Why general relativity: gravity as geometry

Newton's law of gravitation is one of the most successful ideas in the history of science. With a single inverse-square force it explains falling apples, the tides, the orbits of the planets and the paths of spacecraft that reach Pluto after a nine-year flight. Yet by the early twentieth century physicists knew that it could not be the whole story. It acts instantly across any distance, which clashes with special relativity's rule that no signal travels faster than light. It contains an unexplained coincidence: the mass that measures how hard an object is to accelerate is exactly the mass that measures how strongly gravity pulls on it. And it fails, by a tiny but measurable amount, to predict the orbit of Mercury. General relativity, which Albert Einstein completed in November 1915, resolves all three problems with a single radical idea: gravity is not a force at all but the curvature of spacetime. This course develops that idea from special relativity to black holes, gravitational waves and the expanding universe.

Three cracks in Newtonian gravity

The first crack is instantaneous action at a distance. In Newton's theory the force on the Earth depends on where the Sun is now, not where it was eight minutes ago when the light we see left it. If the Sun were suddenly moved, the Earth would feel the change at once. Special relativity, published by Einstein in 1905, forbids this: the word "now" has no absolute meaning for distant events, because observers in relative motion disagree about which events are simultaneous. A law that depends on the simultaneous positions of two bodies cannot hold in the same form for every inertial observer. Electromagnetism had already shown the way out. In Maxwell's theory a moving charge does not affect a distant charge instantly; the change propagates through the field at the speed of light, as PHYS-382 showed with retarded potentials. A relativistic theory of gravity needs a field of its own, with its own waves.

The second crack is the equality of inertial and gravitational mass. In \(F = m_I a\) the mass \(m_I\) measures inertia. In \(F = G M m_G / r^2\) the mass \(m_G\) measures how strongly the body responds to gravity, rather like a charge. Nothing in Newton's theory requires the two to be equal, yet every experiment finds that they are. Galileo noted that bodies of different weight fall together, Newton tested it with pendulums of different materials, Loránd Eötvös reached about one part in \(10^{9}\) with a torsion balance around 1900, and the MICROSCOPE satellite, which flew from 2016 to 2018, found the ratio equal for titanium and platinum to about one part in \(10^{15}\). Einstein took this equality not as a coincidence but as a clue. If every body falls the same way regardless of what it is made of, then perhaps falling is not something a force does to a body but a property of the stage on which all bodies move.

The third crack is Mercury. Each planet's orbit is an ellipse whose long axis slowly turns, mostly because the other planets tug on it. For Mercury the observed turning of the closest point, the perihelion, is about 574 arcseconds per century. Careful Newtonian calculations, begun by Urbain Le Verrier in 1859, account for about 531 of these. The remaining 43 arcseconds per century, about the width of a coin seen from 100 m, resisted every explanation, including an unseen planet called Vulcan inside Mercury's orbit that astronomers searched for and never found. In lesson 7 we will derive, from the geometry of curved spacetime around the Sun, exactly the missing 43 arcseconds. MATH-430 computed this number from the curvature of a surface; here we will derive it from the orbit equation itself.

Einstein's road from 1907 to 1915

In 1907, while writing a review of special relativity, Einstein had what he later called the happiest thought of his life: a person falling freely from a roof does not feel their own weight. Inside a freely falling box, everything floats; gravity has disappeared locally. Conversely, a box accelerating in empty space reproduces exactly the effects of a gravitational field. This equivalence principle, the subject of lesson 3, immediately predicted that clocks run slower deeper in a gravitational field and that light bends as it passes a massive body. In 1911 Einstein estimated the bending of starlight by the Sun at 0.87 arcseconds, half the value his final theory would give.

The hard part was finding the mathematics. If gravity is geometry, the geometry must be able to vary from place to place and from moment to moment, and the laws of physics must be written in a form that does not depend on the coordinates used to label events. With his friend the mathematician Marcel Grossmann, Einstein learned the differential geometry of Bernhard Riemann, Elwin Bruno Christoffel, Gregorio Ricci-Curbastro and Tullio Levi-Civita, the same tools MATH-430 developed for curved surfaces. After several false starts, including a theory he defended for two years and then abandoned, he presented the final field equations to the Prussian Academy of Sciences on 25 November 1915. A week earlier he had computed the perihelion of Mercury with the nearly finished theory and found 43 arcseconds per century; he wrote that he was beside himself with joy for days. David Hilbert arrived at the same equations independently from a variational principle at almost the same time.

