bernhard.rothenstein said:Textbooks I know derive the formula that accounts for the time dilation by measuring simultaneously the space coordinates of a moving rod.
Is there another way to derive it?
[tex]ds^2=dt^2-dx^2[/tex] is an invariant.bernhard.rothenstein said:Textbooks I know derive the formula that accounts for the time dilation by measuring simultaneously the space coordinates of a moving rod.
Is there another way to derive it?
Please tell me if that can be done in the case of length contraction?clem said:[tex]ds^2=dt^2-dx^2[/tex] is an invariant.
Choosing dx=vdt in one frame and dx=0 in another gives the TD formula.
kdv said:I am guessing you meant "lenght" instead of time in your first sentence. And I am assuming that you mean "deriving from the Lorentz transformations".
That's the most direct way to measure the length of a moving object: measuring the positions of its extremities at the same time in the frame of the measurement. Butthis requires two observers (it could be done with a single observer but then th eobserver is not local to the two events and must take into account the finite speed of light and that makes things much more complicated).
A second way requiring only a single observer is to do the following: Notice the time it takes for the moving object to pass in front of you (i.e. measure the time elapsed between the front of the object being aligned with you and the back of the object being aligned with you. Use the fact that this time you measured must be equal to the length of the object (in your frame) divided by the speed of the object. The speed and the lengths are two unknowns. Now use this in the Lorentz transformations and you can solve for the lenth contraction formula.
bernhard.rothenstein said:Please tell me if that can be done in the case of length contraction?
No. [tex]dx'^2-dt'^2=dx^2-dt^2[/tex].bernhard.rothenstein said:Please tell me if that can be done in the case of length contraction?
pam said:No. [tex]dx'^2-dt'^2=dx^2-dt^2[/tex].
In the moving frame, classical intuition is used to assume that setting dt'=0 gives the correct length of a moving rod. No formula results from the equation because dt can be anything in the rest system. The Lorentz transformation is needed to discuss length.
pam said:No. [tex]dx'^2-dt'^2=dx^2-dt^2[/tex].
In the moving frame, classical intuition is used to assume that setting dt'=0 gives the correct length of a moving rod. No formula results from the equation because dt can be anything in the rest system. The Lorentz transformation is needed to discuss length.
Length contraction is a phenomenon in which an object's length appears to decrease when it is moving at a high speed relative to an observer. This is a consequence of Einstein's theory of special relativity.
The formula for length contraction is L' = L * √(1-v^2/c^2), where L' is the contracted length, L is the original length, v is the velocity of the object, and c is the speed of light.
Length contraction can be derived mathematically using the principles of special relativity. It involves using the Lorentz transformation equations to transform the coordinates of an object in one frame of reference to the coordinates in another frame of reference that is moving at a constant velocity relative to the first frame.
Yes, length contraction has been observed and confirmed through various experiments and observations. It is a fundamental aspect of special relativity and has been validated by numerous scientific studies.
Length contraction has important implications in fields such as particle physics, where it helps explain the behavior of subatomic particles moving at high speeds. It is also a crucial factor in the design of technologies such as particle accelerators and high-speed spacecrafts.