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6.6 — The Equivalence Principle
Special relativity has a gaping hole in it, and Einstein saw it immediately.
Newton's gravity is instantaneous. In F = Gm_1m_2/r^2, the distance r is the separation now. Move the Sun and the Earth's orbit changes immediately, eight minutes before anybody could see the Sun move. Chapter 6.5 showed that instantaneous action at a distance across a spacelike separation lets you build a causal loop. So Newtonian gravity is not merely inelegant alongside relativity — it is inconsistent with it.
The obvious repair is to write a field theory of gravity that propagates at c, in the way Maxwell did for electromagnetism. Einstein tried, and it failed, for a reason worth understanding: gravity has energy, energy has mass, mass gravitates. The gravitational field is itself a source of gravity, which makes the equations nonlinear in a way electromagnetism never is. The repair took him ten years.
The way in was a single observation.
The happiest thought
In 1907, sitting in the patent office in Bern, Einstein noticed something that had been in front of physics since Galileo:
"If a person falls freely, he will not feel his own weight."
He later called this the happiest thought of his life. It sounds like nothing. It is the foundation of general relativity.
Think about what it means. A person in free fall — falling off a roof, in a lift with a cut cable, orbiting the Earth — feels no gravity at all. Release an object beside them and it hovers. Everything inside their frame behaves exactly as if gravity had been switched off.
Gravity can be made to vanish by choosing your frame. No other force does that. You cannot get rid of an electric field by falling.
Why the equivalence principle exists at all
The reason is an old coincidence that nobody had explained.
Newton's second law uses inertial mass: how hard a thing is to accelerate.
F = m_i a
Newton's gravity uses gravitational mass: how strongly a thing responds to gravity, the gravitational analogue of electric charge.
F = m_g g
Combine them for a falling object:
a = \frac{m_g}{m_i}g
Everything falls at the same rate if and only if m_g/m_i is the same for every material. And it is, so it does. Galileo established it with inclined planes; Newton tested it with pendulums of different materials; Eötvös in 1908 confirmed it to one part in 10^{9}; modern torsion-balance experiments reach 10^{-13}, and the space mission MICROSCOPE reached 10^{-15} in 2017.
Nothing in Newtonian physics requires this. Charge-to-mass ratio varies wildly between particles; there is no reason gravitational-charge-to-mass should not. It is an unexplained numerical coincidence, tested to fifteen decimal places.
Einstein's move was to stop treating it as a coincidence and make it a principle.
The equivalence principle. A uniform gravitational field is locally indistinguishable from a uniformly accelerating reference frame. No experiment performed in a small enough region can tell them apart.
The two boxes. You wake in a sealed box with no windows and feel a downward force of your usual weight. Two possible explanations:
- The box sits on Earth's surface.
- The box is in deep space, accelerating at 9.81 m/s².
No experiment inside can distinguish them. Drop a ball: it accelerates at 9.81 m/s² downward either way. Weigh yourself: same reading. Watch a pendulum, a spring, a gyroscope, a chemical reaction — everything identical.
The word locally carries weight. In a large enough box you could tell: a real planet's field points towards its centre, so two balls dropped far apart would converge slightly, and one dropped high would fall slightly faster than one dropped low. Those differences are tidal effects, and they are what genuine gravity is, once the part that can be transformed away has been removed. Chapter 6.7 makes this precise: tidal effects are curvature.
Consequence 1: light bends
Take the accelerating box and shine a laser horizontally across it.
In the frame of a distant inertial observer, the light travels in a straight line. But the box is accelerating upwards, so by the time the light reaches the far wall, the wall has moved up. The light strikes lower on the far wall than where it entered.
Inside the box, this is described as: the light beam curved downward.
Now apply the equivalence principle. If the accelerating box is indistinguishable from a box in a gravitational field, then light must bend in a gravitational field too.
Compute the deflection. Box width L, acceleration g. The light crosses in t = L/c, and the box moves up:
\Delta y = \frac{1}{2}gt^2 = \frac{gL^2}{2c^2}
The angle of deflection is:
\theta \approx \frac{d(\Delta y)}{dL}\Big/1 = \frac{gL}{c^2}
For starlight grazing the Sun, integrating this along the path with g varying as GM/r^2 gives:
\theta = \frac{2GM}{c^2R}
\theta = \frac{2(6.674\times10^{-11})(1.989\times10^{30})}{(9\times10^{16})(6.96\times10^{8})} = \frac{2.655\times10^{20}}{6.264\times10^{25}} = 4.24\times10^{-6}\ \text{rad}
Converting to arcseconds: 4.24\times10^{-6}\times206265 = 0.87''.
Einstein published this number in 1911 and it is wrong by a factor of two. The correct value, which he obtained in 1915 with the full theory, is 1.75″.
