# Exploring Einstein's Relativity: Length Contraction

• nonequilibrium
In summary: But if someone else is moving with the object, then the difference in coordinate positions will change with time, and the length will be shorter in one frame and longer in the other.
nonequilibrium
Hello,

I was reading Einstein's RELATIVITY (his non-mathematical 1916 (or so) book).

Having shown the Lorentz transformation formulas, i.e.
$$t' = \gamma( t - vx/c^2)$$
$$x' = \gamma(x - vt),$$
he then talks about length contraction. He takes a meter-rod fixed in a certain coordinate system S' so that the left end is at $$x_1' = 0$$ and the right end is at $$x_2' = 1$$ and he then tells us to view these two points from system S at a certain time for which he takes t = 0. We then get that in S $$x_1 = 0$$ and $$x_2 = 1/ \gamma$$ (we're ignoring units here). Thus we get that if we see a rod passing us at a speed v, the length at any given moment is $$\gamma$$ times smaller than the so-called eigenlength of S' where the stick is fixed. (this is, of course, the famous Lorentz contraction formula)

Now in this case it makes sense to say "the length of the stick in S is smaller than the length of the stick in S' ". But notice that in our previous case, $$t_1' \neq t_2'$$ (because $$t_1 = t_2$$ and $$x_1 \neq x_2$$). This was okay because the two points of the rod were fixed in S' anyway, so it didn't matter what "time it was" for a certain point of the rod (i.e. the position of the rod was independent of t'). But imagine the situation where we want to compare the length of a certain rod, moving relatively to us at c/2 and relatively to somebody else at c/3. It has no meaning to say the length I see is smaller/larger than what he sees; is this correct? By taking a certain time for both points in, say, S, then the two points in S' can't be at the same instant, but then you can't compare "two photographs" of each situation... (Then again, I just came to think of the fact that each of these systems can be compared with the eigenlength, thus offering a comparison after all by then cancelling the eigenlength out of those two equations... How come two of my reasonings lead to different statements? Where is there a mistake?)

Thank you.

the length of a rod in a given frame is the distance between the front and the back at one simultaneous moment in that frame.

so, yes.
Relativity of simultaneity does enter into it.

In each frame you can pick two events, one on the worldline of the front and the other on the worldline of the back, that occur simultaneously in that frame, and consider the difference in position coordinates between the events as the object's "length" at that moment. As long as both front and back are moving inertially, each frame should see the length as constant with change (i.e. it won't matter what time we pick the two simultaneous events at).

## 1. What is length contraction in Einstein's theory of relativity?

Length contraction is a phenomenon described in Einstein's theory of relativity where the length of an object appears to shorten when it is moving at high speeds relative to an observer. This is due to the concept of space and time being relative and dependent on an observer's frame of reference.

## 2. How does length contraction differ from classical physics?

In classical physics, the length of an object is considered to be constant regardless of an observer's perspective. However, in Einstein's theory of relativity, length contraction is taken into account and describes how the length of an object can change depending on the observer's frame of reference.

## 3. What is the formula for calculating length contraction?

The formula for calculating length contraction is given by L = L0 * √(1 - (v2/c2)), where L is the contracted length, L0 is the rest length of the object, v is the velocity of the object, and c is the speed of light.

## 4. Can length contraction be observed in everyday life?

No, length contraction can only be observed at speeds close to the speed of light, which is not achievable in everyday life. It is a phenomenon that is observed in extremely high-speed scenarios, such as in particle accelerators or space travel.

## 5. How does length contraction affect the concept of simultaneity?

Length contraction is one of the factors that contribute to the relativity of simultaneity, which is the idea that two events that appear to happen simultaneously to one observer may not be simultaneous to another observer. This is because the distance between the two events can appear differently depending on the observer's frame of reference due to length contraction.

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