The half of the bend a slow body never feels

The deflection of starlight at the edge of the Sun is 1.75 arcseconds, and the famous thing about the number is that Newton’s mechanics, applied to a particle moving at the speed of light, predicts exactly half of it. The usual gloss is that light is special — massless, not a particle in the ordinary sense, and so not bound by the ordinary rule. The gloss is wrong in a way that is worth following all the way down. Send a body of any mass past the same Sun at any speed, and it too is bent by more than Newton says. The excess is not a constant factor. It grows with the speed, from nothing at walking pace to exactly the Newtonian amount again at the speed of light, and light is simply the last point on a curve that every body is on.

One rule with the speed left in #

The factor of two for light was found by splitting the effect of a mass on a passing ray into two halves. Outside a spherical mass the metric differs from flat spacetime in two places. Its time part is stretched, which is the same stretching that makes a lower clock run slow and which by itself produces Newtonian gravity. Its space part is stretched as well, which has no Newtonian counterpart at all. For light the two contribute equally, $2GM/bc^2$ each, and the sum is the measured $4GM/bc^2$.

A body with a rest mass follows a geodesic of the same metric, and the calculation for it is no harder. With the body arriving from far away at speed $v$ and impact parameter $b$, and with everything kept to first order in the field’s strength, the deflection is

$$\delta = \frac{2GM}{b v^2}\left(1 + \frac{v^2}{c^2}\right).$$

The first term is Newton’s. It is the angle a slow body is turned through by the ordinary pull of gravity, and it is the familiar result from the mechanics of hyperbolic orbits in the limit where the turn is small. The second term is new, and it has a remarkable property: multiply it out and the speed cancels. It is $2GM/bc^2$ for every body at every speed.

The figure is the whole argument in one frame. The ratio of the true deflection to Newton’s is $1 + v^2/c^2$ exactly to the order the weak-field expansion is good for, and the points computed from the exact geodesic sit on the curve to about a fifth of a per cent at the distance used — a discrepancy that is the next term of the expansion showing, not a failure of the rule. At a tenth of the speed of light the ratio is 1.01. At half the speed of light it is 1.25. Light is the point at the far right, where the ratio has reached 2 and the famous factor has appeared. Nothing changes character there. The curve does not jump, the rule does not switch, and a neutrino at 0.99999 of the speed of light is bent by a hair less than a photon for exactly the reason a slow proton is bent by a good deal less.

Two halves, and only one of them cares about speed #

Written as a sum rather than a product, the deflection separates into two terms that behave completely differently:

$$\delta = \underbrace{\frac{2GM}{b v^2}}_{\text{curvature of time}} + \underbrace{\frac{2GM}{b c^2}}_{\text{curvature of space}} .$$

The first grows without limit as the body slows. The second does not depend on the body at all.

The reason the first half depends on speed is the reason every Newtonian deflection does. A body passing a mass is pulled sideways for as long as it is near it, and a slower body is near it for longer. The sideways velocity it picks up is the pull multiplied by the time, which goes as $1/v$; the angle is that sideways velocity divided by the forward one, which brings in a second factor of $1/v$. So the time half falls as $1/v^2$, which is why a comet grazing the Sun at a few hundred kilometres a second is swung round through a large angle while a proton at nearly the speed of light is barely nudged.

The second half has no such story because it is not produced by a pull acting over a time. It is produced by the geometry of the space the body is crossing. A mass makes the space around it slightly larger inside than its surface area implies — a sphere drawn round the Sun has a little more radius than its circumference would suggest — and a line that is as straight as the geometry allows, laid past such a mass, comes out turned. How much it turns is a property of the geometry and the impact parameter, not of how quickly anything travels along it. A surveyor’s taut line stretched past the Sun, if one could be made, would be bent by the same 0.876 arcseconds that the space term gives a photon.

That statement needs a qualification before it can be trusted, and the qualification is not a small one. Splitting a spacetime into “time” and “space” is a choice of how to slice it, and the split used here is the one natural to a mass at rest — the frame in which the Sun is not moving and the metric does not change with time. In another frame the two terms mix. What is not a choice is the total deflection a distant observer measures, and the dependence of that total on the speed of the thing being deflected. A theory in which space did not curve would give the first term alone, at every speed; a theory in which the space term had a different coefficient would move the whole curve in the first figure. Measuring the deflection of two bodies at two different speeds would separate the two terms directly, without any appeal to how the metric was written down.

Why a slow body is blind to half of general relativity #

The practical consequence is stark once it is put on a scale that runs from walking pace to light.

