Illustrated Physics
physics, drawn
Relativity

The light that pulls twice as hard as its mass

Energy has mass, E/c², and a box of light weighs that much more than an empty one. But the gravity something produces is not set by its energy alone. Pressure counts too, three times over, and light's pressure is a third of its energy — so a beam of light pulls on things beside it twice as hard as its mass would suggest. The box weighs only E/c² because its walls, straining to hold the light in, pull with a negative amount that exactly cancels the difference. Two parallel beams of light do not attract each other at all; two opposed ones attract four times as hard. And a substance whose pressure is negative enough does not pull at all — it pushes.
16 min read 5 figures What stays the sameApproximations that lie

Assumes: The box of light that weighs something · The bend Newton got half right

Mass is a form of energy read E=mc2E = mc^2 as a statement that energy already is mass, and the invariant that survives a boost found that mass is the invariant length of the four-vector of energy and momentum. The mass that is missing weighed binding energy in nuclei, and the box of light that weighs something sealed light in a mirrored box and found the box heavier by exactly E/c2E/c^2. The fuel a starship needs and the rocket that leaves its fuel at home then used the same bookkeeping for propulsion.

All of that concerns inertia: how hard a system is to accelerate, which is what E/c2E/c^2 measures. Gravity is a different question. The pull that has to equal the pull back found that in Newton’s theory a body’s mass appears twice in gravitation, once as what is pulled and once as what pulls, and that their equality is a law with experimental support. In general relativity the pulling side is richer than a mass. The source of gravity is the whole flow of energy and momentum, and in the part that produces the familiar attraction, pressure appears alongside energy with a weight of three. For almost everything ever weighed that makes no difference. For light it doubles the pull.

Energy plus three times pressure

In Newton’s theory the gravitational potential obeys Poisson’s equation, ∇2Φ=4πGρ\nabla^2\Phi = 4\pi G\rho, with ρ\rho the mass density. In general relativity, for a slowly varying field produced by a fluid at rest, the corresponding equation is

∇2Φ=4πG(ρ+3pc2),\nabla^2\Phi = 4\pi G\left(\rho + \frac{3p}{c^2}\right),

where ρc2\rho c^2 is now the energy density and pp the pressure. The three is one for each direction in which the pressure pushes; a stress that pushed in only one direction would contribute once. For cold matter the pressure is negligible and the equation is Newton’s.

How hard a substance pulls, per unit of its energy. The strength of the gravity a uniform substance produces, ρ + 3p/c², relative to its energy density ρ alone, against its pressure as a fraction of its energy density, w = p/ρc². Cold matter, with no pressure worth counting, pulls in proportion to its energy: 1. Radiation, with a pressure a third of its energy density, pulls twice as hard. The stiffest matter relativity allows, w = 1, pulls four times as hard. At w = −1/3 a substance pulls not at all, and below it — a tension rather than a pressure — it pushes: a vacuum energy with w = −1 repels with twice its energy, which is what makes an expanding universe filled with one accelerate.
Fig. 1 The gravity a uniform substance produces, ρ+3p/c2\rho + 3p/c^2, relative to its energy density alone, against its pressure as a fraction of its energy density, w=p/ρc2w = p/\rho c^2. Dust: 1. Radiation, w=1/3w = 1/3: 2. The stiffest matter relativity allows, w=1w = 1: 4. At w=−1/3w = -1/3: nothing. A vacuum energy, w=−1w = -1: −2, a push.

The figure plots how hard a substance pulls, per unit of its energy, against the ratio of its pressure to its energy density. A gas of cold particles sits at the left of the positive range and pulls exactly as its energy says. A gas of photons, or of any particles moving near the speed of light, has a pressure a third of its energy density and pulls twice as hard. Matter as stiff as relativity allows — a sound speed equal to light’s — has pressure equal to its energy density and pulls four times as hard. The pressure that weighs down what it holds up found this extra weight of pressure deciding the heaviest possible neutron star.

The line crosses zero at w=−1/3w = -1/3 and goes negative below it. A substance under tension rather than pressure — with a negative pp — reduces the gravity it produces, and one whose tension exceeds a third of its energy density produces a net push. A vacuum energy, the energy of empty space, has p=−ρc2p = -\rho c^2 by the requirement that it look the same to every observer, and it repels with twice the strength of its energy. That is why a universe dominated by such an energy expands faster and faster, and why the discovery in 1998 that the expansion is accelerating was taken as evidence that it is.

