Gravity as geometry: a short excursion
The clocks on GPS satellites are deliberately built to tick slightly slow before launch: by about 38 millionths of a second per day. Without that correction the error would be the equivalent of kilometres of distance within a day. Why should a perfectly good clock need adjusting just because it is high up and moving fast?
This essay is an excursion. The Dirac essay that follows does not depend on it. It takes one idea from Einstein's general theory of relativity, that gravity is felt only through comparisons between different places, and follows it to a prediction you can check on a satellite: clocks run at different rates at different heights. It stops short of the full theory, which needs mathematics well beyond this book, and it says where the simple argument stops working.
Falling boxes feel no gravity
Everything falls with the same acceleration: a feather and a hammer, in a vacuum, hit the ground together. Einstein took this as the clue. Imagine a closed laboratory falling freely. Everything in it, including you, falls together, so nothing presses on anything: objects let go of hang in mid-air, exactly as they would in a laboratory drifting in empty space far from any star. Conversely, a laboratory in empty space being accelerated upwards by a rocket feels exactly like one standing on the Earth: objects fall to the floor.
Einstein's equivalence principle turns this into a law: in a small enough region, and over a short enough time, the effects of gravity cannot be told apart from the effects of acceleration, by any experiment whatever. The qualification matters, and we will come back to it.
Clocks higher up run faster
Put two clocks in a rocket accelerating upwards at $g$ in empty space, one on the floor and one a height $h$ above it. The floor clock sends a light signal up. The light takes a time $h/c$ to arrive, and meanwhile the rocket has gained speed $u = gh/c$. The upper clock is therefore moving away from where the light was emitted, faster than the rocket was moving at the time, and by the Doppler effect it receives the signal at a frequency lower by the fraction
$$\frac{\Delta\nu}{\nu} = \frac{u}{c} = \frac{gh}{c^2}.$$The upper clock sees the lower one ticking slow. The same happens for a signal sent downwards, reversed: the lower clock sees the upper one ticking fast. By the equivalence principle, the same must hold for two clocks at rest in a gravitational field of strength $g$. A clock lower down in gravity runs slower, by the fraction $gh/c^2$, than one higher up. This is the gravitational redshift: light climbing out of gravity arrives with lower frequency.
The effect is tiny near the Earth: $g/c^2 = 1.1\times10^{-16}$ per metre of height. In 1959–60, Robert Pound and Glen Rebka sent gamma rays 22.5 m up a tower at Harvard and measured the predicted shift, $gh/c^2 = 2.5\times10^{-15}$. Today optical atomic clocks can detect the difference in rate made by raising one of them by a fraction of a metre.
Worked example: GPS
GPS satellites orbit at a radius of about 26,560 km, some 20,200 km above the ground. For height differences that large, $gh/c^2$ is replaced by the difference in gravitational potential divided by $c^2$, which for a sphere of mass $M$ is $(GM/c^2)(1/R_E - 1/r)$, with $GM = 3.986\times10^{14}$ m³/s² for the Earth and $R_E = 6371$ km.
Gravity: $(GM/c^2)(1/R_E - 1/r) = 5.3\times10^{-10}$. Over a day of 86,400 s, the satellite clock gains 45.7 μs on a ground clock.
Motion: the orbital speed is $v = \sqrt{GM/r} = 3.87$ km/s, so the special-relativity slowing is $v^2/2c^2 = 8.3\times10^{-11}$, a loss of 7.2 μs a day.
Net: the satellite clock gains about 38.5 μs per day. GPS works by timing radio signals, which cover $c\times38.5\ \mu\text{s} = 11.5$ km in that time. That is the error a day's uncorrected gain would put into the measured distance to one satellite. It is not by itself the error in a position fix: a receiver combines several satellites and solves for its own clock error at the same time, so how much reaches the position depends on the details. But the satellite clocks would no longer keep the common time on which the system relies. That is why the satellite clocks are set to run slow by this amount before launch. (This simple calculation ignores the Earth's rotation and the slight ellipticity of the orbits, both of which the real system corrects for.)
