# Lorentz contraction, what is it?

Can we describe our universe without it?

Example; Explaining a muon able to hit Earth, by its time dilation only? Or a spinning disc contracting?

Does it exist, and if it does, does it do so from all frames of existence, that means, if you observe a Lorentz contraction, will the person in that frame, 'experience' it too?

By that I also mean that arguing that no 'frames of reference' can be 'simultaneous' considering a 'event', won't cut to the cheese/chase whatever, here :)

If we assume a shared universe, there should be a 'real' contraction, or not be a 'real' contraction. To have it both ways would definitely hurt my head :)

?

• It's not at all clear to me what you're asking. The only actual question I see is asking whether a Lorentz contraction observed in one frame of reference is also observed in another frame - is that what you mean to ask? – David Z Feb 13 '11 at 21:32
• Yes. And the muon example is often used to describe it how you will need both to define how it's possible for a muon to reach Earth.. By the way, what's this with negatives without explanations? A way to prove that you have a view without having to stand by it? – user1907 Feb 13 '11 at 21:44
• I can't speak for anyone else, but I left an explanation for my downvote, namely that your question is unclear. – David Z Feb 13 '11 at 21:53
• The other down-vote was me. And the reason is that I don't see any question at all. Only the one in the title but it's too elementary and you can look it up on wikipedia. – Marek Feb 13 '11 at 22:14
• Its a little late to close this question so I'll join @Marek and @David in down-voting it. The question contains some serious misconceptions which likely needs a course in theoretical or modern physics - at the level of an American undergraduate class - for their resolution. – user346 Feb 14 '11 at 5:25

The Lorentz contraction is the reduction of the length of a moving object that follows from its motion - one of the main effects implied by Einstein's special theory of relativity. If the length of the object in the direction of motion is $L_0$, if measured at rest, then the other observer who sees the object as moving at speed $v$ will see the length reduced to $$L = L_0 / \gamma$$ where the Lorentz factor $\gamma=1/\sqrt{1-v^2/c^2}$ is greater than one.

Muons' average lifetime is something like 2 microseconds. By the speed of light, they would travel 600 meters in average. However, their Lorentz factor is of order $\gamma\approx 10+$. This means that a very large fraction of the muons produced at the top of the atmosphere are easily able to get through the atmosphere, something like 8 kilometers, before they decay.

This "increased ability" to get very far is interpreted in the following ways:

1. In the Earth's frame, the muons' life processes are slowed down by time dilation, by the factor of $\gamma$. This effectively increases their lifetime by an order of magnitude, so they're able to fly not for 2 microseconds times $c$ but for 20+ microseconds times $c$ which is equal to 8 kilometers or so.

2. In the muon's inertial frame, where the muon is moving down, the vertical thickness of the atmosphere is Lorentz-contracted by the factor of $\gamma$, so even though the muon only lives for 2 microseconds in this frame, it will also be able to get through the 8 kilometer atmosphere because the thickness seems to be less than a kilometer for the muon.

As you can see, the effect exists, is real, can't be made unreal, can't be forgotten, can't be denied, and so on. You have asked the same question many times, and I hope that I answered your question many times, too. The Lorentz contraction is a basic consequence of the special theory of relativity and there's no doubt that this prediction is valid.

But no, the object itself doesn't see itself contracted - just like it doesn't see itself moving. In its inertial frame, the rest frame, it can't experience the Lorentz contraction because it's at rest. The speed is $v=0$ which means $\gamma=1$. The whole point of the Lorentz contraction is that it only appears because of the relative motion of the object and the inertial frame that keeps track of the object.

(I wanted to avoid the verb "observe" to emphasize that this is not an optical effect: the Lorentz contraction is about how long the object actually is, not how long it looks like when the finite speed of light emitted or reflected by this object is taken into account.)

Using your terminology, the very point of the special theory relativity - the point that gave it its name - is that the Universe is not shared. The standard terminology is that the space and time are not absolute. Many quantities - such as the length of the objects or the duration of a process - depend on the observer, especially his velocity. That's why we say that those quantities - including space and time - are relative.

(You may say that "everything in the Universe" is shared, but this "everything" has to be defined as the whole spacetime. A particular slicing of the spacetime to space and time, which is needed to measure the length at a given moment, or the duration of a process, is not shared.)

