# Distance dilation and length contraction

1. Apr 23, 2014

### ANvH

I am trying to get another insight in what I think is currently for me a contradiction. I have searched the forums, and the web but come to the following assertion:

The coordinate distance is the computation of the proper distance times the gamma-factor, while the "coordinate" length is a contraction according to the proper length divided by the gamma-factor. Is it the result of the delay in arrival time of light signals between the two ends of a length, and distance is based on a different concept?

I understand that there is no difference between distance and length when gamma equals unity, i.e., length and distance are measured in a IRF where the objects are at rest and then proper length is synonym to proper distance.

2. Apr 23, 2014

### PAllen

This is not correct. If a distance (between two objects) or length (of some rod) are measured in frame A, then another frame (B), will measure both of those as reduced by gamma for the relative velocity of A and B (assuming the distance/length is parallel to the relative motion).

3. Apr 23, 2014

### ghwellsjr

I've never heard of "distance dilation". Did you just make that up or do you have a reference for it?

I'm not quite sure what your issue is but I'll make a stab at it. Here is a spacetime diagram for the rest frame of an object with a Proper Length of 5 five feet OR it could be two separate particles that are separated by a distance of 5 feet. I'm defining the speed of light to be 1 foot per nanosecond. The dots represent 1 nanosecond of Proper Time for each worldline:

Now if we transform to another IRF moving at 0.6c with respect to the first one, gamma is 1.25:

If you look at the Coordinate Distance between the bottom two dots (events), you see that it is gamma times the corresponding distance in the first frame (5*1.25=6.25). Is that what you are calling "distance dilation"?

However, when we want to measure or specify lengths, we need to do it at the same Coordinate Time for the two ends. So the length of the object OR the distance between the particles is 4 feet at the Coordinate Time of 0 or any other Coordinate Time you want to pick. This is the Proper Length divided by gamma (5/1.25=4).

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Last edited: Apr 23, 2014
4. Apr 23, 2014

### Bill_K

No, it's not the arrival time of light signals. What does it is the difference in simultaneity.

Δx = γ(Δx' - v Δt')
Δt = γ(Δt' - v/c2 Δx')

For events which are simultaneous in your frame, we set Δt' = 0 and get Δx = γ Δx'. In this case Δx is bigger than Δx'.

But for events which are simultaneous in my frame, in the second equation set Δt = 0 and get Δt' = v/c2 Δx'. Then from the first equation, Δx = γ(Δx' - v2/c2 Δx') = γ(1 - v2/c2) Δx', or Δx = (1/γ) Δx'. and Δx is smaller than Δx'.

5. Apr 23, 2014

### ANvH

I am referring to the wikipedia page where Proper length or rest length is set against proper distance http://en.wikipedia.org/wiki/Proper_length#See_also. It tells us that proper length and proper distance are different.

The proper distance is defined as
$Δ\sigma=\sqrt{Δr^{2}-c^{2}Δt^{2}}$​
and working this out
$Δ\sigma=\frac{cΔt}{\gamma}\sqrt{-1}$​
if $v<c$ and
$Δ\sigma=\frac{cΔt}{\gamma}$​
if $v>c$. Since $cΔt=Δr$, $Δr=\gamma Δ\sigma$. I am aware that the proper distance only makes sense when the square root is taken from a real value, i.e., $v$ should be larger than $c$. "Distance dilation" is the interpretation.

I appreciate the responses, but there is a difference between proper length and proper distance. Furthermore, if
$Δ\sigma=\sqrt{-Δr^{2}+c^{2}Δt^{2}}$​
then "distance dilation" is real when $v<c$.

Last edited: Apr 23, 2014
6. Apr 23, 2014

### Bill_K

If you read it again, you'll sse that they are indeed totally different. One is the proper length of an object, and the other is the proper distance between two events.

7. Apr 23, 2014

### Staff: Mentor

All proper lengths are proper distances, but not all proper lengths are proper distances.

