# Important note concerning relativity of time

1. Oct 22, 2006

### myoho.renge.kyo

Resnick on pages 470 and 471 in Physics (4th edition) states that

T = t/sqrt(1 - v^2 / c^2), where T is the time value (of an event) obtained by an observer using a moving clock, and t the value of the same event using a stationary clock.

But A. Einstein on page 49 in The Principle of Relativity states that

T = t*sqrt(1 - v^2 / c^2), where T is again the time value (of an event) obtained by an observer using a moving clock, and t the value of the same event using a stationary clock.

suppose that i have two synchronous stationary clocks. let one of them depart (the earth) with a velocity v = 0.5c at a specified direction so that i can use either Resnick's formula or Einstein's formula.

using Resnick's formula, if the current period of time (t) is 10/22/2006, 20:00:00 thru 21:00:00 universal time, in burbank, california, the current period of time (T) would be 20:00:00 thru 21:09:17. this does not mean at all that the moving clock is slower. it actually means the opposite. for example, if i am going to bed at 5:00:00 (the alarm is off, but it is also set at 5:00:00) universal time, and i want to set the alarm clock to wake up at 13:00:00, i see that in the process of setting the alarm, the period of time 5:00:00 thru 5:00:20 corresponds to 5:00:00 thru 13:00:00. this means that while setting the alarm, the clock moved much faster than the real time. the same analogy can be used here; the period of time (20:00:00 thru 21:00:00) measured using the stationary clock corresponds to the time period (20:00:00 thru 21:09:17) measured using the moving clock.

using Einstein's formula, we see indeed that the moving clock is slower. for example, the current period of time t (20:00:00 thru 21:00:00) corresponds to the current period of time T (20:00:00 thru 20:51:58).

Last edited: Oct 22, 2006
2. Oct 23, 2006

### Staff: Mentor

I suspect that these are your own loose translations of what Resnick and Einstein have written. If you want to discuss something they wrote, please transcribe it exactly as given.

In any case, this is standard stuff. The "time dilation" formula tells you that moving clocks run slow. That means if Alice is moving at a speed v with respect to Bob, then Bob's measurements (using Bob's clocks and metersticks) will find Alice's clocks running slowly. That means that if one of Alice's clocks measures a time period of [itex]\Delta T_0[/tex], then Bob will measure the time interval according to his clocks to be:
$$\Delta T = \frac{\Delta T_0}{\sqrt{1 - v^2/c^2}}$$.

Of course, this time dilation effect is completely symmetric: Alice will measure (according to her clocks) that Bob's clocks are running slow.

(Note: One must apply this "time dilation" formula carefully. It does not say that if Alice measures the time interval between two separated events, then Bob will measure the time interval between those same events according to that formula. For that kind of general translation between measurements made in different frames one must use the full Lorentz transformations. The "time dilation" formula is a special case of the more general Lorentz transformations.)

3. Oct 26, 2006

### myoho.renge.kyo

using the same example you used to explain what Resnick meant (with Alice and Bob), can you do the same to explain what Einstein meant when he states on page 49 in The Principle of Relativity that

T = t*sqrt(1 - v^2 / c^2). this will help me compare what Resnick said with what Einstein said and see where they are actually saying the same thing. thanks.

4. Oct 26, 2006

### JesseM

Einstein said that T is the time on the moving clock, so if we treat Bob as stationary and Alice as moving, then if Bob measures the time between two events on Alice's worldline to be $$\Delta t$$, then Alice will have measured the time between those same two events to be $$\Delta T = \Delta t \sqrt{1 - v^2/c^2}$$. So, you can see that Alice has measured a smaller time interval between the two events on her worldline, which is consistent with the fact that her clock is running slower in Bob's frame.

