---
title: A Taste of General Relativity
module: Foundations of Relativity
moduleNumber: 1
lessonNumber: 5
order: 105
summary: >
  Einstein's happiest thought was that a freely falling observer feels no
  gravity: a uniform gravitational field is locally indistinguishable from an
  accelerating frame. That equivalence principle predicts that light bends near
  a mass, that clocks run slow deep in a gravitational well, that Mercury's
  orbit precesses, and that radar echoes are delayed. Every prediction has been
  confirmed, and pushing the redshift to its limit gives the black hole.
topics: [Foundations of Relativity]
draft: false
sources:
  - book: Tipler & Llewellyn
    ref: "Ch. 2 — Relativity II; §2-5 General Relativity, Principle of Equivalence"
  - book: Tipler & Llewellyn
    ref: "§2-5 Deflection of Light, Gravitational Redshift, Perihelion of Mercury, Black Holes"
---

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.[^tl-gr]

## The equivalence principle

In Newtonian mechanics the mass in $\vec{F} = m_{\text{i}}\vec{a}$ (the
inertial mass, resisting acceleration) and the mass in $\vec{F} = G M
m_{\text{g}}/r^2$ (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 $10^{12}$. Einstein took it as fundamental.

> **Postulate (Principle of equivalence).** A uniform gravitational field is
> completely equivalent to a uniformly accelerated reference frame. No
> experiment performed inside a closed compartment can distinguish standing at
> rest in a gravitational field $g$ from accelerating through empty space at
> $a = g$.

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.

$$
% caption: The equivalence principle. A ball released in a box accelerating
% upward at a (left) falls to the floor exactly as it does in a box at rest in a
% gravitational field g equal to a (right); no internal experiment tells them
% apart.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % left box: accelerating
  \draw[black, thick] (0,0) rectangle (2.6,3.4);
  \draw[->, acc, very thick] (1.3,3.7) -- (1.3,4.5) node[above] {a};
  \fill[black] (1.3,2.2) circle (3pt);
  \draw[->, black] (1.3,2.0) -- (1.3,1.2);
  \node[anchor=north] at (1.3,-0.05) {accelerating box};
  % right box: gravity
  \begin{scope}[xshift=5cm]
    \draw[black, thick] (0,0) rectangle (2.6,3.4);
    \fill[black] (1.3,2.2) circle (3pt);
    \draw[->, black] (1.3,2.0) -- (1.3,1.2);
    \draw[->, acc, very thick] (3.0,3.0) -- (3.0,2.2) node[right, midway] {g};
    \node[anchor=north] at (1.3,-0.05) {box at rest in g};
  \end{scope}
\end{tikzpicture}
$$

## 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,

$$
\d s^2 = c^2\,\d t^2 - (\d r^2 + r^2\,\d\phi^2),
$$

is modified near a spherical nonrotating mass $M$ to

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

The factor $A(r)$ plays a role loosely analogous to the $\gamma$ 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

$$
\alpha = \frac{4GM}{c^2 R}.
$$

For a ray grazing the Sun, $R = R_\odot = 6.96 \times 10^8\ \mathrm{m}$ and $M =
1.99 \times 10^{30}\ \mathrm{kg}$, giving $\alpha = 1.75$ 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.

$$
% caption: Eddington eclipse geometry. Starlight grazing the Sun is bent by
% angle alpha, so the star appears shifted away from the Sun. The bending goes
% as one over R, the closest approach.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % Sun
  \draw[black, thick, fill=black!10] (0,0) circle (0.55);
  \node[anchor=north] at (0,-0.6) {Sun};
  % Earth
  \fill (7.2,0) circle (3pt);
  \node[anchor=north] at (7.2,-0.1) {Earth};
  % true star direction (straight, upper)
  \coordinate (star) at (-1.2,2.3);
  % actual bent path: star -> grazing point -> earth
  \coordinate (graze) at (0,0.75);
  \draw[acc, thick] (star) -- (graze);
  \draw[acc, thick] (graze) -- (7.2,0);
  % apparent straight line back from earth
  \draw[black, dashed] (7.2,0) -- (-1.9,1.9) node[left, black] {apparent position};
  \node[anchor=south east] at (star) {star};
  % closest approach R
  \draw[black, <->] (0.08,0.05) -- (0.02,0.72);
  \node[black, anchor=west] at (0.12,0.4) {R};
  % deflection angle marker near earth
  \node[acc] at (5.4,0.55) {alpha};
\end{tikzpicture}
$$

