Illustrated Physics
physics, drawn
Concept

Light deflection — where it appears

The bending of a light ray by a gravitational field, 4GM/c²b at an impact parameter b in the weak field. It is twice what a Newtonian calculation gives, and because the amount falls with distance off axis rather than rising, it produces rings and pairs of images instead of a focus.

Named by 9 essays across 2 fields — each of them below, with the objects they name alongside it.

A radar echo past the Sun, delayed by 233 microseconds. The extra time a round-trip radar signal takes when its path passes close to the Sun, against how close, for a reflector 0.723 AU away. Grazing the Sun's limb the delay is 233 microseconds — about 70 kilometres of light travel, on a path of hundreds of millions — and it falls only as the logarithm of the impact parameter, so the effect is still tens of microseconds ten solar radii out. That slow falloff is what makes the measurement possible: the delay can be watched building and fading as the geometry changes, rather than having to be caught at one instant. The dashed curves are the delay the solar corona's plasma adds at three radio frequencies. It is the competing effect, it is larger than the gravitational one close in, and it falls as the square of the frequency while the gravitational delay does not depend on frequency at all — which is how the two are separated, and why the sharpest measurement of this was made with a spacecraft carrying three radio links instead of one.

The delay that is not a bend

The same metric that bends a ray also slows it, and the two are different tests. A radar echo from Venus arrives 233 microseconds late when its path grazes the Sun — and half of that delay is accumulated more than twenty solar radii away, in a field thousands of times weaker.

astrophysics · Light deflection A lens with no focal length. Where a ray crosses the axis, against how far off the axis it passed the deflecting body, for 1 solar mass of radius 1 solar radius. A glass lens deflects a ray by an angle proportional to its distance off axis, which is precisely the condition for every ray to arrive at one point — the flat dashed line. Gravity deflects by 4GM/c²b, which grows smaller further out, so the crossing distance goes as b² and each ray has its own focus. The grazing ray crosses at 548 astronomical units and a ray passing at 12 radii crosses at 78857; the square law is verified on the drawn curve to 1.5e-16. So there is no image plane at all, only a half-line of foci beginning at the first of those and running outward for ever. Anything placed on that line sees not an image but a ring, and moving along it does not refocus anything — it selects which rays are being seen.

The lens with no focal length

A glass lens bends a ray by an angle proportional to how far off the axis it passes, which is precisely the condition for every ray to arrive at one point. Gravity bends by an angle that falls with distance off the axis, so every ray has its own focus and there is no image plane anywhere — only a half-line of foci, beginning 548 astronomical units from the Sun and running outward for ever.

astrophysics · Light deflection Five rays, and the one that decides which side everything falls on. Light traced past a non-rotating mass at five impact parameters, by integrating the exact null geodesic rather than the weak-field formula. The shaded disc is the horizon and the dashed circle is the photon sphere at three masses. Rays passing closer than 5.196 masses are captured — 2 of the five here — and rays passing further escape, however far they are bent on the way. The two nearest the critical value behave in the most striking way: one wraps 1.06 times round before leaving and the other wraps round before falling in, and between them lies a ray that would circle for ever on the unstable orbit. Far away the same integration reproduces the familiar weak-field deflection: at sixty masses it gives 0.07021 radians against 4M/b = 0.06667, so the strong field and the weak one are one calculation.

The circle light cannot leave

A black hole has three radii and they are all sometimes called its size. The horizon is where nothing can come back from; the photon sphere, half again as far out, is where light can circle and cannot keep circling; and the shadow a distant observer sees is larger than either, because the ray that just escapes was bent on its way out.

astrophysics · Horizons Four bodies sent past one mass at 30 masses, at four speeds. Paths traced from the exact geodesic of a non-rotating mass, every one aimed at the same impact parameter of 30 masses and differing only in speed. The slower the body, the harder it is turned: 0.4c by 32.9°, 0.6c by 16.6°, 0.8c by 11.1°, light by 8.5°. The weak-field rule (2GM/bv²)(1 + v²/c²) gives 27.7°, 14.4°, 9.8°, 7.6° — close for the fast bodies and increasingly short for the slow ones, which pass near enough to feel the field's strong part. The angles are the true ones: near a compact mass nothing has to be exaggerated.

The half of the bend a slow body never feels

Light passing the Sun is bent by twice Newton's angle, and the factor of two is usually read as a fact about light. It is the end point of a curve every body lies on. A body at speed v is bent by (2GM/bv²)(1 + v²/c²) — the 1 from the curvature of time, which a slow body feels in full, and the v²/c² from the curvature of space, which is the same absolute angle for a comet as for a photon and is simply swamped when the body is slow.

astrophysics · Light deflection The slice through a mass that a ruler has to climb. A plane through the centre of a uniform body whose compactness 2GM/Rc² is 0.34 — about a neutron star's — drawn as a curved surface on which distances along the surface are the distances rulers measure. The body's edge is a circle whose circumference is 2πR; a ruler laid from the centre to that edge along the surface measures 1.068 R, 6.8 per cent more. Outside the body the surface is Flamm's paraboloid; inside it is a cap of a sphere. The flat line beneath is what Euclid would give. The height of the surface means nothing; only lengths along it are measured. For the Earth the same excess is 1.5 mm in 6,371 km and the surface would be indistinguishable from the flat line.