Why pressure belongs in the source

Pressure appears in the source of gravity for a reason that becomes clear once it is seen for what it is. Pressure is a rate of arrival found that the pressure of a gas is the momentum its molecules carry across a surface per unit area per second: a flow of momentum. In relativity energy and momentum are parts of one object, and so are their densities and their flows. The quantity that sources gravity is the whole array — energy density, energy flow, momentum density and momentum flow — and for a body at rest in the slow-field limit the part that produces the ordinary attraction is the energy density plus the momentum flow summed over the three directions. That sum is ρc2+3p\rho c^2 + 3p.

For a gas the two terms come from the same molecules. Each molecule carries energy, which counts in ρ\rho, and momentum, whose flow counts in pp. The speeds in a still room found air molecules moving at five hundred metres per second, a millionth of the speed of light, so the momentum flow they carry is a part in a trillion of their rest energy, and air’s pressure adds that much to its pull. Heat the gas and the molecules move faster; heat it until they approach the speed of light and their momentum flow approaches a third of their energy, which is where the radiation line in the first figure sits. A gas of photons is simply the limit in which every particle moves at cc.

A box of light, and its walls

The equation seems to contradict the result that a box of light weighs E/c2E/c^2. Radiation of energy EE in a volume VV has a pressure p=E/3Vp = E/3V, so the light’s contribution to the source is E+3pV=2EE + 3pV = 2E, divided by c2c^2 — twice the mass it adds to the box.

The weight of a box of light, term by term. A closed box with mirrored walls holding light of energy E, and the contributions to the gravity it produces, in units of E/c² (the walls' own rest mass left out). The light counts twice: once for its energy and once for its pressure, since for radiation 3pV = E. The walls hold the light in, and to do so they are under tension, a negative pressure, whose contribution is exactly −E: in any body in equilibrium the stresses sum to zero over the whole, so the walls' tension cancels the light's pressure. The whole box weighs E/c², as mass–energy equivalence says it must. Light on its own, with no walls, weighs 2E/c² for the purpose of attracting a slow body.
Fig. 2 The contributions to the gravitating mass of a closed mirrored box holding light of energy E, in units of E/c2E/c^2, the walls’ rest mass left out: the light’s energy, +1; its pressure, 3pV, +1; the walls’ tension, −1; the whole box, +1.

The resolution is the walls. They hold the light in, which means the light pushes on them and they are stretched: the material of the walls is under tension, a negative pressure. Its contribution to the source of gravity is negative and, as the figure shows, exactly cancels the light’s pressure. This is not a coincidence of the box’s design. Max von Laue proved in 1911 that in any body in static equilibrium the stresses, integrated over the whole body, sum to zero — the pushes and pulls inside must balance, or the body would not be in equilibrium. So for any closed system at rest the pressure terms cancel between its parts, and its total gravity is set by its total energy alone: E/c2E/c^2, the same as its inertia. The mass a charge’s field gets wrong by a third met the same theorem on the side of inertia, where the field of a charged sphere seemed to have four-thirds of the right mass until the stresses holding the charge together were counted.

The theorem explains why nothing in everyday experience shows pressure’s weight: every body anyone has put on a scale is a closed system in equilibrium, and its internal pressures are cancelled by internal tensions. Pressure’s weight shows up only where the pressure is not balanced by stresses in something else within the system — in a star, where it is balanced by gravity instead, or in a beam of light that no walls contain.

A beam of light, pulling

A beam of light without walls is exactly such a case, and its gravity has some odd properties that Richard Tolman, Paul Ehrenfest and Boris Podolsky worked out in 1931.

A beam of light pulls hardest on what comes towards it. The sideways pull of a long, thin beam of light on a body moving parallel to it at velocity βc, as a multiple of the pull the same body would feel from a static mass with the same energy per unit length, against β: 2(1 − β)²/(1 + β²). A body at rest beside the beam is pulled twice as hard as by a static mass — the beam's pressure along its length counts. A body moving with the beam is pulled less, and a ray of light running alongside it is not pulled at all (β = 1). A ray running against it is pulled 4 times as hard as a static mass of the same energy would pull it. Tolman, Ehrenfest and Podolsky found this in 1931. Two parallel laser beams do not attract each other; two opposed ones do.
Fig. 3 The sideways pull of a long, thin beam of light on a body moving parallel to it at βc, relative to the pull of a static mass with the same energy per unit length: 2(1−β)2/(1+β2)2(1-\beta)^2/(1+\beta^2). A body at rest: 2. A ray of light running with the beam: 0. A ray running against it: 4.

