Illustrated Physics
physics, drawn
Concept

Schwarzschild metric — where it appears

The geometry of spacetime around a static, spherical mass found by Schwarzschild in 1916. It fixes both how fast clocks run at each radius and how long radial rulers are, and contains the horizon at 2GM/c².

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

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 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

Named alongside it

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

Light deflectionBlack hole shadowCapture cross-sectionEffective potentialEvent horizonExcess radiusGeneral relativityGravitational lensingGravitational redshiftImpact parameterNull geodesicPhoton sphere

All concepts