# Why not length contraction?

#### Shawn Garsed

Hi everybody,

I have a question concerning length contraction/time dilation.

I read about how time dilation is calculated using transverse motion and how length contraction is calculated using longtitudinal motion in order to preserve lightspeed and it got me thinking: why not use length contraction for both types of motion. So instead of dilating time for transverse motion, why not contract the length by the same amount you contract it in the longtitudinal directon.

I know this may sound weird and I'm not sure if I'm formulating it very clear, but I was thinking about it and I couldn't figure out why we need TD and LC to preserve lightspeed.

Can you tell me if there's something I'm missing or that maybe I'm looking at it the wrong way?

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#### Doc Al

Mentor
I read about how time dilation is calculated using transverse motion and how length contraction is calculated using longtitudinal motion
Perhaps you're thinking of typical "thought experiments" used to derive time dilation and length contraction. For example, a "light clock" has light moving sideways (transverse) to the direction of motion. Is this what you're talking about? (If I'm way off, please provide details of what you read and where you read it.)

In any case, length is only contracted parallel to the direction of motion, not transverse to it. (It's easy to show that length cannot contract perpendicular to the direction of motion.) Time dilation, on the other hand, applies to any moving clock, not just to "transverse" motion.

#### JesseM

Science Advisor
This does show a weakness in using the light-clock thought experiment to derive time dilation though--if imagine the clock did contract in the direction perpendicular to the axis of motion, then both frames could agree the light moved at c between the two mirrors without any need for time dilation. You need to appeal to arguments beyond this thought-experiment to explain why this wouldn't satisfy the two postulates of relativity.

#### robphy

Science Advisor
Homework Helper
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This does show a weakness in using the light-clock thought experiment to derive time dilation though--if imagine the clock did contract in the direction perpendicular to the axis of motion, then both frames could agree the light moved at c between the two mirrors without any need for time dilation. You need to appeal to arguments beyond this thought-experiment to explain why this wouldn't satisfy the two postulates of relativity.
The light clock argument assumes that there is no transverse contraction, just like a "common sense" Galilean analysis would. If necessary, just start with the "nails on a meterstick"-type thought experiment that there can be no length contraction in the transverse direction.

#### JesseM

Science Advisor
If necessary, just start with the "nails on a meterstick"-type thought experiment that there can be no length contraction in the transverse direction.
What do you mean by "nails on a meterstick type thought experiment"? Are you talking about something like the idea of putting a series of vertical rulers in the frame where the light clock is moving and noting which marks on these rulers line up with which marks on a vertical ruler moving along with the light clock? If there was length contraction in this direction, the 1-meter mark on the moving ruler would line up with some mark at less than 1 meter on the resting rulers, which means an observer in the moving frame would have to agree that it was his own ruler that was shrunk in the vertical direction, and that's incompatible with the first postulate of relativity which implies that each frame should see objects moving relative to that frame behave in the same way (i.e. if one frame sees rulers moving relative to that frame shrink rather than expand, then all frames should see the same thing).

#### robphy

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JesseM, Yes, that's the one.

#### Shawn Garsed

Hi again folks.

I'm sorry doc al, I should've been a little bit clearer, but the thought experiment you're referring to is the one I was talking about.

It's easy to show that length cannot contract perpendicular to the direction of motion.
How is it easy to show this? Are you also talking about the "nails on a meterstick" type thought experiment? If so, is there a website where I can read about it.

P.S.
I don't doubt SR, it's just that I try to understand it intuitively (I'm an amateur when it comes to physics) by using the (absolute) constancy of the speed of light and then trying to deduce what "has" to happen in order to preserve lightspeed. The light-clock thought experiment helped me alot, but it also brought up this problem for me. And like I said, I don't doubt SR and that's why I asked you guys if I'm missing something.

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#### JesseM

Science Advisor
How is it easy to show this? Are you also talking about the "nails on a meterstick" type thought experiment? If so, is there a website where I can read about it.
I don't know of a website discussing why there can't be length contraction perpendicular to the direction of motion...but do you understand the first postulate of relativity, which says that the laws of physics are supposed to work exactly the same in all inertial frames? So if Alice is in a frame where the light clock is moving, and Bob is in a frame where the light clock is at rest, and if both are holding up rulers vertically (perpendicular to the direction they see one another moving), then if Alice sees Bob's ruler shrunk relative to her own, the first postulate implies that Bob must see Alice's ruler shrunk relative to his own.

