Illustrated Physics
physics, drawn
Concept

Impact parameter — where it appears

The perpendicular distance between a projectile's undeflected line and the centre it passes. It is the variable every scattering law is written in, and the way a deflection depends on it decides whether a system focuses, spreads or leaves a caustic.

Named by 7 essays across one field — each of them below, with the objects they name alongside it.

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.

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 A target that grows as the traveller slows. The capture cross-section of a non-rotating black hole for particles arriving with speed v, in units of the horizon's own cross-sectional area π(2GM/c²)², against v/c, both on logarithmic axes (solid). For light it is 27/4 = 6.75 times the horizon's area. It grows as the particle slows, following 4c²/v² (dashed) — 404 at a tenth of light speed and 7.43·10⁶ at the 220 km/s of the Galaxy's dark matter. The dotted curve is a Newtonian ball of radius 8GM/c² — four horizon radii — capturing by ordinary gravitational focusing, and for slow particles the hole and the ball cannot be told apart.

The target that grows as the traveller slows

Light aimed at a black hole is swallowed if it passes within about five of the hole's gravitational radii, a disc six and three-quarter times the horizon's own. A particle arriving slowly is swallowed from much further out, and the disc grows as the inverse square of its speed: dark matter drifting at 220 kilometres a second sees a target seven million times the horizon. The reason is that slow matter carries almost no angular momentum for a given aim, and angular momentum is the only thing that keeps anything out. For slow traffic the hole behaves exactly like an ordinary ball four times its own size.

astrophysics · Horizons 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.

Light deflectionEffective potentialGeneral relativityGeodesicSchwarzschild radiusCross-sectionDeflection angleEvent horizonGravitational lensingPhoton sphereSpacetime curvatureAberration

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