Distance dilation and length contraction

[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.

 

[PAllen]

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.

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).

 

[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).

Does that clarify your "contradiction"?

 

[Bill_K]

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?

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'.

 

[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 $v<c$.

 

[Bill_K]

I appreciate the responses, but there is a difference between proper length and proper distance.

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.

 

[Nugatory]

I appreciate the responses, but there is a difference between proper length and proper distance.

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.

 

[ANvH]

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.

...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.

 

[ANvH]

Thank you for the clarification.

Sorry, Thanks to all of you ;-)

 

[ghwellsjr]

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.

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 anyone of the blue dots (events) and anyone 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

 

[ANvH]

I think you are still mixed up.

No, I wrote

Since $cΔt=Δr$, $Δr=\gamma Δ\sigma$.

in post #5. The $Δr$ is the dilated distance. I know that proper distance is invariant, so is the proper time and proper length.

 

[ANvH]

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?

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 $\gamma$.

 

[ghwellsjr]

No, I wrote

Since $cΔt=Δr$, $Δr=\gamma Δ\sigma$.

in post #5. The $Δr$ is the dilated distance. I know that proper distance is invariant, so is the proper time and proper length.

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:

$Δ\sigma=\sqrt{Δr^{2}-c^{2}Δt^{2}}$​

to:

$Δ\sigma=\frac{cΔt}{\gamma}\sqrt{-1}$​

if $v<c$

and:

$Δ\sigma=\frac{cΔt}{\gamma}$​

if $v>c$.

Make sure you explain what v is.

 

[ANvH]

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.

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.

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

 

[ghwellsjr]

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.

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.

Here is a spelled out derivation.

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

 

[ANvH]

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

yes, again a typo.

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

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.

 

[ghwellsjr]

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.

Now we are back to my first post #3.

 

[ANvH]

Now we are back to my first post #3.

yes,

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"?

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.

 

[ghwellsjr]

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"?

yes,

and Bill_K acknowledged the difference in length and distance, which was a big relief ;-), his response did hit a trigger.

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 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.

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.

 

[ANvH]

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"

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

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".

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.

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.

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...

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.

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.

that's what you should learn.

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.

not some new invention that doesn't make sense or that implies a contradiction.

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.

 

[Nugatory]

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.

An event is a single point in space-time. It cannot be stretched, contracted, or anything else of the sort, and it is completely unaffected by the Lorentz transforms. A few examples of events:
- The hands of my wristwatch point to noon.
- I opened my hands, dropping an object that I had been holding.
- The object I dropped hits the floor.
- The Earth's shadow passes across a particular point on the moon during a lunar eclipse.
- A car driving down the road passes mile marker 63.

It is sometimes convenient to label events with coordinates, by saying things like "The point in space-time that is the surface of the earth, at latitude X and longitude Y, at 9:30 AM GMT" or "the event happened at position (x=X, y=Y, z=Z and time t=T) using coordinates in which I am at (x=0, y=0, z=0, and time zero is 3:30 AM according to my wristwatch). Note that I always need four numbers, three spatial and one temporal, to label an event completely, but there are many different coordinate systems that I can use, and the choice is somewhat arbitrary - I'd use latitude and longitude to label a point in the middle of the ocean, but street and avenue to label a point in midtown Manhattan.

The Lorentz transformations have nothing to do with events, they just give us the mathematical relationship between certain (especially interesting) coordinate systems. The events are out there whether we assign coordinates to them or not, and if we do, whether or not the coordinates that we choose are of the type for which the Lorentz transforms apply.

 

[Fredrik]

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.

Time dilation is really just a measure of how much the scale on the t' axis differs from the scale on the t axis. The scale on the x' axis differs from the scale on the x-axis by the same factor. It would make sense to call this "distance dilation", but no one actually uses that term.

Length contraction is pretty different from time dilation.

The two vertical lines describe the motion of a rod. Since they're vertical, the rod is at rest in the (t,x) coordinate system. In the (t',x') coordinate system, the rod has velocity -v, where v is the velocity of the primed coordinate system in the unprimed. The length that the observer using the (t,x) coordinates assigns to the rod is a property of the line segment from the origin to A. The length that the observer using the (t',x') coordinates assigns to the rod is a property of the line segment from the origin to B.

Time dilation and length contraction both involve a change of scale (and that change is the same, not the opposite), but length contraction also involves the fact that the observers disagree about which line segment in spacetime is "the rod, right now".

 

[ANvH]

Time dilation is really just a measure of how much the scale on the t' axis differs from the scale on the t axis. The scale on the x' axis differs from the scale on the x-axis by the same factor. It would make sense to call this "distance dilation", but no one actually uses that term.

Length contraction is pretty different from time dilation.

The two vertical lines describe the motion of a rod. Since they're vertical, the rod is at rest in the (t,x) coordinate system. In the (t',x') coordinate system, the rod has velocity -v, where v is the velocity of the primed coordinate system in the unprimed. The length that the observer using the (t,x) coordinates assigns to the rod is a property of the line segment from the origin to A. The length that the observer using the (t',x') coordinates assigns to the rod is a property of the line segment from the origin to B.

Time dilation and length contraction both involve a change of scale (and that change is the same, not the opposite), but length contraction also involves the fact that the observers disagree about which line segment in spacetime is "the rod, right now".

Thanks, and this is exactly how I understand it.

 

[ghwellsjr]

Thanks, and this is exactly how I understand it.

Good, since you understand it exactly, is the length of the rod from the origin to B an example of contraction or dilation?