# Benefits of time dilation / length contraction pairing?

1. Mar 15, 2009

### neopolitan

There have been more than a few threads where there clearly is confusion about the use of time dilation and length contraction.

People initially think that:

1. in an frame which is in motion relative to themselves, time dilates and lengths contract; and
2. velocities in a frame which is in motion relative to themselves are contracted lengths divided by dilated time.

I admit that it stumped me for a long time, because of what I see as inconsistent use of primes and for me a much more useful pair of equations would have a more consistent use of primes, similar to the Lorentz transformations.

I was told during a long discussion that time dilation and length contraction are used, even though they pertain to different frames, because they have greater utility. I took that at face value, but now I wonder again.

What exactly is the greater utility of time dilation and length contraction equations which prevents the use of two contraction equations which would do away with the confusion I mentioned above?

(And by the way, introducing arguments that t in time dilation is the period between tick and tock doesn't really help, because this is more indicative of the confusion since we use clocks everyday to measure the time between events in terms of the number of ticks and tocks rather than in terms of the duration of pause between each tick and tock. Reinterpreting how we use time to make the equation work is not indicative of any greater utility.)

If it is a purely historical thing, then I would be far happier with it if that little tidbit were taught at the same time as the equations are introduced. But it isn't.

There is also the potential argument that they are only useful right at the beginning of one's odyssey into relativity, so it doesn't really matter. Sure, ok, then it doesn't matter if you use a more intuitive pairing does it?

Bottom line: what is so great with time dilation?

cheers,

neopolitan

2. Mar 15, 2009

### ZapperZ

Staff Emeritus
Would it surprise you to learn that these "time dilation" and "length contraction" are merely consequences of the Lorentz transformation coupled with Einstein's SR postulates?

So do you really have a problem with the fundamental theory, or do you simply have a problem with the consequences of the theory? If you do not have a problem with the fundamental theory (because you didn't even mention it), then where in the logical derivation of the consequences did we lose you and they become "inconsistent"? For example, start with a standard derivation of time dilation as presented in standard physics textbooks. Point out exactly where such a thing becomes logically inconsistent.

What is so "great" about time dilation? It explains a whole bunch of empirical observations. What more can you ask for?

Zz.

3. Mar 15, 2009

### JesseM

They are consistent. Unprimed is used in these equations to represent times and lengths in the frame where the ruler/clock is at rest; primed is used to represent the corresponding times and lengths in the frame where the ruler/clock is moving. Can you specify what alternative you're proposing?

4. Mar 15, 2009

### Saw

I would put it the other way round, ZapperZ. I've looked at the transversal light clock example and the longitudinal light clock example by themselves and wondered how one can derive from those examples the consequence that there is RECIPROCAL TD and LC , if you consider those derivations isolatedly.

A different thing is that, if you consider jointly RECIPROCAL TD, LC and RS, the whole system of SR really seems to make sense. Thus the "standard derivation of time dilation as presented in standard physics textbooks" may have a limited use as a way to derive the quantitative aspect of those elements, which however only make sense (as a RECIPROCAL measurement) when they are put together and integrated in a "global system", which by the way does not contemplate any duality of opinions about reality, but, on the contrary, full agreement on what has really happened.

But maybe I am imagining something you are not saying. In your view, the "standard derivation of time dilation as presented in standard physics textbooks" (for instance, the light clock example)..., does it logically prove that there is RECIPROCAL time dilation?

Last edited: Mar 15, 2009
5. Mar 15, 2009

### neopolitan

No, if L=x' and L'=x (or perhaps $$L = \Delta x'$$ and $$L' = \Delta x$$).

Is that what you are saying?

But then where is the consistency of prime notation that JesseM claims in his post?

I do understand that, with the awkward interpretation, time dilation does explain a lot of empirical observations. No problems there. But so would a time equation in the same form as the length equation, and it would not lead to the problem with people thinking "speed of light in another inertial frame is contracted length divided by dilated time ... hang on, that's not invariant!" Then they visit here and have someone metaphorically yelling at them "you are mixing frames!" without actually explaining why you can't use time dilation and length contraction that way.

I know you can't, I just want to know what advantages exist with having time dilation and length contraction expressed the way they are?

Jesse, I can't believe you say the frames are all consistent. We went over it for days, in emails with diagrams. How about I post your very own diagram here to help clarify?

cheers,

neopolitan

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6. Mar 15, 2009

### ZapperZ

Staff Emeritus
Er.. come again?

There are two aspects to this. One is the logical derivation, i.e. mathematical derivation, based on SR's postulates. It has to based on that because there's nothing mathematically that can derive c being a constant in all frames.