Check: for a clock in low orbit, the speed term wins. The International Space Station, 420 km up and moving at 7.7 km/s, loses about 24 μs a day on a ground clock, and the two effects cancel at an altitude of about 3,200 km.
Where the equivalence stops: tides
The equivalence principle holds only in a small region, and the reason points to the full theory. Drop two balls side by side, 1 m apart, in a freely falling laboratory. Each falls towards the centre of the Earth, and the two directions are not quite parallel, so the balls drift towards each other: slowly, by about 77 micrometres after ten seconds of falling, but measurably. Two balls one above the other drift apart, because the lower one is pulled slightly harder. No uniform acceleration of the laboratory can produce this squeezing and stretching. These are tidal effects, the same that raise the ocean tides.
So gravity can be removed by free fall only locally, one small region at a time, and the regions do not fit together into one flat frame. Einstein's general theory, completed in 1915, makes this precise: tidal effects are the curvature of spacetime, and matter and energy determine that curvature. Free-falling bodies follow the straightest possible paths through curved spacetime. Uniform gravity, which the equivalence argument handled, is exactly the part that free fall removes; curvature is the part it cannot remove. The full theory predicts more than the redshift: the extra bending of starlight passing the Sun, first measured in 1919; the slow turning of Mercury's orbit; and gravitational waves, detected directly in 2015.
Two questions
1. A mountain observatory. A clock stands on a mountain 3,000 m above sea level. By how much does it gain on a sea-level clock in a year, from gravity alone?
Worked answer
The fractional rate difference is $gh/c^2 = 9.81\times3000/(3.00\times10^8)^2 = 3.3\times10^{-13}$. A year is $3.16\times10^{7}$ s, so the gain is about $1.0\times10^{-5}$ s: ten microseconds a year.
The easy slip is to include the orbital-speed term from the GPS example. The mountain clock is not in orbit; it moves with the Earth like the sea-level clock, so, apart from a tiny difference in rotation speed, only the height matters.
2. Is gravity an illusion? Someone argues: "The equivalence principle shows that gravity can always be made to disappear by falling freely, so gravity is not a real force, only a matter of the frame you choose." What is right and what is wrong?
Worked answer
What is right: the uniform part of gravity in a small region can be removed by free fall, and in that sense the "force" of gravity depends on the frame. What is wrong: the tidal part cannot be removed by any choice of frame. Two falling balls still drift together, whatever you do. That residue is spacetime curvature, and it is as real as anything in physics: it raises the tides and bends starlight.
The tempting wrong turn is to apply the equivalence principle beyond its scope. It is a statement about small regions. The general theory is the account of how the small regions fit together, and the fitting is where gravity lives.
Further reading
- Einstein, Relativity: The Special and the General Theory (1916, tr. Lawson), §XX "The Equality of Inertial and Gravitational Mass as an Argument for the General Postulate of Relativity" (the accelerated chest), §XXIII "Behaviour of Clocks and Measuring-Rods on a Rotating Body of Reference", and Appendix III (c) on the displacement of spectral lines towards the red.
Where this leaves us
A GPS clock needs adjusting because clocks run at different rates depending on where they are and how they move: slower when moving, by special relativity, and slower lower down in gravity, by the equivalence principle. For GPS the gravitational gain wins, by 38.5 μs a day. The equivalence argument explains that much; the tides, which no choice of frame removes, are where the full general theory, and curved spacetime, begin.
General relativity and quantum mechanics have never been fully joined, and that remains one of the open problems of physics. Special relativity and quantum mechanics were joined in 1928, for the electron, by Paul Dirac, using the relation between energy, momentum and mass from the previous essay. The result accounted for the electron's spin and its magnetism, and predicted a particle nobody had yet seen. That is the last optional essay.