What are the physics of length contraction?

[CKH]

Objects in motion relative to an inertial frame S are measured to have contracted lengths in S in the direction of motion. How does Special Relativity view this contraction physically?

For a concrete example, suppose I have a bar of steel moving lengthwise and, as it moves past me (in frame S) I tap it in the center. How do I determine how it will vibrate? (It is shorter than at rest and more massive. Perhaps it's elasticity has changed. etc.?)

 

[ghwellsjr]

Objects in motion relative to an inertial frame S are measured to have contracted lengths in S in the direction of motion. How does Special Relativity view this contraction physically?

SR views contracted lengths on an equal basis with uncontracted (Proper) lengths. They all depend on the selected reference frame.

For a concrete example, suppose I have a bar of steel moving lengthwise and, as it moves past me (in frame S) I tap it in the center. How do I determine how it will vibrate? (It is shorter than at rest and more massive. Perhaps it's elasticity has changed. etc.?)

SR by itself cannot answer your question but if you can answer your question in one frame, say, the rest of the bar, then you can use the Lorentz Transform to answer your question in another frame where the bar is moving.

 

[CKH]

So in a sense we don't worry about what happens because the bar is shorter in motion, but instead treat that as only a perspective or view of the bar (like distant objects appearing smaller)? Thus if we know what happens in the bar's rest frame we can just transform that to S?

What then is the meaning of "Lorentz invariance" of a physical law? How is such a thing tested?

(Experiments on objects moving at significant fractions of light speed are limited; the only examples that come to mind are particle accelerators and cosmic rays.)

 

[Blazejr]

Lorentz invariance means roughly that if given law holds in one inertial frame it should hold unchanged in any other inertial frame and transformation law of physical quantities is given by Lorentz transformation.

For example if 4-momentum is conserved in some physical system according to some intertial observer it would also be conserved for any other inertial observer. Moreover, if we know components of 4-momentum vectors of all particles in that system in first frame we could calculate components in new frame using Lorentz transformation.

 

[Meir Achuz]

For a concrete example, suppose I have a bar of steel moving lengthwise and, as it moves past me (in frame S) I tap it in the center. How do I determine how it will vibrate? (It is shorter than at rest and more massive. Perhaps it's elasticity has changed. etc.?)

If you determine how the bar vibrates its rest system, then SR tells you how to Lorentz transform the rest frame trajectories of points on the bar to any moving frame. That's all you need to know. Transforming other things, like elastic constants, is not necessary.

 

[ghwellsjr]

So in a sense we don't worry about what happens because the bar is shorter in motion, but instead treat that as only a perspective or view of the bar (like distant objects appearing smaller)?

No, Length Contraction is not something that you can see or view. It's the result of applying Einstein's second postulate (that light propagates at c) to measurements that are taken after the object has already passed and then doing some calculations. It's how we define length.

Thus if we know what happens in the bar's rest frame we can just transform that to S?

Yes, or if we know what the bar's length is in S, we can transform that to its rest frame (or any other frame).

What then is the meaning of "Lorentz invariance" of a physical law?

The mathematical equations that express a physical law need to remain unchanged after undergoing the Lorentz Transformation.

How is such a thing tested?

(Experiments on objects moving at significant fractions of light speed are limited; the only examples that come to mind are particle accelerators and cosmic rays.)

Probably the first test to show that the Galilean Transformation was not correct was the Michelson Morley Experiment which was done at extremely low speeds. Relativistic effects are present at all speeds but they are very slight at low speeds so an experiment needs to have very high precision to manifest them.

 

[CKH]

No, Length Contraction is not something that you can see or view. It's the result of applying Einstein's second postulate (that light propagates at c) to measurements that are taken after the object has already passed and then doing some calculations. It's how we define length.

I'm not sure I understand yet. What I was getting at is whether length contraction is merely a matter of "perspective" from the rest frame? In other words, can we view Lorentz transformation as simply a "distortion" of what is actually happening in the frame of a moving object?

On the other hand we can in principle perform physical measurements on the object in motion from the rest frame. Ignoring the practical difficulty of such measurements, can we use physical laws in the rest frame to derive the vibrations of the bar when tapped while in motion?

I ask because such an approach actually asks what are the properties of the moving object in the rest frame. What is it's mass, length and elasticity for example so that we may directly calculate the vibrations independently from the rest frame of the moving object. Then we must show that our calculations are in fact Lorentz invariant.

Perhaps this is what is meant when we say that our physical laws themselves must be "Lorentz invariant" for consistency of the physical analysis of the moving object with the corresponding analysis in the object's rest frame.