The second is the experimental verification. A logical derivation of anything in physics is no guarantee that it is valid. It is the experimental observation consistent with such result that elevates its validity. It somehow appears as if you want to start from the tail end of it to justify itself, which is fine if you are trying to formulate a new theory. But considering that SR is such a well-established theory with solid foundation, and every one of the consequences can be derived from such foundation, it would be logical to start from there. And that's where I do not get the OP. Is there a problem with the foundation in the first place or is he/she only do not get the consequences? I don't think that is such an unreasonable query.

Zz.

7. Mar 15, 2009

### ZapperZ

Staff Emeritus
I don't see it.

Forget SR/Lorentz transformation. Start with Galilean transformation. Do you have a problem with that as well? Write down the velocity and displacement of a moving object in two different inertial frames via Galilean transformation. My guess is that you have a problem with that as well, because fundamentally, none of what you wrote above really has anything to do with SR.

At what point do you acknowledge about the empirical verification of these things?

Zz.

8. Mar 15, 2009

### Mentz114

Hard to believe. Can't you see that this is an irrelevance. How else could you express time dilation except as an increase the gap between ticks ? Counting ticks still means you have to multiply the number of ticks by the time between ticks.

When you talk about primes, are you making a point about notation ? You could use some other way to distiguish two frames, but it wouldn't make any difference !

9. Mar 15, 2009

### JesseM

Yes, and never did you offer any convincing argument that there was something "inconsistent" about the standard definitions. My diagram shows that they are consistent, given the usual definition of "length" and "time interval"--the "spatial analogue of time dilation" and the "temporal analogue of of length contraction" in the diagram don't refer to any commonly-used or intuitive physical quantities (the 'spatial analogue of time dilation' refers to the distance in the primed frame between two events that are simultaneous in the unprimed frame; the 'temporal analogue of length contraction' is even weirder, it refers to the time in the primed frame between two surfaces of simultaneity that cross through the events in the unprimed frame).

10. Mar 15, 2009

### neopolitan

Yes, I know that is what time dilation is (and must be to keep everything right).

What is difficult for people new to SR, the very ones who are being taught time dilation, is that the equation involving t, ostensibly the same t which they may well be used to in Gallilean boosts (x'=x-vt) and the kinetics equations (v = (si-so)/t and so on), is using a quite different definition of t.

Jesse mentions "time interval", which is fine, I just wonder why we don't use a different symbol for time dilation ($$\tau$$ perhaps) to highlight the difference between "time interval" - time between ticks - and "(measured) time elapsed" - number of ticks.

But none of this answers the original question, what are the benefits of having time dilation and length contraction rather than a pair of equations which would not lead to the continual confusion I alluded to in an earlier post?

(It seems the only responses so far are "you can't do it any other way" and "you are confused". Is there really no other way?)

cheers,

neopolitan

11. Mar 15, 2009

### ZapperZ

Staff Emeritus
This complain has nothing to do with SR. Look at your kinematics problem. You use the SAME thing there! This simply re-enforces my earlier assertion that this isn't about "time dilation" at all. You are simply confused on what we call 'time' in any dynamical system.

Zz.

12. Mar 15, 2009

### Mentz114

Neopolitan,
There are four equations, one for time each dimension. It's hard to see what you mean.
I don't think most people are 'continually confused'.

13. Mar 15, 2009

### neopolitan

Ok, I accept that I may be confused. Let's look at kinematics first.

Say I do a simple experiment with a small car designed to move at a set speed (I don't what that speed is). I run it past two posts (so and si) and measure the time.

To work out the speed, I use the equation I mentioned before.

How do I work out the time t?

This is my suggestion. I have a stop watch, I start it when the car passes the first post and I stop it when it passes the second post. The value on the watch is then t, which shows me the result in "ticks" each of which will probably be 1ms or 10ms long.

Therefore, t= (number of ticks on my stopwatch).

Not (time between each tick on my stopwatch).

Then I notice there was a separation between me and post si and cleverly work out that that means that I don't get to see the car pass the post instantaneously, that there are now simultaneity issues, and therefore I shouldn't really be using gallilean equations, but rather lorentz based ones.

How do I work out the speed now?

I can use the value on my stopwatch, plus the knowledge that the information about the car passing si travelled to me at c (approximately, because I am not in a vaccuum).

Still, my value of t is in terms of ticks of my stopwatch, t = (number of ticks of my stopwatch before I receive information about the car passing si minus the number of ticks of my stopwatch which elapsed while that information was in transit).

Then, I decide to get more tricky. Because I have been told that for things in motion relative to me time dilates, I want to see some empirical evidence for it.

I put a video camera on post si and a stopwatch on the car along with a mechanism which starts both my stopwatch and the stopwatch on the car as the car passes post so. I call the car "Prime". I call myself "Unprime". I call the result I calculate from my stopwatch t and the result captured on the video camera as the car passes t'.