The share climbs as the square of the speed, so every factor of ten in speed is a factor of a hundred in sensitivity. At the Earth’s orbital speed of thirty kilometres a second, a hundredth of a per cent of the speed of light, the curvature of space contributes one part in a hundred million to any deflection. A spacecraft passing the Sun at two hundred kilometres a second gets a part in two and a half million. Even an electron with ten kilo-electronvolts of kinetic energy, which is moving at a fifth of the speed of light, takes less than four per cent of its bend from the space term.

That is why the curvature of space went unnoticed through three centuries of celestial mechanics, and why the first test of it was a measurement on starlight rather than on a planet. It is not that planets are insensitive to general relativity. Mercury’s orbit precesses by 43 arcseconds a century more than Newtonian perturbations allow, and the curvature of space contributes to that too. But the precession is itself an effect of order $v^2/c^2$ per revolution, a part in a hundred million, and it becomes measurable only because it accumulates over hundreds of orbits. A single pass of a slow body shows the space term at a part in $10^8$. A single pass of light shows it at one half. The factor of two is the one place in weak-field gravity where the curvature of space is not a small correction to something much larger, and that is the reason the eclipse of 1919 could decide the question with photographic plates and a ruler.

The opposite sign from electricity #

The falling curve in the second figure deserves a paragraph of its own, because it is the comparison that shows what is unusual about gravity.

An electric charge passing another charge is also deflected, and its small-angle deflection is set by the sideways impulse it receives divided by its forward momentum. The impulse is fixed by the charges and the impact parameter: $2kqQ/bv$. The momentum is $\gamma m v$, which grows faster than $v$ as the speed approaches $c$ — the same growth that makes a push at an angle fail to point where the body goes. So the deflection is

$$\delta_{\text{Coulomb}} = \frac{2kqQ}{b\,\gamma m v^2},$$

which is the non-relativistic answer divided by $\gamma$. A fast charged particle is stiffer than Newtonian mechanics says, because its inertia has grown and its charge has not. That is why particle accelerators need ever larger magnets as the energy climbs — a magnetic field turns a charge without speeding it up, and turning a heavier momentum takes a stronger field — and it is the familiar sense in which “relativistic mass” makes things harder to push.

Gravity reverses the sign of the correction, and the reason is the equivalence principle. The gravitational charge of a body is not a separate number like an electric charge; it is the body’s energy. When the inertia grows by $\gamma$, so does the pull, and the $\gamma$ cancels exactly — which is why the time half of the deflection, $2GM/bv^2$, contains no $\gamma$ at all. Then the curvature of space adds a positive term on top. The electric curve falls from 1 toward 0; the gravitational curve rises from 1 toward 2. Both start from the same Newtonian value, and they part company for two separate reasons, one of which is the equivalence principle and the other of which is the geometry.

Close in, speed decides what is caught #

Everything so far has been the weak field, where the deflection is small and the expansion in $GM/bc^2$ converges quickly. The exact geodesic does not need that restriction, and close to a compact mass it shows the same dependence on speed in a much stronger form.

Each speed has an impact parameter inside which the body falls in, and outside which it escapes. Near that threshold the body is carried round the mass on a nearly circular path before it leaves — more than once, and as many times as the aim is fine enough to allow — because the threshold is the top of a barrier in the effective potential, and a body arriving with exactly the energy of the top of the barrier can sit on it indefinitely. For light the barrier’s top is the photon orbit at three masses, and the threshold is the impact parameter $3\sqrt 3\,GM/c^2$ that sets the size of a black hole’s silhouette — two and a half times the horizon’s own radius. For a massive body the top of the barrier moves outward as the body slows, and the threshold impact parameter grows much faster than the barrier does.

The reason is again the time half of the geometry. A slow body is focused toward the mass from a long way off — gravitational focusing is the name for the whole Newtonian part of the deflection acting on a stream of bodies — so a body aimed wide of the mass is bent onto a course that brings it close. What matters for capture is the angular momentum per unit energy the body brings in, and for a slow body that is $bv/c$; the threshold on it is fixed near four masses, which gives a threshold impact parameter of $4GM/c v$ and a target area of $16\pi (GM/c^2)^2 (c/v)^2$.

So the target a black hole presents is not one number. For light it is 27 times $\pi (GM/c^2)^2$ — the photon sphere’s shadow, and the reason the dark region in an image of a black hole is larger than the horizon. For a body arriving at a tenth of the speed of light it is sixty times larger, and for the slow gas and dust that actually surround most black holes it is larger again by the square of however much slower they are. Capture of slow matter is overwhelmingly the time half’s business, which is why the rate at which a black hole sweeps up gas from its surroundings is a Newtonian calculation to excellent accuracy, while the size of its shadow on the sky is not.

