Length contraction and time dilation

[eok20]

I'm having trouble understanding how length contracts while time dilates when the 2 equations in the lorentz transformation dealing with these are nearly identical (which makes me think that length and time should transform in the same way).

Thanks in advance.

 

[jdavel]

I'm having trouble understanding how length contracts while time dilates when the 2 equations in the lorentz transformation dealing with these are nearly identical (which makes me think that length and time should transform in the same way).

Thanks in advance.

They do. It's just that the word's are confusing because one refers to a length and one refers to a rate. "Length contraction" means the length of rods is less. "Time dialation" means the rate that clocks tick is less.

 

[JesseM]

They do. It's just that the word's are confusing because one refers to a length and one refers to a rate. "Length contraction" means the length of rods is less. "Time dialation" means the rate that clocks tick is less.

I don't think that's the right way to think about it. Say L represents the length of a ruler in its own rest frame and T represents the time elapsed between two events according to a clock at rest in that frame, and l and t represent the length of the ruler and the time elapsed between the events as seen in a frame moving at velocity v relative to the ruler. In that case the transformations are:

$$l = L/\gamma$$

$$t = \gamma T$$

So, one equation involves dividing by $$\gamma$$ and the other involves multiplying by it.

 

[jdavel]

JesseM,

But the difference comes about because, as you said, "L represents the length of a ruler" and "T represents the time elapsed between two events". Those aren't analagous. If you want to talk about "the length of the ruler" then you have to talk about "the tick rate of the clock". If you want to talk about "the time elapsed between two events" then you have to talk about "the distance between two events.

Using your notation, if you let L and l be the measured distances between two events and let T and t be the measured times elapsed between the two events, then if L = gamma*l, T = gamma*t.

The symmetry between space and time in relativity is one of the the most beautiful relationships in all physics. And the way the "time dilation" and "length contraction" formulas are constructed is, in my opinion, an abomination. They're a relentless source of confusion, to say nothing of all the "I've proven that Einstein was wrong" claims that they've spawned.

Anyway, that's what I think.

 

[JesseM]

But the difference comes about because, as you said, "L represents the length of a ruler" and "T represents the time elapsed between two events". Those aren't analagous. If you want to talk about "the length of the ruler" then you have to talk about "the tick rate of the clock". If you want to talk about "the time elapsed between two events" then you have to talk about "the distance between two events.

Using your notation, if you let L and l be the measured distances between two events and let T and t be the measured times elapsed between the two events, then if L = gamma*l, T = gamma*t.

Those formulas aren't correct if you apply them to the problem you seem to be describing. If you take two events which are simultaneous in one frame, so that t=0 in that frame, the time T between them in another frame will not also be zero. Similarly, if you look at the distance between two events which happen at the same location but different times in one frame, so that l=0 in that frame, the distance L between those events in another frame is not also 0.

The way it's normally used, the time dilation formula is only meant to apply when you have two events which happen at the same location but different times in one frame, and you want to know the time between them in another frame. And the length contraction formula isn't even meant to give you the distance between the same two events in different frames, instead it tells you that if you look at the distance L between two events representing the position of the front and back of an object "at the same moment" in its own rest frame, then if you want to know the distance l between two events representing the position of the front and back of the object "at the same moment" in a different frame (since the frames disagree about simultaneity, these can't be the same two events), the relation between the distances is given by l = L/gamma.

The symmetry between space and time in relativity is one of the the most beautiful relationships in all physics.

What symmetry are you talking about, exactly? Time cannot simply be treated as a fourth spatial dimension, for example.

 

[jdavel]

Those formulas aren't correct if you apply them to the problem you seem to be describing...

All I can say to that is that you're right, What I meant to say is that if the magnitude of the distance between two events is greater in one frame, then so is the magnitude of the elapsed time greater in that frame. This has to be true in order for the total interval to be invariant. A cursory look at the time dilation and length contraction formulas might give the impression that it's the other way around.

