Illustrated Physics
physics, drawn
Concept

Rindler horizon — where it appears

The surface behind an observer with constant acceleration from which no signal can ever reach it, a distance c²/a away. Light from near it is redshifted exponentially, which makes the vacuum appear warm to the observer.

Named by 3 essays across one field — 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 The crests of one wave, meeting an observer who never stops accelerating. A spacetime diagram, time up and distance across, both scaled by the acceleration: the worldline of an observer with constant proper acceleration (a hyperbola), its horizon (the dashed light line it approaches and never crosses), and the crests of a plane wave of a single frequency travelling in the same direction, as parallel lines at 45°. Within the drawing the observer meets crests at proper times −0.18, −0.05, 0.11, 0.29, 0.51, 0.80, 1.20, 1.90 in units of c/a: the gaps lengthen, each crest having to chase the observer further, and after the last of them, at 1.90, it meets no more — every later crest lies beyond its horizon. One wave of one frequency therefore arrives as a signal whose frequency falls as e^(−aτ/c) for ever. Nothing about the wave is random, and the observer's clock is perfectly regular.

The heat hidden in a Doppler shift

An observer who accelerates without stopping sees empty space as warm. The usual derivation takes quantum field theory and several pages. There is a shorter one, and it needs no quanta at all. Watch a single wave of a single frequency from a steadily accelerating rocket: it is redshifted, and because the rocket keeps speeding away, the redshift grows exponentially. Take the spectrum of that fading signal, and it is exactly a Planck spectrum, at a temperature set by the acceleration and nothing else — the same whatever frequency the wave started with. The thermal distribution was hidden in the geometry of the horizon all along.

relativity · Accelerated frames 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

Named alongside it

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

Equivalence principleProper accelerationUnruh effectDetailed balanceDoppler effectFermat's principleFourier transformGeodesicGravitational time dilationHawking radiationHawking temperatureHorizon

All concepts