Where the formula stops being the answer #

The body is a test particle. It has no size, no spin and no gravity of its own, and it does not disturb the mass it passes. A body with spin does not follow a geodesic exactly: the coupling between its spin and the curvature adds a force that depends on the orientation of the spin, and for a spinning body passing close to a compact mass the path is measurably different from the one drawn here. A body comparable in mass to the deflector is not a test particle at all, and the problem becomes the two-body problem, whose relativistic version has no closed solution.

The mass is spherical and not rotating. A rotating mass drags the local frames round with it and deflects a prograde body differently from a retrograde one; the thresholds in the last two figures split into a pair for each speed.

The weak-field formula is first order. The rule $(1 + v^2/c^2)$ is the leading term of an expansion in $GM/bc^2$ and in the deflection itself. For slow bodies the deflection is large whenever the impact parameter is not enormous, and then the exact hyperbolic orbit replaces the small-angle formula even in Newtonian mechanics — which is why the paths in the first figure, at thirty masses, turn by more than the formula says. The exact geodesic used in every figure has no such limitation; the formula is a reading of it, not a replacement for it.

The split between the two halves is a statement in one slicing. The clean separation into a time term and a space term is exact in the frame where the mass is at rest and the field is static, and the physical content is carried by the speed dependence of the total rather than by the separation. A distant observer measures one angle, not two.

A stream of bodies bent into a caustic #

The paths are drawn in a plane, the one containing the mass and the body’s approach, and a single path at each speed. What is missing is the family: a stream of bodies at one speed arriving from one direction at every impact parameter. Such a stream is bent into a caustic behind the mass, a line along which paths cross — the half-line of foci that light forms behind the Sun, beginning at 548 astronomical units — and the start of that line moves toward the mass as the stream slows, because the slower stream is bent harder at every impact parameter. For a stream at three hundred kilometres a second, the typical speed of dark matter through the Solar System, the grazing paths are swung through more than eighty degrees and cross the axis within a few solar radii, and paths passing thirty solar radii out cross it at the Earth’s distance — against 548 astronomical units for light. A slow stream of dark-matter particles passing the Sun, focused in exactly this way, would produce a density enhancement behind it along the direction of the stream; the figure of a single path cannot show the enhancement, because a single path has no density.

The paths also carry no clocks. A body on each of these paths ages at a rate that depends both on its speed and on its depth in the field, and the body that whirls round the mass before escaping arrives, on its own clock, well behind one that passed wide — which is the delay a mass imposes read for a massive traveller, and a different number from light’s.

Still open: whether a massive body has ever been seen to bend by the relativistic amount #

The deflection of light by the Sun has been measured to better than a part in ten thousand, and the result is exactly the space term plus the time term with equal weight. The corresponding measurement for a body with a rest mass has not been made at a level that sees the space term. Nothing that can be followed past the Sun moves fast enough for the space term to be more than a part in a million of its deflection, and nothing fast enough can be followed. Neutrinos cross almost anything and arrive from a distant source on paths bent by intervening masses, and the prediction for them differs from light’s by less than a part in $10^{14}$ at their measured masses, which is far beyond any conceivable measurement of their directions.

So the rule $(1 + v^2/c^2)$ is tested at its two ends — at $v \to 0$ by all of celestial mechanics, and at $v = c$ by every measurement of starlight and radio sources near the Sun — and in between it rests on the principle that every body without spin follows a geodesic of the same metric. That principle is also what underwrites the universality of free fall, tested to parts in $10^{15}$, so the gap is not one anybody expects to open. But it is a gap, and what would close it is a deflection or a delay measured for something that is neither slow nor light.

The next thing to ask of the geodesic principle is where it genuinely fails. A spinning body is not a point, and the coupling between spin and curvature makes its path depend on which way it spins — a small effect for a gyroscope orbiting the Earth and a large one for a spinning compact object passing close to another. Past that the body is not a test particle, and the problem stops being a path in a given geometry and becomes two geometries acting on each other. The habit worth carrying from here is to read a famous factor as a value of a function rather than as a constant. The two in $4GM/bc^2$ is $1 + v^2/c^2$ evaluated at the one speed where the second term has caught up with the first, and asking what the number would have been for a slower traveller is what shows which part of it was geometry and which part was falling.