We might attempt to derive from physical law (in the rest frame) the orthogonal elasticity of the moving object (how?), and then, using its dilated mass and contracted length, calculate the frequency of vibration of the moving object directly in the rest frame. We would expect the result to be Lorentz invariant, but the analysis in motion seems very difficult since it raises questions about the properties of the object in motion.

 

[Nugatory]

What I was getting at is whether length contraction is merely a matter of "perspective" from the rest frame? On the other hand we can in principle perform physical measurements on the object in motion from the rest frame.

That's a false dichotomy. Some quantities are frame dependent, and when observers moving at different speeds relative to one another measure these quantities, they get different results. Other quantities are frame-invariant so all observers get the same result when they measure these quantities. However, it's a mistake (unless you are going to be much more precise about exactly what you mean) to say that the frame-dependent ones are "merely a matter of perspective". For example: The frequency of a light signal is frame-dependent. Thus, an observer at rest relative to a light source might see ultraviolet light that gives him an attractive suntan, while an observer moving rapidly towards that light source will observe that the light is blue-shifted into the gamma spectrum... It will be very hard to convince that observer that his immenent death by acute radiation poisoning is "merely a matter of perspective".

Ignoring the practical difficulty of such measurements, can we use physical laws in the rest frame to derive the vibrations of the bar when tapped while in motion?

We can in principle, but the problem is excrucuatingly complicated and very error-prone because it's so easy to lose sight of which properties of the bar are not frame-independent. Indeed, some of the more entertaining relativity "paradoxes" (Bug-rivet and Bell's spaceship paradox come to mind) are constructed by slipping a frame-dependent quantity from one frame into another and hoping the audience won't notice the sleight of hand.

 

[ghwellsjr]

No, Length Contraction is not something that you can see or view. It's the result of applying Einstein's second postulate (that light propagates at c) to measurements that are taken after the object has already passed and then doing some calculations. It's how we define length.

I'm not sure I understand yet. What I was getting at is whether length contraction is merely a matter of "perspective" from the rest frame? In other words, can we view Lorentz transformation as simply a "distortion" of what is actually happening in the frame of a moving object?

To say that something is a "distortion" of what is actually happening suggests that you believe that there is a reality beyond what we can determine by observation, experiment or measurement. But all efforts to nail down that kind of reality have proven fruitless. You are better off accepting the fruitful efforts of scientists for over a century as embodied in the Theory of Special Relativity.

On the other hand we can in principle perform physical measurements on the object in motion from the rest frame. Ignoring the practical difficulty of such measurements, can we use physical laws in the rest frame to derive the vibrations of the bar when tapped while in motion?

I ask because such an approach actually asks what are the properties of the moving object in the rest frame. What is it's mass, length and elasticity for example so that we may directly calculate the vibrations independently from the rest frame of the moving object. Then we must show that our calculations are in fact Lorentz invariant.

Perhaps this is what is meant when we say that our physical laws themselves must be "Lorentz invariant" for consistency of the physical analysis of the moving object with the corresponding analysis in the object's rest frame.

We might attempt to derive from physical law (in the rest frame) the orthogonal elasticity of the moving object (how?), and then, using its dilated mass and contracted length, calculate the frequency of vibration of the moving object directly in the rest frame. We would expect the result to be Lorentz invariant, but the analysis in motion seems very difficult since it raises questions about the properties of the object in motion.

I think you have hit the nail on the head when you say there is a practical difficulty to doing this sort of analysis. And your example of striking a rod in the center makes it even more difficult because now it is a two-dimensional scenario. Did you consider that the rod would be set in motion?

Consider a simpler scenario where you have a long thin rod that you strike on its end and you want to analyze how the sound wave propagates through the rod and how the rod accelerates and how it may end up with a different Proper Length. It should be obvious that Special Relativity, by itself, is inadequate to do the analysis. You have to know a lot more information about the characteristics of the material of both the rod and the hammer that strikes it. When you realize that it's the electromotive forces that are operating at the speed of light on untold trillions of atoms and molecules you may appreciate why nobody bothers to attempt such an analysis. Instead, we go for the macroscopic approximations that are easier to analyze and give us a good enough answer.

So if you want a rigorous analysis for striking a moving rod with a moving hammer, you are just going to compound the analysis even further. Rather than do that, we already know that we will get the same answer if we do any kind of analysis in the rest frame of the rod and Lorentz Transform to the frame in which it is moving, as long as all the physics remain intact under the Lorentz Transform. It's just math. It can't fail.

 

[CKH]

Yes the rod would receive a traverse impulse and it's momentum would change meaning it is no longer moving parallel to the bar. The length is unaffected (by vector decomposition of velocity?) but now we have some width contraction, yuck!

