Gravity as Curved Spacetime

Beginner

Drop a hammer and a feather where there is no air, as an astronaut once did on the Moon, and they land together. Every object falls with the same acceleration, whatever its mass or material. Einstein, in 1915, took it as a clue: if everything falls alike, perhaps falling is not something done to objects but a property of space and time themselves.

A spacecraft in orbit is not beyond gravity: it falls around the Earth all the time, and the crew and their pencils fall with it, so they float. Now picture a closed cabin. If it falls freely, you float as if gravity had vanished. If it is a rocket speeding up far from any planet, you are pressed to the floor as if by gravity. From inside you cannot tell the difference: this is the equivalence principle.

Einstein's answer is general relativity: mass and energy curve space and time, and a body left alone follows the straightest path it can through that curved spacetime. The rubber sheet in the animation pictures this: the star dents the grid, the blue ball keeps circling the dent, and the gold light ray bends away from the dashed straight line.

Time is curved too. The slow clock on the star turns three times while the fast clock far away turns four. On Earth the effect is tiny, but GPS satellites must correct for it, or positions would drift by about 10 km a day. Pushed to extremes, the theory predicts black holes, which trap light, and gravitational waves, first detected in 2015.

Check yourself: an astronaut floats inside the Space Station. Is there no gravity up there? Answer: there is, almost as much as on the ground. She floats because she falls together with the station.

Intermediate

What changes, and what we measure

Every body falls with the same acceleration, whatever its mass or material: experiments confirm it to about one part in $10^{15}$. Einstein's general relativity (1915) takes this literally: if the motion does not depend on the body, it belongs to space and time. Mass and energy curve spacetime, and a freely falling body, feeling no force, follows the straightest path it can through it. Standing on the ground, you are the exception: the floor pushes you off your free-fall path, and you feel that push as weight.

Two kinds of time appear in the formulas. The proper time $\tau$ is what a clock shows at its own place, and every process there keeps step with it. The far-away time $t$ is the reading of a clock at rest far from all masses, a common reference. The distance $r$ from the centre of a mass $M$ is defined by circumference, $r=C/2\pi$, because in curved space a ruler laid from the centre gives slightly more. The constants $G$ and $c$ enter through $GM/c^2$, which is a length.

The Schwarzschild radius sets the scale

Twice that length is the Schwarzschild radius,
$r_s=\frac{2GM}{c^2}$

It is 2.95 km for the Sun and 8.9 mm for the Earth, growing in proportion to the mass: 29.5 km for a black hole of 10 solar masses. A body squeezed inside its own $r_s$ lets nothing out again, not even light, and $r_s$ becomes the radius of its event horizon. For the Sun, whose radius is 696,000 km, the 2.95 km is only a scale, not a surface.

The ratio $r_s/r$ measures how strong gravity is at a distance $r$: $1.4\times 10^{-9}$ at the Earth's surface, $4.2\times 10^{-6}$ at the Sun's, about 0.34 on a neutron star and 1 at a horizon. Every relativistic correction is of this order, which is why Newton's theory serves so well in the Solar System.

The working formulas

Four formulas answer most practical questions. The first is exact; the other three hold only in weak gravity, and the third only for speeds far below $c$:

• $\tau =t\sqrt{1-\frac{r_s}{r}}$
• $\frac{\Delta \tau }{\tau }\approx \frac{gh}{c^2}$
• $\frac{\Delta \tau }{\tau }\approx -\frac{v^2}{2c^2}$
• $\delta =\frac{4GM}{c^{2}b}$

The first gives the rate of a clock at rest at distance $r$, compared with far-away time; it falls to zero at the horizon. For small $r_s/r$ the square root is close to $1-GM/(r{c}^2)$, so near a planet a clock at $r+h$ gains on one at $r$ the fraction $GMh/(r^2{c}^2)=gh/c^2$, the second formula. A clock 1 m higher runs fast by $1.1\times 10^{-16}$; 1 km higher, it gains 9.4 ns a day.

