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8/21/03 |
Simultaneity |
Consider two arbitrary events located in space and time. Let Dx and Dt be the distance and time between the two events as measured in the rest frame and Dx’ and Dt’ the distance and time between the two events as measured in the moving frame. Then the first and last inverse Lorentz transformation equation gives the relationship
Dx = g(Dx’
+ vDt’)
Dt = g
(Dt’ + vDx’/c2).
If we now let the two events under consideration be separated by a distance Dx in the rest frame but occur at the same instant of time in the moving frame, then we obtain:
This equation shows that simultaneity is a relative quantity. Two events that are simultaneous in the moving frame are not simultaneous in the rest frame. In fact, the time difference between the events in the rest frame increases with both distance and velocity.
Notice that the relativity of simultaneity occurs only in the direction of motion. Events in front of the moving observer shift forward in time while those behind the moving observer shift backward in time. Events to the side of the moving observer do not shift in time because the coordinate differences Dy and Dz have no effect upon Dt.
The presence of time shifts in front of and behind a moving observer (which never occur in classical physics) suggests that relativistic physics may encounter major difficulties with the principle of causality. This principle asserts that cause always precedes effect, or that decisions can never affect the past.
But if time can be shifted forward and backward simply by changing from one reference frame to another, what is to prevent a person from violating the principle of causality or using time travel to visit the past? If time can be shifted into the past, what prevents someone from changing the past?
The speed of light is what prevents this from happening.
According to relativity, the only way to preserve the principle of causality is to limit the velocity of information to the speed of light. Therefore, the theory of relativity asserts that nothing – no person, no object, no information – can travel faster than the speed of light.
See Serway pp.13-14 and Tipler pp.17-20 for more information on simultaneity. Also look at the problems at the end of Chapter 1 in both books for additional problems on this topic.
P1. Length Contraction and Simultaneity
The distance across our galaxy is approximately 100,000 light years and the radius of the earth’s orbit around the sun is about 1.5x1011 meters. (a) What is the orbital velocity of the earth? (b) What is the distance across our galaxy relative to the earth if one assumes that the sun is at rest in our galaxy? (c) If an event occurs on the opposite side of the galaxy simultaneous with an event on the sun (as measured by the sun), what is the difference in time between these events according to the earth if the earth is moving toward the other side of the galaxy? (d) Six months later, when the earth is moving in the opposite direction, what is the difference in time between the two events? (e) According to the earth, how much time has elapsed on the other side of the galaxy during this six-month time period on earth?
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