The term "proper length of <something>" is a convenient shorthand for "the proper distance between the positions of the two ends of <that something> at the same time, where 'at the same time' is defined using the simultaneity convention of an inertial frame in which the object is at rest", which in turn is a long-winded way of saying "the rest length of the object".

It is particularly easy to calculate the proper length using coordinates in which the object is at rest, because then the positions of the ends are not changing with time and the proper length is just the coordinate distance between the two ends. That situation corresponds to our intuitive understanding of "length", the one that I'm using every day when I grab a tape measure to see how long a piece of wood is before I cut it to length or whatever.

Last edited: Apr 23, 2014
8. Apr 23, 2014

### ANvH

...of the endpoints of the object. Ok, so the coordinate distance between the endpoints of an "objects distance", subject to $\gamma>1$, is a transform where the events are not simultaneous anymore. They were simultaneous when $\gamma$ was unity. In the diagram from ghwellsjr one can read this "dilated distance" (6.5 feet) when comparing event-leftend(0) and event-rightend(0').

Yes, this is different from the objects length (4 feet).

Thank you for the clarification.

9. Apr 23, 2014

### ANvH

Sorry, Thanks to all of you ;-)

10. Apr 23, 2014

### ghwellsjr

I think you are still mixed up.

The Proper Distance between the two events that I called out in the second diagram is:

√(6.252 - 3.752) = √(39.0625 - 14.0625) = √25 = 5 feet

In the first diagram the Proper Distance is:

√(52 - 02) = √(25 - 0) = √25 = 5 feet

The point of the Proper Distance, as the wiki article said, is that it is invariant--it doesn't matter what frame you use to do the calculation. It isn't 5 feet (not 4) in one frame and dilated in another frame by multiplying 5 by gamma to get 6.25 feet (not 6.5). The point I was making in my previous post is that 6.25 is an invalid measurement of distance since the two events are not at the same Coordinate Time.

The wiki article never says anything about "dilated distance". I ask you again, did you make that term up on your own or can you provide an online reference for it?

Here are a couple more examples of Proper Distance:

If you look at my diagrams, you can pick any one of the blue dots (events) and any one of the red dots (events) and calculate the Proper Distance between them in either of the two diagrams and you will get the same answer.

Let's take the top blue dot and the second red dot from the top. In the first diagram the calculation would be:

√(52 - 12) = √(25 - 1) = √24 ≈ 4.9 feet

In the second diagram it is:

√(72 - 52) = √(49 - 25) = √24 ≈ 4.9 feet

Another example: Let's take the bottom blue dot and the second from the top red dot. In the first diagram it is:

√(52 - 32) = √(25 - 9) = √16 = 4 feet

In the bottom diagram it is:

√(42 - 02) = √(16 - 0) = √16 = 4 feet

11. Apr 23, 2014

### ANvH

No, I wrote
in post #5. The $Δr$ is the dilated distance. I know that proper distance is invariant, so is the proper time and proper length.

12. Apr 23, 2014

### ANvH

In post #5 I also said that "Distance dilation" is the interpretation of the $Δr$. Thus, I made it up, given $Δ\sigma$ is multiplied by $\gamma$.

13. Apr 23, 2014

### ghwellsjr

Yes, you wrote a lot of stuff in post #5 that I cannot figure out including the above quote. Why do you say $cΔt=Δr$? That doesn't appear to me to be an equality. Not even close. In my three examples that I gave you in post #10, it is not true in any of them.

It would also be helpful if you would show how you got from:

to:

and:

Make sure you explain what v is.

Last edited: Apr 23, 2014
14. Apr 23, 2014

### ANvH

I should have said that $Δr=v^{2}Δt^{2}$, that was unfortunately a typo, thanks for catching this. I apologize for making this mistake. Here is a spelled out derivation.

