Time dilation implies that when a clock moves relative to a frame S, careful measurements made by observers in S will find that the clock is running slow. This is not at all the same thing as saying that a single observer in S will see the clock running slow, and this latter statement is not always true. To understand this, remember that what we see is determined by the light as it arrives at our eyes. Consider an observer standing close beside the x axis as a clock approaches her with speed V along the axis. As the clock moves from position A to B, it will register a time
, but as measured by the observer's helpers, the time between the two events ("clock at A" and "clock at B") is
However, since B is closer to the observer than A is, the light from the clock at B will reach the observer in a shorter time than will the light from A. Therefore, the time
between the observer's seeing the clock at A and seeing it at B is less that Δt. (a) Prove that
(which is less than
). Prove both equalities. (b) What time will the observer see once the clock has passed her and is moving away?
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Alarm clock 1 is positioned at x = -d and alarm clock 2 at x = +d in reference frame A. According to a clock in reference frame A, both go off at t = 0. An observer in reference frame B, which is moving relative to A, measures alarm clock 2 as going off first. In what direction is B moving relative to A?