The third is the slowing of moving clocks from Special Relativity. The fractional rates add with their signs: height makes a clock gain, speed makes it lose. The fourth gives the bending of a ray whose straight-line path would pass the centre at distance $b$, the impact parameter; with $r_s$ it reads $\delta =2r_s/b$, in radians. Grazing the Sun that is 1.75″, as measured during the eclipse of 1919. Newton's law, applied to a particle at $c$, gives half as much; the other half comes from curved space.

Worked example: the animation's clocks and ray

The animation exaggerates: its star's radius is only $1.8r_s$. The star clock stands at $r_1=2.16r_s$ and the far clock at $r_2=22r_s$, so their rates are
$\sqrt{1-1/2.16}=\sqrt{0.537}\approx 0.733$
$\sqrt{1-1/22}=\sqrt{0.955}\approx 0.977$

The ratio is $0.733/0.977=0.750$, exactly 3/4. In one loop the far-away time advances by $12.0/0.977=12.28\mathrm{s}$, so the far clock shows ${\tau }_2=12.0\mathrm{s}$ and the star clock ${\tau }_1=0.733\cdot 12.28=9.0\mathrm{s}$: the final values of the red ${\tau }_1$ and green ${\tau }_2$ readouts. Each hand turns once per 3 s of its own time: three turns on the star, four far away.

The gold ray passes at $b=8r_s$. The weak-field formula gives $\delta \approx 2/8=0.25\mathrm{rad}$, or 14.3°, but the exact path drawn bends by 17.8°: this close to the star the formula falls short. The blue ball circles at $r=10.4r_s$. Since $GM=r_s{c}^2/2$, its speed $\sqrt{GM/r}$ is $c/\sqrt{20.8}\approx 0.22c$, Newton's circular-orbit value, exact here when time is counted in far-away seconds.

Worked example: GPS clocks

Navigation satellites orbit 20,200 km above the ground, at $r=6371+20200=26571\mathrm{km}$ from the Earth's centre, while the ground clock sits at the Earth's radius $R=6371\mathrm{km}$. For the Earth $GM/c^2=4.435\mathrm{mm}$. Comparing the rates at the two radii, the satellite clock gains on a ground clock the fraction
$\frac{GM}{c^2}\left(\frac{1}{R}-\frac{1}{r}\right)=4.435\mathrm{mm}\times \left(\frac{1}{6371\mathrm{km}}-\frac{1}{26571\mathrm{km}}\right)\approx 5.29\times 10^{-10}$
which over a day of 86,400 s adds up to $+45.7\mu \mathrm{s}$.

The satellite also moves at $v=\sqrt{GM/r}=\sqrt{3.986\times 10^{14}/(2.657\times 10^7)}\approx 3.87km/s$, which slows its clock by $v^2/(2c^2)\approx 8.3\times 10^{-11}$, or $-7.2\mu \mathrm{s}$ a day. The net effect is $+45.7-7.2=+38.5\mu \mathrm{s}$ a day. Positions come from signal travel times, and light covers $c\cdot 38.5\mu \mathrm{s}\approx 11.5\mathrm{km}$ in that time, so uncorrected fixes would drift by kilometres within a day. The satellite clocks are therefore built to tick slow by about $4.5\times 10^{-10}$.

Reading the rubber-sheet picture

The grid is the orbital plane, and the dent's depth is drawn in proportion to the Newtonian potential $\Phi =-GM/r$ (smoothed near the star). The depth also shows how much clocks slow down, since in weak fields $d\tau /dt\approx 1+\Phi /c^2$. The dashed grey line is light's path without the star, the black bar marks $b$, and $\delta$ labels the angle between that line and the gold ray. The expert view adds Newton's prediction as a dashed purple path.