That's why this fundamental insight due to Einstein, one that represents one of the two main pillars of modern physics, is called relativity, even though this theory also depends on some quantities' being invariant - and Einstein himself used to prefer the name "theory of invariants" for relativity, at least in the early days. However, he also adopted the name "relativity" later and this name has a good justification, one that was described above. Apologies for the headaches that this fact may cause to you - but it's Nature's "fault", not mine and not even Einstein's.

• Nice one Luboš. Don't know if I've asked it that many times though, not here :) But I like the way you present it. – user1907 Feb 13 '11 at 21:56
• You're such a great teacher, I hope someone offers you a job and a salary to match or.. do you enjoy your time here and your blog too much? ;) – John McVirgo Feb 13 '11 at 21:58
• Hmm, I know this is a touchy subject, and as you said Luboš, it do gives me somewhat of a headache :) Not in the way it may be real, but in its further implications. And that's why I would have liked the 'philosophy' tag on it. Making it possible for folks to answer with their opinions without needing to feel that they have to stake their reputations on it. That said I actually agree with you Luboš. Although the implications created by doing so does not diminish my headache :) – user1907 Feb 13 '11 at 22:07
• Thanks for your interest, @yoron, and compliments, @John. I've added the philosophy tag. – Luboš Motl Feb 14 '11 at 7:03
• @yoron and @Lubos: I actually removed the philosophy tag initially because this really seemed like a question about the Lorentz transformation. Since you saw fit to override that and re-add it, I'll let it stay... although I will say that the philosophical nature of the question is a significant part of why I don't think it's a good one. – David Z Feb 14 '11 at 7:25

If there were a spaceship that is 200m long when you view it at rest, and then the ship accelerated to about 87% of the speed of light, you would view the spaceship's length to be 100m. That is Lorentz contraction, but it is not a mechanical effect. It is purely kinematic - it's an artifact of the way we make measurements. The ship is not being crushed.

The passengers inside the spaceship would still believe that the ship's length was 200m because the body of the ship is not the only thing getting shorter. Everything inside the ship gets shorter by the same amount. For example, a ruler would get shorter by a factor of 2. Therefore, if people inside lined up a bunch of rulers to measure the ship, they'd get the value 200m, because from your point of view they're measuring something 100m long using rulers that are half their normal length.

We might wonder why the ship doesn't suffer any ill-effects from being squashed like this. If I flatten a soda can by a factor of two by putting it in a vice and squeezing, it bursts and crumples. Why don't the soda cans in the spaceship burst?

When I put a soda can in a vice and squeeze, the front and back of the soda can get closer together. If I had a ruler next to the can, the distance between the front and back, measured by the ruler, would decrease. But that is not so in the spaceship. The soda can shrinks, and so does the ruler, and so as far as the ruler and soda can are concerned, there's no contraction going on at all. You might be clever and say, "What if I put the ruler in the vice along with the soda can?", but this would not make everything shorter - just those two. For example, the vice has plates that are getting closer to each other when measured against the size of the vice.

The source of length contraction, and the intuitive problems it causes, can be traced back to our intuition about the relativity of simultaneity.

Let's consider how you would make a measurement of the spaceship's length. What you would do is lay down a giant measuring tape, and wait until the spaceship comes up to it. Then, right when it gets up to your tape, you would measure where the front of the spaceship is, where the back is, and subtract them to get the length of 100m.

To get around this simultaneity disagreement, maybe you and the passengers agree on a different procedure. You'll set up a stopwatch, and measure how much time it takes for the ship to pass you. Then you multiply by the speed of the ship to get the length. Doing this, you still get 100m.

This time, the passengers still call foul, but for a different reason. They think your stop watch is running slow due to time dilation. In fact it's running slow by a factor of two, so you recorded half as much time as you should have, and incorrectly measured their 200m ship as 100m.

So length contraction is unavoidable and it's real, but it's not a matter of things "really getter shorter" or not. It's a matter of disagreement on how to measure lengths and times when you're moving in different reference frames.