I'm beginning to see how complicated the physics can get, but as you both point out, the physics is real in any frame.

I've read a few articles (to the best of my limited abilities) concerning rotation of a disc; it seems even the trained physicist disagree on the analysis.

Rather than do that, we already know that we will get the same answer if we do any kind of analysis in the rest frame of the rod and Lorentz Transform to the frame in which it is moving, as long as all the physics remain intact under the Lorentz Transform. It's just math.

It is still a requirement that the physics applies correctly to the moving object. You cannot assume that the physics of the moving object is actually equivalent to the Lorentz transformation of the rest frame physics. You need to prove that for consistency of your physical laws.

We know length is frame dependent, width is not, mass is and time is. It's not immediately obvious (to me) for example how forces transform. How do we figure that out? Isn't mass itself defined by the force required to accelerate? How do we address such questions in SR? SR per se only makes rules about length and time.

These issues concern Lorentz invariance of physical analysis. Some measurements require the use of the transformation and some don't. We need some kind of proof that the analysis in the object's rest frame must be Lorentz equivalent to a corresponding analysis in the frame in which the object moves (which is very complicated).

(By the way, is the "excess" mass of a moving object equivalent to it's kinetic energy?)

 

[Nugatory]

Isn't mass itself defined by the force required to accelerate?

You can save yourself some grief by using $F=\frac{dp}{dt}$ where $p$ is the momentum, instead of $F=ma$ - the two are pretty much equivalent even in classical physics - and relating mass, energy, and momentum with $E^2=(m_0c^2)^2+(pc)^2$ so that the only mass involved is the invariant rest mass $m_0$.

These issues concern Lorentz invariance of physical analysis. Some measurements require the use of the transformation and some don't. We need some kind of proof that the analysis in the object's rest frame must be Lorentz equivalent to a corresponding analysis in the frame in which the object moves (which is very complicated).

You can write down your physical law, grind away at it until everything you can is expressed in terms of distance, time, and frame-invariant quantities, then Lorentz transform the times and positions. If when you're done with this exercise all the $\gamma$ have dropped out, you've found a Lorentz-invariant law. This is generally sufficiently difficult for complex properties of solid bodies that it easier to solve the problem using coordinates in which the object is at rest, then Lorentz-transform the result - that's what everyone is telling you to do with your vibrating steel rod.

(By the way, is the "excess" mass of a moving object equivalent to it's kinetic energy?)

Yes, if by the "excess mass" you mean the quantity $(\gamma-1)m_0$. This value is frame-dependent so it's often not especially convenient to work with.

 

[CKH]

You can save yourself some grief by using $F=\frac{dp}{dt}$ where $p$ is the momentum, instead of $F=ma$ - the two are pretty much equivalent even in classical physics - and relating mass, energy, and momentum with $E^2=(m_0c^2)^2+(pc)^2$ so that the only mass involved is the invariant rest mass $m_0$.

Is $p=m_0v$? Somehow I don't think so.

You can write down your physical law, grind away at it until everything you can is expressed in terms of distance, time, and frame-invariant quantities, then Lorentz transform the times and positions.

Yes, but what are the frame invariant quantities besides the obvious such as width? Is force a frame invariant quantity simply because it is fundamental (not defined via time and length)? How does mass end up being frame dependent? Does it depend on how it is defined?

Yes, if by the "excess mass" you mean the quantity $(\gamma-1)m_0$. This value is frame-dependent so it's often not especially convenient to work with.

Yes. So kinetic energy is actually measurable as a change in mass and the same is true of gravitation potential energy and electrical potentials of charged objects?

 

[PAllen]

Is $p=m_0v$? Somehow I don't think so.

No, given E² = p²c² + (mc²)²

(where I use m for m0), solve for p and you certainly do not get mv. Note that E = γmc². In fact, you get:

p = γmv

 

[PAllen]

Yes, but what are the frame invariant quantities besides the obvious such as width? Is force a frame invariant quantity simply because it is fundamental (not defined via time and length)? How does mass end up being frame dependent? Does it depend on how it is defined?

Now you are at a point where a little systematic study would serve you well. The only invariant quantities are scalars (formed from contraction of vectors and tensors). On the other hand, vectors are often considered to be frame independent (not invariant) objects, in the sense that their expressions changes from one frame to another, but only in such a way as to preserve all possible scalar invariants that can be constructed from them.

In special relativity, you need to use 4-vectors if you want frame independence.

 

[PAllen]

Yes. So kinetic energy is actually measurable as a change in mass and the same is true of gravitation potential energy and electrical potentials of charged objects?

If you bring in gravity, you general relativity, and the concept of mass is quite complex in general relativity.

