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Time in Relativity Help

By — McGraw-Hill Professional
Updated on Sep 12, 2011

No Absolute Time

One of the first results of Einstein’s speed-of-light axiom is the fact that there can be no such thing as an absolute time standard. It is impossible to synchronize the clocks of two observers so that they will see both clocks as being in exact agreement unless both observers occupy the exact same point in space.

In recent decades we have built atomic clocks, and we claim they are accurate to within billionths of a second (where a billionth is 0.000000001 or 10 −9 ). But this has meaning only when we are right next to such a clock. If we move a little distance away from the clock, the light (or any other signal that we know of) takes some time to get to us, and this throws the clock reading off.

The speed of EM-field propagation, the fastest speed known, is approximately 3.00 × 10 8 m/s (1.86 × 10 5 mi/s). A beam of light therefore travels about 300 m (984 ft) in 1.00 × 10 −6 s (1.00 μs). If you move a little more than the length of a football field away from a superaccurate billionth-of-a-second atomic clock, the clock will appear to be in error by 1.00 μs. If you go to the other side of the world, where the radio signal from that clock must travel 20,000 km (12,500 mi) to reach you, the time reading will be off by 0.067 s. If you go to the Moon, which is about 4.0 × 10 5 km (2.5 × 10 5 mi) distant, the clock will be off by approximately 1.3 s.

If scientists ever discover an energy field that can travel through space instantaneously regardless of the distance, then the conundrum of absolute time will be resolved. In practical scenarios, however, the speed of light is the fastest possible speed. (Some recent experiments suggest that certain effects can propagate faster than the speed of light over short distances, but no one has demonstrated this on a large scale yet, much less used such effects to transmit any information such as data from an atomic clock.) We can say that the speed of light is the speed of time. Distance and time are inextricably related.

Point Of View

Suppose that there are eight clocks in space arranged at the vertices of a gigantic cube. Each edge of the cube measures 1 light-minute, or approximately 1.8 × 10 7 km (1.1 × 10 7 mi), long, as shown in Fig. 20-1. We are given a challenge: Synchronize the clocks so that they agree within the limit of visibility, say, to within 1 second of each other. Do you suppose that this will be easy?

Relativity Theory Simultaneity Point Of View

Fig. 20-1 . A hypothetical set of eight clocks arranged at the vertices of a cube that measures 1 light-minute on each edge. How will we synchronize these clocks?

Because the clocks are so far apart, the only way we can ascertain what they say is to equip them with radio transmitters that send time signals. Alternatively, if we have a powerful enough telescope, we can observe them and read them directly by sight. In either case, the information that tells us what the clocks say travels to us at the speed of light. We get in our space ship and maneuver ourselves so that we are in the exact center of the cube, equidistant from all eight clocks. Then we proceed to synchronize them using remote-control, wireless two-way data communications equipment. Thank heaven for computers! The task is accomplished in a just a few minutes. It can’t be done instantaneously, of course, because our command signals take the better part of a minute to reach the clocks from our central location, and then the signals coming back from the clocks take just as long to get to us so that we can see what they say. Soon, however, everything is in agreement. Clocks A through H all tell the same time to within a fraction of a second.

Satisfied with our work, we cruise out of the cube. We take a look back at the clocks. What do we see? The clocks have already managed to get out of sync. We take our ship back to the center of the cube to correct the problem. When we get there, however, there is no problem to correct! The clocks are all in agreement again.

You can guess what is happening here. The clock readings depend on how far their signals must travel to reach us. For an observer at the center of the cube, the signals from all eight clocks, A through H , arrive from exactly the same distance. However, this is not true for any other point in space. Therefore, the clocks can be synchronized only for that one favored point; if we go somewhere else, we will have to synchronize them all over again. This can be done, but then the clocks will be synchronized only when observed from the new favored vantage point. There is a unique sync point —the spot in space from which all eight clocks read the same—for each coordination of the clocks.

No sync point is more valid than any other from a scientific standpoint. If the cube happens to be stationary relative to some favored reference point such as the Earth, we can synchronize the clocks, for convenience, from that reference point. However, if the cube is moving relative to our frame of reference, we will never be able to keep the clocks synchronized. Time depends on where we are and on whether or not we are moving relative to whatever device we use to indicate the time. Time is not absolute, but relative, and there is no getting around it.

Time in Relativity Practice Problem

Problem

Suppose that there is an atomic clock on the Moon (clock M ), and that its time signals are broadcast by a powerful radio transmitter. This clock is set to precisely agree with another atomic clock in your home town on Earth (clock E ), and it is also equipped with a radio transmitter. If you travel to the Moon, what will be the relative readings of the two clocks, as determined by listening to the radio signals?

Solution

Radio signals travel through space at about 3.00 × 10 5 km/s. The Moon is about 4.0 × 10 5 km, or 1.3 light-seconds, away from Earth. The reading of clock M will be shifted approximately 1.3 s ahead in time (that is, earlier) because the time lag for its signals to reach you will be eliminated. The reading of clock E will be shifted about 1.3 s behind in time (that is, later) because a time lag will be introduced where previously there was none. When you get to the Moon, clock M will be approximately 2.6 s ahead of clock E .

Practice problems of these concepts can be found at: Relativity Theory Practice Test

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