Read the sheet as a picture, not as the theory. A ball rolls into a real dent only because the Earth's gravity pulls it down, so the analogy uses gravity to explain gravity. It also shows only curved space, while for slow bodies nearly all of gravity comes from the curving of time; curved space matters mainly for fast bodies such as light.

Common mistakes

• Expecting a slow clock to feel slow. Beside the star clock at $2.16r_s$ your watch, heart and atoms all run normally; only a comparison shows that you age 9 years while a far observer ages 12.
• Getting the GPS signs wrong. Weaker gravity makes the satellite clock gain 45.7 µs a day and speed makes it lose 7.2 µs, so the net is +38.5 µs, not 52.9 µs.
• Using $gh/c^2$ for large heights. For GPS it gives $9.81\cdot 2.02\times 10^7/c^2\approx 2.2\times 10^{-9}$, four times too much, because $g$ weakens with height; use $GM/c^2(1/R-1/r)$ instead.
• Treating $r_s$ as the size of a body, or a black hole as a vacuum cleaner. The Sun's 2.95 km lies deep inside it, and a black hole of the Sun's mass would leave the Earth's orbit unchanged, because outside the mass only $M$ matters.

Where you meet it

Every satnav fix depends on these corrections, and optical atomic clocks now notice a lift of a few centimetres. Gravitational lensing magnifies distant galaxies and maps invisible dark matter. Black-hole shadows have been imaged since 2019, and gravitational waves from merging black holes have been recorded since 2015.

This lesson joins Special Relativity, which supplies proper time and the speed term, to Gravitation and Orbits, its weak-field limit. And because energy has mass, as Fission, Fusion and E = mc² shows, every form of energy curves spacetime, light and pressure included.

Advanced

Measuring curvature from inside

How could beings who live inside a space tell that it is curved, without stepping outside it? They can measure. Draw a sphere around some matter and measure its surface area $A$. In flat space the area fixes the radius: $A=4\pi r^2$, so the radius must be $\sqrt{A/4\pi }$. Now measure the radius directly, by laying rulers end to end from the centre out to the sphere. If the rulers read more than $\sqrt{A/4\pi }$, space is curved, and the difference, called the excess radius, measures by how much.

Legend

$M$	Mass inside the sphere.	$\mathrm{kg}$
$A$	Area of the sphere that the ring stands for.	${\mathrm{m}}^2$
$\sqrt{A/4\pi }$	Radius that flat space would give for the area $A$.	$\mathrm{m}$
$r$	Radius measured with rulers laid from $M$ along the sheet to the ring.	$\mathrm{m}$
$\Delta r$	Excess radius, $r-\sqrt{A/4\pi }$; for matter of roughly uniform density it is $GM/(3c^2)$.	$\mathrm{m}$

Figure 1 shows the same test one dimension lower, so that it can be drawn. The curved sheet is the space, the ring stands for the sphere of area $A$, and the mass $M$ sits at the bottom of the dip. The sheet is drawn as part of a sphere, because that is the shape space takes inside matter of uniform density.

Look first at the dashed line: it is the horizontal distance from the axis to the ring, the radius $\sqrt{A/4\pi }$ that flat space would give for a ring of this size. Now follow the curve marked $r$: a ruler laid from $M$ to the ring cannot cut across the dip, it has to follow the slope of the sheet, so it reads a longer distance. The bar underneath sets the two lengths side by side, $r=\sqrt{A/4\pi }+\Delta r$. The extra piece $\Delta r$ is the excess radius that beings living in the sheet would measure.

Einstein's law of gravitation, in its simplest form, puts a number on this excess. Take a small sphere around matter of roughly uniform density, with mass $M$ inside it. Then the measured radius exceeds the flat-space value by
$\Delta r=r-\sqrt{A/4\pi }=\frac{GM}{3c^2}$

Here $G$ is Newton's gravitational constant and $c$ the speed of light. Because $G/c^2$ is so small, the effect is tiny. For the Earth, treated as uniform, $GM/c^2=4.435\mathrm{mm}$, so the excess radius is only about 1.5 mm; for Jupiter it is 0.47 m. No survey could detect these. The power of the rule lies in its generality.

