# Why does time dilation apply for objects moving with a velocity less than c?

I am confused with some of the aspects of time dilation. I read the derivation (the one with the mirrors) and I understand it and it seems to make sense to me. In the usual derivation (http://www.drphysics.com/syllabus/time/time.html) , person A who moves with the apparatus (mirror-light source), sees the light travel a smaller distance while person B who is stationary sees the light move a larger distance with the same speed and hence the time dilation formula is obtained.

However, suppose person C moves at a rate of 0.8c wrt person D in a space rocket, why does he/she experience time dilation? Say they both starts their clocks at the same instant, when person C passes by person D. If person C sees person D move a distance of d = 1.20 x 10^8 m/s, at time 0.5s, shouldnt person D see person C moving at the same rate in the opposite direction (From her ref. frame) and conclude that the distance between them is d, and hence t = 0.5s simultaneously? But the solved solution I have applies time dilation and says that its been only 0.3 s for person D when 0.5s have passed for person C. What is dont understand is where this physical difference is coming from. In the derivation, due to the speed of light having to be the same in both frames, different values of time were obtained. In this case however,
a) person C is moving at a speed less than c and
b) in either one's reference frame, the other person moves at the 0.8c

Why does time dilation apply then? =[

I can easily solve the question by blindly applying the formula, but it doesnt make sense to me. Also, this is a solved problem and not a homework question.

I greatly appreciate any help!

PS: I understand the lightening example for simultaneity, but again, the example shows one observer moving at a finite speed wrt to the other. I am more confused with examples where one of the observers move at speeds close to c.

PS: I understand the lightening example for simultaneity, but again, the example shows one observer moving at a finite speed wrt to the other. I am more confused with examples where one of the observers move at speeds close to c.
This is possibly the root of the confusion. There is no such thing as speeds close to c in any absolute sense. There is only relative motion. In your example you say one observer is moving at some finite speed wrt the other and of course the converse is true, but there is no concept of which observer is "really" moving. Let us say A is moving at 0.8c relative to B and B is moving at -0.8c relative to A, then that is all there is. We cannot say either A or B is the one that is really moving at close to the speed of light and the other is really stationary, that is just a subjective view. Whether you assume A or B is the one that is really stationary, the calculations work out the same. A sees B's light clock ticking slower than his own clocks and B sees A's light clock ticking slower than her clocks. That is the relative nature of relativity.

P.S. Your thread title implies that time dilation should only apply to objects that are moving at c or greater, but that makes no sense in relativity, as all objects that we can detect or measure can only move at c or less, relative to other objects.

This is possibly the root of the confusion. There is no such thing as speeds close to c in any absolute sense. There is only relative motion. In your example you say one observer is moving at some finite speed wrt the other and of course the converse is true, but there is no concept of which observer is "really" moving. Let us say A is moving at 0.8c relative to B and B is moving at -0.8c relative to A, then that is all there is. We cannot say either A or B is the one that is really moving at close to the speed of light and the other is really stationary, that is just a subjective view. Whether you assume A or B is the one that is really stationary, the calculations work out the same. That is the relative nature of relativity.
Ah, bad way of wording it, I meant one observer moving at a speed close to c wrt to the other xD And I am confused because I dont understand how time dilation applies to such a situation. I found derivations of time dilation with the mirror-light source situation, but couldnt find any where one observer moves at a fraction of the speed of light wrt the other.
Thanks!

Ah, bad way of wording it, I meant one observer moving at a speed close to c wrt to the other xD And I am confused because I dont understand how time dilation applies to such a situation. I found derivations of time dilation with the mirror-light source situation, but couldnt find any where one observer moves at a fraction of the speed of light wrt the other.
Thanks!
The derivations are always done with one observer moving at a fraction of the speed of light wrt the other. See http://en.wikipedia.org/wiki/Time_dilation .

In the equation:

$$\Delta t ' = \frac{\Delta t}{\sqrt{1-v^2/c^2}}$$

the fraction (v/c)^2 is the fraction of the speed of light that the primed observer is moving relative to the other. In this example the other (unprimed) observer is at rest wrt the light clock and the primed observer is moving at v relative to the light clock. Therefore the two observers are moving at v relative to each other.

Ah, bad way of wording it, I meant one observer moving at a speed close to c wrt to the other xD And I am confused because I dont understand how time dilation applies to such a situation. I found derivations of time dilation with the mirror-light source situation, but couldnt find any where one observer moves at a fraction of the speed of light wrt the other.
Thanks!
Do you mean if A is at rest with the light clock and sees the elapsed time as t, and if B is moving at v relative to A and sees the elapsed time on the light clock as t', what will C measure the elapsed time to be (t'') if C is moving at v' relative to B?