What everyone will agree on is the proper time $\tau$ between two events, defined by

$$\tau = \sqrt{\Delta t^2 - \left(\frac{\Delta x}{c}\right)^2}$$

$\Delta t$ is the time separation between two events (you start your stopwatch, you stop it), and $\Delta x$ is the space separation between them. Different observers will have different time or space separations, and therefore different lengths of the ship, but this proper time (or for spacelike-separate events the spacetime interval $s^2 = -(c\tau)^2$) will be agreed on.

There's a nice parable at the beginning of Taylor and Wheeler's Spacetime Physics. Imagine two surveyors. One of defines the cardinal directions based on geographic north, and the other based on magnetic north. They would tell you different things when you asked how far north New York is of Los Angeles. However, if you asked them the total distance, you'd get the same answer. The surveyors have different coordinate systems to measure the same thing, and so the components of the separation are different.

In relativity "distance" becomes just one component of the spacetime separation, just like in space "north/south distance" is just one component of the space separation. Observers in different reference frames measure different distance components because they are using different coordinates for the same underlying spacetime events.

• Lovely explanation, although it invites to a 'SpaceTime' that not only have this 'time dilation' but also 'Lorentz contraction' and defined as one or the other depending on your 'frame of reference'. So, in some weird way, they then becomes the same, don't they, even if expressed differently depending on where I stand observing? – user1907 Feb 13 '11 at 22:31
• Rereading you again, I'm not totally sure if I got it right? Maybe I jumped the gun a little assuming that you placed a time dilation and a Lorentz contraction as equally true? Are you thinking of it as a result of different positional systems, and looking at the 'arrow of time' as something illusionary if so? Because it seems to me that you have to look at both the same way if so? Having a 'time dilation' as real but a Lorentz contraction as a kinematic effect would make them unequal, wouldn't it? And also invalidate the muon & spinning disc example where you expect the rim to 'crack up' – user1907 Feb 13 '11 at 22:40
• @yoron Yes, time dilation and Lorentz contraction are very similar effects. They are both consequences of viewing the same events in different reference frames. You can read the formulas for both of them off of the formula for a Lorentz transformation (en.wikipedia.org/wiki/…), which encodes all the information about how to go between boosted reference frames. – Mark Eichenlaub Feb 13 '11 at 22:40
• You might see it as a result of a time dilation though? In where the Lorentz contraction will be a time dilations direct result? (keep forgetting that as I hit 'enter' it will place my post as finished:) I will need to think more about this. It do give me a headache :) – user1907 Feb 13 '11 at 22:52
• @yoron I prefer to view both time dilation and length contraction as results of the invariance of the spacetime interval rather than one being a result of the other. Alternatively, you can view them as results of the constancy of the speed of light. You might use time dilation to derive length contraction, but time dilation itself would be based on the constant speed of light. If you wanted, you could use the constant speed of light to derive length contraction, then use length contraction to derive time dilation. The constant speed of light/invariant interval underlie it all. – Mark Eichenlaub Feb 13 '11 at 23:02

You're making an issue of something that can be seen in everyday life.

Someone in a car driving past you will say they waved to you at (x1',y1', z1', t1') and crashed at (x1',y1', z1', t2') using their space and time coordinates, whereas you claim these events occured at (x1, y1, z1, t1 = t1') and (x2, y2, z2, t2 = t2') in your frame.

Can you see the difference?

There is a difference in the space interval you both measure between these two events, and so you could view it as a space contraction of some sort.

Special relativity takes it a stage further with the introduction of a time contraction between events, between different frames.

• "There is a difference in the space interval you both measure between these two events, and so you could view it as a space contraction of some sort." I agree that they seem to go together, relatively speaking, but I find it harder than you to get my head around a Lorentz contraction. But I really appreciate the tries from you all to explain how you see it. I'll reread it and see if I can put it into the SpaceTime I observe myself. Doing it your way is one possibility, although what that would make of a Lorentz contraction I'm not sure? Do you see it as everything 'contracting', at all speeds? – user1907 Feb 14 '11 at 4:19
• @yoron the world consists of events, which are located by time and space measurements within a frame. To understand that there is no inconsistency within special relativity, you need to use the Lorentz transformations, rather that the Lorentz contraction forumla which is only true for events that are simultaneous in one frame, the time dilation formula is only true for events at the same location in one frame. To understand special relativity takes time and practise, and not the causal read of a popular book on the subject. – John McVirgo Feb 14 '11 at 23:32