Relativistic length contraction

In summary: The contraction is real, even if that theory was wrong. Finally, Einstein's contribution was the "physical" interpretation of the Lorentz Contraction. The contraction is not in the physical arm of the experiment but in the measuring rods and clocks of the observer.In summary, the concept of relativity was first introduced by Galileo, stating that the laws of mechanics are the same for all inertial observers. However, this was challenged by Maxwell's equations, which showed that the force exerted by a moving electron is proportional to its speed, not acceleration. This led to the Michaelson-Morley experiment, which gave a null result and suggested that the motion of electrons caused the arms
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How do you find relativistic length contraction?
 
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  • #2
L' = L * root(1 - v^2/c^2)
 
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  • #3
i mean how is it you analyze the way on how to get relativistic length contraction~
sorry, i didn't make myself clear...
 
  • #4
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.
 
  • #5
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.

[tex]\Delta x'=\gamma (\Delta x-v\Delta t)[/tex]
[tex]\Delta t'=\gamma (\Delta t-v\Delta x/c^2)[/tex]

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 [itex]\Delta x=L[/itex] (the length of the rod in our frame) and [itex]\Delta t=0[/itex]. Then we call the [itex]\Delta x'=L_o[/itex]. Inserting these expressions into the first of the two equations I stated above gives:

[tex]L_o=\gamma (L-v(0))[/tex]
[tex]L_o=\gamma L[/tex]
[tex]L=\frac{L_0}{\gamma}[/tex]

This is the length contraction formula.
 
  • #6
Tom Mattson said:
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?
 
  • #7
spuij - if you knew a little more about physics, and were a little more mature, you wouldn't make such statements
 
  • #8
Tom Mattson said:
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."
 
  • #9
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
[tex]\frac{d_1}{c+v}+ \frac{d_1}{c-v}= [/tex]
[tex]\frac{2cd_1}{c^2- v^2}[/tex]

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 [itex]\sqrt{c^2- v^2}[/itex] (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 [itex]\frac{2d_2}{c^2- v^2}[/itex].

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.
 
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  • #10
wow! thank you very much! :)
 

What is relativistic length contraction?

Relativistic length contraction is a phenomenon in which the length of an object appears to decrease when it is moving at high speeds relative to an observer. This is a consequence of Einstein's theory of special relativity, which states that the laws of physics are the same for all observers in uniform motion.

How does relativistic length contraction occur?

Relativistic length contraction occurs because as an object moves at high speeds, its velocity approaches the speed of light. According to Einstein's theory of special relativity, as an object approaches the speed of light, its time slows down and its length in the direction of its motion appears to decrease.

What is the formula for relativistic length contraction?

The formula for relativistic length contraction is L = L0 * √(1 - v^2/c^2), where L0 is the object's rest length, v is its velocity, and c is the speed of light. This formula shows that as an object's velocity increases, its length appears to decrease, approaching zero as its velocity approaches the speed of light.

Is relativistic length contraction observable in everyday life?

Relativistic length contraction is not observable in everyday life because it only becomes significant at speeds close to the speed of light. In our daily experiences, objects do not move at such high speeds, so the effects of length contraction are not noticeable.

What are the practical applications of relativistic length contraction?

Relativistic length contraction has important practical applications in fields such as particle physics, where particles are accelerated to near-light speeds. It also plays a role in GPS technology, as the clocks on GPS satellites have to be adjusted for the effects of time dilation and length contraction in order to accurately measure positions on Earth.

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