Confirmation followed quickly. Karl Schwarzschild, serving in the German army on the Russian front, found the exact solution for the field outside a spherical mass in December 1915; it is the subject of lesson 6. In 1919 expeditions led by Arthur Eddington and Frank Dyson photographed stars near the Sun during a total eclipse and reported deflections consistent with Einstein's 1.75 arcseconds rather than the Newtonian half value. The result made Einstein world famous. The 1919 measurements were rough, but modern radio measurements confirm the full value to better than one part in ten thousand.

The central idea in one sentence

The physicist John Wheeler summarized general relativity in one sentence: spacetime tells matter how to move, and matter tells spacetime how to curve. The first half is the geodesic principle. A freely falling body, one acted on by no forces other than gravity, follows the straightest possible path through curved spacetime, called a geodesic. Planets orbit the Sun not because a force pulls them sideways but because they move as straight as they can through a spacetime that the Sun has curved. The second half is the Einstein field equation, which relates the curvature of spacetime at each point to the energy, momentum and pressure there. Lessons 4 and 5 make both halves precise.

An analogy helps, as long as it is not taken too literally. On the surface of the Earth, two walkers set off due north from different points on the equator, each walking in a straight line. They start parallel, yet they meet at the North Pole. An observer who did not know the Earth was curved might say that a force pulled them together. In fact each walked a geodesic, a great circle, and their convergence reveals the curvature of the surface. In general relativity, two freely falling bodies released side by side above the Earth also drift together as they fall toward its middle. Their relative acceleration, the tidal effect, is the true signature of curvature, and it cannot be removed by any choice of falling frame.

The analogy has two limits worth stating at once. First, it is spacetime that is curved, not just space: most of the familiar effects of gravity on slow objects come from the way time runs at different rates at different heights, not from any bending of space. Lesson 3 shows that a planet's orbit is mostly a consequence of time running slower closer to the Sun. Second, the familiar picture of a heavy ball on a rubber sheet is misleading, because it uses gravity under the sheet to explain gravity. Curvature in general relativity is intrinsic, measured from inside by the readings of rulers, clocks and freely falling particles, exactly as Gauss's Theorema Egregium in MATH-430 measured the curvature of a surface from within.

When does general relativity matter?

Newtonian gravity is the limit of general relativity when gravitational fields are weak and speeds are small compared with light. A convenient measure of field strength is the dimensionless ratio of gravitational potential energy to rest energy for a test mass at the surface of a body of mass \(M\) and radius \(R\):

This compactness is about \(7 \times 10^{-10}\) at the Earth's surface and \(2 \times 10^{-6}\) at the Sun's. Relativistic corrections to Newtonian predictions are of this order, which is why Mercury's anomaly is only 43 arcseconds per century and why the Global Positioning System must correct its clocks by a few parts in \(10^{10}\). For a white dwarf the ratio is about \(10^{-4}\), for a neutron star about 0.2, and at the horizon of a black hole it reaches one half. Where the ratio approaches unity, Newtonian intuition fails completely and only general relativity can be trusted.

It helps to work in geometrized units, in which \(G = c = 1\). Then mass, length and time are all measured in the same unit. The conversion factors are \(G/c^2 = 7.42 \times 10^{-28}\) m per kg and \(G/c^3 = 4.93 \times 10^{-36}\) s per kg. The Sun's mass becomes 1.48 km or 4.93 microseconds, and the Earth's mass becomes 4.4 mm. These are not arbitrary: half of the Schwarzschild radius \(r_s = 2GM/c^2\) of a body is its mass in geometrized units. The Sun would have to be compressed to a radius of 2.95 km to become a black hole, the Earth to about 9 mm. Throughout the course we will write formulas with \(G\) and \(c\) restored when computing numbers and set them to one when the algebra would otherwise be cluttered.

