Why Does Gravity Slow Time?
Imagine a pair of twins, Alice and Bob, who will live to exactly the same age. Rather than giving this age in years, which might be confusing in what follows, lets say each will live for one billion heart beats, and their hearts beat at 60 beats per minute. Alice, a hurricane hunter by trade, has become bored with Earths puny storms and has moved to Jupiter to chase its Great Red Spot, a centuries old cyclone of truly mammoth proportions.
Now gravity is stronger on Jupiter than on the Earth, one consequence being that Alice weighs more. But more interestingly, Albert Einsteins theory of general relativity his theory of space, time and gravity says that, due to the lower gravitational potential on Jupiter than on Earth, time as experienced by Alice is moving more slowly relative to time experienced by Bob back on the Earth. What does this mean? First, the word relative is crucial here: it means that as far as Alice is concerned, nothing in her own experience indicates to her that time is moving more slowly. The point is, more slowly relative to what? Alice herself feels nothing out of the ordinary, for instance her heart still beats at 60 beats per minute according to her wristwatch. It is only when Alice and Bob compare their experiences of the passage of time that they notice something very strange.
For example, when they speak with each other over the satellite link, Bob notices that Alices voice is a bit deeper and she is speaking more slowly exactly like a tape recording replayed at a slightly slower speed. But Alice does not feel that she is speaking slowly, or thinking slowly, or anything else for her is happening more slowly. And from Alices point of view, she notices that Bobs voice is higher pitched than she remembers, and he is talking (and thinking, and doing everything else) a bit faster exactly like a tape recording played back at a faster speed. More to the point, when Bob puts the microphone next to his heart, Alice hears it beating at faster than 60 beats per minute according to her wristwatch (and her heart); conversely, Bob hears Alices heart beating more slowly than his. Both agree that Alice could return to Earth before her billion heart beats run out and attend Bobs funeral.
How is this possible? Why does gravity affect the rate at which time moves? One of the simplest ways to understand this begins with Einsteins equation E = mc2, which says that mass is a (very concentrated) form of energy. As a consequence of this, Einstein reasoned that it would be possible to build a perpetual motion machine (and thus get something energy for nothing) unless gravity slows time. The machine in question is show in the figure. It consists of a vertical conveyor belt stretched between two pulleys which, to keep things simple, we imagine to be frictionless.
Attached to the belt are a number of identical, equally spaced buckets, each of which contains a single atom of mass m. Now suppose each of the atoms on the left side of the belt have absorbed a photon of energy ÄE.
These excited atoms have more energy, and hence more mass. Using E = mc2, the increase in mass is m = ÄE/c2. With the whole apparatus in the gravitational field of the Earth, the force of gravity will be greater on the excited atoms (each with mass m+Äm) than the ground state atoms (each with mass m), resulting in a net force causing the conveyor belt to tend to rotate counter clockwise.
To keep the conveyor belt moving we arrange for the excited atoms to emit a photon of energy ÄE as they reach the bottom, returning them to their ground state (with mass m). We then use a mirror to direct the emitted photons up to the top, and use these photons to re-excite the ground state atoms as they arrive at the top of the conveyor belt. Assuming the energy of the photons received at the top is ÄE (the same energy they had at the bottom), we clearly have a perpetual motion machine: the heavier, excited atoms will always be on the left, the lighter ground state atoms on the right, resulting in a constant, net tendency for the belt to rotate counter clockwise. We could use this tendency to make our machine do useful work. Energy for nothing! All the worlds energy problems solved!
Of course, this must be impossible. But where is the mistake in our reasoning? A bit of thought reveals that the mistake is in assuming that the energy of the photons received at the top is ÄE. In fact it must be less than ÄE; call it ÄE. How much less? In being lowered from the top to the bottom (through a height h, say), each excited atom loses a gravitational potential energy of (m+Äm)gh, where g = 9.8 m/s2. However, for each excited atom thus lowered, there is a ground state atom raised, which gains an amount of energy mgh. The net gain in energy is the difference: (m+Äm)gh mgh = Ämgh. To avoid this net gain (energy for nothing), it must be true that Ämgh is precisely equal to the energy lost by the photon as it travels from the bottom to the top, i.e. ÄE [bottom] ÄE[top] = Ämgh. One may imagine the photon losing energy as it climbs against the Earths gravitational field much like a rock thrown upward loses kinetic energy as it slows down, the main difference being that the photon does not slow down; it always moves at the speed of light. Using Äm = ÄE/c2, the previous equation reads: ÄE ÄE = ÄEgh/c2. Dividing both sides by ÄE and rearranging terms we get:
ÄE / ÄE = 1 gh/c2.