The full law demands the same for every small sphere, around every event (a point in space at one instant) and for every observer, however they move. Writing all of these conditions at once needs three ingredients. The metric is the table that turns steps in the coordinates into the times and lengths that clocks and rulers actually measure. The Einstein tensor $G_{\mu \nu }$ collects the curvature; it is built from second derivatives of the metric, just as the bending of a curve comes from the second derivative of its graph. The stress–energy tensor $T_{\mu \nu }$ collects the density and flow of energy and momentum. The law then reads
$G_{\mu \nu }=\frac{8\pi G}{c^4}{T}_{\mu \nu }$

Curvature stands on the left, matter on the right. The coupling $8\pi G/c^4\approx 2.1\times 10^{-43}{\mathrm{N}}^{-1}$ is extremely small, so spacetime is very stiff: enormous energy densities produce only slight curvature. In empty space $T_{\mu \nu }=0$, so $G_{\mu \nu }=0$, yet spacetime there is not flat, because $G_{\mu \nu }$ holds only part of the curvature. The part that remains is the tidal field, which stretches a falling cloud of dust along the line towards the mass and squeezes it sideways.

Clock rates from the equivalence principle

Gravity does not only curve space; it also changes the rate of time, so that clocks higher up run faster. This follows from the equivalence principle alone: inside a small closed room, no experiment can tell a uniform gravitational field from a steady acceleration.

Legend

$g$	Acceleration of the rocket (left); strength of the gravitational field on the ground, pointing down (right).	$m/{\mathrm{s}}^2$
$c$	Speed of the light pulse sent from the floor clock to the ceiling clock.	$m/s$
$h$	Height of the cabin: the distance between the two clocks.	$\mathrm{m}$
$gh/c$	Speed the ceiling gains while the pulse climbs; it sets the Doppler shift $gh/c^2$.	$m/s$

Figure 2 tells the argument twice. Start on the left. Far from any planet, a rocket cabin of height $h$ accelerates upwards at $g$, with one clock on the floor and one on the ceiling. The floor clock sends a light pulse up at speed $c$. The pulse needs a time of about $h/c$ to arrive, and during that time the ceiling speeds up by $g\cdot (h/c)$. So when it receives the pulse, the ceiling is moving away faster than the floor was when it sent it, by the speed $gh/c$ (the short arrow on top). A receiver moving away sees light Doppler-shifted to a lower frequency, by the fraction (speed)/$c$, here $gh/c^2$.

Suppose the floor clock sends one pulse per tick. No pulses are lost on the way, yet the ceiling receives them at a rate lower than its own clock ticks, by the fraction $gh/c^2$. Pulses sent down from the ceiling arrive at a higher frequency, because the floor is moving towards them. So everyone aboard agrees: the ceiling clock runs fast and the floor clock slow, by the fraction $gh/c^2$.

On the right the same cabin stands on the ground, in a gravitational field $g$ pointing down. By the equivalence principle nothing inside can tell this room from the rocket, so the result must be the same: in a gravitational field, a clock higher by $h$ runs fast by $gh/c^2$.

Near the ground $gh$ is the rise in the gravitational potential $\Phi$, the potential energy per unit mass. The rule therefore holds for any static potential, taken as zero far away. While a clock far away records a time $dt$, a clock at rest at potential $\Phi$ records its own, proper, time $d\tau$, and
$\frac{d\tau }{dt}\approx 1+\frac{\Phi }{c^2}$

Since $\Phi$ is negative near a mass, clocks there run slow. Around a mass $M$, $\Phi =-GM/r$, and the rule is the first-order form of the exact rate found below, $\sqrt{1-r_s/r}\approx 1-GM/(r{c}^2)$, where $r_s=2GM/c^2$ is the Schwarzschild radius of the mass.

