Illustrated Physics
physics, drawn
Concept

Unruh effect — where it appears

The finding that an observer accelerating through the vacuum sees it as a thermal bath at a temperature proportional to the acceleration. The constant is tiny — 4 × 10⁻²¹ kelvin at one gravity — and the effect is the flat-spacetime relative of the temperature a horizon has.

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 The turnaround, made gentler and gentler, and the difference that does not move. A round trip to a star 4 light-years away at 0.6c, with the turnaround done at nine different accelerations from a tenth of a gravity to a thousand. The upper curve is the age difference between the twins and the lower one is how much of that difference the turnaround itself contributes. At 0.1 g the turn accounts for 51 per cent of it; at 1000 g it accounts for 0.00 per cent, and it keeps falling. The total does not follow it down: it tends to 2.67 years, which is what the instantaneous-turnaround cartoon gives. So the acceleration is not what makes the twins differ. It is what makes one twin's path the bent one, and a bent path through spacetime is shorter for the same reason a bent path on a map is longer — but the amount is in the legs, not in the corner, and the corner's contribution can be made as small as anyone likes without the difference going away.

The clock that does not feel the turn

Proper time is the integral of dt over gamma, which presumes that a clock's rate depends on its speed and on nothing else — not on its acceleration, not on how long it has been accelerating. That is an assumption about clocks rather than a theorem about spacetime, and the twin result is empty without it.

relativity · Accelerated frames 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 Equilibrium is where the entropy peaks, and there the temperatures differ. Two cavities of radiation, one high in a gravitational field and one low, free to exchange energy, with the clock at the bottom running at 0.8 of the rate of the one at the top. What is conserved is the energy either would deliver to a distant observer, so energy held at the bottom counts for 0.8 of its local value. Across: the share of that conserved energy held at the top. Above: the total entropy of the two gases. Below: the temperature a thermometer in the top cavity reads, as a fraction of one in the bottom cavity. The entropy peaks at a share of 0.339, found by search, and there the top cavity is at 0.800 of the bottom's temperature — the clock-rate ratio exactly. Where the two local temperatures are equal, at a share of 0.556, the entropy is 1.82 per cent below its peak and energy still flows downward, into the deeper cavity.

The column that is hotter at the bottom

Two bodies in equilibrium have the same temperature — that is what equilibrium was supposed to mean. In a gravitational field it is false. A column left alone until nothing in it changes is warmer at the bottom by exactly the factor by which clocks there run slow, a part in ten million billion per metre on the Earth and more than a per cent across the outer kilometre of a neutron star, and near a black hole's horizon the equilibrium temperature grows without limit.

relativity · Relativistic thermodynamics 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

Named alongside it

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

AccelerationEquivalence principleHawking radiationHorizonProper accelerationRindler horizonBlackbodyClock hypothesisDetailed balanceDoppler effectEnergy conservationEntropy

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