The radius a mass adds that no circumference shows

The clock that runs slow lower down derived the gravitational redshift from a photon climbing a tower. The parallelogram that will not close showed that the redshift alone forbids spacetime from being flat. The clock that measures a height turned the shift into a surveying instrument, and the clocks that must all slow together asked whether every kind of clock slows by the same fraction. Every one of those arguments is about clocks. Each reads one number from the geometry of spacetime around a mass: how fast time passes at a given place, the component that the notation of general relativity calls $g_{00}$.

The geometry has a second part, and no clock can see it. It is the part that says how long a ruler is. Around a spherical mass the two parts are tied together, one growing exactly as the other does, so a body that slows clocks by a part in a thousand million also stretches radial distances by a part in a thousand million. For the Earth that stretch is a couple of millimetres over the whole radius. It is not measurable with a tape, but it is not small in the sense that matters: in every experiment with light it is as large as the slowing of clocks, and half of the famous bending of starlight by the Sun is this and not the other thing.

Two radii of one sphere #

There are two honest ways to measure the radius of a planet. One is to walk round it, measure the circumference and divide by $2\pi$. The other is to dig to the centre and lay rulers end to end out to the surface. In flat space they agree, because that is what $2\pi$ means. Near a mass they do not.

The metric outside a spherical mass, found by Schwarzschild in 1916, can be written in terms of a coordinate $r$ chosen so that a sphere of coordinate $r$ has area $4\pi r^2$ and circumference $2\pi r$. That choice is what makes $r$ the radius a surveyor would report after walking round. With it, the proper distance a ruler measures between two nearby spheres is not $dr$ but

$$d\ell = \frac{dr}{\sqrt{1 - 2Gm(r)/rc^2}},$$

where $m(r)$ is the mass inside the sphere. Outside the body $m$ is the whole mass $M$; inside, it is whatever the density profile puts there. The square root is less than one wherever there is mass inside, so each step outwards measured with a ruler is longer than the growth in circumference over $2\pi$. Add the steps from the centre to the surface and the radius measured with rulers exceeds the radius measured by walking round.

The figure makes the geometry visible by a trick of Ludwig Flamm’s from the same year. Take the plane through the body’s centre and bend it into a third dimension, so that distances along the bent surface are the distances rulers measure in the plane. The circles of constant $r$ remain circles of circumference $2\pi r$, but the surface between them is tilted, and a path climbing the tilt is longer than its horizontal extent. Outside the body the surface is a paraboloid of revolution; inside a uniform body it is part of a sphere, joined to the paraboloid with no kink. At a neutron star’s compactness the ruler from centre to edge is 6.8 per cent longer than the circumference allows.

The height of the surface has no meaning at all. Nothing in space is displaced into a fourth dimension, and the same geometry could be drawn with the surface bent downwards. Only the lengths along it are real, and only the lengths in the plane — the picture shows one slice of the space around the mass, not spacetime, and it says nothing about clocks. That makes it a picture of exactly the half of gravity the earlier arguments could not reach.

The excess radius #

Richard Feynman, in his lectures on gravitation, gave the difference a name: the excess radius, the ruler radius minus the circumference over $2\pi$. For a body whose field is weak everywhere the square root can be expanded, and the excess is

$$\delta \approx \int_0^R \frac{Gm(r)}{rc^2}\,dr .$$

For a body of uniform density, $m(r) = M r^3/R^3$, the integral is $GM/3c^2$ exactly. It does not depend on the radius of the body at all, only on its mass. A uniform Earth, with $GM/c^2 = 4.4$ millimetres, has an excess radius of 1.48 millimetres. A uniform Sun has one of 0.49 kilometres. Feynman used the number to suggest that Einstein’s field equations can be stated in one sentence: the excess radius of a small sphere is proportional to the mass-energy inside it. The factor that makes that sentence exact is the whole content of the theory’s spatial part.