A body at rest beside the beam is pulled twice as hard as it would be by a static mass of the same energy: the beam’s pressure, all along its direction of travel, counts. A body moving along the beam is pulled less, and a ray of light running alongside it is not pulled at all. A ray running the opposite way is pulled four times as hard. The figure draws the whole curve. Two parallel laser beams therefore do not attract each other, however intense; two opposed ones do.

The factor of two for a body at rest has a mirror image that is famous. The bend Newton got half right found that a static mass bends a passing ray of light twice as much as Newton’s theory predicts. Here a passing ray of light pulls a static body twice as hard as its mass predicts. The two factors of two are the same fact read in two directions: gravity couples to light with twice the strength that a Newtonian counting of its energy would give, whether the light is the thing pulled or the thing pulling. The zero for parallel beams has a simple reading too. In the frame of one beam, the other is not moving relative to it, yet both are light and have no rest frame; the combination of energy flow, momentum flow and pressure that makes the source of gravity cancels exactly for two things travelling together at the speed of light.

Where pressure’s weight matters

In ordinary circumstances the correction is too small to notice.

Where pressure's weight is worth counting. The pressure as a fraction of the energy density, p/ρc², on a logarithmic axis — three times which is the fractional extra gravity the pressure adds — in six places: air at sea level, 9·10⁻¹³; the Earth's centre, 3·10⁻¹⁰; the Sun's centre, 2·10⁻⁶; a white dwarf's centre, 10⁻⁴; a neutron star's centre, 0.2; light, or any radiation, 0.3. In the Earth and the Sun the pressure's contribution is a few parts in a million or less, and in a white dwarf a part in ten thousand; at the centre of a neutron star it is comparable with the energy itself, and for radiation it is the same size as the energy by definition. Pressure's weight is negligible in every body a Newtonian can visit and decisive in the two a Newtonian cannot describe.
Fig. 4 The pressure as a fraction of the energy density, on a logarithmic axis, in six places: air at sea level, 9×10−139\times10^{-13}; the Earth’s centre, 3×10−103\times10^{-10}; the Sun’s centre, 2×10−62\times10^{-6}; a white dwarf’s centre, 10−410^{-4}; a neutron star’s centre, about 0.2; any radiation, one third.

The figure lists the ratio of pressure to energy density in six places. At the centre of the Earth the pressure is three hundred and sixty gigapascals, enormous by any laboratory standard, and still less than a part in a billion of the energy density of the rock, because the energy density includes the rest energy of every nucleus. At the Sun’s centre it is two parts in a million. Only in a neutron star, where matter is squeezed until the pressure competes with its rest energy, and in radiation, whose pressure is a fixed third of its energy, does the correction reach order one.

That is why the extra gravity of pressure is invisible in every body a Newtonian can visit and decisive in the two a Newtonian cannot describe: collapsed stars and the early universe. For the first fifty thousand years after the Big Bang the universe’s energy was mostly radiation, and the deceleration of its expansion, which goes as ρ+3p/c2\rho + 3p/c^2, was twice what the same energy in cold matter would have produced; the expansion rate itself, which depends on the energy density alone, is what the abundances of the light elements made in the first minutes test.

A star’s mass, counted from inside

Tolman found a way to write the mass of any static, bounded body as an integral over its interior that makes the role of pressure explicit.

A star's mass, counted as energy plus three times its pressure. For a star of uniform density, against its compactness 2GM/Rc²: Tolman's integral for its mass split into the part from its energy density and the part from its pressure, each weighted by the local clock rate and the stretch of radial distance, as fractions of the mass measured from outside. The two always add to exactly one. At a compactness of 0.1 the energy supplies 96.8 per cent and the pressure 3.2; at 0.5, 78 and 22; at 0.85, 41 and 59. The pressure's share grows as the star is squeezed, and the energy's share falls to compensate, because its clocks run slower. Neither part alone is the mass; the external field sees only the sum.
Fig. 5 For a star of uniform density, against its compactness 2GM/Rc22GM/Rc^2: Tolman’s integral for the mass, split into the part from energy density and the part from three times the pressure, each weighted by the local clock rate and the stretch of radial distance, as fractions of the mass seen outside. At 0.1: 96.8 and 3.2 per cent; at 0.5: 78 and 22; at 0.85: 41 and 59. They always sum to one.

