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zenterix
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- Let's say we have a meter stick in frame A. Frame B sees it as a stick with length ##\sqrt{1-v^2}##.
How many sticks are there? Just one?
Why does it/does it not make sense to speak of frame A "seeing" this shorter length stick that frame B sees (thus seeing it even shorter), and so on, infinite times? What's wrong with this reasoning?
I have a question about the concept of length contraction.
The black line from (0, 0) to (1, 0) represents a meter stick in my stationary frame, call it frame A. The blue axes represent my coordinate system with coordinates x and t.
The green axes represent the coordinate system of a moving frame, call it frame B, with coordinates x' and t'.
I assume frame B moves along the positive x direction and there is no movement along y and z directions.
Since the meter stick is at rest in frame A, we can easily determine its position in spacetime in frame A as time increases. Considering just the endpoints of the meter stick, x=0 and x=1 represent the so-called world lines of these endpoints: they stay in the same position forever.
What about from frame B's perspective?
O and Q do not have the same time coordinates in frame B and thus are not simultaneous.
On the other hand, one end of the meter stick at a certain t>0 represented by point P (at which t'=0) is simultaneous with the other end of the meter stick in frame B.
This is still difficult to imagine for me, but I think I understand the argument.
If we calculate the x' coordinate of point P we get
Hence, the meter stick as "seen" from frame B is shorter. I put "seen" in quotes because, again, it is still difficult to digest this idea of seeing a shorter meter stick.
Also, this length seems to have nothing to do with the geometry of the picture I drew: the purple segment should be longer than the black segment. From my previous questions on this subject, I gather this is because the geometry of spacetime is not Euclidean.
Now, what happens if we think about a meter stick in frame B as seen from frame A? From B's perspective, the meter stick is at rest, but it is moving from A's perspective.
The world line in frame B is x'=1: the meter stick stays at x'=1 forever.
Pictorially, we have
with the meter stick in B being OP and in A being OQ.
If we calculate the length of OQ we reach
My question is the following.
Let's say we have a meter stick in frame A. Frame B sees it as a stick with length ##\sqrt{1-v^2}##.
How many sticks are there? Just one?
Why does it/does it not make sense to speak of frame A "seeing" this shorter length stick that frame B sees (thus seeing it even shorter), and so on, infinite times? What's wrong with this reasoning?
In addition, what happens at t'=1?
It seems it would be
The stick would seem to be represented by the segment between the two pink points. If this is so, then the stick just keeps having the same length forever in frame B, it's just a bit shorter than in A.
The black line from (0, 0) to (1, 0) represents a meter stick in my stationary frame, call it frame A. The blue axes represent my coordinate system with coordinates x and t.
The green axes represent the coordinate system of a moving frame, call it frame B, with coordinates x' and t'.
I assume frame B moves along the positive x direction and there is no movement along y and z directions.
Since the meter stick is at rest in frame A, we can easily determine its position in spacetime in frame A as time increases. Considering just the endpoints of the meter stick, x=0 and x=1 represent the so-called world lines of these endpoints: they stay in the same position forever.
What about from frame B's perspective?
O and Q do not have the same time coordinates in frame B and thus are not simultaneous.
On the other hand, one end of the meter stick at a certain t>0 represented by point P (at which t'=0) is simultaneous with the other end of the meter stick in frame B.
This is still difficult to imagine for me, but I think I understand the argument.
If we calculate the x' coordinate of point P we get
Hence, the meter stick as "seen" from frame B is shorter. I put "seen" in quotes because, again, it is still difficult to digest this idea of seeing a shorter meter stick.
Also, this length seems to have nothing to do with the geometry of the picture I drew: the purple segment should be longer than the black segment. From my previous questions on this subject, I gather this is because the geometry of spacetime is not Euclidean.
Now, what happens if we think about a meter stick in frame B as seen from frame A? From B's perspective, the meter stick is at rest, but it is moving from A's perspective.
The world line in frame B is x'=1: the meter stick stays at x'=1 forever.
Pictorially, we have
with the meter stick in B being OP and in A being OQ.
If we calculate the length of OQ we reach
My question is the following.
Let's say we have a meter stick in frame A. Frame B sees it as a stick with length ##\sqrt{1-v^2}##.
How many sticks are there? Just one?
Why does it/does it not make sense to speak of frame A "seeing" this shorter length stick that frame B sees (thus seeing it even shorter), and so on, infinite times? What's wrong with this reasoning?
In addition, what happens at t'=1?
It seems it would be
The stick would seem to be represented by the segment between the two pink points. If this is so, then the stick just keeps having the same length forever in frame B, it's just a bit shorter than in A.
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