# What space contracts in Special Relativity?

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1. Jan 22, 2014

### dubiousraves

Hi. I know that space for an object in motion will contract, relative to an observer. We generally read that space contracts for the object in the direction of motion. But doesn't space around the object also contract? Can someone clarify this? Thanks.

2. Jan 22, 2014

### dEdt

I don't think this is a good way of thinking about length contraction.

According to the theory of relativity, the length of an object in motion (along the direction of motion) is shorter than if the object was at rest. This is least ambiguous way of describing length contraction.

The problem with characterizing length contraction as a contraction of space itself is that space doesn't contract. Space is always defined relative to an observer. Suppose someone named Alice is observing an object. If that object starts moving, Alice's description of space doesn't change at all. Rather, the object just shrinks.

3. Jan 22, 2014

### HallsofIvy

Hendrik Lorentz, for whom the Lorentz transforms are named, had a theory that would explain the contraction of a moving object- since the magnetic field of a charged object was NOT invariant under motion, he suggested that in some, as yet unknown, way, the magnetic field in a moving object was stronger in the direction of motion and so caused it to contract. That would cause a material object to contract, not the space around it. But later experiment (in particular the "Kennedy experiment" which was like the Michaelson-Morley experiment but with arms of differing length, showed that this is not true. Contrary to what dEdts says, if you have two poles, one after the other, moving in the direction of their lengths, the space between them also contracts. "Length contraction" is a contraction of space, not just material objects.

4. Jan 22, 2014

### dubiousraves

OK thanks. Actually, I meant to say "length" contraction. But let's take the example of a rocket making a round trip to a nearby star at close to c. According to the rocket's clocks, maybe 50 years pass (I'm not doing math here, just giving a rough estimate), but back on Earth thousands of years have elapsed. My question is, in the length-contraction part of this journey, what exactly is seen as contracting? Obviously, it can't just be the rocket shrinking that causes all the duration difference, so I'm assuming that the lengths of space are also contracting. Or, is it just that the vast majority of the difference is due to time dilation? Thanks. Feel free to give a formula for the length contraction.

5. Jan 22, 2014

### dubiousraves

Ah, thanks Hallsoflvy, I didn't see your comment until after I posted my response above. I would still like to see the formula(s) that delineate which lengths/spaces are contracting and in what proportion these are to the time dilation.

6. Jan 22, 2014

### dEdt

HallsofIvy

Two points:
1) It's a fact that the intermolecular forces in a moving object are different than in a stationary object. Fitzgerald used this observation to predict length contraction. There's no reason to think that his derivation or reasoning were wrong, considering that he used Maxwell's equations to get his result and Maxwell's equations are Lorentz invariant.

2) If you have two poles separated by some distance, how the distance changes with time depends on how exactly you accelerate the two poles. It's possible to accelerate them so that the distance grows, shrinks, or stays the same.

dubiousraves

Let's analyze the rocket trip from two reference frames: the Earth's and the rocket's.

From the Earth's reference frame, the only thing that's shrinking is the rocket. The distance to the star does not change in the slightest. From the perspective of someone on Earth, the reason that the astronauts on the rocket think the trip only took 50 years is time dilation.

According to the theory of relativity, if a process at rest takes a time $T$ to complete, then the same process moving with speed $v$ will take a longer time to complete, namely $\frac{T}{\sqrt{1-v^2/c^2}}$. This is known as time dilation, and explains why the astronauts would only think that 50 years elapsed: their clocks were slowed down!

From the astronaut's reference frame, it's the Earth and the star that are moving. Hence the distance between them contracts, which makes the journey that much shorter. That's the reason, from the astronaut's perspective, why the trip only took 50 years.

7. Jan 22, 2014

### dubiousraves

Thanks. I am aware of the time dilation equation. However, if you say the distance between the Earth and the star contracts from the astronaut's reference frame, is there not a formula that would express that exact contraction?

Last edited: Jan 22, 2014
8. Jan 23, 2014

### ghwellsjr

The formula for Length Contraction is the inverse of the one for Time Dilation, but those are just short-cuts that can get you into trouble. The equations that will never get you into trouble are called the Lorentz Transformation process. I like to use units for a simplified version of those equations where c has the value of 1, for example, 1 foot per nanosecond. I also like to express velocity as a fraction of c which we call beta and assign the Greek letter beta (β).