The way it's normally used, the time dilation formula is only meant to apply when you have two events which happen at the same location but different times in one frame, and you want to know the time between them in another frame. And the length contraction formula isn't even meant to give you the distance between the same two events in different frames, instead it tells you that if you look at the distance L between two events representing the position of the front and back of an object "at the same moment" in its own rest frame, then if you want to know the distance l between two events representing the position of the front and back of the object "at the same moment" in a different frame (since the frames disagree about simultaneity, these can't be the same two events), the relation between the distances is given by l = L/gamma.

I agree. But don't you think that's a very confusing paragraph. I don't think that someone new to SR could understand it. But that's the kind of description you get into when you talk about LC and TD instead distance intervals and time intervals.

What symmetry are you talking about, exactly? Time cannot simply be treated as a fourth spatial dimension, for example.

No, and if it could it would be quite boring. The symmetry I'm talking about is most clearly seen when you convert the time coordinate to length by multiplying by c. Then instead of Galileo's

x' = x - vt

t' = t

you get this,

x' = Y(x - Bct)

ct' = Y(ct - Bx)

Don't you think that's great?

 

[JesseM]

All I can say to that is that you're right, What I meant to say is that if the magnitude of the distance between two events is greater in one frame, then so is the magnitude of the elapsed time greater in that frame. This has to be true in order for the total interval to be invariant. A cursory look at the time dilation and length contraction formulas might give the impression that it's the other way around.

Yes, that's true.

I agree. But don't you think that's a very confusing paragraph. I don't think that someone new to SR could understand it. But that's the kind of description you get into when you talk about LC and TD instead distance intervals and time intervals.

It's confusing because I tried to state it in the language of "events", and because someone new to SR wouldn't find the notion of different definitions of simultaneity to be very intuitive. What is intuitive is that if the back of an object is at position x=3 meters at the same time that the front is at position x=5 meters in my frame, the "length" in should be 2 meters. Likewise, it's intuive that if a moving clock reads 3:00 at the moment it passes one of my clocks that reads 3:00, and then later it reads 4:00 at the moment it passes one of my clock that reads 5:00, then it has elapsed only one hour in what was really two hours in my frame. Those are the notions that the Lorentz contraction and time dilation formulas are based on.

No, and if it could it would be quite boring. The symmetry I'm talking about is most clearly seen when you convert the time coordinate to length by multiplying by c. Then instead of Galileo's

x' = x - vt

t' = t

you get this,

x' = Y(x - Bct)

ct' = Y(ct - Bx)

Don't you think that's great?

That is pretty cool. Another cool trick is to just treat time as an imaginary space dimension, so that 1 year is really i light-years--then the formula for the spacetime interval between events is just like the pythagorean formula for distance between points in space, and the Lorentz transform looks just like the formula for converting between two cartesian coordinate systems whose axes are rotated relative to each other. Apparently treating time as an imaginary space dimension doesn't work in general relativity, though.

 

[Mortimer]

I can't help to point out again that Euclidean relativy so much simplifies this that I can even explain it to a complete layman. It is not a denial of classical relativity but just an alternative mathematical framework that replaces the Minkowski framework. I know even of a teacher who uses it in his class against mainstream approval because he noticed that then at last his students began to show an understanding of relativity.

In Euclidean relativity Lorentz transformations become rotations in SO(4). A moving rod has rotated towards the negative axis of the time dimension according to an observer at rest. This rotation gives the rod a length component in the timedimension, hence the non-simultaneity of points on the rod. The spatial length component decreases, hence its length contraction in space.
Something similar happens to the 4D velocity vector c of the rod. It can be decomposed in a spatial velocity component and a time velocity component . The latter decreases with increasing spatial velocity, hence the slowdown of clocks in moving objects.
This animated GIF file shows how it works: http://www.rfjvanlinden171.freeler.nl/simplified/length.gif