The figure plots the excess, as a fraction of the radius, against the compactness $2GM/Rc^2$ — the Schwarzschild radius over the body’s radius — for uniform bodies from the Earth to a neutron star. The fraction is one-sixth of the compactness while that is small, a straight line of slope one on logarithmic axes. The exact integral for a uniform body can be done in closed form, $R(\arcsin\sqrt{x}/\sqrt{x} - 1)$ with $x$ the compactness, and it peels away upwards once the compactness reaches a few tenths. At a neutron star the weak-field estimate is 16 per cent short; the ruler radius of a 1.4-solar-mass star twelve kilometres round is 12.83 kilometres. A uniform body cannot be made more compact than eight-ninths, where the pressure needed at its centre becomes infinite, and near that limit the excess becomes a large fraction of the radius.

The line covers ten orders of magnitude and the four bodies sit on it in order, which is itself worth noticing. The same curvature that is a part in six hundred million of the Earth’s radius is a part in fourteen of a neutron star’s. Nothing about the physics changes along the line. What changes is how deep the body sits in its own potential, and that is the same number, $GM/Rc^2$, that sets how slowly its surface clocks run. A body whose clocks run slow by a fraction $\epsilon$ has rulers stretched, on average across its interior, by a fraction of the same order.

Where the extra length is gathered #

The integrand $Gm(r)/rc^2$ says which parts of the body contribute. At each radius the stretch depends on the mass inside and on the radius itself, so mass concentrated near the centre makes $m(r)/r$ larger at every radius outside it and lengthens the total, while mass spread towards the surface makes it smaller.

The real Earth is not uniform. Its iron core holds a third of the mass inside a sixth of the volume, and the figure replaces the uniform model with a two-layer one: a core 3,480 kilometres in radius at the density that gives it its measured mass, and a mantle carrying the rest. The layered Earth’s excess radius is 2.00 millimetres, a third more than the uniform value, and nearly a quarter of it is gathered inside the core, where the uniform model gathers only a tenth. For the Sun, whose density falls a hundredfold from centre to edge, the effect is larger still: modelled as a polytrope of index three, the standard first approximation to its structure, the Sun’s excess radius is 1.95 kilometres, four times the uniform value.

That dependence on the interior is the difference between this and every clock argument in the earlier essays. A clock at the surface of a body responds to the potential there, which depends only on the total mass and the radius — the field outside a sphere cannot find its core, and neither can a clock on its surface. The ruler from the centre to the surface passes through the interior and responds to how the mass is arranged inside it. Two bodies with the same mass and circumference, one with a dense core and one uniform, have identical exteriors, identical surface clocks and different ruler radii.

The rulers outside the body #

Outside the body $m(r)$ is the constant $M$, and the stretch goes as $1/r$. The excess between two spheres, $r_1$ and $r_2$, both outside the mass, is then

$$\Delta \approx \frac{GM}{c^2}\ln\frac{r_2}{r_1},$$

a logarithm, which grows without limit however slowly. The space around a mass is not flat at any distance; it only becomes flat in each small region.

The figure plots the extra distance outward from the Sun’s surface. Out to the Earth’s orbit the rulers find 7.9 kilometres more than the difference of the two circumferential radii; out to Neptune, 13 kilometres. On a logarithmic axis of radius the excess is a straight line, with the slope $GM/c^2$ — the Sun’s gravitational radius, 1.48 kilometres, and nothing else. Every decade of distance adds the same 3.4 kilometres.

A radar pulse sent from the Earth’s distance down to the Sun’s surface and back crosses the extra length twice: 15.9 kilometres, 53 microseconds of travel. That is half of the Shapiro delay for such a path. The other half comes from the clocks: a pulse travelling deep in the Sun’s potential moves, by the reckoning of a distant clock, more slowly than $c$, by exactly the fraction that the clock there runs slow. The delay that is not a bend measured the total — 233 microseconds for an echo from Venus grazing the Sun — and found half of it gathered more than twenty solar radii out. The logarithm is why: the space half of the delay has no favoured scale, and each decade of the path contributes as much as the last.