Energy conservation gives the same answer. Light of energy $E$ has mass $E/c^2$, so in falling through a height $h$ it gains the energy $Egh/c^2$, the fraction $gh/c^2$ of its energy; climbing, it loses that fraction. A photon's energy is proportional to its frequency, so its frequency drops by the same fraction.

Pound and Rebka measured the shift in 1960 over a 22.5 m tower: a fraction of only $2.5\times 10^{-15}$. In 1976 a hydrogen maser clock flown to about 10,000 km confirmed it far more precisely.

The orbiting-clock graph

Satellites carry clocks, and for them both effects matter: height makes a clock run fast, speed makes it run slow. The expert graph applies both to a clock in a circular orbit of radius $r$, measured from the Earth's centre, compared with a clock on the ground at the Earth's radius $R$.

Height: the potential rises from $-GM/R$ on the ground to $-GM/r$ in orbit, so by the rule above the orbiting clock gains the fraction $GM/c^2(1/R-1/r)$ of every second (green curve). Speed: Special Relativity slows a moving clock by the factor $\sqrt{1-v^2/c^2}\approx 1-v^2/(2c^2)$, and in a circular orbit $v^2=GM/r$, so the clock loses $GM/(2r{c}^2)$ of every second (blue curve). The total (red curve) is
$\frac{GM}{c^2}\left(\frac{1}{R}-\frac{3}{2r}\right)$

The graph multiplies these fractions by the 86,400 seconds of a day. A grazing orbit, at $r=R$, gains nothing from height and loses 30.1 µs a day, all of it from speed. With height the green gain rises towards 60.1 µs a day, its value for a clock infinitely far away, while the blue loss shrinks as $1/r$. The red curve crosses zero where $1/R=3/(2r)$, that is at $r=1.5R$, about 3,190 km up: lower orbits lose time, higher ones gain.

Particular heights can be read off the red curve. The Space Station, 400 km up, loses 24.7 µs a day, so six months aboard leave an astronaut about 4.5 ms younger. GPS clocks, about 20,200 km up, gain 38.5 µs a day, and the right edge of the graph, at 25,000 km, reaches 41.8 µs.

Free fall as the longest proper time

Newton describes falling with forces. General Relativity replaces them with one rule about clocks: between two events, a free body follows the worldline (its path through space and time) along which its own clock records the most time. In flat spacetime that worldline is straight, at constant velocity; this is why the travelling twin of Special Relativity, who turns round, comes home younger.

Near the ground two effects compete, as in the last section: height makes a clock run fast by $gh/c^2$, and speed makes it run slow by $v^2/(2c^2)$. So along a path of height $h(t)$ and speed $v(t)$ the proper time is
$\tau =\int \left(1+\frac{gh}{c^2}-\frac{v^2}{2c^2}\right)dt$

Height pays and speed costs. The best path climbs high enough to gain time, but not so fast that its speed eats the gain.

Legend

$t$	Time since the throw.	$\mathrm{s}$
$h$	Height of the ball above the ground.	$\mathrm{m}$
$T$	Time from the throw to the catch.	$\mathrm{s}$
$H_1$	Peak height of a low path, $H_2/2$.	$\mathrm{m}$
$H_2$	Peak height of the free-fall path, $g{T}^2/8$.	$\mathrm{m}$
$H_3$	Peak height of a high path, $2H_2=g{T}^2/4$.	$\mathrm{m}$
$H$	Peak height of a path.	$\mathrm{m}$
$\Delta \tau$	Proper time a clock on the path gains over a clock left on the ground.	$\mathrm{s}$

Figure 3 tests the rule on a ball thrown up and caught a time $T$ later. The left panel shows three paths between the same two events, all arches $h=4H(t/T)(1-t/T)$ that differ only in their peak height $H$: a low one, $H_1$; the path of free fall, the one the ball actually follows, $H_2$; and a high one, $H_3$. Which gives a clock carried along it the most time?