The radius a mass adds that no circumference shows

Walk round the Earth's equator and divide by 2π, and the number is the Earth's radius. Dig to the centre and measure the same radius with a tape, and the tape comes out about two millimetres longer. Nothing is wrong with either measurement. A mass does not only slow the clocks near it; it stretches the rulers too, and the stretch is exactly as large as the slowing. Clocks cannot see it. Light can, and half of every bend and every delay that light suffers near the Sun is this.

astrophysics · Gravitational redshift Six ways a wave of gravity could move a ring of free masses. Sixteen free masses on a ring (dashed) and where a passing gravitational wave of each of the six possible polarisations moves them at its crest, drawn at a strain of 0.32, some 10²⁰ times larger than a real wave's. Top row: plus and cross, the two general relativity allows, which stretch one direction across the wave while squeezing the other and change no area; and breathing, which swells the ring evenly. Bottom row, drawn in planes containing the direction of travel (arrow): longitudinal, which stretches along the travel; and the two vector modes, which shear those planes. Any metric theory of gravity permits these six and no others; which of them a theory keeps is decided by the field that carries gravity.

The four shapes a wave of gravity leaves out

A passing gravitational wave squeezes a ring of free masses in one of two patterns, and that is usually presented as the whole menu. It is general relativity's choice from a longer one. The tide a wave carries is a symmetric table of six numbers, so any theory in which gravity is geometry allows six shapes; a theory keeps two only if its graviton has no mass and nothing else rides along with it. Give the graviton the smallest mass imaginable and it gains three more states, and one of them changes how much the Sun bends light — by a quarter, however small the mass.

astrophysics · Gravitational waves Light inside an accelerating rocket travels on circles. Rays of light leaving one point inside a rocket with constant acceleration, drawn in the rocket's own coordinates: height above the horizon up the page, in units of c²/a (one light-year for one gravity), and distance across. Each ray is a straight line in an inertial frame, transformed event by event. Launched across the rocket (0°) or at +30°, +60°, −30°, −60°, every ray is an exact semicircle centred on the horizon, with radius 1.000, 1.155, 2.000, 1.155, 2.000: the rays bend down towards the horizon and meet it at right angles. A ray sent straight up stays straight. These are the straight lines of the hyperbolic plane, although the spacetime inside the rocket is flat.

The rocket in which light travels on circles

Inside a rocket that accelerates for ever at a constant rate, a beam of light sent across the cabin sags towards the floor, as Einstein argued it must if acceleration and gravity are indistinguishable. Followed further than any cabin, the sag turns out to be exact geometry: every ray inside the rocket is a perfect semicircle centred on the horizon below, and the rays obey the rules of a non-Euclidean plane in which the angles of a triangle add to less than two right angles. The spacetime in the rocket is flat; the curvature belongs to the clocks.

relativity · Accelerated frames Rays that leave the far side of a compact star. Light rays leaving the surface of a star of radius 2.9 Schwarzschild radii — a neutron star of 1.4 solar masses and about 12 km — and travelling to an observer far off to the right, traced along the exact paths of Schwarzschild's geometry, shown above and below the line of sight. The last ray to escape leaves the surface tangentially from a point 121° round from the point facing the observer, so the bright arc — everything the observer sees — covers 76 per cent of the surface rather than the half that straight rays would show. The dashed circle is the horizon a star of the same mass would have if it were compressed inside it.

The star whose far side can be seen

Look at a ball and half of it faces you; the other half is hidden behind it. That half is a fact about straight lines, and near a neutron star the lines are not straight. Light leaving the surface is bent round the star on its way out, so a distant observer sees three-quarters of a typical neutron star at once — and a star squeezed to 1.76 times its own Schwarzschild radius would show every point of its surface, the point directly behind it included.

astrophysics · Light deflection The bend that depends on when the planet was there. The difference, in microarcseconds, between the quasar's deflection computed from Jupiter's position when the light passed it — its position as seen from the Earth, 53 minutes earlier — and from its position at the moment of observation, over the day of the 2002 pass. In those 53 minutes Jupiter moves 40998 km, 0.57 of its own radius, and the deflection, a vector of 1.14 mas pointing away from the planet, turns and shrinks by up to 46 µas — about one twenty-fifth of itself. That small term was what the experiment was designed to see. The VLBA measured 51 ± 9 µas, the value general relativity predicts for a deflection pointing away from the position light reveals.

The bend a moving planet gives

Jupiter bends starlight by sixteen thousandths of an arcsecond at its edge, a hundredth of what the Sun manages, and in 2002 radio telescopes spread across a continent measured the bend on a quasar three and a half arcminutes from the planet. The experiment was designed to see something smaller still: the fact that Jupiter moves, by half its own width in the time light takes to reach the Earth, so that the bend depends on where the planet was when the light went past. The measured term was announced as the speed of gravity, and the argument over whether it was anything of the kind is the clearest lesson there is in what an experiment's number is a number of.

astrophysics · Light deflection

Named alongside it

The objects these essays reach for when they reach for this one.

General relativityImpact parameterGeodesicEffective potentialFermat's principleGravitational lensingGravitational redshiftGravitational time dilationPhoton sphereRefractive indexSchwarzschild metricSchwarzschild radius

All concepts