleration and thus is not an inertial frame, assuming there is no motion with Earth/Betelgeuse (not true, but just for simplicity) then you can consider them as inertial frames, assuming you also ignore their gravity wells, and set your definition of a rest frame as one in inertial free fall around them.

This is how the twin paradox is resolved, the twin that took the trip to Betelgeuse underwent acceleration, breaking the symmetry between his frame and any other.


A ship that accelerated to Betelgeuse can not claim that it is in a perfectly symmetrical frame compared to all other frames, so there is no reason to say the measurement of the distance is unaltered by relativistic effects.


My issue was with the guy claiming that his frame underwent no changes due to relativity, that the distance simply became shorter when he went faster, and that he underwent no time dilation, but only had to cross 2 light years in a bit over 2 years real time.


With the unaccelerated twin back at home, it is obvious that the one who went to Betelgeuse had accelerated, so he would only be 4 years older, while the one at home would be nearly 1300 years older.

 

[Dale]

Why are the frames of Earth or Betelgeuse better than the ship frame? The ship frame has undergone acceleration and thus is not an inertial frame

I already addressed this.

You are correct that the ships rest frame is a non-inertial frame, however you can still analyze the ship's motion from the inertial frame where it is at rest during the outbound leg. That frame is just as valid as Earth's frame, and in that frame the distance is <640 ly.

Although the ship's frame is non-inertial there exists a perfectly inertial frame where the distance between Earth and Betelgeuse is <2 ly.

 

[DaveC426913]

The ship frame has undergone acceleration and thus is not an inertial frame,

It is only non-inertial while it is accelerating. It is inertial when the ship is coasting. And while it is coasting, all measurements of distances and times still apply.

In short, ignore the accceleration part of the trip. Pretend you start all your obesevations after the ship shuts off its engine.

 

[Max™]

Yeah, during that part of the journey you can claim your frame is the same as any other inertial frame.

Unfortunately there is no way you can have a ship start at rest on Earth, then instantly hit .9999~whatever c, then instantly decelerate at Betelgeuse.


If you were moving already but not accelerating, you could claim an inertial frame, and when you whizzed past the Earth (or when the Earth whizzed past you, etc), then whizzed past Betelgeuse, yes, you would measure that distance as being 2 ly or so.

If you went from Earth to Betelgeuse and/or came back, it isn't an inertial frame, and trying to claim the shortened distance caused the time dilation is just... backwards, isn't it?

 

[Dale]

trying to claim the shortened distance caused the time dilation is just... backwards, isn't it?

I wouldn't claim that length contraction causes time dilation, nor vice versa.

 

[Max™]

Nor would I, which is why I was so disturbed by the other guy insisting that time dilation is caused by covering a contracted distance.

 

[Dale]

I would say that both the length contraction and the time dilation are caused by the same thing: the Poincare symmetry of the laws of nature.

 

[DaveC426913]

Yeah, during that part of the journey you can claim your frame is the same as any other inertial frame.

Unfortunately there is no way you can have a ship start at rest on Earth, then instantly hit .9999~whatever c, then instantly decelerate at Betelgeuse.


If you were moving already but not accelerating, you could claim an inertial frame, and when you whizzed past the Earth (or when the Earth whizzed past you, etc), then whizzed past Betelgeuse, yes, you would measure that distance as being 2 ly or so.

If you went from Earth to Betelgeuse and/or came back, it isn't an inertial frame, and trying to claim the shortened distance caused the time dilation is just... backwards, isn't it?

Not sure what you're missing here. The time dilation has nothing to do with the initial acceleration. Ignore it.

Heck, pretend the spaceship started 100ly behind Earth if you want, so its coating as it passes Earth. Nowits entire journey between Earth and Betelguese is at .99995c.

Or better yet (as I keep mentioning) the ship is stationary and Earth/Betelguese is on the move. You get the same results, yet the spaceship has been in an interial frame the entire time.

 

[Max™]

Yes, I understand that part just fine, thanks.

My point was that he was stating if he left here, going there, then he could go fast enough to make the actual distance between here and there contract to 2 light years, such that he actually traveled 2 light years in any frame, not 640 light years in the Earth/Betelgeuse resting frame.

Then he stated that he would only experience 2 years of time because of how short the journey was, and said that is why time dilation happens.

 

[DaveC426913]

Yes, I understand that part just fine, thanks.

My point was that he was stating if he left here, going there, then he could go fast enough to make the actual distance between here and there contract to 2 light years, such that he actually traveled 2 light years in any frame, not 640 light years in the Earth/Betelgeuse resting frame.

Then he stated that he would only experience 2 years of time because of how short the journey was, and said that is why time dilation happens.