Compute the time $\Delta \tau$ that the travelling clock gains on a clock left on the ground by averaging the two terms over the flight. The average height of such an arch is two-thirds of its peak, $2H/3$. The speed is $v=dh/dt=(4H/T)(1-2t/T)$, and the average of $(1-2t/T{)}^2$ over the flight is $1/3$, so the average of $v^2/2$ is $8H^2/(3T^2)$. Multiplying each average by the flight time $T$ and dividing by $c^2$ gives
$\Delta \tau =\frac{T}{c^2}\left(\frac{2}{3}gH-\frac{8H^2}{3T^2}\right)$

The right panel plots this gain against $H$: an upside-down parabola. It peaks where its slope, $2g/3-16H/(3T^2)$, is zero, at $H_2=g{T}^2/8$. That is exactly the height of free fall: a ball that rises for $T/2$ under gravity climbs $\frac{1}{2}g(T/2{)}^2=g{T}^2/8$. For $T=3\mathrm{s}$ the peak is 11.0 m and the gain is $g^2{T}^3/(24c^2)\approx 1.2\times 10^{-15}\mathrm{s}$. The low path $H_1$, half as high, gains three-quarters of that. The path $H_3=g{T}^2/4$, twice as high, gains nothing: its extra speed cancels exactly the benefit of height.

The same works for any potential, with $gh$ replaced by $\Phi$. Multiply the integrand by $-m{c}^2$: the integral becomes $\int (-m{c}^2+\frac{1}{2}m{v}^2-m\Phi )dt$. The first term adds the same amount for every path, so making $\tau$ as large as possible is the same as making $\int (\frac{1}{2}m{v}^2-m\Phi )dt$ as small as possible. That is the principle of least action of Newtonian mechanics, whose equation of motion is $\vec{a}=-\nabla \Phi$. The mass $m$ cancels from it, so every body falls the same way: the universality of free fall is built into the geometry. Strictly, $\tau$ is a maximum only against nearby paths.

The Schwarzschild geometry

The rules so far are weak-field approximations. Outside a spherical, non-rotating mass, such as a star or a black hole, the field equation has an exact solution, found by Karl Schwarzschild in 1916. It is a metric: it gives the proper time $d\tau$ between two nearby events from their coordinate separations,
$c^{2}d{\tau }^2=\left(1-\frac{r_s}{r}\right)c^{2}d{t}^2-\frac{d{r}^2}{1-r_s/r}-r^{2}d{\Omega }^2$

Here $t$ is the time of a clock far away, $r$ is chosen so that a circle around the centre has circumference $2\pi r$, $d\Omega$ is the small angle between the two events as seen from the centre, and $r_s=2GM/c^2$. Two readings follow at once. For a clock at rest, $dr=d\Omega =0$, and the metric gives $d\tau =\sqrt{1-r_s/r}dt$: the exact clock rate. For a ruler laid along the radius, $dt=d\Omega =0$, and the length it measures is $dr/\sqrt{1-r_s/r}$, more than $dr$. This stretching of radial distances is the excess radius of the first section.

Bending light

Light passing a mass is bent, and the size of the bend sets General Relativity apart from Newton's theory: Newtonian reasoning gives only half of what is observed.

Legend

$M$	Mass that bends the ray.	$\mathrm{kg}$
$b$	Impact parameter: the distance from $M$ to the incoming line of the ray.	$\mathrm{m}$
$c$	Speed of light, along the ray.	$m/s$
$\delta$	Deflection in General Relativity, $4GM/(c^{2}b)=2{\delta }_N$.	$\mathrm{rad}$
${\delta }_N$	Deflection from the time part alone, as in Newton's theory: $2GM/(c^{2}b)$.	$\mathrm{rad}$

Figure 4 follows a ray of light past the mass $M$. Far to the left the ray travels in a straight line at speed $c$; the impact parameter $b$ is the distance from $M$ to that incoming line. Far to the right both drawn paths are straight again, along the dashed lines, but turned: one through the angle ${\delta }_N$, the other through $\delta$, twice as much. The rest of this section explains the factor of two.

