Illustrated Physics
physics, drawn
Concept

Horizon — where it appears

A surface beyond which no signal can reach a given observer, whether it is made by a mass, by an acceleration, or by an expanding universe. It is a property of the observer as much as of the geometry, and different observers have different ones.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

The temperature of an acceleration. The Unruh temperature against proper acceleration, both axes logarithmic. An observer accelerating through empty space finds it is not empty: the state that an inertial observer calls the vacuum, an accelerated one finds populated, with a thermal spectrum at T = ħa/2πck — which is 4.06e-21 kelvin for every metre per second squared. The line is straight because the relation is exactly proportional, and the numbers on it are what make the effect so hard to see: one gravity gives 3.98·10⁻²⁰ K, a centrifuge at 10⁵ g gives 3.97·10⁻¹⁵ K, an electron in a strong laser gives 40.6 K, the surface of a solar-mass hole gives 6.17·10⁻⁸ K. Reaching one kelvin requires 2.5·10²⁰ m/s², which is 2.5·10¹⁹ gravities and beyond anything that can be sustained. What makes the effect worth taking seriously despite that is not its size but its structure: it says the number of particles present is not a property of the field alone but of the observer as well, and the same expression with a black hole's surface gravity in place of the acceleration is the Hawking temperature exactly, checked here to a part in 10¹².

The temperature of an acceleration

Empty space is empty for an observer who is not accelerating. For one who is, the same state of the same field is a thermal bath at a temperature proportional to the acceleration — and the constant of proportionality is 4 × 10⁻²¹ kelvin for every metre per second squared, which is why nobody has felt it. What the effect changes is not what can be measured but what a particle is.

relativity · Accelerated frames Colder the bigger it is. The Hawking temperature against mass, on logarithmic axes, with the microwave background drawn across it. The slope is minus one exactly, so a heavier hole is colder — a negative heat capacity, which is the fact everything else here follows from. The two lines cross at 4.50e+22 kg, about a hundredth of the Moon's mass. Anything heavier than that is colder than the sky it sits in and absorbs more than it emits, so it grows rather than evaporates. A stellar-mass hole is at 6.2e-8 kelvin and will not begin to lose mass until the background has cooled below that, which takes something like 10¹² years. Evaporation is not something happening now to any hole anybody has observed.

The hole that outlives everything and then does not

A black hole radiates at a temperature that rises as it shrinks, so losing energy makes it lose faster. The whole history follows from that one sign: a life proportional to the cube of the mass, nearly nothing happening for almost all of it, and an end that arrives in a second.

astrophysics · Horizons The power at stake, which is none. The power an electron radiates by Larmor's formula, against its acceleration, with 5 cases marked. a charge on a table: 9.8e+0 m/s², 5.49e-52 W; a laboratory centrifuge: 1.0e+6 m/s², 5.71e-42 W; a proton at the LHC: 1.9e+16 m/s², 2.06e-21 W; an electron in a linac: 2.0e+19 m/s², 2.28e-15 W; an electron in a hydrogen atom: 9.0e+22 m/s², 4.62e-8 W. The slope is two, measured on the drawn line. A charge held at one gravity radiates 5.49e-52 watts, which over the whole age of the universe comes to 2.39e-34 joules — far less than one photon of any kind. So the question of whether it radiates is not an experimental question about a charge on a table, and never has been. Every number here is a straight line on logarithmic axes with an exponent the figure measures.

Whether a charge on a table glows

The equivalence principle says a charge at rest in a gravitational field is a charge accelerating in empty space, and an accelerating charge radiates. Nothing is supplying the energy. The argument has run for eighty years, and its resolution is that radiation is not something a single observer can define.

astrophysics · Radiating charge Two terms, and the distance neither can beat. The smallest distance a probe of a given momentum can resolve, as the sum of two terms. The falling one is the uncertainty relation: more momentum, shorter wavelength, finer resolution. The rising one is gravity: the probe's own energy curves the region it is probing, and past a point it makes a horizon larger than the thing being looked at. With the gravitational term at 1× the Planck area, the least resolvable distance is 2.286e-35 m — each located by scanning the drawn curve over four hundred thousand momenta rather than by substituting into a formula. There is no momentum at which the resolution is better than that, so the ordinary procedure for measuring a distance has a floor, and the floor is the Planck length up to a factor of order one.

The length no experiment can resolve

Measuring a small distance needs a short wavelength, a short wavelength needs a large energy, and a large energy in a small region makes a horizon. Past a point, pushing harder makes the probe bigger — and the distance where that turns round is the Planck length.

astrophysics · Planck scale The quantity that went down, and the one that went up. The books for GW150914: two black holes of 36 and 29 solar masses merging into one of 62, with a final spin of 0.67. 3.0 solar masses left as gravitational waves, so the mass fell by 4.6 per cent. The total horizon area rose, from 107417 to 168330 in units of the Sun's gravitational radius squared — an increase of 57 per cent. The two progenitors are taken as non-spinning, which is the assumption that makes the test hardest to pass: a spinning hole of the same mass has a smaller horizon, so any spin they actually had would only widen the gap. Mass is the quantity that behaves the way energy usually does and it is not the one with a direction. Area is, and it is the reason the area has been read as an entropy ever since.

The area that is not allowed to shrink

Two black holes merge and the result weighs less than the sum, because three solar masses left as gravitational waves. The horizon area went up by more than half. Mass is the quantity that behaves like energy and it is not the one with a direction; area is, and the theorem saying so has been tested against a real merger.

astrophysics · Horizons

Named alongside it

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

Black holeMeasurementEntropyEquivalence principleIrreversibilityUnruh effectAccelerationAngular momentumBlackbodyConservationCosmic microwave backgroundDimensional analysis

All concepts