I think that the result of my empirical experiment will be that time does not dilate, but rather contracts, since the car's t' will be less than my t.

I can get around that by changing the definition of time in my dynamical system. I don't think that is such a fabulous idea, but I could do it.

Am I simply confused here?

cheers,

neopolitan

14. Mar 15, 2009

### JesseM

What would be the pair of equations you're proposing, that wouldn't involve quantities which are far more unintuitive to students than "length" and "time interval", and which wouldn't be much more difficult to actually apply to the types of introductory problems found in textbooks? Please write them down.

15. Mar 15, 2009

### JesseM

Is the watch riding in the car or at rest relative to the posts on the ground? If at rest on the ground, then it must be far from at least one of the posts when the car passes it...how do you ensure that the watch is stopped "when" it passes the post it's not next to? Let's say the watch is next to the first post, so you start it when the car passes...do you stop it when you see the light from the car passing the second post, or do you prearrange things so that it will stop simultaneously with the event of the car passing the second post, relative to the rest frame of the watch and posts? If the latter, then this isn't a novel suggestion, it's the time interval that's always used when calculating speed=distance/time.
The time in speed=distance/time is always the number of ticks on your stopwatch, not "time between each tick on your stopwatch". We'd only be interested in the "time between each tick on your stopwatch" if we knew the number of ticks in the watch's frame and then wanted to figure out the time for the watch to elapse this number of ticks in a different frame where the watch was in motion.
Yes, then in this case you are really calculating the time on your watch between the car passing the first post and the time on your watch that is simultaneous with the car passing the second post in the watch's frame...as I said this is the time interval you'd always want to use when calculating speed=distance/time for the car in the ground frame.
But that's just because you have "primed" and "unprimed" backwards from the normal convention. The usual convention is that the unprimed t represents the time interval between two events on the worldline of a clock as measured in the clock's rest frame (so both events happen at the same spatial location in the unprimed frame, and t will be equal to the time interval as measured by the clock itself), whereas the primed t' represents the time interval between the same two events in a frame where the clock is moving (so the events happen at different locations in the primed frame). This is consistent with the convention for length contraction, where the unprimed L represents the distance between two ends of an object in the object's own rest frame, and the primed L' represents the distance between the ends of the same object in the frame where the object is in moving.

16. Mar 15, 2009

### neopolitan

$$t'=\gamma t$$

$$L_{x}'=L_{x} / \gamma$$

$$L_{y}'=L_{y} / \gamma$$

$$L_{z}'=L_{z} / \gamma$$

Then you are confused. I don't think you mean that though.

$$t'=\gamma t$$

$$L'=L / \gamma$$

$$t'=\gamma (t-x.\frac{v}{c^{2}}$$

$$x'=\gamma (x-vt)$$

then fair enough.

The pair I am talking about is:

$$t'=\gamma t$$

$$L'=L / \gamma$$

And the (to me more intuitive) pair is:

$$\Delta t'=\Delta t / \gamma$$

$$\Delta x'=\Delta x / \gamma$$

With the explanation that lengths contract in a frame in motion (no different to existing situation) and that clocks slow down in a frame in motion. If you like, you could then talk about how this is equivalent to time intervals dilating, with each tick taking longer in a frame in motion.

Then, when people like chrisc come along (and myself, many moons ago), they won't get into trouble for doing the obvious calculation $$x' / t'$$ and ending up with $$v / \gamma^{2}$$ or worse ... $$c / \gamma^{2}$$.

Hopefully this also answers Jesse's question.

17. Mar 15, 2009

### neopolitan

I know this is the convention. What is the benefit of that convention?

Note that I clearly specified thing so that I would have one primed frame (that of the car) and one unprimed frame (mine) and values from the primed frame would be primed and values from the unprimed frame would be unprimed. Is that not consistent?

The convention is to not really talk about a single frame, but to have a length for which both ends are simultaneous and clock for which ticks are colocated.

The thing that got me thinking about this most recently is chrisc's concern which I think was about light clocks (whether his issue was or was not about light clocks is immaterial either way).

Consider a single light clock. This has both a length (between a tick mirror and a tock mirror) $$\Delta x$$ and consecutive ticks $$\Delta t$$. Lie it down on a carriage of some sort so that the vector $$\Delta x$$ is parallel with the direction of the carriage's potential motion.

The only speed the carriage can have for which the conditions behind time dilation and length contraction can be simultaneously applied to the dimensions of the light clock (I'm only talking little 's' version of simultaneous here, not the strict definition) is zero. Only when the carriage is at rest (relative to are observer) are the ends of the light clock simultaneous (relative to the observer) and consecutive ticks are colocal (relative to the observer).