Tolman’s integral adds up energy density plus three times pressure throughout the body, each weighted by the local rate of clocks — the redshift factor the clock that runs slow lower down derived — and by the stretch of radial distances that the radius a mass adds that no circumference shows described. For a static body the result is exactly the mass measured from outside by orbiting satellites. The figure evaluates the two parts for a star of uniform density, where everything is known in closed form, and they sum to one at every compactness to within the numerical integration. In a weakly compact star the pressure contributes a few per cent. In a very compact one it contributes more than half, and the energy’s contribution falls to compensate, because deep inside the clocks run so slowly that the energy there is worth less to the outside.

So a star is held together partly by the gravity of the very pressure that holds it up, and the total that an outside observer weighs is a fixed number that the two parts share differently as the star is squeezed. Neither share is separately measurable from outside. Only the sum is.

A string that does not pull

The same bookkeeping predicts something stranger for an object under tension along one direction only. A hypothetical cosmic string — a thin line of trapped field energy left over from the early universe, proposed in several theories — has a tension along its length exactly equal to its energy per unit length, as a relativistic string must. Tension is negative pressure, and here it acts in one direction only, so the source of the Newtonian pull — the energy density plus the pressures in the three directions — is ρc2+0+0−ρc2\rho c^2 + 0 + 0 - \rho c^2, which is zero. A straight cosmic string exerts no gravitational pull at all on bodies at rest beside it.

It is not without effect. The space around it is cut, like a sheet of paper with a wedge removed and the edges joined: a circle round the string has a circumference slightly less than 2π2\pi times its radius, by an angle proportional to the string’s energy per unit length. Nothing beside it is pulled, but two rays of light passing on either side are bent towards each other, and a string crossing the sky would show double images of the galaxies behind it, identical and undistorted. Searches for such doubled images, and for the gravitational waves oscillating loops of string would emit, have found none so far, and bound the energy per unit length. The prediction that a massive object can have no pull is the same rule as everything else in this essay: gravity counts energy and pressure together, and a tension can cancel an energy.

The same accounting settles a question that sounds as if it should have a simpler answer: how much does a neutron star weigh compared with the neutrons it is made of? Counting the neutrons and multiplying by the mass of a free neutron gives more than the star’s gravitational mass, measured from an orbiting companion, by ten to twenty per cent. The difference is the star’s gravitational binding energy, negative and large, and in Tolman’s integral it appears as the reduced weight of energy deep in a potential where clocks run slow. When a neutron star forms from the collapse of a stellar core, that difference is the energy carried off, almost all of it by neutrinos, in the ten seconds of the collapse — some 3×10463\times10^{46} joules, more than the Sun will radiate in its whole life.

What the pictures leave out

The equation with ρ+3p/c2\rho + 3p/c^2 applies to a fluid whose pressure is the same in every direction and to fields that change slowly. For anything else — a beam of light, a stretched wire, a spinning body — the source of gravity is the full stress–energy tensor, with ten independent components, and the pull depends on how the source and the body being pulled are moving relative to each other, as the beam figure shows. The numbers in the figure of pressures are order-of-magnitude values, and the neutron star’s is uncertain by a factor of two, reflecting the unknown equation of state. And none of the figures says anything about the gravitational field of the pressure in a beam at a distance comparable with its length, where the beam’s ends matter; the results quoted are for a long beam and a body near its middle.

No laboratory has measured the gravitational field of a beam of light directly, and none is likely to: a kilowatt beam carries energy equivalent to about 4×10−204\times10^{-20} grams per metre. The effects are established because they follow from the same equations that predict the bending of starlight, the Shapiro delay and the orbits of binary pulsars, all of which are measured.

Still open: what vacuum energy weighs

The one place where a negative pressure’s gravity is thought to have been observed is the accelerating expansion of the universe, attributed to a vacuum energy or something like it with ww close to −1-1. Whether ww is exactly −1-1, as a true vacuum energy must be, or differs from it slightly and changes with time, as some models of a dynamical dark energy propose and as some recent survey results have hinted, is one of the most actively measured quantities in physics. The value matters because it decides the universe’s future: a ww below −1-1 would make the repulsion grow without limit, and one above it would let the acceleration fade.

The habit worth carrying away is to ask what a source contains besides its energy. Gravity is sourced by energy density plus three times pressure, so radiation pulls twice as hard as its energy and a vacuum energy pushes; a box of light weighs only E/c2E/c^2 because its walls’ tension cancels the light’s pressure, as the stresses in any body in equilibrium must. Every scale ever made weighs systems whose pressures are balanced inside them, which is why E/c2E/c^2 has always been enough, and why it is not the whole of what gravity counts.