So let's say we have a rod of length 5 feet laying motionless along the x-axis. A spacetime diagram showing this simple scenario is:

The blue and red lines represent the two ends of the rod. The rod is spread out all along the space between it's two endpoints but we don't care about that, we only care about the endpoints that are used in determining its length. I'm only showing the positions of the endpoints for just a short time but we understand that the lines really extend upward and downward. Now let's see what happens if we transform the coordindates of this scenario into one that is moving at 0.6 toward the left. In this new diagram, the rod will be moving to the right at 0.6c and we can see how long it is.

The first thing we have to do when using the Lorentz Transformation is calculate the value of gamma, the Lorentz factor (the same value used to determine Time Dilation), as a function of β, in this case -0.6:

γ = 1/√(1-β2) = 1/√(1-(-0.6)2) = 1/√(1-0.36) = 1/√(0.64) = 1/0.8 = 1.25

Now we take the value of the t and x coordinates of each event (dot) in the diagram and calculate new values for the new diagram. The two equations for the new primed values are:

t' = γ(t-βx)
x' = γ(x-βt)

We can take another shortcut and just calculate the two endpoints of each of the two worldlines and fill in the extra events (dots) proportionally. So for the top of the blue line, x=0 and t=4:

t' = 1.25(4-(-0.6)*0) = 1.25(4) = 5
x' = 1.25(0-(-0.6)*4) = 1.25(2.4) = 3

For the bottom end of the blue line we have x=0 and t=0:

t' = 1.25(0-(-0.6)*0) = 1.25(0) = 0
x' = 1.25(0-(-0.6)*0) = 1.25(0) = 0

For the top of the red line we have x=5 and t=4:

t' = 1.25(4-(-0.6)*5) = 1.25(4+3) = 1.25(7) = 8.75
x' = 1.25(5-(-0.6)*4) = 1.25(5+2.4) = 1.25(7.4) = 9.25

For the bottom of the red line we have x=5 and t=0:

t' = 1.25(0-(-0.6)*5) = 1.25(3) = 3.75
x' = 1.25(5-(-0.6)*0) = 1.25(5) = 6.25

From those calculations and by interpolating (or by more calculations) we can make this diagram:

Now it's important to know that when we want to determine the length of an object (or the space between objects), we have to do it along a line where time is constant. So look at the horizontal grid line where time is 5 nanoseconds. The top of the blue line is on that grid line at x=3 feet and immediately to its right is the second dot up on the red line with an x coordinate value of 7 feet. The difference between these to events (dots) is 4 feet. This is the contracted length of the rod from its Proper Length in its rest frame divided by gamma (5/1.25=4).

You can also see the Time Dilation in this diagram where the dots representing 1 nsec intervals of time in the rest frame of the rod are now stretched out to 1.25 nsec of Coordinate Time.

The other important feature of Special Relativity call Relativity of Simultaneity is also shown where events that were simultaneous in the first frame occur at different times in the transformed frame.

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9. Jan 23, 2014

### Bill_K

He didn't have the slightest idea what he was doing, pure guesswork. Working in 1890, Lorentz knew nothing of Quantum Mechanics or the nature of intermolecular forces or what actually determines the size of an object.

10. Jan 23, 2014

### dubiousraves

Thanks for that elegant explanation, GHWellsJr., but I'm still wondering if the the space AROUND a moving objects contracts.

11. Jan 23, 2014

### nitsuj

Is space itself ever measured? Or is it the length between points

12. Jan 23, 2014

### dEdt

I don't think that's fair. True, his derivation was based on a simplified and classical model, but so was all of 19th century physics. Do we begrudge Clausius and Maxwell of their kinetic theory of gases because it used a simplified and classical model?

13. Jan 23, 2014

### Bill_K

There is no such model, and Lorentz did not pretend to have one. You can't build a solid object out of Maxwell's Equations.

How would you explain in classical terms the Lorentz contraction of a gas?

14. Jan 23, 2014

### ghwellsjr

I did try to hint at that when I said, "...when we want to determine the length of an object (or the space between objects)...", so the answer is yes. The blue and red lines can represent either the two ends of a single object or two separate objects, it doesn't matter.

But remember, length (or distance) contraction is merely a coordinate effect. When we change reference frames and use the Lorentz Transformation process, the time and length (or distance) coordinates change. Nothing has actually changed to the objects or the space. The same can be said for Time Dilation.