Note that rearranging this equation to give the time measured by Bob in terms of Alice's time gives $$\Delta t = \Delta T / \sqrt{1 - v^2/c^2}$$, which is what Resnick's equation was probably saying (although your paraphrase of what the equation meant seemed wrong, I'd like to see Resnick's exact quote). Again though, it's important that the two events be along Alice's worldline, or at least two events on the worldline of an object that's at rest in Alice's frame--the time interval between the two events in Alice's frame can be measured by a single clock at rest in her frame (because in her frame the events happen at the same position but different times), while Bob requires two synchronized clocks at rest in his frame to measure the time between the events (because in his frame the two events occur at different positions as well as different times). If you wanted to look at two events on Bob's worldline, and compare the time intervals in each frame, then you'd use the same equations above but with $$\Delta t$$ now referring to the time-interval measured by Alice using two synchronized clocks, and $$\Delta T$$ now referring to the time interval measured by Bob using a single clock.

Last edited: Oct 26, 2006
5. Oct 27, 2006

### myoho.renge.kyo

moving clocks do indeed run slower

thanks so much. i understand now. let me demonstrate with what Resnick wrote:

THE RELATIVITY OF TIME

We consider two observers: S is at rest on the ground, and S’ is in a train moving on a long straight track at constant speed u relative to S. The observers carry identical timing devices, consisting of a flashing light bulb F attached to a detector D and separated by a distance L0 from a mirror M. The bulb emits a flash of light that travels to the mirror. When the reflected light returns to D, the clock ticks and another flash is triggered. The time interval t0 between ticks is just the distance 2*L0 traveled by light divided by the speed of light c:

t0 = 2*L0/c. (1)

The interval t0 is observed by either S or S’ when the clock is at rest with respect to that observer.

We now consider the situation when one observer looks at the clock carried by the other.

According to S, the light beam travels a distance 2*L, where L = sqrt[L0^2 + (u*t/2)^2]. The time interval measured by S for the light to travel this distance at a speed c (the same speed measured by S’!) is

t = 2*L/c = 2*sqrt[L0^2 + (u*t/2)^2]/c. (2)

Substituting for L0 from Eq. 1 and solving Eq. 2 for t gives

t = t0/sqrt(1 - u^2/c^2). (3)

The factor in the denominator of Eq. 3 is always less than or equal to 1, and thus t is always greater than or equal to t0. That is, the observer relative to whom the clock is in motion (observer S) measures a greater interval between ticks. This effect is called time dilation. The time interval t0 measured by an observer (S’ in this case) relative to whom the clock is at rest is called the proper time. The proper time interval between events is the smallest interval between them that any observer can measure; all observers in motion relative to the clock measure longer intervals.

For example, a pion at rest decays in a time interval of 26.0 ns; this interval is a proper time interval and is designated t0. (The pion is in effect a clock, and the interval from formation to decay of the pion can be regarded as a tick of the clock.) An observer in the laboratory, relative to whom the pion is in motion at a speed of u = 0.913*c, would be expected to measure a time interval of

t = t0/sqrt(1 - u^2/c^2) = (26.0 ns)/sqrt[1 - (0.913)^2] = 63.7 ns.

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Let’s say that I am moving along with the pion, and I measure the interval of time t0 = 26.0 ns from formation to decay of the pion with my clock next to me. After I obtain that measurement, I adjust the tick-rate of my clock 2.45 faster than what it was, such that if the previous tick-rate was r, then the new tick-rate is 2.45*r. When I measure again the interval of time t0’ from formation to decay of the pion, t0’ = 63.7 ns.

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Let’s say now that I am stationed in the laboratory; relative to me the pion is in motion at a speed of u = 0.913*c, and I measure the interval of time t = 63.7 ns from formation to decay of the pion with my clock next to me (the tick-rate of my clock in this case is not 2.45*r; the tick-rate of my clock is r).

I can see that the measurement obtained using the clock next to me when I was stationed in the lab, is the same as the measurement obtained using the clock next to me when I was moving along with the pion (after I adjusted the tick-rate of my clock 2.45 faster than what it was). This indicates that t0 (T according to what Einstein wrote on page 49 in The principle of Relativity) is indeed t*sqrt(1 - u^2/c^2). Thanks. I understand now.

Last edited: Oct 27, 2006