The problem is that these two claims would lead to inconsistent predictions...do you understand the idea that all frames are supposed to agree on their predictions about local events like which mark on one ruler passes right next to which mark on another ruler? If so, suppose Alice is in a frame where the light clock is moving at 0.6c, and Bob is in a frame where the light clock is at rest (so Alice is moving at 0.6c in this frame). Suppose Bob is holding a ruler vertically, and Alice is holding her own vertical ruler positioned so that the 0-meter marks on both rulers line up when they pass next to one another. Given this, if Alice predicts that Bob's ruler should shrink along the vertical direction by a factor of $$\sqrt{1 - 0.6^2}$$ = 0.8 in Alice's frame, that should mean Alice would predict the 1-meter mark on Bob's ruler would line up with the 0.8-meter mark on Alice's ruler. But if in Bob's frame it is Alice's ruler that's supposed to shrink by a factor of 0.8 relative to his own, that would mean Bob would predict the 1.25-meter mark on Alice's ruler would line up with the 1-meter mark on Bob's ruler (since 1.25 * 0.8 = 1). Here we have two clearly inconsistent predictions about the local fact of what mark on Alice's ruler passes next to the 1-meter mark on Bob's ruler--they can't both be right! So the laws of physics couldn't work symmetrically in both Alice's frame and Bob's frame, which would violate the first postulate of relativity.

#### Shawn Garsed

I do know the first postulate and I understand your example, but then my first reaction is; why doesn't this problem arise when length contraction is applied in the longtitudinal direction?

#### JesseM

Science Advisor
I do know the first postulate and I understand your example, but then my first reaction is; why doesn't this problem arise when length contraction is applied in the longtitudinal direction?
Because when two rulers are oriented perpendicular to the axis with which they're moving relative to one another, there's only going to be a single moment when both rulers are next to one another, so any given mark on one ruler will only ever be next to a single mark on the other ruler. When they're oriented parallel to the axis of motion, in this case any given mark on one ruler will pass next to every mark on the other ruler in sequence, so the question of the timing of when marks on one ruler pass marks on another becomes important to deciding which ruler is longer, and since the two frames synchronize their clocks differently, it actually is possible for each frame to conclude that the other ruler is shorter without this leading to any disagreements in their predictions about local events. Take a careful look at the example I posted in this thread to see how this works.

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#### phyti

Hi everybody,

I have a question concerning length contraction/time dilation.

I read about how time dilation is calculated using transverse motion and how length contraction is calculated using longtitudinal motion in order to preserve lightspeed and it got me thinking: why not use length contraction for both types of motion. So instead of dilating time for transverse motion, why not contract the length by the same amount you contract it in the longtitudinal directon.

I know this may sound weird and I'm not sure if I'm formulating it very clear, but I was thinking about it and I couldn't figure out why we need TD and LC to preserve lightspeed.

Can you tell me if there's something I'm missing or that maybe I'm looking at it the wrong way?
We don't need TD and LC to preserve lightspeed. It seems you're looking at this from the wrong end. Time dilation is a consequence of a finite constant light speed. That's why it's postulate 2 in SR.

#### JesseM

Science Advisor
We don't need TD and LC to preserve lightspeed. It seems you're looking at this from the wrong end. Time dilation is a consequence of a finite constant light speed. That's why it's postulate 2 in SR.
But that's not really a physical fact about the universe--it's just a historical accident that Einstein included the constant speed of light as a postulate and derived time dilation and length contraction from that, in principle there's no reason you couldn't start out by treating time dilation and length contraction as fundamental postulates.

#### neopolitan

Shawn,

this reply is modified from another thread:

Note that the two equations:

Time Dilation: $$t' = \frac{t}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$$

Length Contraction: $$L' = {L}.{\sqrt{1-\frac{v^{2}}{c^{2}}}}$$

have assumptions behind them which are rarely stated explicitly.

The time dilation equation applies in the frame in which two events happen at the same location. The length contraction applies in the frame in which two events happen simultaneously. Think about that, take two events which are at the same place and happen simultaneously ... they are the same event.

You can't apply time dilation and length contraction to the same frame without being trivial (ie v=0, then $${\sqrt{1-\frac{v^{2}}{c^{2}}}} = 1$$, then t' = t and L' = L).

I hope this helps.

cheers,

neopolitan

#### phyti

But that's not really a physical fact about the universe--it's just a historical accident that Einstein included the constant speed of light as a postulate and derived time dilation and length contraction from that, in principle there's no reason you couldn't start out by treating time dilation and length contraction as fundamental postulates.
He treated light propagation as a physical phenomena, thus requiring it to be the same for all frames, in keeping with postulate 1, the one you so eagerly defend. His confidence was supported by the evidence from the MM experiment.
If the effects of motion on clocks is not a physical fact, why does science keep publishing papers to that effect?