The missing factor is worth understanding, because it is the whole difference between this chapter and the rest of the Part. The calculation above accounts only for the fact that time runs differently at different heights. The full theory says that space is curved as well, and the curvature of space contributes an exactly equal amount. Half the deflection is time, half is space.
Newtonian gravity applied to a light "particle" also gives 0.87″, which Johann von Soldner computed in 1801. So the 1919 measurement was a genuine three-way test: 0″ if light is unaffected, 0.87″ if Newton, 1.75″ if Einstein.
Arthur Eddington's expedition to the eclipse of 29 May 1919 measured 1.98'' \pm 0.16'' from Príncipe and 1.61'' \pm 0.40'' from Sobral. Consistent with 1.75″, inconsistent with 0.87″. The result made Einstein famous overnight — the London Times of 7 November 1919 ran "Revolution in Science / New Theory of the Universe / Newtonian Ideas Overthrown".
The 1919 error bars were uncomfortably large and the analysis has been argued over ever since. It hardly matters now: radio interferometry measures the deflection to better than 0.02 %, and it agrees with 1.75″.
Consequence 2: time runs slower lower down
This one is derived most cleanly from the Doppler effect, and it does not need any curved-space machinery.
Setup. A rocket of height h accelerates upward at g in deep space. A clock at the bottom emits light pulses at frequency f_0, received by a detector at the top.
The light takes t = h/c to reach the top. In that time, the top of the rocket has gained speed:
\Delta v = gt = \frac{gh}{c}
So the receiver is moving away from where the light was emitted, and sees a Doppler redshift (Chapter 6.3, non-relativistic limit since \Delta v \ll c):
\frac{\Delta f}{f} = -\frac{\Delta v}{c} = -\frac{gh}{c^2}
Now apply equivalence. The same must happen in a gravitational field:
\boxed{\frac{\Delta f}{f} = -\frac{gh}{c^2}}
Light climbing out of a gravitational field is redshifted. And because the receiver counts fewer cycles per second than the emitter produced, and the number of cycles emitted is a physical count that everyone must agree on, the only consistent conclusion is that the lower clock is running slow.
\boxed{\frac{\Delta t_{\text{high}}}{\Delta t_{\text{low}}} = 1 + \frac{gh}{c^2}}
More generally, in terms of gravitational potential \Phi:
\frac{\Delta t_1}{\Delta t_2} = \sqrt{\frac{1+2\Phi_1/c^2}{1+2\Phi_2/c^2}} \approx 1 + \frac{\Phi_1-\Phi_2}{c^2}
Time runs slower where gravity is stronger. Not the clocks — time.
An energy argument that gives the same answer
Here is a second route that some find more convincing.
A photon of energy E = hf has effective mass E/c^2 by E = mc^2. Climbing height h costs gravitational potential energy:
\Delta E = \frac{E}{c^2}gh
So the photon arrives with energy E(1 - gh/c^2), and since E = hf, its frequency has dropped by the same fraction. Same result, and it shows why the effect is unavoidable: a photon cannot slow down to pay the toll, since its speed is fixed, so it must give up frequency instead.
The Pound–Rebka experiment
Robert Pound and Glen Rebka measured gravitational redshift in 1959 in the Jefferson tower at Harvard, over a height of just 22.5 m.
Predicted shift:
\frac{\Delta f}{f} = \frac{gh}{c^2} = \frac{(9.81)(22.5)}{9\times10^{16}} = 2.45\times10^{-15}
Two and a half parts in a thousand million million. Measuring that seems impossible, and the trick that made it possible is the Mössbauer effect, discovered the year before.
Normally a nucleus emitting a gamma ray recoils, and the recoil takes some energy and — worse — varies randomly, smearing the emission line. Mössbauer found that a nucleus locked in a crystal lattice can emit with the whole crystal taking the recoil, which because the crystal is enormously more massive means essentially no recoil at all. The emission line becomes extraordinarily sharp: for iron-57's 14.4 keV gamma ray, a fractional width of about 3\times10^{-13}.
That is still a hundred times wider than the effect. Pound and Rebka got round it by moving the source on a loudspeaker cone, adding a controlled ordinary Doppler shift, and finding the speed that exactly cancelled the gravitational one. The required speed is tiny:
v = \frac{gh}{c} = \frac{(9.81)(22.5)}{3\times10^{8}} = 7.4\times10^{-7}\ \text{m/s}
0.74 micrometres per second — a millimetre every twenty-two minutes.
They measured the shift to 10 % in 1959 and to 1 % in 1964. General relativity confirmed in a lift shaft, using a loudspeaker moving at less than a micrometre per second.
Modern precision
Optical clocks now measure this on a laboratory bench. In 2010, a NIST team compared two aluminium-ion clocks and detected the redshift from raising one by 33 centimetres. In 2018 the effect was measured across a single millimetre-scale cloud of atoms.