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 $\Delta t = h/c$ the
detector gains speed $v = a\,\Delta t = gh/c$, so it recedes from the emission
point and measures a Doppler-redshifted frequency,

$$
\frac{\Delta f}{f_0} \approx -\frac{v}{c} = -\frac{gh}{c^2} = -\frac{\Delta\Phi}{c^2},
$$

where $\Delta\Phi = gh$ 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

$$
\frac{f}{f_0} = 1 - \frac{GM}{c^2 R} \quad(\text{redshift, light leaving}),
\qquad
\frac{f}{f_0} = 1 + \frac{GM}{c^2 R} \quad(\text{blueshift, light arriving}).
$$

$$
% caption: A photon climbing out of a gravitational well loses frequency: it is
% blue at the bottom near the mass and redshifted by the time it reaches the top,
% because clocks run slower lower in the well.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % mass at bottom
  \draw[black, thick, fill=black!10] (0,0) circle (0.5);
  \node at (0,0) {M};
  % vertical climb path
  \draw[->, black] (0,0.55) -- (0,4.6) node[above] {height};
  % wave near bottom (short wavelength)
  \draw[thick] (0.9,0.8) sin (1.05,1.0) cos (1.2,0.8) sin (1.35,0.6) cos (1.5,0.8)
    sin (1.65,1.0) cos (1.8,0.8) sin (1.95,0.6) cos (2.1,0.8);
  \node[anchor=west] at (2.3,0.8) {emitted, high f};
  % wave near top (long wavelength)
  \draw[black, thick] (0.9,3.9) sin (1.2,4.15) cos (1.5,3.9) sin (1.8,3.65) cos (2.1,3.9);
  \node[black, anchor=west] at (2.3,3.9) {received, lower f};
  % arrow up
  \draw[->, acc, thick] (0.55,1.4) -- (0.55,3.5) node[midway, right] {photon rises};
\end{tikzpicture}
$$

Pound and Rebka confirmed the shift in 1960 in Earth's own field, measuring the
fractional frequency change of gamma rays falling $h = 22.5\ \mathrm{m}$, a
predicted $gh/c^2 = 2.45 \times 10^{-15}$, 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.

$$
% caption: Perihelion precession. Each orbit advances the point of closest
% approach, so the ellipse slowly rotates and the path traces a rosette. The
% advance is greatly exaggerated here.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % Sun
  \fill[black] (0,0) circle (2.5pt);
  \node[anchor=north east] at (0,0) {Sun};
  % rotating ellipses
  \draw[acc, thick, rotate=0] (0.7,0) ellipse (2.0 and 1.2);
  \draw[black!55, thick, rotate=40] (0.7,0) ellipse (2.0 and 1.2);
  \draw[black!55, thick, rotate=80] (0.7,0) ellipse (2.0 and 1.2);
  % perihelion markers along the rotating major axis
  \fill[black!70] (2.7,0) circle (1.8pt);
  \fill[black!70] (40:2.7) circle (1.8pt);
  \fill[black!70] (80:2.7) circle (1.8pt);
  \node[black!70, anchor=west] at (2.75,0) {perihelion};
  \draw[->, black] (2.9,0.5) arc (10:75:1.4);
  \node[black] at (2.7,2.1) {advance};
\end{tikzpicture}
$$

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**

$$
R_G = \frac{2GM}{c^2}.
$$

> **Definition (Schwarzschild radius).** The radius $R_G = 2GM/c^2$ at which the
> gravitational field of a mass $M$ becomes strong enough that nothing,
> including light, can escape from within it. A mass compressed inside its
> Schwarzschild radius is a black hole; for the Sun, $R_G \approx 3\
> \mathrm{km}$.

Curiously, setting the Newtonian escape speed $v_e = \sqrt{2GM/R}$ 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 | $\alpha = 4GM/c^2 R$ | Eddington eclipse, 1919 |
| Gravitational redshift | $\Delta f/f = GM/c^2 R$ | Pound-Rebka, 1960 |
| Perihelion advance | $43''$/century for Mercury | Einstein, 1916 |
| Radar (Shapiro) delay | light slowed near a mass | Shapiro, 1971 |
| Black hole horizon | $R_G = 2GM/c^2$ | 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](/quantum-mechanics) chapters that come next.

[^tl-gr]: **Tipler & Llewellyn**, _Modern Physics_, §2-5 — General Relativity: the principle of equivalence, curved spacetime and the modified interval, the deflection of light ($\alpha = 4GM/c^2R$, Eddington), gravitational redshift ($\Delta f/f = GM/c^2R$, Pound-Rebka), the perihelion advance of Mercury, the Shapiro delay, and black holes with the Schwarzschild radius $R_G = 2GM/c^2$.