# Why never any length expansion under Lorentzian Relativity?

1. Feb 8, 2015

### hkyriazi

Lorentz believed in the ether, and developed his version of Relativity Theory based on interactions of moving charged particles (electrons, specifically) with the ether. What I don't understand is how one always gets length contraction for movement in an inertial system moving with respect to one's own inertial system. For a mechanical theory of interaction of matter with the ether, one would (naively, it seems) think that increasing one's rate of movement against the stationary ether would cause a length contraction, but that moving more with the ether would result in a length expansion.

I realize this can't be the case, because it would allow one to identify the stationary ether reference frame without having to look, say, at the Cosmic Microwave Background Radiation, but I don't see how one can reconcile this aspect of relativity theory with a mechanical ether (which presumably was Lorentz's view).

What am I missing? (This seems to be simply another way of saying, "If person A's yardstick A in inertial frame A sees a yardstick in moving inertial frame B as being shrunk, why does person B in frame B not see A's yardstick as enlarged?")

Last edited: Feb 8, 2015
2. Feb 8, 2015

### stevendaryl

Staff Emeritus
I'm not going to get into Lorentz ether theory, but let's look at how it works in SR. Suppose Alice and Bob are traveling relative to each other at some speed $v$. From the point of view of Alice, Bob's meter stick is contracted relative to hers. But when Bob measures Alice's meter stick, he thinks that it's Alice's meter stick that is contracted. How does Alice explain why Bob got the "wrong" (from her point of view) impression about how who had the longest meter stick?

Well, think about how Bob might measure Alice's meter stick. Here's one way he might do it: He notes the time, $\tau_1$ (according to his clock) that he passes one end of Alice's meter stick. Then he notes the time, $\tau_2$ that he passes the other end. He computes the length of Alice's meter stick by:

$L = v \delta \tau$

However, from Alice's point of view, Bob's clock is running slow, by a factor of $\gamma$. So $\delta \tau = \frac{1}{\gamma} \delta t$. So Bob computes Alice's meter stick to be shortened by a factor of $\gamma$.

3. Feb 9, 2015

### ghwellsjr

If you believe in the ether, then you consider only its inertial system to be the one that establishes the length of an object. So if you are moving with respect to the ether then objects moving even faster than you will be more length contracted than you are and objects moving slower than you will be length contracted less than you so you could say that in this last case the object is expanded compared to you.

What you are missing is the fact that you can't see length contraction so even if you believe in the ether, you cannot tell which inertial system it is at rest in.

4. Feb 13, 2015

### harrylin

As stevendaryl's answer already clarified, that "paradox" of mutual length contraction and time dilation exists in SR with or without ether. What you are perhaps missing (and this is the most common oversight) is the crucial role relativity of simultaneity plays in determining moving length. The length of a moving object is determined based on simultaneous measurement of the extremes. If we assume or pretend one reference frame to be in rest (or even in "true rest"), then reference frames that are in relative motion to it necessarily have a different (or "wrong") simultaneity calibration of distant clocks along the line of motion.

By the way it's a basic student exercise to derive the mutual time dilation and length contraction equations from the Lorentz transformations. Try it for yourself and see that you need to impose those frame dependent simultaneity conditions in order to obtain these equations. (Seriously, you really should try to derive this yourself, if you did not already do so; it's an eye opener for understanding SR. Choose the other frame's simultaneity and you obtain expansion instead of contraction).

Last edited: Feb 13, 2015
5. Feb 16, 2015

### hkyriazi

I like this way of determining length, as it seems (but see below) to eliminate any funny business about relativity of simultaneity; by measuring the time by difference, any error in one measurement will be exactly canceled by equal errors in the 2nd measurement.

Just to make sure we're on the same page, is Alice saying "Oh, Bob thinks my meter stick is short only because his clock is running slow, so he think less time elapsed during his measurement than actually did elapse. If he used the correct time, he'd get the same answer as me - exactly one meter"? This brought another question immediately to mind, and which relates to something in HarryLin's post (#4), which is "How does Bob measure Alice's velocity?" (HarryLin stated "The length of a moving object is determined based on simultaneous measurement of the extremes.")

To measure Alice's velocity, then, it seems we must do what HarryLin says, and measure the ends of some yardstick in her reference frame simultaneously, because we can't assume that we know the correct length of anything in her reference frame in order to measure her velocity by elapsed time - that'd be a circular bit of reasoning.

6. Feb 16, 2015

### stevendaryl

Staff Emeritus
Yes, you're right. For a moving object, we have two interrelated quantities: (1) The length of the object, and (2) its speed. If we know the speed, then we can compute the length, and if we know the length, we can compute the speed. If we don't know either one, then we need a standard of simultaneity.

The same thing is true of the "rate" of a moving clock. If the clock is moving at a constant velocity (no turning around), there is no way to measure the "rate" of the clock without a standard for simultaneity.

7. Feb 16, 2015

### hkyriazi

Taking Stevendaryl's example, could I not set Alice in motion in various directions, and measure her yardstick in each of those cases? Then I would be "seeing" length contraction. Let's assume that F=ma is valid regardless of direction (perhaps not a valid assumption), and that we use some spring-loaded device for precisely accelerating Alice to some constant velocity, and repeating that procedure over and over until we get a good sampling of her motion - and yardstick length - in all directions. Relating to my original question, one would naively think, based on Lorentz's contraction hypothesis, that if we looked at movement in all different directions, there'd be one direction in which contraction was greatest (movement most "upstream" against the ether), and that the opposite direction (movement most with the ether) would yield maximal length expansion. We could repeat this exercise using various spring tensions to create a wide range of accelerations and resulting velocities. At some velocity we'd find that the maximal length expansion peaks and then begins to go down, and that velocity peak would identify the ether rest frame. (I realize that Lorentz also introduced a quantity called "local time," so there's that added wrinkle. I also realize that this is not the way such an experiment would actually play out.)

8. Feb 16, 2015

### hkyriazi

I have Robert Resnick's 1968 text, "Introduction to Special Relativity", and it has such exercises. I went through them once, many years ago, and will repeat the exercise. Thanks for the suggestion. :-)