Relativistic length contraction

[asdf1]

How do you find relativistic length contraction?

 

[El Hombre Invisible]

L' = L * root(1 - v^2/c^2)

 

[asdf1]

i mean how is it you analyze the way on how to get relativistic length contraction~
sorry, i didn't make myself clear...

 

[El Hombre Invisible]

I don't get what you mean. Do you mean how does length contraction occur? Any observable that is not at rest within the frame of reference is length contracted in that frame compared to another frame in which it is at rest. However, the phenomenon is only observale on a macroscopic level at speeds comparative with that of the speed of light in a vacuum. As objects large enough to have a measurable length are unlikely to be found traveling at such speeds, it is not a phenomenon that is likely to be observed.

 

[quantumdude]

I think he means, "From what more fundamental relation is the length contraction formula derived?"

In that case, let's briefly follow Einstein's steps. Consider Newtonian mechanics. Galileo's principle of relativity states that the laws of mechanics should be the same for all inertial observers. And indeed, Newtonian mechanics is unchanged under Galilean transformations. The problem is that Maxwellian electrodynamics is not the same in every inertial frame under that transformation. So what to do? Find a set of transformations under which both mechanics and electrodynamics are the same for all inertial frames. This leads to the Lorentz transformation, whcih reads as follows for boosts along the x-axis.

$$\Delta x'=\gamma (\Delta x-v\Delta t)$$
$$\Delta t'=\gamma (\Delta t-v\Delta x/c^2)$$

This shows how spacetime intervals transform between inertial frames in special relativity. So let Event 1 be "measurement of the spacetime coordinates of the left end of a rod that is moving along the x-axis with velocity v" and Event 2 be "measurement of the spacetime coordinates of the right end of a rod that is moving along the x-axis with velocity v". Let the rod be in the "primed" frame and let us be in the "unprimed" frame.

If the measurements are made simultaneously, then $\Delta x=L$ (the length of the rod in our frame) and $\Delta t=0$. Then we call the $\Delta x'=L_o$. Inserting these expressions into the first of the two equations I stated above gives:

$$L_o=\gamma (L-v(0))$$
$$L_o=\gamma L$$
$$L=\frac{L_0}{\gamma}$$

This is the length contraction formula.

 

[asdf1]

I think he means, "From what more fundamental relation is the length contraction formula derived?"

yes, you're right! thanks! but why doesn't maxwell equations aren't the same in the different inertial frames?

 

[yogi]

spuij - if you knew a little more about physics, and were a little more mature, you wouldn't make such statements

 

[robphy]

So what to do? Find a set of transformations under which both mechanics and electrodynamics are the same for all inertial frames.

I probably would have said
"Find a set of transformations under which electrodynamics is the same for all inertial frames. Then, find a mechanics and kinematics that are compatible with those transformations."

 

[HallsofIvy]

Here is my take on it: "Relativity" was first proposed by Gallileo! Well, Gallilean relativity, anyway. He argued that if you were enclosed in a carriage moving in a straight line at a constant speed, there was no experiment you could do that would tell you your speed, or even if you were moving. Essentially that's "F= ma". We can feel forces but forces are proportional to acceleration, not speed. Speed has to be "relative" to some outside reference. Of course, Gallileo didn't know anything about magnetic-electric forces.

Maxwell's equations show that the force exerted by a moving electron is proportional to its speed not acceleration! Based on that, we should be able to do some kind of electro-magnetic experiment inside a "closed carriage" and so arrive at an "absolute" speed, not relative to the ground outside that has no bearing on our experiment.

That was exactly what the Michaelson-Morley experiment was designed to do: an electro-magnetic (light) experiment inside a "closed carriage" (the earth) to find our "absolute" speed. The experiment gave a null result- even with an electro-magnetic experiment, you can't find an "absolute" speed.

It was Lorentz who suggested that perhaps the motion of the electrons produced exactly enough "pull" that the arms of the scale in the direction of motion were contracted exactly enough to give the null result. That's a lovely theory! That's exactly the kind of subtle interaction we often see. Lorentz, purely ad-hoc, calculated how much the arm must have contracted.

Here's one way to see that: imagine a man who can swim at speed c, in still water, swimming in a river with current v. If he swims a distance d₁, downstream, he moves (relative to the shore) at speed c+ v and takes time d₁/(c+v) to do that. He then turns and swims the same distance up stream. Now his speed is c- v (notice that v must be less than c in order for to be able to do this) and so he takes time d₁/(c-v). His total time for the round trip is
$$\frac{d_1}{c+v}+ \frac{d_1}{c-v}=$$
$$\frac{2cd_1}{c^2- v^2}$$

Now, he turns and swims a distance d₂ across the river and back.
In order to be able to swim directly across the river, he must "angle" up stream. If he swims at an angle to the current with speed c and the current is still v, it's comparatively easy to calculate that his "speed made good" through the water is $\sqrt{c^2- v^2}$ (draw the right triangle of vector speeds and use the Pythagorean Theorem). The time to go a distance d₂ across the river and back is $\frac{2d_2}{c^2- v^2}$.

Finally: Suppose the two times are the same? What relation does d₁ have with d₂? You guessed it: the Lorentz Contraction formula.

A later experiment, called, I think, the "Kennedy experiment", showed that Lorentz's basic idea- that the electrons in the physical arm of the experiment cause the contraction was wrong- but the contraction formula, based on experimental results, was still valid. Einstein was among those who argued that it must be space itself which contracts, not just physical objects.

 

[asdf1]

wow! thank you very much! :)