Finally, we also know that the energy of a photon is proportional to its frequency (the proportionality constant being Plancks constant). So we can rewrite our equation as:
f / f = 1 gh/c2,
where f is the frequency of the photon at the bottom and f is the frequency of the photon at the top. Since 1 gh/c2 is less than one it says that the frequency of the photon received at the top (as measured by a clock at the top) is less than the frequency of the photon that left the bottom (as measured by a clock at the bottom).
This is a very strange result. To appreciate why, it will help to switch from the particle picture of light (photons) to the wave picture, and recall that light, which is just electromagnetic waves, can be produced by an oscillating electric charge. Our result says that if Bob is standing on the surface of the Earth holding an electric charge in his hand, and waving his hand back and forth once per second (f = 1), then Alice, standing at the top of a tower of height h, will receive electromagnetic waves of frequency f < 1. Said more directly, wave crests of light leaving Bobs hand once every second (one second according to his wristwatch), will arrive at Alices position at a rate of less than once every second (one second according to her wristwatch). See the figure.
But wouldnt this require the number of wave crests between Bob and Alice to be continually increasing with time? Cant be! With a fixed distance h between the two, such a continual bunching up of wave crests would require the wavelength of the light (distance between crests) to be getting smaller and smaller as time goes on. Since the speed of light is always the same, smaller wavelength means higher frequency, which means Alice would be seeing light of higher and higher frequency as time goes on.
And this is definitely not happening: Alice sees light of constant frequency, f. (Also notice that f < f means the wavelength is longer for Alice than it is for Bob; the wave train stretches out as it moves upwards, as shown in the figure.) But if there is no bunching up of wave crests, how is it possible that Alice sees fewer than one wave crest per second according to her wristwatch (i.e., more than one second of Alice-time elapses between crests), unless time itself is moving more quickly at Alices location than at Bobs? This is the effect of gravity on time! The hands on Alices wristwatch spin around faster than the hands on Bobs wristwatch. Alices heart beats faster than Bobs.
Here is an alternative way to see this. As Bob is waving his hand back and forth, a wave crest leaves the electric charge in his hand each time it reaches the right end of its back and forth arc (see the figure). Any other light illuminating Bobs hand (that allows Alice to see Bob) will travel to Alice at the same speed as the wave crests of light leaving the electric charge in his hand. This means that when Alice looks down at Bob she must see his hand waving back and forth at the same frequency that the crests of light from the electric charge are reaching her, i.e. at the frequency f < 1. Thus, Alice sees Bob waving his hand at a rate of less than once per second according to her wristwatch. Bob appears to be waving his hand in slow motion! Indeed, everything about Bob and his surroundings on the ground will appear to Alice to be happening in slow motion. Conversely, Bob will see Alice in fast motion, like a video on fast forward.
Notice, also, the longer Alice waits at the top of the tower, the greater the difference that will accumulate between the number of seconds (and heart beats) she experiences and the number Bob experiences. Eventually climbing back down from the tower and comparing notes, both will agree that more seconds (and heart beats) have elapsed for Alice than for Bob. Alice will be older than Bob because she has spent time in a place (height h above the Earth) where time itself moves more quickly relative to time on the Earths surface! In the case of Alice going to Jupiter, the gravitational potential will at first increase as she leaves the Earths surface (her time will move more quickly relative to Bobs time, as just discussed), and will continue to increase as she travels away from the Sun towards Jupiter. But eventually the gravitational potential will decrease to a value lower than that on the Earths surface as she descends into Jupiters very strong gravitational field. On Jupiter her time will be moving more slowly relative to Bobs time, as described at the beginning of this essay.
So how big is the effect of gravity on time? Very tiny, at least as far as we are able to experience here on the Earth. For example, taking h = 100 metres gives f / f = 1 gh/c2 0.99999999999999. This means that Bobs clock is running 99.999999999999% as fast as Alices, meaning we would have to wait a very long time for a noticeable difference in age to accumulate. Nevertheless, the effect has been measured using very accurate atomic clocks and the results are in excellent agreement with our formula. Indeed, the Global Positioning System (GPS), which relies on very accurate atomic clocks both on the Earth and carried by satellites high above the Earth, must account for this effect in order to work with the accuracy it does. It should be emphasized, however, that this warping of time, while small anywhere in our solar system, is huge in other more interesting places in our universe. A black hole is an extreme example, where gravity is so strong at the event horizon that time is slowed to a stop relative to anyone outside the horizon! But this is another story