G, c = 6.674e-11, 2.998e8
Msun = 1.989e30
bodies = [                       # name, mass (kg), radius (m)
    ("Earth",          5.972e24, 6.371e6),
    ("Jupiter",        1.898e27, 6.991e7),
    ("Sun",            Msun,     6.957e8),
    ("white dwarf",    0.6*Msun, 8.7e6),
    ("neutron star",   1.4*Msun, 1.2e4),
    ("Sgr A* horizon", 4.3e6*Msun, 2*G*4.3e6*Msun/c**2),
]
print("body             GM/(Rc^2)     r_s           v_esc/c")
for name, M, R in bodies:
    phi = G*M/(R*c**2)
    rs = 2*G*M/c**2
    vesc = min((2*phi)**0.5, 1.0)
    rs_txt = f"{rs*1e3:9.3f} mm" if rs < 1 else f"{rs/1e3:9.3g} km"
    print(f"{name:15s}  {phi:10.2e}   {rs_txt}   {vesc:6.3f}")
print("Sun's mass in geometrized units:", round(G*Msun/c**2/1e3, 3), "km =",
      round(G*Msun/c**3*1e6, 3), "microseconds")

body             GM/(Rc^2)     r_s           v_esc/c
Earth              6.96e-10       8.869 mm    0.000
Jupiter            2.02e-08     0.00282 km    0.000
Sun                2.12e-06        2.95 km    0.002
white dwarf        1.02e-04        1.77 km    0.014
neutron star       1.72e-01        4.14 km    0.587
Sgr A* horizon     5.00e-01    1.27e+07 km    1.000
Sun's mass in geometrized units: 1.477 km = 4.926 microseconds

The table shows the enormous range of field strengths in nature. At the surface of the Earth the escape speed is about 11 km/s, less than one part in twenty thousand of light speed. At a neutron star the escape speed is about three fifths of light speed; light leaving the surface loses about a fifth of its energy, and the star's own gravity bends the light so strongly that more than half of its surface is visible from one side. At a black hole's horizon the escape speed formally equals the speed of light, which is the Newtonian way of saying that nothing comes out.

What general relativity predicts

Before diving into the mathematics, it is worth listing what the theory predicts and what has been observed. Each item below appears in a later lesson.

It is a remarkable record. A theory built in 1915 from a thought experiment about falling and a demand for mathematical consistency predicted phenomena, such as black holes and gravitational waves, that its author doubted and that took most of a century to observe. Every test so far has agreed with it, from laboratory measurements at the scale of a fraction of a millimeter to the collision of black holes more than a billion light-years away. At the same time, the theory points beyond itself. It predicts singularities where it breaks down, it does not fit with quantum mechanics, and applied to the universe as a whole it requires dark matter and dark energy whose nature is unknown. The capstone returns to these open questions.

Gravity and everyday technology

General relativity is not only a theory of exotic objects. Satellite navigation depends on it every day. A GPS satellite orbits at about 20,200 km altitude, where gravity is weaker, so its clock runs fast compared with a clock on the ground by about 45.7 microseconds per day. Its orbital speed of 3.9 km/s makes the clock run slow, by special relativity, by about 7.2 microseconds per day. The net effect is a gain of about 38.6 microseconds per day. Light travels about 11.6 km in that time, so a navigation system that ignored relativity would accumulate errors of kilometers within a day. The satellite clocks are deliberately set to tick slightly slower before launch so that they keep time with the ground. Lesson 3 computes both effects from first principles; MATH-430's capstone computed the same number from a different direction.

Precise timing goes further. Optical atomic clocks now compare frequencies to about one part in \(10^{18}\), enough to detect the gravitational slowing of time between two clocks whose heights differ by a few centimeters. In 2010 a group at the US National Institute of Standards and Technology demonstrated exactly this with aluminum ion clocks separated by 33 cm vertically. Clocks of this quality are now being used as gravity sensors, mapping the shape of the geoid. The equivalence principle that Einstein found in a thought experiment has become a surveying tool.