#### JesseM

Science Advisor
He treated light propagation as a physical phenomena, thus requiring it to be the same for all frames, in keeping with postulate 1, the one you so eagerly defend. His confidence was supported by the evidence from the MM experiment.
If the effects of motion on clocks is not a physical fact, why does science keep publishing papers to that effect?
I was saying it's not a physical fact that length contraction and time dilation are "consequences" of the constant speed of light as opposed to the other way around--of course I was not denying that either of them are physical facts individually! I'm just saying you could equally well start by postulating time dilation and length contraction along with the first postulate, and from that you could derive the constant speed of light as a "consequence". There is no direction of causality here, logically each one implies the other. You seemed to suggest something else when you said "We don't need TD and LC to preserve lightspeed. It seems you're looking at this from the wrong end. Time dilation is a consequence of a finite constant light speed." Would you deny that logically, it is equally correct to say "a finite constant light speed is a consequence of time dilation and length contraction"?

#### Shawn Garsed

Take a careful look at the example I posted in this thread to see how this works.
I read the example and it seems you're describing a spacetime diagram. It's easy to see how both observers will see each others length contracted using the diagram. What I'm having problems with is how to visualize it.

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#### JesseM

Science Advisor
I read the example and it seems you're describing a spacetime diagram. It's easy to see how both observers will see each others length contracted using the diagram. What I'm having problems with is how to visualize it.
I'm not really depicting a spacetime diagram, but just a series of snapshots of how both rulers look at particular instants in each frame. The first image shows three snapshots from t=0 microseconds, t=1, and t=2 in the A frame; the second shows three snapshots from t'=0, t'=1, and t'=2 in the B frame. You can see in each of the first three snapshots that at each instant the B ruler is shrunk by a factor of two relative to the A ruler in the A frame, and you can see in the second three snapshots that at each instant the A ruler is shrunk by a factor of two relative to the B ruler in the B frame; but you can also check that neither frame ever disagrees about what clocks attached to each ruler-marking read at the moment a given pair of ruler-markings are passing next to one another.

Basically it has to do with the fact that if all the clocks on the A ruler are synchronized in the A frame, and all the clocks on the B ruler are synchronized in the B frame, then in the A frame all the clocks on the B ruler are out-of-sync, and in the B frame all the clocks on the A ruler are out-of-sync. And measuring the "length" of anything in your own frame depends on using clocks which are synchronized in your frame so you can note the difference between the position of the front end of the object and the position of the back end of the object at the "same moment" in your own frame; for example, if the back end of a rod moving at 0.6c relative to you is at the x=0 light-second mark on your ruler when the clock there reads t=0 seconds, and the front end of the moving rod is at the x=20 light-second mark on your ruler when the clock there also reads t=0 seconds, then you can conclude the rod is 20 light-seconds long in your frame. An observer in the rod's rest frame will agree about what these two clocks of yours read at the moment the back and front of the rod passed them, but in his frame these events were not simultaneous so they don't imply the same thing about the length of the rod relative to your ruler. In his frame, the rod is actually 25 light-seconds long (so the length in your frame is shrunk by a factor of $$\sqrt{1 - 0.6^2}$$ = 0.8 relative to the length in his frame), while the distance between the x=0 and x=20 mark on your ruler is only 16 light-seconds (so he sees the distance between marks on your ruler shrunk by the same factor of 0.8). However, in his frame the clocks attached to these two marks on your ruler, which are synchronized in your frame, are actually out-of-sync by 12 seconds (in general, if two clocks are synchronized and a distance L apart in their own rest frame, then in a frame where both clocks are moving at speed v in a direction parallel to the axis joining them, they will be out-of-sync by vL/c^2; in this example L=20, v=0.6 and c=1). So, in his frame at the moment the front end of the rod is next to the x=20 mark on your ruler with the clock there reading t=0, the back end of the rod is still 25-16=9 light-seconds away from the x=0 mark on your ruler, and the clock at that mark reads t=-12 seconds at that moment. Then since your ruler is moving at 0.6c in his frame, it will take 9/0.6 = 15 seconds for the x=0 mark to catch up with back end of the rod, and because of time dilation your clock at that mark will only have ticked forward by $$15*\sqrt{1 - 0.6^2}$$ = 15*0.8 = 12 seconds; so, the clock will have gone from t=-12 seconds to t=0 seconds by the time it reaches the back end of the rod. So you can see from this that in spite of the fact that he thinks the distance between ends of the rod is 25 light-seconds while the distance between the x=0 and x=20 mark on your ruler is only 16 light-seconds, he nevertheless ends up agreeing that the front end of the rod is next the the x=20 mark when your clock there reads t=0, and the back end of the rod is next to the x=0 mark when your clock there also reads t=0.