Yes, he actually travels 2 light years between Sol and Betelguese. That is how far apart they actually are in his FoR. This is not just an illusion, they really are that close.


No, he does not experience time dilation in his own FoR.

However, observers in other FoRs will observe the time dilation, just as he will observe theirs.

Then he stated that he would only experience 2 years of time because of how short the journey was, and said that is why time dilation happens.

This doesn't make sense. Unless I'm misunderstanding, you (he) is saying that, essentially time dilation isn't really occurring, what's really occurring is that the distance is shrinking to 2 ly, so that his hop is a mere 2ly, which would only take 2 years. The implication of that though is that (well, to repeat myself) there's no time dilation. As in,
a] he could communicate with Earth in real-time (because neither is moving slower), and
b] when he got back, he'd be a mere four years older and so would Earth.

Here's the thing: even if the spaceship is not traveling away from Earth, the time dilation still occurs. If Earth watched through a telescope as the spaceship raced past Earth, they would indeed see him moving very slowly within his spacehip.

 

[Max™]

This doesn't make sense. Unless I'm misunderstanding, you (he) is saying that, essentially time dilation isn't really occurring, what's really occurring is that the distance is shrinking to 2 ly, so that his hop is a mere 2ly, which would only take 2 years. The implication of that though is that (well, to repeat myself) there's no time dilation. As in,
a] he could communicate with Earth in real-time (because neither is moving slower), and
b] when he got back, he'd be a mere four years older and so would Earth.

Here's the thing: even if the spaceship is not traveling away from Earth, the time dilation still occurs. If Earth watched through a telescope as the spaceship raced past Earth, they would indeed see him moving very slowly within his spacehip.

Exactly, he was claiming that the time dilation was just a result of him crossing a shorter distance.

I know that in his FoR it is a shorter distance, but that only holds for suitably chosen frames.

He still undergoes time dilation, as I tried to explain to him, but he insists that since he doesn't "observe" any himself, it is just a result of him crossing a shorter distance.

Which I tried to point out is only valid from his frame, while other observers would disagree with him, and that he could resolve his observations with other frames if he understood that his measurement was a result of him experiencing those relativistic effects, and corrected his calculations accordingly.

 

[DaveC426913]

Exactly, he was claiming that the time dilation was just a result of him crossing a shorter distance.

I know that in his FoR it is a shorter distance, but that only holds for suitably chosen frames.

He still undergoes time dilation, as I tried to explain to him, but he insists that since he doesn't "observe" any himself, it is just a result of him crossing a shorter distance.

Which I tried to point out is only valid from his frame, while other observers would disagree with him, and that he could resolve his observations with other frames if he understood that his measurement was a result of him experiencing those relativistic effects, and corrected his calculations accordingly.

Explain to him the scenario where
a] the spaceship flies by Earth, and Earth trains a telescope on him and sees him moving 320X slower, and
b] he tries to communiate with Earth in Morse Code via a laser comm, firing a pulse a second, and Earth receives the pulses only once every 320 seconds.

 

[harrylin]

I had a big epic discussion on another board after someone stated that "if I were to travel to Betelgeuse at a sufficient velocity I would reduce the distance between myself and Betelgeuse until it is say, 2 light years, which means I would only experience 2 years or so during my journey", this jumped off the page at me as a huge error.

He and others insisted that I must not know what I'm talking about if I don't agree with him on this, and completely ignored when I explained that this would violate the invariance of the spacetime interval.

I pointed out that yes, the distance would appear shorter from your frame, but you wouldn't claim the distance actually contracted and time dilation is a result of you having to cross 2 light years rather than 640~ light years to Betelgeuse.

I've never seen someone actually claim that it is a real physical effect, or that length contraction means the rest of the universe shrinks due to your motion, have I been terribly mistaken all this time about relativity?

There are several issues here; not sure if they all have been covered now. So here are my 2cts.

- length contraction cannot cause time dilation: they are both effects from a common cause. They are both a function of relative speed; and obviously your or someone else's motion will not cause a shrinking of the universe.
- however, the proper experience of the traveller will be that the distance appears to be reduced to 2 light years because it only takes a little more than 2 years to travel that distance at near light speed. And other co-moving observers who use a co-moving reference frame may agree with him and hold that it is you who is experiencing relativistic effects. This symmetry of experience is at the heart of relativity.

 

[Max™]

>.<

I checked in on them, and noticed one of them saying this:

"Of course it's only 2 light years, because we measure light taking 2 years to get here from there, if it wasn't a shorter distance, we'd have to be moving FTL"...

...or you could be moving slower through time.

Then they tried to justify it by saying "Euclidean distances are the ones we have to use in normal life, so we have to use them to explain relativity too"... wut?