Light moves with $d\tau =0$. Describe its path by $u=1/r$ as a function of the angle $\phi$ around the mass, with a prime meaning $d/d\phi$. Setting $d\tau =0$ in the Schwarzschild metric and using the conserved energy and angular momentum gives
$u^{\prime \prime }+u=\frac{3GM}{c^2}{u}^2$

Without the right-hand side the solution is a straight line; the small term on the right bends it. This is the equation the animation integrates for its gold ray.

To see the factor of two, look at the metric in a weak field. There the factor in front of $c^{2}d{t}^2$ is $g_{tt}\approx 1+2\Phi /c^2$, and the factor in front of the squared space step is $g_{rr}\approx -(1-2\Phi /c^2)$: both differ from flat space by the same amount, $2\Phi /c^2$.

A slow body covers very little space per unit of time, so only the time part acts on it, and that part alone gives Newton's gravity. Applied to light, the time part alone would give the path bent through ${\delta }_N=2GM/(c^{2}b)$, the angle Newton's theory gives for a particle moving at $c$. But light crosses as much space as time, $|d\vec{x}|=cdt$, so the space part acts on it as strongly as the time part, and the bending doubles:
$\delta =\frac{4GM}{c^{2}b}=2{\delta }_N$

Since $4GM/c^2=2r_s$, this is also $\delta =2r_s/b$. In the animation the ray passes at $b=8r_s$: Newton's purple path is bent by 7.2°, this formula gives $0.25\mathrm{rad}=14.3°$, and the exact gold ray turns through 17.8°, because so close to the mass the weak-field formula falls short. At the Sun's edge, $b\approx 2.4\times 10^5{r}_s$, the formula gives 1.75″, and radio telescopes confirm it to better than 0.1 %.

Mercury and the strong-field radii

For a planet the same extra term appears in its orbit equation, $u^{\prime \prime }+u=GM/{\ell }^2+3GM{u}^2/c^2$, where $\ell$ is the angular momentum per unit mass. Without the last term the solutions are Newton's ellipses, which close after one turn. The extra term makes the ellipse turn forward a little on every lap: its closest point to the Sun advances by
$\Delta \phi =\frac{6\pi GM}{c^{2}a(1-e^2)}$

where $a$ is the semi-major axis of the ellipse and $e$ its eccentricity. For Mercury, with $a=5.79\times 10^{10}\mathrm{m}$ and $e=0.206$, this is $5.0\times 10^{-7}\mathrm{rad}$ per orbit, or 43″ in the 415 orbits of a century: exactly the part of Mercury's advance that the pulls of the other planets could not explain.

In strong fields the same equations mark out three radii. At the horizon, $r_s$, the rate $\sqrt{1-r_s/r}$ of a clock at rest falls to zero. At the photon sphere, $1.5r_s$, light can circle the mass, but unstably. At the innermost stable circular orbit, $3r_s$, stable orbits end; it forms the inner edge of the discs of hot gas around black holes.

Worked problem: a clock riding the ball

How fast, exactly, does a clock in orbit run? Put a clock on the blue ball of the animation, which circles the star at $r=10.4r_s$ with angular speed $\omega$. In far-away time, Kepler's law holds exactly for circular orbits in this geometry: ${\omega }^2=GM/r^3$. On the orbit $r$ is constant, so $dr=0$, and the angle advances by $d\Omega =\omega dt$. Putting these into the metric and dividing by $c^{2}d{t}^2$ gives
$\frac{d\tau }{dt}=\sqrt{1-\frac{r_s}{r}-\frac{r^2{\omega }^2}{c^2}}$

With Kepler's law and $GM=r_s{c}^2/2$, the last term is $r^2{\omega }^2/c^2=GM/(r{c}^2)=r_s/(2r)$, so
$\frac{d\tau }{dt}=\sqrt{1-\frac{3r_s}{2r}}$
At $r=10.4r_s$ this is $\sqrt{1-1.5/10.4}=\sqrt{0.856}\approx 0.925$.

