Clocks in a gravitational field.

When comparing a clock under the influence of gravitational forces with one very far from such influences it is found that the first clock is slow compared to the second. To see this consider the same clock we used in the Special Theory of Relativity. For this experiment, however, imagine that the clock is being accelerated upward, being pulled by a crane. The clock gives off a short light pulse which moves towards the mirror at the top of the box, at the same time the mirror recedes from the pulse with even increasing speed (since the box accelerates). Still the pulse eventually gets to the mirror where it is reflected, now it travels downward where the floor of the box is moving up also with ever increasing velocity (see Fig. 7.8).

**Figure 7.8:** An accelerated clock. The circle denotes a pulse of light which at the initial is sent from a source; after a time it reaches the top of the the box and is reflected. The time it takes to do the trip is *longer* than for a clock at rest.
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On the trip up the distance covered by light is larger than the height of the box at rest, on the trip down the distance is smaller. A calculation shows that the whole distance covered in the trip by the pulse is larger than twice the height of the box, which is the distance covered by a light pulse when the clock is at rest.

Since light always travels at the same speed, it follows that the time it takes for the pulse to go the round trip is longer when accelerating than when at rest: clocks slow down whenever gravitational forces are present.

This has an amazing consequence: imagine a laser on the surface of a very massive and compact planet (so that the gravitational field is very strong). An experimenter on the planet times the interval between two crests of the laser light waves and gets, say, a millionth of a second. His clock , however, is slow with respect to the clock of an observer far away in deep space, this observer will find that the time between two crests is larger. This implies that the frequency of the laser is larger on the planet than in deep space: light leaving a region where gravity is strong reddens. This is called the gravitational red-shift (see Fig. 7.9).

**Figure 7.9:** The gravitational redshidft. Since clocks slow down in a strong gravitational field then light, whose oscillations can be used as clocks, will be shifter towards the red as it leaves a region where gravity is strong.
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As for time dilation, the slowing down of clocks in the presence of gravitational forces affects all clocks, including biological ones. A twin trveling to a region where gravity is very strong will come back a younger than the twin left in a rocket in empty space. This is an effect on top of the one produced by time dilation due to the motion of the clocks. The gravitational forces required for a sizable effect, however, are enormous. So the twin will return younger...provided she survives.