If that is what you are asking, the easy way to calculate it is to use the relativistic velocity addition equation to work out the speed of C relative to A and then use the regular time dilation equation. Even better, use the Lorentz transformation for time. It might be an idea to give an example with numerical parameters for what you have in mind and then make it clear what you want to calculate.

ah sorry for taking so long to reply, I was thinking about the stuff. Thanks very much for helping me!
I am actually confused with the cause of time dilation. The only "reason" I know is the derivation of the formula; however, my brain thinks that the derivation is based on two observers moving wrt to each other trying to figure out how long it takes for a light beam (moving with one observer) to move from a source to a mirror and get reflected back. In this case, because the speed of light is the same in all reference frames, one observer assumes that time is dilated in the other's frame.
In the case where again one observer is moving with respect to the other however, if there is no such situation with the light beam, how does one know time dilation occurs?
There is no conflict with exceeding the speed of light in this case.
Sorry maybe I am thinking the wrong way =S

by "situation" I mean that there is no light clock apparatus.

btw, referring to post 90, proper time is the smallest time that can be measured right?

ah sorry for taking so long to reply, I was thinking about the stuff. Thanks very much for helping me!
No problem It is good to think before replying. (I am sometimes guilty of not doing that. :tongue:). Delays of several days between posts are fairly normal here and we understand that people have other stuff to do in their regular lives and sometimes it is good to take time to reflect on a problem.

I am actually confused with the cause of time dilation. The only "reason" I know is the derivation of the formula; however, my brain thinks that the derivation is based on two observers moving wrt to each other trying to figure out how long it takes for a light beam (moving with one observer) to move from a source to a mirror and get reflected back. In this case, because the speed of light is the same in all reference frames, one observer assumes that time is dilated in the other's frame.
In the case where again one observer is moving with respect to the other however, if there is no such situation with the light beam, how does one know time dilation occurs?
The light clock is a simple way of demonstrating time dilation. With no light clock, two digital wrist watches will suffice, but explaining how one watch slows down due to the flow of electrons and the changing shape of the circuits on the printed circuit and changes in the resonant frequency of a crystal are more difficult to explain in simple terms. Even two biological entities will suffice as crude clocks. Time dilation affects all time related processes. Perhaps you have heard of relativistic mass? This concept is discouraged these days but sometimes it has its uses as I will demonstrate here. Imagine you have a flywheel with mass m. When this flywheel is moving in a straight line relative to you parallel to its axis of rotation, its relativistic mass and angular moment of inertia increases. To conserve angular momentum, the rotation rate of the flywheel slows down. In this case the flywheel acts as a crude clock. To a person at rest with the flywheel axis, the flywheel is rotating (ticking) faster than it appears to be rotating to an observer with relative motion. With any physical evolving system, if you analyse it deeply enough you will find it is consistent with time dilation.
There is no conflict with exceeding the speed of light in this case.
This statement confuses me very much. Where do you imagine that there is a conflict with exceeding the speed of light?

As I said before, it might be helpful if you can describe a simple situation precisely, where you feel a conflict might arise.

btw, referring to post 90, proper time is the smallest time that can be measured right?
Yes, in natural coordinate systems in Special Relativity, that is normally the case. Here I am assuming that you mean the "smallest time" interval between two events, where the proper time is defined as the interval between two co-located events. People can define weird "unnatural" coordinates where coordinate time intervals are shorter than proper time intervals if they wish, but they only serve to confuse.

The light clock is a simple way of demonstrating time dilation. With no light clock, two digital wrist watches will suffice, but explaining how one watch slows down due to the flow of electrons and the changing shape of the circuits on the printed circuit and changes in the resonant frequency of a crystal are more difficult to explain in simple terms. Even two biological entities will suffice as crude clocks. Time dilation affects all time related processes. Perhaps you have heard of relativistic mass? This concept is discouraged these days but sometimes it has its uses as I will demonstrate here. Imagine you have a flywheel with mass m. When this flywheel is moving in a straight line relative to you parallel to its axis of rotation, its relativistic mass and angular moment of inertia increases. To conserve angular momentum, the rotation rate of the flywheel slows down. In this case the flywheel acts as a crude clock. To a person at rest with the flywheel axis, the flywheel is rotating (ticking) faster than it appears to be rotating to an observer with relative motion. With any physical evolving system, if you analyse it deeply enough you will find it is consistent with time dilation. .
AHH!! That helps a lot and the relativistic mass concept is interesting! xD

This statement confuses me very much. Where do you imagine that there is a conflict with exceeding the speed of light?.
I really need to word myself more carefully.
I thought the concept of time dilation arose because the speed of light is the same in all reference coords. What I meant by the quoted statement is that in the situation I described with the time clock, because the speed of light is the same in the ref. frame of both observers, the speed of their respective motion could not be added subtracted using the rules of Galilean transforms. But in the situation without the light lock, there was no such situation to consider xD so I got confused on how to consider the time dilation effect. But it makes sense now, because of the way you described different methods for measuring time, all of which are affected by the effects of time dilation.

Yes, in natural coordinate systems in Special Relativity, that is normally the case. Here I am assuming that you mean the "smallest time" interval between two events, where the proper time is defined as the interval between two co-located events.
yes, that is basically what I meant. =]

Thanks very much for all the help! I really appreciate it.