One loop takes 12.28 s of far-away time, so the ball's clock reads $0.925\cdot 12.28\approx 11.4\mathrm{s}$. Compare the two clocks that stand still in the animation. The far clock, at $22r_s$, runs at $\sqrt{1-1/22}=0.977$ and reads 12.0 s; the star clock, at $2.16r_s$, runs at 0.733 and reads 9.0 s. The ball's clock falls behind the far clock but stays well ahead of the star clock.

The rate splits into one factor for each effect. The gravity factor, $\sqrt{1-1/10.4}=0.951$, is the rate of a clock at rest at that radius. The motion factor, $\sqrt{1-0.231^2}=0.973$, is the slowing of Special Relativity for the speed $0.231c$ that a static observer beside the orbit measures. Their product is $0.951\cdot 0.973\approx 0.925$. That speed exceeds the Newtonian $\sqrt{GM/r}=0.219c$ because the static observer's own clock runs slow, so it times the ball over fewer seconds.

Now compare an orbiting clock with one standing still at radius $R$, whose rate is $\sqrt{1-r_s/R}$. The orbiting clock runs faster when $1-3r_s/(2r)>1-r_s/R$, that is when $r>1.5R$. This is the zero of the red Earth curve, 3,190 km up, but here the result is exact and holds in strong fields too. For the star clock at $2.16r_s$ the break-even orbit lies at $3.24r_s$, just outside the last stable orbit at $3r_s$.

Limits of validity

The weak-field formulas need $r_s/r\ll 1$ and $v\ll c$. Near the animation's star they fail visibly: for the star clock the weak-field rate $1-r_s/(2r)$ gives 0.769 instead of the exact 0.733, and the deflection formula misses the exact 17.8° by 3.5°. In the Solar System the errors are of order $(r_s/r{)}^2$, about $2\times 10^{-11}$ even at the Sun's surface. A neutron star is different: with 1.4 solar masses inside a 12 km radius it has $r_s/r=0.34$, and a clock on its surface runs at 0.81. The Earth graph also leaves out one small effect, the rotation of the ground clock, worth about 0.1 µs a day.

The theory has passed every test so far: light bending and radio delays near the Sun, Mercury's orbit, clocks, the shrinking orbits of binary pulsars and, since 2015, gravitational waves. The first wave detected came from two merging black holes of about 36 and 29 solar masses; it changed the 4 km arms of the LIGO detectors by about $4\times 10^{-18}\mathrm{m}$.

The real limit is quantum. The theory predicts singularities, inside black holes and at the start of the expanding universe, where curvature grows without bound. Long before that, near the Planck length, $\sqrt{\hslash G/c^3}\approx 1.6\times 10^{-35}\mathrm{m}$, quantum effects of gravity must take over, and no tested theory yet describes them.

Where this leads

Special Relativity is the flat, gravity-free limit, and its twin effect returns here as the rule of longest proper time. Gravitation and Orbits is the weak-field, slow-motion limit. Just as accelerated charges radiate in Electromagnetic Waves, accelerated masses radiate gravitational waves at $c$, though only through changing quadrupoles, since momentum conservation forbids dipole radiation. Fission, Fusion and E = mc² explains why every form of energy counts in $T_{\mu \nu }$.

Combined with quantum theory, a black hole should glow like the blackbodies of Blackbody Radiation and the Quantum, at the temperature $T=\hslash c^3/(8\pi GM{k}_B)$, about $6\times 10^{-8}\mathrm{K}$ for one solar mass. Particles and Forces: the Standard Model describes every known interaction except gravity; joining the two is the great open problem.