Half of every ray #

What makes the spatial part more than a curiosity is that light feels it in full. A slow body moving past the Sun feels almost only the time part of the metric, because its equation of motion is dominated by how clocks run at different heights — which is Newtonian gravity, read correctly. Light moves as fast through space as it does through time, and the two parts enter its path with equal weight.

The figure sorts five effects by the share each takes from the two halves. The redshift is all clocks, which is why every argument about the redshift was blind to the rulers. The bending of starlight at the Sun’s edge, 1.75 seconds of arc, is half each: the bend Newton got half right is Newton’s gravity supplying the time half, which is all a falling particle knows about, and the missing half is the space curvature this essay has been describing. The Shapiro delay is half each. A gyroscope carried in orbit precesses a third from the time part and two thirds from the space part, which is why its measurement by Gravity Probe B in 2011 was a test of the rulers as much as of anything. A comet at fifty kilometres a second is bent almost entirely by the time part; the half of the bend a slow body never feels showed that the space part’s share is $v^2/c^2$, three parts in a hundred million.

This is why the spatial curvature was measured first with starlight, in 1919, and not with rulers, and why it is still measured with light. Theories of gravity that agree with general relativity about clocks can disagree about rulers, and the disagreement is carried by one number in the parametrised post-Newtonian framework, $\gamma$, the amount of space curvature produced by unit mass. General relativity says $\gamma = 1$. A theory in which gravity only slows clocks would have $\gamma = 0$, and would bend starlight by Newton’s 0.87 seconds of arc. The best measurement, from the delay of radio signals to the Cassini spacecraft passing behind the Sun in 2002, found $\gamma - 1 = (2.1 \pm 2.3)\times 10^{-5}$.

What a surveyor would need #

It is natural to ask whether the rulers could be measured directly. The answer is a useful lesson in what the effect is.

The excess radius of the Earth is two millimetres out of 6,371 kilometres, a part in three thousand million, and it concerns a distance nobody can lay a tape along. A survey on the surface cannot find it by comparing circumferences, because the circumference is what defines the coordinate. What a surface survey could in principle find is the curvature itself, as a failure of the angles of a triangle to add up to two right angles. For a triangle lying in a plane through the Earth’s centre, outside the Earth, the angles fall short of $\pi$ by the area of the triangle times $GM/c^2r^3$. At the Earth’s surface that factor is $1.7\times10^{-23}$ per square metre, so a triangle a hundred kilometres on a side, with an area of $4\times10^{9}$ square metres, has angles short by $7\times10^{-14}$ of a radian. Gauss surveyed a triangle of mountain tops in the 1820s, with sides of about a hundred kilometres, and his accuracy was about a second of arc, $5\times10^{-6}$ radian. The curvature is eight orders of magnitude below it.

The curvature is not small because the metric is barely different from flat. It is small because the radius of curvature, $1/\sqrt{GM/c^2r^3}$, is about $2.4\times10^{11}$ metres — 1.6 astronomical units — so any triangle that fits on the Earth is tiny compared with it. The term free fall cannot remove found the same scale in the tidal acceleration: the curvature of spacetime near the Earth is set by $GM/r^3$, and the time part and the space part of it have the same size. Light can measure the space part because light crosses a region where the integrated curvature is large — the whole path past the Sun — rather than a small patch where it is negligible.

The geometry inside #

The spherical cap inside the body in the first figure is not a coincidence of the drawing. Inside a uniform body the space around the centre is exactly the three-dimensional surface of a sphere, of radius $\sqrt{R^3c^2/2GM}$, and the body occupies a cap of it. That was found by Karl Schwarzschild in his second paper of 1916, a month after the first, which described the field inside a star of constant density. For a uniform Earth the radius of that sphere is $1.7\times10^{11}$ metres, a little more than the distance to the Sun, and the Earth is a cap of it about four hundred-thousandths of a radian across.