#### phyti

I was saying it's not a physical fact that length contraction and time dilation are "consequences" of the constant speed of light as opposed to the other way around--of course I was not denying that either of them are physical facts individually! I'm just saying you could equally well start by postulating time dilation and length contraction along with the first postulate, and from that you could derive the constant speed of light as a "consequence". There is no direction of causality here, logically each one implies the other. You seemed to suggest something else when you said "We don't need TD and LC to preserve lightspeed. It seems you're looking at this from the wrong end. Time dilation is a consequence of a finite constant light speed." Would you deny that logically, it is equally correct to say "a finite constant light speed is a consequence of time dilation and length contraction"?
Light speed is a property of the physical universe, and observer independent.
Time dilation is observer dependent. Theories such as SR are specifically formed to be free of observer dependent ideas as first principles, and will thus have a more general, even universal application.

#### JesseM

Science Advisor
Light speed is a property of the physical universe, and observer independent.
Light speed is independent of your choice of inertial frame, but it's not truly coordinate-independent like proper time along a given worldline, in a non-inertial coordinate system it can take a value other than c.

Also, although the time dilation experienced by a particular clock is frame-dependent, the general law that a clock moving at velocity v is slowed by a factor of $$\sqrt{1 - v^2/c^2}$$ holds in all inertial frames. So I would say that law of time dilation is independent of your choice of inertial frame in the same sense that any other laws of physics, such as the laws of electromagnetism, are independent of your inertial frame (and in the case of electromagnetism, the fact that different frames disagree on the time dilation of particular clocks is analogous to the way that different frames disagree on the velocity of particular charged particles and therefore disagree on the magnetic fields they create). In this sense, taking the law of time dilation as a postulate is analogous to the first postulate which says that all laws of physics should be independent of your choice of inertial frame.
phyti said:
Theories such as SR are specifically formed to be free of observer dependent ideas as first principles
Says who? Can you quote any textbooks or other sources that state this rule? Even if you consider time dilation and length contraction to be "observer dependent" while you don't say the same about the speed of light or the first postulate (which I disagree with for the reasons above), taking them as postulates does not in any way change the physical content of the theory, so even if physicists chose to follow this rule, it would merely be an aesthetic preference on their part, not something that has any relevance to actually using physical theories to make predictions.

#### Shawn Garsed

Thanks for the example and right now I'm letting it sink in, but do you know if there's a visual representation of this?
Also, in the thread you referred me too in an earlier post I wanted to know how you calculated the time-coordinates on B's ruler as seen from ruler A's frame.

#### robphy

Science Advisor
Homework Helper
Gold Member
Light speed is independent of your choice of inertial frame, but it's not truly coordinate-independent like proper time along a given worldline, in a non-inertial coordinate system it can take a value other than c.
"Light speed" is independent of observer if one defines it [most simply]
in terms of the light cone in the tangent-space of a given event in spacetime.
"Light speed" refers to a null [lightlike] vector [which is tangent to the light cone] at a given event. No reference to coordinates is needed.

Assigning the notion of a speed to something more complicated [like a curve in spacetime] requires more geometrical structure and careful definitions.

#### JesseM

Science Advisor
"Light speed" is independent of observer if one defines it [most simply]
in terms of the light cone in the tangent-space of a given event in spacetime.
"Light speed" refers to a null [lightlike] vector [which is tangent to the light cone] at a given event. No reference to coordinates is needed.
Of course the set of events that lie on the future or past light cone of a given event E is a coordinate-invariant fact--is that what you're saying? If so, how is it relevant to expressing the second postulate of relativity in a coordinate-invariant way, which is what phyti and I were discussing? It seems to me that the second postulate is specifically about the coordinate speed of light being invariant from one inertial frame to another, I can't think of a way of rephrasing the second postulate in a way that doesn't refer to inertial coordinate systems.

#### robphy

Science Advisor
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Of course the set of events that lie on the future or past light cone of a given event E is a coordinate-invariant fact--is that what you're saying? If so, how is it relevant to expressing the second postulate of relativity in a coordinate-invariant way, which is what phyti and I were discussing? It seems to me that the second postulate is specifically about the coordinate speed of light being invariant from one inertial frame to another, I can't think of a way of rephrasing the second postulate in a way that doesn't refer to inertial coordinate systems.
Since Minkowski spacetime is a vector space,
there is a blurring in the distinction
between using the light cone
to refer to sets of events in spacetime and
to refer to directions in the tangent space at an event.