$Δ\sigma = \sqrt{Δr^{2}-c^{2}Δt^{2}}= \sqrt{v^{2}Δt^{2}-c^{2}Δt^{2}}=Δt\sqrt{v^{2}-c^{2}}=cΔt\sqrt{\frac{v^{2}-c^{2}}{c^{2}}}=cΔt\sqrt{-\gamma^{-2}}=\frac{cΔt}{\gamma}\sqrt{-1}$​

Last edited: Apr 23, 2014
15. Apr 23, 2014

### ghwellsjr

You still have a mistake. You left off a squared term.

But the whole idea of associating a velocity with events is a mistake and so the rest of your derivation is bogus.

Last edited: Apr 23, 2014
16. Apr 23, 2014

### ANvH

yes, again a typo.

I guess you are right because it looks like the $Δ\sigma$ in the derivation is the $Δ\tau$, proper time. Your diagrams always depict the coordinate distance, and not the coordinate length as the space axis. Proper distance is the distance between events, and given events become non-simultaneous when $v>0$, one is tempting to figure how the event distance transforms under a Lorentz transformation.

17. Apr 24, 2014

### ghwellsjr

Now we are back to my first post #3.

18. Apr 24, 2014

### ANvH

yes,

yes,

and Bill_K acknowledged the difference in length and distance, which was a big relief ;-), his response did hit a trigger. I do appreciate the diagrams you often provide, yet a diagram with change of metrics, stretching the units of $t$ and $x$ can be very helpful too.

19. Apr 25, 2014

### ghwellsjr

No he didn't. He, and I, and the wikipedia article, and everyone else acknowledges that there is a difference between "Proper Length" and "Proper Distance" but you want there to be a difference between "length" and "distance" and that "Proper Length" gets divided by gamma to produce "length contraction" while "Proper Distance" gets multiplied by gamma to produce a "distance dilation". The first of these is correct but the second one makes no sense because there is no gamma associated with Proper Distance since there is no speed associated with it. It doesn't matter what frame you do the calculation in or whether the two events that you want to determine the Proper Distance for are associated with a single object (at some speed) or two different objects (at different speeds) or not even associated with any objects at all. It works for any two spacelike separated events and there is no speed associated with events and therefore there can be no way to determine a gamma for them. I gave you numerous examples to show this.

The unqualified terms "length" and "distance" can be used interchangeably so you are trying to draw a distinction (and a contradiction) which doesn't exist. Your term "distance dilation" is your own invention and should not be promoted as a significant or important concept.

I'm glad you appreciate my diagrams but I disagree that stretching (dilating) the units is helpful. My diagrams depend on the Lorentz Transformation process and all the concepts associated with it are well accepted and established and that's what you should learn, not some new invention that doesn't make sense or that implies a contradiction.

20. Apr 25, 2014

### ANvH

Pretty nitpicking ;-). Ok, I should have said "proper"

You know, if you guys have invented the word "Proper Distance", and I overlooked "of events", then you should bet on it that the word "distance" in relativity is very ambiguous. So don't blame me for raising a contradiction, until is was resolved by Bill_K's answer.

I was just contemplating, nothing more or less. You are blowing it up out of proportions, which I think is not fair to someone who is just contemplating. As if it is wrong to do so...

Yes I try to draw a distinction but just for me. It is simply based on the notion that a wave's angular frequency and wavenumber are both "dilated" or "contracted" by a Lorentz transformation. It does not undergo a temporal dilation together with a wavelength contraction.

I know what length contraction is, I know what time dilation is, I know that Proper distance is analogous to proper time (see wikipedia). I know now the term "Proper Distance of Events", and I am not a kid that should learn what to do or what not to do. Let's take it to understanding, to the meaning, and to contemplation.

I am not implying anything. I'd like to know all the intricacies, including the notion that the events are stretched spatially and temporally by a Lorentz transform. At least that is my current impression.

And I am aware that "distance dilation" is completely wrong because of all of the above.