A Taste of General Relativity

Special relativity treats inertial frames. Gravity resists that framing, because a gravitational field cannot be transformed away everywhere by moving to a frame in uniform motion. Einstein's route to including gravity was the observation that it can be transformed away locally: an observer in free fall feels no gravity at all. General relativity, published in 1916, builds on that single idea and predicts effects that Newtonian gravity does not.¹

The equivalence principle

In Newtonian mechanics the mass in $F = m_{i} a$ (the inertial mass, resisting acceleration) and the mass in $F = GM m_{g} / r^{2}$ (the gravitational mass, sourcing weight) are numerically equal, which is why all bodies fall with the same acceleration $g$ regardless of mass. Classical theory offers no reason for the equality; experiment confirms it to better than one part in $1 0^{12}$ . Einstein took it as fundamental.

A person in a windowless box accelerating at $a$ in deep space feels a floor push identical to weight, and dropped objects fall with acceleration $a$ ; the same person standing in a gravitational field $g = a$ has identical experiences. The equivalence makes the equality of inertial and gravitational mass a requirement rather than a coincidence, and it extends the principle of relativity to accelerated frames: there is no absolute acceleration any more than there is absolute velocity.

The equivalence principle. A ball released in a box accelerating upward at a (left) falls to the floor exactly as it does in a box at rest in a gravitational field g equal to a (right); no internal experiment tells them apart.

Curved spacetime

Applying the equivalence principle to light already shows that gravity bends light: in an accelerating box, a horizontal light beam entering one wall strikes the far wall slightly lower, tracing a parabola, so by equivalence light must fall in a gravitational field. Encoding this consistently requires that mass alters the geometry of spacetime itself. The flat interval of special relativity, written in polar coordinates for motion in a plane,

d s^{2} = c^{2} d t^{2} - (d r^{2} + r^{2} d ϕ^{2}),

is modified near a spherical nonrotating mass $M$ to

d s^{2} = A^{2} c^{2} d t^{2} - \frac{d r ^{2}}{A ^{2}} - r^{2} d ϕ^{2}, A (r) = 1 - \frac{2 GM}{c ^{2} r} .

The factor $A (r)$ plays a role loosely analogous to the $γ$ of special relativity: the $d t$ term describes gravitational time dilation and the $d r$ term gravitational length contraction. Because $A < 1$ near the mass, light effectively slows to $A c$ there and the wavefronts bend toward $M$ , exactly as light bends toward regions of higher refractive index. The full theory replaces the gravitational force with this curvature, along which free particles and light travel the straightest available paths.

Deflection of light

Integrating the light's path through the modified geometry as it grazes a mass $M$ at closest approach $R$ gives the total deflection angle

α = \frac{4 GM}{c ^{2} R} .

For a ray grazing the Sun, $R = R_{⊙} = 6.96 \times 1 0^{8} m$ and $M = 1.99 \times 1 0^{30} kg$ , giving $α = 1.75$ arc seconds. The effect is invisible against the Sun's glare except during a total eclipse, when stars near the limb can be photographed and their apparent positions compared with their true ones.

Eddington eclipse geometry. Starlight grazing the Sun is bent by angle alpha, so the star appears shifted away from the Sun. The bending goes as one over R, the closest approach.

Eddington's 1919 expeditions measured the deflection during a solar eclipse from two sites, obtaining $1.98 \pm 0.12$ and $1.61 \pm 0.30$ arc seconds, averaging to the predicted $1.75$ and confirming the $1/ R$ dependence. Modern radio measurements, needing no eclipse, agree with general relativity to about $0.1$ percent. The same bending focuses light from distant galaxies through intervening masses, the gravitational lensing now used to map dark matter.