Observers in a scenario cannot tell that we switched to a different reference frame when we did the above exercise and so nothing changes either in what they see, what they measure, or what they observe. In fact, Length Contraction and Time Dilation are completely unobserverable. They can only be determined by observers by proactively sending out radar signals, waiting for their echoes, observing clocks on other objects, logging all the data and the times they occurred according to their own clock and then doing a lot of computation. Only after the entire scenario is over will they be able to go back and make diagrams like I did and see those coordinate effects.

15. Jan 23, 2014

### dEdt

If you assume that intermolecular forces are electrical in nature then you can deduce, based on how the electric field of a moving point charge compares to a stationary charge, that the size of molecules as well as the intermolecular distance will shrink by a factor of gamma. I won't pretend that this is a precise derivation or anything (and as you mention, a precise derivation is classically impossible because classical physics does not permit the existence of solid bodies), but nonetheless my original point sill stands: contrary to what HallsofIvy stated, the contraction of a moving body is ultimately due to the behavior of the forces holding that body together, whether or not it's practically feasible to calculate length contraction by analyzing these forces. Special relativity doesn't change this fact. It only allows us to efficiently calculate what the length contraction will be without having to analyze, say, intermolecular forces.

As for the length contraction of a box of gas, the explanation is the same as for a rod: the box containing the gas will contract because of the behavior of the forces holding the box together.

16. Jan 23, 2014

### Mentz114

This is wrong. Molecular forces don't come into it. As ghwells said

17. Jan 23, 2014

### WannabeNewton

This entire discussion is going to lead down a road that has already seen blow after blow after blow in the past. Some people try to separate length contraction into two classes: the Lorentz contraction obtained from boosting between inertial frames, which these people would label a "coordinate artifact" and the Lorentz-Fitzgerald contraction which is what you are referring to and which said people would label a "real effect". Therefore these people would qualify Lorentz contraction as a different phenomenon from Lorentz-Fitzgerald contraction. Others would argue that they are not different phenomena but rather just different "components" of the length contraction relativized to a given observer (which is, IMO, the better viewpoint). With that terminological issue out of the way, it should be clear now that neither your posts nor those of others in this thread have encompassed both "components" of length contraction. Everyone is speaking past one another by exclusively referring to one or the other.

18. Jan 23, 2014

### JVNY

At the risk of bringing blows down upon myself, can I ask some questions to clarify the two views?

Consider the basic barn and pole paradox, where the barn has a front door and a solid rear wall. A pole has greater proper length than the proper length of the barn; the pole and barn are in inertial motion with respect to each other; in the barn frame, the pole is length contracted and enters the barn through the front door; the door closes, and the pole is entirely within the barn (at least for a very short period of time in the barn frame).

dEdt and ghwellsjr, do you each agree that the pole is entirely within the barn in the barn frame for at least a short period of barn time?

dEdt: if yes, does this occur only if the rod is in absolute motion (such that its molecules are contracted due to the behavior of the forces holding the rod together), or do you conclude that there is a behavior of forces on the rod regardless of whether the rod had been accelerated, the barn had been accelerated, or even if one does not know which one was accelerated (given that we are told that there is no absolute motion in inertial motion)?

ghwellsjr: if yes, what do you mean by the rod's length contraction being merely a coordinate effect, or as WannabeNewton states a "coordinate artifact"? If the rod is contracted and fits in the barn in the barn's frame, isn't its length contractedness in the barn frame a "real effect," using WBN's phrase?

19. Jan 23, 2014

### dEdt

Yes, absolutely.

I'm not 100% sure I understand your question, but I'll answer as best as I can.

Let's say we're in the barn frame and happen to see a rod flying past us. We don't know how the rod reached that speed -- maybe it was travelling that way for its entire existence, maybe it was recently accelerated to that speed, maybe it was travelling even faster in the past and recently decelerated -- nor do we care. Further suppose we have the divine knowledge that the same rod would have a length $L$ along the direction of motion if it were stationary. Then, necessarily, the length of the rod will be $L\sqrt{1-\beta^2}$ in the barn frame.

20. Jan 23, 2014

### ghwellsjr

No one gets blows for asking questions here.

That's not the basic barn and pole paradox because it doesn't have a back door which opens at just the right time to let the pole out unharmed. Nevertheless, we can deal with either type of situation.

Yes.

No, it's based on the convention of simultaneity. Conventions are man-made concepts. What's real is the fact that the pole will smash into the rear of the barn and explode it unless something is done to stop it. If you do stop it, the other issue of "contractedness" comes into play.