A clock on your desk runs measurably faster than one on the floor. Over a lifetime the difference is tens of nanoseconds, and it is real.
The gravitational field of the whole Earth
For a spherical body, the potential is \Phi = -GM/r, so:
\frac{\Delta t_\infty}{\Delta t_r} = \frac{1}{\sqrt{1-\frac{2GM}{rc^2}}}
Expanding for weak fields:
\frac{\Delta t}{\Delta t_\infty} \approx 1 - \frac{GM}{rc^2}
On Earth's surface:
\frac{GM}{Rc^2} = \frac{(6.674\times10^{-11})(5.972\times10^{24})}{(6.371\times10^{6})(9\times10^{16})} = \frac{3.986\times10^{14}}{5.734\times10^{23}} = 6.95\times10^{-10}
Earth's surface runs slow by 7 parts in 10^{10} compared with deep space — about 22 milliseconds per year, or 1.7 seconds over a human lifetime.
On the Sun's surface, the factor is 2.12\times10^{-6}, so solar clocks run slow by about 66 seconds a year. This shifts the Sun's spectral lines measurably, and it was one of the three classical tests Einstein proposed. It is harder to measure than it sounds, because convective motions in the solar photosphere produce Doppler shifts of comparable size, and the clean confirmation came from white dwarfs, which have far stronger fields — Sirius B's redshift was measured in 1925 and matches prediction.
At the surface of a neutron star, 2GM/rc^2 \approx 0.4, so clocks run about 23 % slow. And at a black hole's event horizon it goes to zero, which is what an event horizon is. Chapter 6.9 derives it.
Consequence 3: the equivalence principle is only local
Everything above works in a small box. In a large one, the fiction breaks down, and how it breaks is the clue to the next chapter.
Release two balls side by side in a falling lift, a few metres apart. Both fall towards the Earth's centre, so their paths converge. To an observer in the lift, the balls drift towards each other for no visible reason.
Release two balls one above the other. The lower one is closer to Earth, so it feels slightly stronger gravity and falls slightly faster. To the observer, the balls drift apart.
These relative accelerations cannot be transformed away. They are the difference between a real gravitational field and a uniform acceleration, and they are what physically remains once the equivalence principle has removed everything it can.
They are tidal forces — the same effect that raises the ocean tides (Chapter 1.9), stretching the Earth along the Earth–Moon line and squeezing it across.
The pattern is exactly what Chapter 6.7 will call curvature. Two people who set off north from different points on the equator, each walking in a perfectly straight line, will find themselves drifting together — not because a force is pulling them but because the surface they are walking on is curved. Einstein's proposal is that the drift of freely falling objects is the same phenomenon in spacetime.
What Einstein still needed
The equivalence principle gives three real predictions — light bends, time dilates with height, light redshifts climbing out — and it gave the wrong answer for the first one by a factor of two. It cannot, by itself, produce a complete theory, because:
- It is only local. It says nothing about the field of an extended body.
- It gives no equation relating mass to the field it produces.
- It has no way to describe the tidal effects that survive after the transformation.
What is needed is a mathematics of curvature: a way to describe a space whose geometry varies from place to place, to say what a straight line means in it, and to write down how much curvature a given amount of mass produces.
That mathematics already existed. Gauss, Riemann and others had built it in the nineteenth century as pure geometry, with no thought of physics. Einstein spent from 1912 to 1915 learning it, largely with the help of his friend Marcel Grossmann, and it was the hardest part of the whole enterprise.
Where this shows up in your life
GPS. Satellite clocks run fast by 45 μs/day from the reduced gravity at their altitude, computed exactly as above. Without correction, positions drift by kilometres a day. Chapter 6.9 does the arithmetic.
Anything involving precise time. Financial trading timestamps, the international time standard, and geodesy — measuring the shape of the Earth — all now have to account for the height of the clock.
Chronometric levelling turns the effect into a tool: since clock rate depends on gravitational potential, comparing two optical clocks measures their difference in height, to about a centimetre over any distance. This is becoming a practical surveying method, and it measures the height difference along a geoid rather than a geometrical height, which is what geodesists actually want.
And you are ageing faster than someone at sea level if you live in the mountains — by about 5 microseconds per year per kilometre of altitude. It is entirely real and entirely useless.
What the next chapter fixes
Einstein needed a language for curved space, and Chapter 6.7 builds it from nothing. It starts with a two-dimensional surface you can picture, defines what distance means on it, shows how to say "straight line" without leaving the surface, and works up through the metric tensor, index notation, Christoffel symbols and the Riemann tensor. Every symbol is introduced in words before it is used, and the goal throughout is the single equation in Chapter 6.8 that ties all of it to matter.