Gravitational redshift

The equivalence principle predicts that clocks run slow deep in a gravitational well. Consider light rising a height $h$ from source $A$ to detector $B$ in a box accelerating at $a = g$ . During the light's travel time $Δ t = h / c$ the detector gains speed $v = a Δ t = g h / c$ , so it recedes from the emission point and measures a Doppler-redshifted frequency,

\frac{Δ f}{f _{0}} \approx - \frac{v}{c} = - \frac{g h}{c ^{2}} = - \frac{ΔΦ}{c ^{2}},

where $ΔΦ = g h$ is the gravitational potential difference between $A$ and $B$ . By equivalence the same shift must occur in a real gravitational field at rest, where no Doppler effect exists, so it can only mean the clock at the lower potential $A$ genuinely runs slow relative to the one at $B$ . For a spherical mass the general result is

\frac{f}{f _{0}} = 1 - \frac{GM}{c ^{2} R} (redshift, light leaving), \frac{f}{f _{0}} = 1 + \frac{GM}{c ^{2} R} (blueshift, light arriving) .

A photon climbing out of a gravitational well loses frequency: it is blue at the bottom near the mass and redshifted by the time it reaches the top, because clocks run slower lower in the well.

Pound and Rebka confirmed the shift in 1960 in Earth's own field, measuring the fractional frequency change of gamma rays falling $h = 22.5 m$ , a predicted $g h / c^{2} = 2.45 \times 1 0^{- 15}$ , to within one percent using the Mössbauer effect. Atomic clocks flown on aircraft and satellites confirm it at larger scales, and the Global Positioning System must correct for it to keep time.

Precession of Mercury's orbit

General relativity's first success was Mercury. Newtonian gravity predicts a closed elliptical orbit; the observed orbit does not close, its perihelion advancing by an anomalous $43$ arc seconds per century after all known Newtonian perturbations are removed. Einstein's 1916 paper computed exactly that advance from the curved-spacetime correction to the orbit, with no adjustable parameters. The orbit traces a slowly rotating rosette rather than a fixed ellipse.

Perihelion precession. Each orbit advances the point of closest approach, so the ellipse slowly rotates and the path traces a rosette. The advance is greatly exaggerated here.

A related prediction, the delay of light passing through a gravitational field, was confirmed by Shapiro in 1971 using radar echoes from planets: signals grazing the Sun return measurably late because light slows in the curved geometry near the mass.

Black holes

Pushing the gravitational redshift to its limit gives an object from which light cannot escape. From $f / f_{0} = 1 - GM / c^{2} R$ , the emitted frequency redshifts all the way to zero when the radius shrinks to $R = GM / c^{2}$ . That estimate uses the low-field approximation; the exact treatment gives the Schwarzschild radius

R_{G} = \frac{2 GM}{c ^{2}} .

Curiously, setting the Newtonian escape speed $v_{e} = 2 GM / R$ equal to $c$ gives the same $R_{G}$ ; Laplace obtained it in the eighteenth century through two canceling errors. Astronomers have since identified many black holes, including one at the center of the Milky Way.

The theory also predicts gravitational waves, ripples in spacetime radiated by accelerating masses and propagating at $c$ . The gradual decay of the Hulse-Taylor binary pulsar's orbit matched the predicted energy loss to gravitational radiation, and interferometers of the Michelson type, scaled to kilometer arms, now detect the waves directly.

Prediction	Formula	Test
Light deflection	$α = 4 GM / c^{2} R$	Eddington eclipse, 1919
Gravitational redshift	$Δ f / f = GM / c^{2} R$	Pound-Rebka, 1960
Perihelion advance	$4 3^{''}$ /century for Mercury	Einstein, 1916
Radar (Shapiro) delay	light slowed near a mass	Shapiro, 1971
Black hole horizon	$R_{G} = 2 GM / c^{2}$	Milky Way center and others
Gravitational waves	ripples at speed $c$	Hulse-Taylor pulsar; interferometers

General relativity closes the relativity module. Its curved spacetime governs astrophysics and cosmology, the final destinations of the modern-physics sequence, while the special theory of the earlier lessons underlies every high-energy process in the quantum chapters that come next.

Tipler & Llewellyn, Modern Physics, §2-5 — General Relativity: the principle of equivalence, curved spacetime and the modified interval, the deflection of light ( $α = 4 GM / c^{2} R$ , Eddington), gravitational redshift ( $Δ f / f = GM / c^{2} R$ , Pound-Rebka), the perihelion advance of Mercury, the Shapiro delay, and black holes with the Schwarzschild radius $R_{G} = 2 GM / c^{2}$ . ↩