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bookofproofs

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Hi, I'm a (hobby) mathematician and only an amateur physicist, so maybe below there are only trivia questions. Thank you in advance for conversation and clarification.

All mathematical proofs of "time dilation" I have looked at so far were based on the Pythagorean theorem. In all such proofs, there is a step of "dividing" both sides of the equation by the term (1-v^2/c^2), leading to the Lorentz factor 1/√(1-v^2/c^2), which is the time dilation factor. My questions are:

1) Why can we "without loss of generality" assume that the observed light signal is sent

2) Why is it necessary at all to divide by the term (1-v^2/c^2)? I mean, this needs the assertion that this calculation step only holds, if the velocity v does not equal the speed of light c. By how about the case v=c (e.g. if two photons move relatively to each other and experience a time dilation)? Why is it not more convenient to use the more simple form √(1-v^2/c^2) x t1 = t2 (which would allow the case v=c) instead of the classical Lorentz factor version of time dilation t1=t2 x 1/√(1-v^2/c^2), where we must exclude the case (v=c), because otherwise we would divide by zero?

3) It is known from particle accelerators that the energy needed to speed up massive particles (like protons) grows exponentially as their speed v approaches c from below. Why is this argument often used to "show" that "no massive particle can travel faster than light"? I mean, it only shows that it cannot travel at a speed exactly equal to the speed of light (v=c), but it does not (!) show that it cannot travel "faster" than light.

4) Please note that for the case v > c, the Lorentz factor is still well-defined and produces a complex number (square root of a negative real number). Because the time dilation factor would be a complex number in this case, my question is, could we interpret this as time having "two dimensions" (just like a complex number can be interpreted as a two-dimensional real number)? The first dimension of time would be its "real part" and this can be "observed" for all speeds v < c. The second dimension of time would be its "imaginary part", which would be "observed" in inertial frames of reference moving relatively to each other at a constant speed v > c?

<<mentor note: link to homepage removed>>

All mathematical proofs of "time dilation" I have looked at so far were based on the Pythagorean theorem. In all such proofs, there is a step of "dividing" both sides of the equation by the term (1-v^2/c^2), leading to the Lorentz factor 1/√(1-v^2/c^2), which is the time dilation factor. My questions are:

1) Why can we "without loss of generality" assume that the observed light signal is sent

__perpendicularly__to the direction of movement so that we can rely on the "right angle" and use the Pythagorean theorem in the calculation? My point is that we could also assume a more general case (which I wonder why it should not be possible), in which the light signal would be sent in an__arbitrary angle to the direction of movemen.__In this case, the proof would require the general "law of cosines" rather than the Pythagorean theorem. But then, the Lorentz factor and time dilation formula would look different, leading to different time dilation effects, wouldn't they? (Actually, I was trying to check out but was not able to simplify my general "law of cosine" factor to the known Pythagorean-case Lorentz factor)?2) Why is it necessary at all to divide by the term (1-v^2/c^2)? I mean, this needs the assertion that this calculation step only holds, if the velocity v does not equal the speed of light c. By how about the case v=c (e.g. if two photons move relatively to each other and experience a time dilation)? Why is it not more convenient to use the more simple form √(1-v^2/c^2) x t1 = t2 (which would allow the case v=c) instead of the classical Lorentz factor version of time dilation t1=t2 x 1/√(1-v^2/c^2), where we must exclude the case (v=c), because otherwise we would divide by zero?

3) It is known from particle accelerators that the energy needed to speed up massive particles (like protons) grows exponentially as their speed v approaches c from below. Why is this argument often used to "show" that "no massive particle can travel faster than light"? I mean, it only shows that it cannot travel at a speed exactly equal to the speed of light (v=c), but it does not (!) show that it cannot travel "faster" than light.

4) Please note that for the case v > c, the Lorentz factor is still well-defined and produces a complex number (square root of a negative real number). Because the time dilation factor would be a complex number in this case, my question is, could we interpret this as time having "two dimensions" (just like a complex number can be interpreted as a two-dimensional real number)? The first dimension of time would be its "real part" and this can be "observed" for all speeds v < c. The second dimension of time would be its "imaginary part", which would be "observed" in inertial frames of reference moving relatively to each other at a constant speed v > c?

<<mentor note: link to homepage removed>>

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