Part 7 of 7

This essay is one argument about Mass-energy. The others:

What a beam's energy buys, three ways. The speed reached against the energy intercepted, measured in the body's own rest energy, for a perfect mirror pushed by a beam, a perfect absorber pushed by the same beam, and a photon rocket that carries the same energy as fuel and throws it out behind. The mirror's curve is γ(1 + β) = 1 + 2E/mc², a rapidity of ln(1 + 2E/mc²); the dots integrate the reflected beam's force, (2P/c)(1 − β)/(1 + β), and agree with it to 10⁻¹⁵. To reach 0.2c the mirror needs 0.1124 of its rest energy, the absorber 0.2500 and the photon rocket 0.2247 — for a 1 g sail, 10.1 terajoules against 20.2. The rocket's rapidity is ln(1 + E/mc²), so the mirror is the rocket with its fuel left at home and each joule used twice, once arriving and once leaving. The absorber does worst, because the energy it keeps becomes rest mass it then has to carry. The rocket that leaves its fuel at home Part 6 — A mirror pushed by a beam from the ground carries no propellant, and relativity gives its speed in closed form: its rapidity is ln(1 + 2E/mc²), the photon rocket's equation with the fuel left behind and every joule used twice. What stops it is not the energy, which can be stored for days, but diffraction, which fixes the distance over which the energy can be handed over — and so demands an acceleration of tens of thousands of g.

Where this goes next

The mass of two things that have none. The invariant mass of a pair of photons of equal energy, in units of E/c², against the angle between them. It is computed from the total energy and the vector sum of the two momenta, and agrees with 2E·sin(θ/2) to 1.0e-14. at 0° the pair weighs 0.000 E/c²; at 30° the pair weighs 0.518 E/c²; at 60° the pair weighs 1.000 E/c²; at 90° the pair weighs 1.414 E/c²; at 120° the pair weighs 1.732 E/c²; at 180° the pair weighs 2.000 E/c². Two photons flying in the same direction have no mass between them at all, because their momenta add to exactly the energy over c; anything else and they do. Nothing has been added: the constituents are massless at every angle, and the mass of the system is a property of the arrangement. At 180° the pair weighs 2E/c², which is every joule it contains — the case of a sealed box of light, where the two beams cancel in momentum and the whole energy shows up on the scales. The box of light that weighs something Two photons flying apart have a mass between them, though neither has one. Seal them in a mirrored box and the box is heavier than it was empty, by exactly the energy inside divided by c². Mass is not a property of stuff and it does not add up — it is a property of a system, and 99 per cent of the mass of everything anybody has ever weighed is of this kind. Two predictions, a factor of two apart. The deflection of light passing a mass, against impact parameter, on logarithmic axes. The lower line is what a Newtonian photon does — it falls while it crosses, and comes out bent by 2GM/bc². The upper line is what a geodesic does in curved spacetime, which is exactly twice that. At the surface of a body of 1.99·10³⁰ kg the two are 0.88″ and 1.75″. Both are straight lines of slope minus one, so the ratio is two everywhere and the measurement is a choice between two theories rather than a fit. The bend Newton got half right A photon treated as a falling body passes a mass and comes out deflected. So does a photon treated as a straight line in curved spacetime — by exactly twice as much. The factor of two is not a refinement; it is the sharpest available statement that gravity is geometry. The heaviest star free neutrons can hold up. Mass against radius for cold stars of free neutrons, each point a different central density, held up by the pressure of the neutrons' exclusion alone: in general relativity and, with the same gas, in Newton's gravity. Relativity's curve reaches a highest mass of 0.71 solar masses at a radius of 9.5 km and turns back; Newton's keeps rising towards 5.8 solar masses as the stars shrink. The measured masses of neutron stars reach about 2.08 (dashed), three times what free neutrons allow, so the matter inside has to be stiffer than a gas of free neutrons: the repulsion between nucleons at short range is holding them up. The pressure that weighs down what it holds up In Newton's gravity a star is held up by pressure, and stiffer matter can hold up more: a star of free neutrons could reach almost six solar masses. In general relativity the pressure itself has weight. Every layer squeezed harder to hold up the layers above adds to the pull it is resisting, and past a point pressing harder makes things worse. The same gas of neutrons then tops out at 0.71 solar masses. Neutron stars twice as heavy exist, which says what their insides must be like — and no matter of any kind can hold a static body smaller than nine-eighths of its Schwarzschild radius.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

Active gravitational massEquation of stateGeneral relativityE = mc²Radiation pressureStress energy tensorTolman mass