![]() ![]() We can show that r cannot be equal to 1, and in fact, has to be given by the time dilation factor. Let r be the rate of Bob's clock, as measured by Alice. Every T seconds (according to Alice's clock), she sends a light signal toward Bob, and every T seconds (according to Bob's clock), he sends a light signal toward Alice. Suppose that we are in a frame where Alice is at rest, and Bob is moving away from Alice at velocity v. This is the clue to developing the time dilation factor. Relativistically, there can't be a difference, since motion is relative you can't say which one of the sender or receiver is moving. So nonrelativistically, there is a distinction in the Doppler formula for the two cases. ![]() Time formula physics plus#It can be derived using Doppler shifts plus the relativity principle (equivalence of inertial frames) plus the fact that the speed of light is constant.įirst, the nonrelativistic Doppler formulas: If you are sending light signals between two observers moving away from each other at relative speed v, then the frequency is shifted lower by a factor of \frac. You might like to think about this question: why can there be no length contraction for distances perpendicular to the motion? distance perpendicular to motion) is the same for both observers?įor example, if length were contracted perpendicular to motion for one observer, then that might equally explain the constant speed of light without time dilation. There is, therefore, an important point about the light clock: how do you know that the height of the clock (i.e. Until you've worked out length contraction, you don't know the relative lengths and you cannot immediately deduce time dilation alone. The problem if the light is shining in the direction of motion is that you also have length contraction to take into account. So, measuring the time light takes to travel a known distance is perhaps the only reliable clock at this stage of the theoretical development of relativity. The reason a light clock is used is that one of the postulates of Special Relativity is that the speed of light is constant in all inertial reference frames. The only alternative is that there is something wrong with your clock. ![]() But, you only need to show time dilation applies in the case of one reliable clock and the conclusion is inevitable: that time itself is dilated. A derivation which just derives the formula for the relationship between proper time and dilated time interval between two events due to relative motion of reference frames. I just need a derivation which doesn't involve any zig-zag motion of light pulse or any other special case. I mean there won't be any zig-zag motion of light pulse if it is sent in the same direction as the motion of the spacecraft, so we can't apply Pythagoras theorem in that case to derive the same formula. Then, we apply Pythagoras theorem to derive the formula.īUT this seems like derivation in a special case, not a universal derivation. I've already read the derivation in which we use the light pulse clock kept in a spacecraft such that the light pulse follows a zig-zag motion due to motion of the spacecraft being perpendicular to motion of the light pulse. ![]()
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