You can only apply time dilation and length contraction to the light clock in a non-trivial way by mixing frames. And if you try to use those, to work out the speed of the photon in the light clock, you end up with a non-sensical result - because of the frame mixing.

I can see that it is useful to be able to work out the amount by which you would need to slow down a clock which is at rest relative to you to match a clock which is in motion relative to you. But I still don't see the huge benefit associated with the pairing of time dilation and length contraction.

Is it mere orthodoxy? Is it historical? Or is there a real concrete advantage?

cheers,

neopolitan

18. Mar 15, 2009

### JesseM

That it would be awfully confusing if primed referred to the rest frame of the clock while unprimed referred to the frame where the ruler was moving, or vice versa.
Consistent with what? I thought we were talking about consistency between notation in the length contraction and time dilation equations, I don't know what it would even mean to ask if the time dilation equation alone is "consistent". You're certainly free to refer to the frame of the clock as the primed frame if you like, but then to be consistent you should also use primed to refer to the rest frame of the object whose length you're talking about in the length contraction equation.
I don't know what you mean by "frame mixing"--don't the time dilation and length contraction equations by definition involve two different frames, one labeled primed and one labeled unprimed? But it's still consistent in the sense that if you use unprimed to refer to the distance between the two mirrors in the clock's rest frame, then unprimed also refers to the time interval between the light going from one mirror to another in the clock's rest frame, and then you can use the time dilation and length contraction equations to get the distance between mirrors and the time between ticks in the frame where the light clock is moving.
I don't understand, the time dilation equation doesn't say anything about slowing down a clock at rest relative to you, it tells you how much time t' will have elapsed in your frame (i.e. how much time elapses on normal unslowed clocks at rest relative to you) when a moving clock ticks forward by some amount t.
Length contraction and time dilation are both just useful for solving basic problems without using the full Lorentz transformation (for example, if a ship is traveling to a location 12 light year away in the Earth's frame at 0.6c relative to the Earth and you want to know what the ship's clock will read when it gets there, you can figure it out either using the time t' it takes in the Earth's frame and then applying the time dilation equation, or using the distance L' between Earth and the destination in the ship's frame, and then use time = distance/speed in that frame). And if you want to write these equations next to each other, it would be confusing if you didn't use the same convention for which notation you use for the rest frame of the clock/ruler you're talking about.

19. Mar 16, 2009

### neopolitan

So, cutting and pasting your words to avoid mistakes:

using "unprimed to refer to the distance between the two mirrors in the clock's rest frame" (L) and noting that "unprimed also refers to the time interval between the light going from one mirror to another in the clock's rest frame" (t) and keeping in mind that this is a light clock where we are using a photon, then c = L / t.

Using your logic, you use the length contraction equation to get "the distance between mirrors" and time dilation to get "the time between ticks in the frame". How far does the photon get in how much time? That would be the speed of light: c = L' / t' = c/$$\gamma^2$$ ?

cheers,

neopolitan

20. Mar 16, 2009

### JesseM

No, because in the primed frame where the light clock is moving, the distance the light travels to get from the left mirror to the right mirror is not equal to the distance between the left and right mirror at a single instant, since both mirrors are moving in this frame. If the whole structure is going from left to right at speed v, and the light is moving at speed c in both directions, then as the light goes from left to right, the distance between the light pulse and the right mirror is shrinking at a "closing speed" of (c - v), while as the light goes from right to left, the distance between the light pulse and the left mirror is shrinking at a closing speed of (c + v). So, the time in this frame for the light to go from left mirror to right and back to left is L'/(c-v) + L'/(c+v) = 2cL'/(c^2 - v^2). So if t' is the time for the light to go from left to right and back in the frame where the light clock is moving, then t' = 2cL'/(c^2 - v^2). So, plugging in $$L' = L*\sqrt{1 - v^2/c^2}$$ and $$t' = t/\sqrt{1 - v^2/c^2}$$ gives:

$$t/\sqrt{1 - v^2/c^2} = 2cL*\sqrt{1 - v^2/c^2}/(c^2 - v^2)$$
multiplying both sides by $$\sqrt{1 - v^2/c^2}$$ gives:
t = 2cL*(1 - v^2/c^2)/(c^2 - v^2)
And since (1 - v^2/c^2) = (1/c^2)*(c^2 - v^2) this simplifies to:
t = 2cL/c^2 = 2L/c, which is exactly what we'd expect to be true in the unprimed frame where the light clock is at rest and the distance between the mirrors is L. Of course you could also reverse this algebra to show that, since the two-way time in the light-clock rest frame is t=2L/c, the two-way time in the frame where the light clock is moving must be t'=2cL'/(c^2 - v^2).

Last edited: Mar 16, 2009