Every uniform body is therefore a cap of a three-sphere, with the rest of the sphere missing and a paraboloid attached instead. The denser the body, the smaller the sphere and the larger the cap. The cap would reach the three-sphere’s equator only at a compactness of one, and well before that, the mass no cold matter can hold up has already taken over: the pressure a static uniform body needs at its centre becomes infinite first. At the surface of a black hole, the surface that only lets things in, Flamm’s paraboloid turns vertical, and the radial distance to the horizon from any sphere outside it is finite while the stretch per unit $r$ grows without limit.

How the two halves are tied #

Why should the stretch of rulers be exactly as large as the slowing of clocks? Nothing in the argument from a photon climbing a tower implied it. The redshift fixes $g_{00}$ from energy conservation and the equivalence principle, and the parallelogram used it to prove that spacetime is curved. The spatial part needs a field equation.

In general relativity the equation that fixes the spatial part, in a static field, is the one Feynman stated as a sentence: the excess radius of a small sphere is $G/3c^2$ times the mass-energy inside it. For a point mass, that gives exactly the $1/\sqrt{1 - 2GM/rc^2}$ written above, and the $2GM/rc^2$ in it is the same combination that appears in the clock rate. Another theory could relate them differently. The Brans–Dicke theory, a rival of the 1960s in which a scalar field is added to the metric, gives $\gamma = (1 + \omega)/(2 + \omega)$ for a coupling constant $\omega$; the Cassini result requires $\omega$ above forty thousand, at which point the theory is indistinguishable from Einstein’s in the solar system. The tie between the two halves is therefore a measured fact to about two parts in a hundred thousand, and it is measured entirely with light.

What the picture cannot show #

Flamm’s surface draws one plane through a static, spherical mass, and every simplification in that sentence matters. A rotating body drags space round with it, and its geometry cannot be drawn as a surface of revolution; the two halves of the metric are then joined by a third, mixed part, which for the Earth is small: in the precession of an orbiting gyroscope it is about a hundred and seventy times weaker than the space part. The picture also cannot show spacetime: it is one instant of a static field, with time removed, and says nothing about the clock rates that every argument about the redshift has been about. Treating the two as one curved surface, with a marble rolling round a dip, is the common popular picture and is wrong in a specific way. The marble in that picture falls because of the height of the surface, which has no meaning; real orbits are set almost entirely by the time part, which the surface does not contain.

The polytrope and the two-layer Earth are models. The real Earth’s core is itself two layers, and the Sun’s density profile is known from helioseismology rather than from a single index; both would move the excess radius by a few per cent. The exact numbers for a neutron star depend on an equation of state that is not known, and the uniform model is used here only to show how the excess grows once the compactness is no longer small.

Still open: whether rulers and clocks stay tied everywhere #

In the solar system the space part and the time part are tied as general relativity says to two parts in a hundred thousand. Whether they stay tied where the field is strong is less certain. Near a neutron star or a black hole both parts are large and nonlinear, and measurements there — of binary pulsars, of merging black holes, of the shadow a black hole casts — test combinations of the two rather than each one separately. Theories that modify gravity at large distances or in strong fields generally predict a $\gamma$ slightly different from one somewhere, and the next improvement in solar-system measurements, from spacecraft ranging and from optical interferometry of stars near the Sun’s edge, aims at a part in a million or better. A difference found there would mean that clocks and rulers respond to mass in different proportions, which no metric theory with a single source can arrange.

The habit worth carrying away is to ask which part of a geometry an instrument can read. A clock reads the rate of time and nothing else, so a test made only with clocks cannot see how space is curved; light reads both halves in equal measure, which is why the curvature of space around the Sun was first measured as the half of a bend that Newton’s gravity did not supply. The Earth’s radius is two millimetres longer than its circumference allows, and no clock on the planet will ever report it.