# Is relativistic effect of length contraction physically unreal ?

Is relativistic effect of length contraction physically "unreal"?

One guru indicate that Length contraction has nothing to do with compression.
It is more correct to view length contraction as a rotation in space time.
If we rotate a box filled with gas or perfect fluid, clearly its density does not change.
Suppose the box is accelerated to speed close to the speed of light. Its length will be shorten due to Lorentz contraction. My question is this: Will the density of the box increase with its speed?

Length contraction has nothing to do with compression. Remember that the observer moving with the rod measures the same length at all times so nothing is being compressed.

It is more correct to view length contraction as a rotation in space time. Have a friend hold a meter stick some distance way from you perpendicular to the line between the 2 of you. From a distance do a measurement. Now have your friend rotate the meter stick 45deg, measure it again. Now from your view point it is shorter then it was before. :

## Answers and Replies

Doc Al
Mentor

It is more correct to view length contraction as a rotation in space time.
A rotation in spacetime, not space. (That bit with the meterstick was just an analogy.)
If we rotate a box filled with gas or perfect fluid, clearly its density does not change.
Why would it?
Suppose the box is accelerated to speed close to the speed of light. Its length will be shorten due to Lorentz contraction. My question is this: Will the density of the box increase with its speed?
Sure. (According to an observer seeing it move.)

Dale
Mentor
2020 Award

Will the density of the box increase with its speed?
How do you define the density of a moving object? (If you use standard definitions then yes)

Fredrik
Staff Emeritus
Science Advisor
Gold Member

It is more correct to view length contraction as a rotation in space time.
A rotation in space is a linear map $x\mapsto Rx$, where R satisfies $R^TR=1$ and can be made to go 1 by continuously changing the parameters that R depends on (3 euler angles). A restricted (i.e. homogeneous, proper and orthochronous) Lorentz transformation is a map $x\mapsto\Lambda x$, where $\Lambda$ satisfies $\Lambda^T\eta\Lambda=\eta$, and can be made to go 1 by continuously changing the parameters that $\Lambda$ depends on (3 euler angles and 3 velocity components). So you can view it as a rotation if you'd like, but it's clearly a generalization of that concept.

$$\eta=\begin{pmatrix}-1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ \end{pmatrix}$$

My question is this: Will the density of the box increase with its speed?

Seems like an increase in density is related to length contraction as well as relativistic mass increase?

Dalespam posted:
How do you define the density of a moving object? (If you use standard definitions then yes)

Could you explain this? I suspect there is a subtley I don't see....in other words, why might one consider a "non standard" definition...

Dale
Mentor
2020 Award

Could you explain this? I suspect there is a subtley I don't see....in other words, why might one consider a "non standard" definition...
Well, it's like anything else in relativity, you can define different meanings for the same terms and come out with different conclusions. For example, you can say "mass" and you can mean "rest mass" or "relativistic mass". You can also say "length" and you can mean "coordinate length" or "proper length" or even "spacetime interval". So something like mass/length³ can mean a lot of subtly different things and thus there is a lot of wiggle-room for different definitions in a word like "density" in relativity.

feynman, suppose we imagine that the entire universe was contained within an area of a finite square "Imagine a square". The interior of the square is the universe. Now draw a line down the middle of the square and imagine that to the left of the line is homogenous matter and that to the right of the line is empty space. now imagine that the entire portion of empty space is a gradient of reference frames (think like... a beaker with a series of fluids, lets say 10 of them of varying densities from greatest to least density). Each of these different layers corresponds to a different frame of reference. Within each frame of reference we understand the concept of 1 meter and 1 meter equals 1 meter so long as we are measuring within that frame of reference. But... If I were to measure an object, which measured 1 meter while measured by a meter stick within a mutual frame of reference (Frame of reference 1) then they would both appear the same length. But... comparing the measurement of another object a particular frame of reference (frame of reference 2) with a meter stick from another frame of reference (frame of reference 3), the object which is in frame of reference 2 would not measure 1 meter RELATIVE to this other frame of reference (frame of reference 3).

"Warped space" is comprised of a continuous "density" gradient of space-time. Where each gradation is a particular reference frame only there are an infinite amount of them.

Dalespam, thanks for your post #6....

Dale
Mentor
2020 Award

You're welcome!

Hopefully, the OP will define the density of a moving object and then the thread can proceed.

A rotation in space is a linear map $x\mapsto Rx$, where R satisfies $R^TR=1$ and can be made to go 1 by continuously changing the parameters that R depends on (3 euler angles). A restricted (i.e. homogeneous, proper and orthochronous) Lorentz transformation is a map $x\mapsto\Lambda x$, where $\Lambda$ satisfies $\Lambda^T\eta\Lambda=\eta$, and can be made to go 1 by continuously changing the parameters that $\Lambda$ depends on (3 euler angles and 3 velocity components). So you can view it as a rotation if you'd like, but it's clearly a generalization of that concept.

$$\eta=\begin{pmatrix}-1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ \end{pmatrix}$$

One of the reason that the rotation in spacetime analogy breaks down is that it is not ordinary rotation of coordinates, sine and cosine function. The Lorentz transformation can be seen as a hyperbolic rotation of coordinates in Minkowski space, where the rapidity φ represents the hyperbolic angle of rotation.

Hyperbolic rotation of coordinates:
http://en.wikipedia.org/wiki/Lorentz_transformation#Hyperbolic_trigonometric_expressions

It is more correct to view length contraction as a rotation in space time. Have a friend hold a meter stick some distance way from you perpendicular to the line between the 2 of you. From a distance do a measurement. Now have your friend rotate the meter stick 45deg, measure it again. Now from your view point it is shorter then it was before.

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In other words length contraction is physically real.

Hurkyl
Staff Emeritus
Science Advisor
Gold Member

One guru indicate that Length contraction has nothing to do with compression.
It is more correct to view length contraction as a rotation in space time.
If we rotate a box filled with gas or perfect fluid, clearly its density does not change.
You're omitting an important part of the analogy -- the 'point of view'. No matter how you orient a meterstick in Euclidean space, it will be a meter long. But when we 'observe' the meterstick by making and studying a two-dimensional image (e.g. a photograph, or the image made in our eyes), the orientation of the meterstick affects its size in the image.

The correspondence with space-time is that no matter how you orient (the worldsheet of) something in 4-dimensional space-time, it will look the same. But if we 'observe' the object by looking at a 3-dimensional picture (e.g. by choosing coordinates, and then looking at the 3-dimensional spatial image taken at a particular moment of coordinate time), it's orientation will affect how it looks.

Incidentally, a better analogy with the meterstick is to look at its cross-sections. Suppose you have a 1 m x 3 cm x 1 cm meterstick. (I don't know what dimensions are normal, so I made them up!) Let's orient it so that the 1m axis is pointing North/South, and the 3cm axis is pointing East/West, and the 1cm axis is pointing Up/Down. If you take a 2-dimensional cross section in tne Up/East plane, you will see a 3 cm x 1 cm rectangle. However, if you change your orientation by rotating the plane 30 degrees about the up axis, your cross sections are now 3.5 cm x 1 cm. Voilà, 'length expansion'. (We get expansion instead of contraction because of the sign difference between Euclidean and Minkowski spaces)

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A.T.
Science Advisor

One of the reason that the rotation in spacetime analogy breaks down is that it is not ordinary rotation of coordinates, sine and cosine function.
The rotation analogy for length contraction works better if you use space-propertime, instead of Minkowski spacetime:
http://www.adamtoons.de/physics/relativity.swf