Uh, no, Relativity is defined as a non-euclidean geometry, you can convert results from that to a euclidean one... but you don't have to do that, last I checked, given the way hyperbolic rotations work in non-euclidean spacetime, I don't see why you would try to force it back into a totally flat geometry, am I missing some amazing benefit from doing that?

 

[harrylin]

>.<

I checked in on them, and noticed one of them saying this:

"Of course it's only 2 light years, because we measure light taking 2 years to get here from there, if it wasn't a shorter distance, we'd have to be moving FTL"...

...or you could be moving slower through time.

Then they tried to justify it by saying "Euclidean distances are the ones we have to use in normal life, so we have to use them to explain relativity too"... wut?

Uh, no, Relativity is defined as a non-euclidean geometry, you can convert results from that to a euclidean one... but you don't have to do that, last I checked, given the way hyperbolic rotations work in non-euclidean spacetime, I don't see why you would try to force it back into a totally flat geometry, am I missing some amazing benefit from doing that?

The space-coordinates of any standard reference system of special relativity (as well as of the GR-based GPS system) are Euclidean or "flat", and there is an obvious practical benefit of describing events in the units with which they are measured.

Now, it may be preferable to choose a reference system in which the visible universe as a whole is approximately in rest, and it seems that that is what you take for granted.
However, as I pointed out earlier, for special relativity (which seems to be what you are discussing here) there is no observable preference for doing so; there is no observational basis for claiming that it's "them" who are moving, and not you, or both you and them. From their perspective - if they choose a reference system that is co-moving with them - the distance between the planets is only 2 light years and it's you who suffers relativistic effects. SR thus explains how both points of view are on equal footing and it sounds as if both you and them don't fully understand that.

Note that so far nothing discussed in this thread is directly related to "explaining relativity" in the sense of "what causes it"; and that would go beyond our knowledge, as it's a matter of opinions.

Harald

PS: see also the parallel thread with similar contents:
https://www.physicsforums.com/showthread.php?t=484798

 

[Max™]

Yeah, I don't have a problem with it being 2 light years in that reference system, and yes I do tend to take for granted that such frames are not as... interesting as ones where the "background stars" are at rest.

My problem is, I know when he measures the time it takes a beam of light to cross a given distance in his frame, he's experiencing significant time dilation, so he's measuring 2 years in his frame for a beam of light to cross 640 light years in the Earth/Betelgeuse/Background frame.

It just seems... inelegant to me to ignore the understanding which can be gained from dealing with the causes of the effect he's describing, rather than simply stating them as brute realities.


More to the point, it feels like to a lie to describe a scenario purely in terms of SR, neglecting the changes which accelerated frames introduce to situations like that.

The whole thing just set off all kinds of alarm buzzers when I read over it, and the worst part was how he was presenting it to the general board crowd. When I pointed out the problems with describing it that way, to an audience which includes people with no real grasp of the subject, they began informing me I must not know what I'm talking about if I was disagreeing with him.

 

[Dale]

More to the point, it feels like to a lie to describe a scenario purely in terms of SR, neglecting the changes which accelerated frames introduce to situations like that.

Note that there is a difference between a frame and an object. In introductory relativity we like to attach frames to every object and associate every frame with some object. That often leaves students with a mistaken impression that frames must always follow some object, which is not correct. In many realistic applications the most convenient reference frame is actually the center of momentum frame, which is an inertial reference in which everything is moving, and often accelerating.

In this case it is not a lie to describe this scenario purely in terms of SR. There is no need to consider an accelerated frame at all. As long as you are neglecting gravity SR may be used regardless of whether or not one or more objects are accelerating.

 

[Max™]

Well, I said feels like, it just doesn't appeal to my sense of aesthetics I guess, how could you not love hyperbolic rotations, right?

I do understand that a frame can be attached to anything, that is generally how I look at it anyways, assigning an arbitrary godseye observer to "peek" into the frame, but I guess that tendency would make one less eager to simply go along with statements like "it's valid in my frame" without asking "but what about other frames of interest??" huh?

My biggest issue with how he phrased it was the declaration that "since I can make the distance shorter by going faster, it takes less time to get there", with the implication that there is no distortion to his motion through time, and that it is simply a matter of warping the universe into a more convenient form to get from here to Betelgeuse by 2013.

THAT is what sent me off in a tizzy, the way he was implying that moving really fast makes distances shorter, and crossing those shorter distances takes less time... which stabs at my brain trying to figure out why he would claim that without clarifying things better.

 

[Passionflower]

"since I can make the distance shorter by going faster, it takes less time to get there",

Max™ that statement is absolutely true.

Frankly I am still not convinced you fully accepted the implications of special relativity.