Sticking to special relativity, the modern approach to mass (as distinct from energy) is to consider invariant mass. For a system of bodies, this is the same as inertial mass of the system in its COM frame. With this background, yes:

- kinetic energy of bodies as measured in the COM frame directly contributes to inertia in that frame = invariant mass (which is frame independent).

- electric potential can contribute in an equivalent way to inertia or invariant mass. You have to be careful to specify 'equal temperature', and non-static cases are more complex, but you can say:

Given 2 identical negatively charged bodies as measured in isolation, and two similar positively charged bodies, and two insulating rods of the same mass but different length, you build two systems:

a) One of each charge separated by the shorter rod
b) One of each charge separated by the longer rod

and you make sure they are at identical temperature, the (a) will have slightly less inertia than (b).

 

[CKH]

OK. Can you recommend an elementary text or website that discusses Lorentz invariance of physical law. There are limits to my mathematics education so diving into advanced mathematics of relativity probably won't get me very far. However, I am curious about how this works in the simplest cases.

 

[ghwellsjr]

We know length is frame dependent, width is not...

All dimensions are frame dependent. Just because a rod is moving in the x direction doesn't mean you have to transform in the x direction. It's just as legitimate to transform in the y or z directions or any combinations of the three dimensions. In our examples, we tend to transform along the direction of motion but that's just because we want to avoid complicated math and because we like to illustrate the results on a spacetime diagram where we are limited to one spatial dimension.

It might be helpful to you if you consider a physics analysis of an object in its rest frame and then transform that to a frame in motion with respect to the rest frame. This is purely a mathematical exercise and doesn't involve any impact on the physics, don't you agree?

So now it would seem to me that your issue is really not one of how scenarios transform but whether the laws of physics need to remain intact under a Lorentz Transformation rather than some other transformation. The best assurance of that is Maxwell's equations which were developed before Lorentz devised his transformation and then after the fact it was discovered that they adhered to the Lorentz Transformation. Furthermore all the other laws of physics that didn't transform intact under LT were found to be defective under the Galilean Transformation. So I don't think it's too hard to accept that the Lorentz Transformation is the standard by which all the laws of physics must adhere.

 

[ghwellsjr]

OK. Can you recommend an elementary text or website that discusses Lorentz invariance of physical law. There are limits to my mathematics education so diving into advanced mathematics of relativity probably won't get me very far. However, I am curious about how this works in the simplest cases.

If you're asking about how the equations of physical law are transformed intact under LT, it's not elementary. I don't think you will find any explanation that you will understand, at least I haven't found any that I understand. I just trust the mathematicians when they say Maxwell's equations work and Newton's don't.

 

[CKH]

All dimensions are frame dependent. Just because a rod is moving in the x direction doesn't mean you have to transform in the x direction. It's just as legitimate to transform in the y or z directions or any combinations of the three dimensions.

That's clear. I was keeping the problem simple by choosing the x direction.

It might be helpful to you if you consider a physics analysis of an object in its rest frame and then transform that to a frame in motion with respect to the rest frame. This is purely a mathematical exercise and doesn't involve any impact on the physics, don't you agree?

The "natural" laws of physics cannot be frame dependent (postulate of SR) while experiment validates the Lorentz transformations. So I agree with your statement in the case of natural law. However for our proposed laws of physics, that is not obvious.

So now it would seem to me that your issue is really not one of how scenarios transform but whether the [our] laws of physics need to remain intact under a Lorentz Transformation rather than some other transformation. The best assurance of that is Maxwell's equations which were developed before Lorentz devised his transformation and then after the fact it was discovered that they adhered to the Lorentz Transformation.

That's exactly what I'm thinking about. Maxwell's equation may be a good example for me to dig into. But Maxwell's equations are not all of physics. I'm guessing we cannot determine the properties of a solid object in motion using only those equations.

So we propose physical laws and then try to determine whether they are in fact Lorentz invariant.

... So I don't think it's too hard to accept that the Lorentz Transformation is the standard by which all the laws of physics must adhere.

Again, that's a postulate of SR for natural law which we accept since SR is experimentally validated.

If you're asking about how the equations of physical law are transformed intact under LT, it's not elementary. I don't think you will find any explanation that you will understand, at least I haven't found any that I understand. I just trust the mathematicians when they say Maxwell's equations work and Newton's don't.

Agreed, a thorough training in the math and physics is required. (It's a little late in life for me.)

My question about forces was answered by some equations that show they are frame dependent. The assumptions behind this seem to include conservation laws, perhaps of momentum and energy. So it seems that to use SR in physics we add additional postulates and work things out from there.

Thanks for everyone's help. It's really improved my lay understanding of SR.