I am using the latter, which will hold true for all spacetimes.

To me the second postulate effectively says that
there is a light-cone for each event in spacetime
which partitions the set of tangent-vectors at that event
into three classes: timelike, spacelike, and null.
All observers [regardless of their state of motion] will agree on that partitioning.
That is, a null vector to one observer is a null vector to all.

#### JesseM

Science Advisor
Thanks for the example and right now I'm letting it sink in, but do you know if there's a visual representation of this?
I would recommend drawing some simple diagrams of the scenario I gave for yourself if you're having trouble following it. You really only need three diagrams:

1. One showing how things look at t=0 in your frame, where the moving rod is 20 light-seconds long and aligned parallel to your ruler, and the back end is lined up with the x=0 mark on your ruler with your clock attached to that mark reading t=0, and the front end lined up with the x=20 mark on your ruler, with your clock attached to that mark reading t=0 as well.

2. Another diagram showing how things look at t'=0 in the rod's rest frame, where the rod is 25 light-seconds long, and the distance between the x=0 and x=20 marks on your ruler is shrunk to only 16 light-seconds. At this moment the front end of the ruler is next to the x=20 mark on your ruler, and your clock at that mark reads t=0; and at the same time in this frame, the back end of the ruler is 9 light-seconds away from the x=0 mark on your ruler, and your clock at the x=0 mark reads t=-12 seconds.

3. A final diagram showing how things look 15 seconds later at t'=15 in the rod's rest frame, when the back end of the rod is lined up with the x=0 mark on your ruler (which means the front end is now 9 light-seconds away from the x=20 mark on your ruler). At this moment, your clock at the x=0 mark reads t=0 seconds, and your clock at the x=20 mark reads t=12 seconds.

Once you have these diagrams, you can verify that since the rod is moving at 0.6c relative to your ruler, the rod is shrunk by a factor of 0.8 in your frame and likewise the distance between the x=0 mark and x=20 mark on your ruler is shrunk by the same factor of 0.8 in the rod's rest frame. You can also see that in the rod's frame, both your clocks are slowed by the same factor of 0.8 (they both tick forward by 12 seconds in the 15 seconds between the first and second diagram), and they are also consistently out-of-sync by 12 seconds.
Shawn Garsed said:
Also, in the thread you referred me too in an earlier post I wanted to know how you calculated the time-coordinates on B's ruler as seen from ruler A's frame.
Mathematically the simplest way is probably just to use the Lorentz transformation, which says that if you know the position x and time t of an event in A's frame, the position x' and time t' of the same event in B's frame must be:

x' = gamma*(x - vt)
t' = gamma*(t - vx/c^2)
with gamma = $$1/\sqrt{1 - v^2/c^2}$$, and v being the velocity that B is moving in A's frame.

Another way would be to use the formulas for length contraction, time dilation and the relativity of simultaneity separately. The length contraction formula says that if two marks on B's ruler are a distance L apart in B's frame, in A's frame the distance between these marks will be only $$L' = L*\sqrt{1 - v^2/c^2}$$. The time dilation formula says that if it takes a time T for a specific clock which is at rest in B's frame (say, the clock with the red hand on B's ruler) to tick forward by a certain number of seconds, then in A's frame that clock will tick slower and therefore take longer to tick forward that amount of seconds, taking a time $$T' = T/\sqrt{1 - v^2/c^2}$$ to do so. Finally, the relativity of simultaneity formula says that if two clocks at rest in B's frame are a distance L apart and synchronized in B's frame, then in A's frame they will be out-of-sync by $$vL/c^2$$ (with the clock in 'front' relative to their direction of motion in A's frame being behind the clock in 'back' by this amount).

If you like I can pick two specific clocks in B's frame and show how the way I drew them in the diagram of A's frame matches up with what you should expect based on both the Lorentz transformation and also based on the separate formulas for length contraction, time dilation and the relativity of simultaneity.

#### Shawn Garsed

If you like I can pick two specific clocks in B's frame and show how the way I drew them in the diagram of A's frame matches up with what you should expect based on both the Lorentz transformation and also based on the separate formulas for length contraction, time dilation and the relativity of simultaneity.
That won't be necessary, but thanks anyway. Actually, the tips you gave me about drawing out a spacetime-diagram helped alot. In fact, it's easy to see the effects of TD, LC and RoS in the diagram. My problem has always been to go from a diagram to actually visualizing how these effects take place in the 'physical' world, but I'm getting there.

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