The temperature of an acceleration

An observer who accelerates for ever has a wall of silence behind them — a horizon, beyond which no signal can ever reach them. This essay is about what that horizon does to the vacuum.

What a particle is, and who decides #

A quantum field’s vacuum is defined as the state with no quanta in it, and “quanta” means excitations of the field’s normal modes. Defining the modes requires a notion of positive frequency, and that requires a time coordinate.

An inertial observer and an accelerated one do not share a time coordinate. The accelerated observer’s natural time is the proper time along a hyperbolic worldline, and the modes that are positive-frequency with respect to it are mixtures of the inertial positive- and negative-frequency modes.

The consequence is that the state containing no inertial quanta contains a thermal population of accelerated quanta. Not a few, and not a random assortment: a Planck distribution, at

$$T = \frac{\hbar a}{2\pi c k_B}$$

So “how many particles are present” is not a property of the state alone. It depends on the observer’s motion, in the same way that “how much energy” or “how long between two events” does — and unlike those, the dependence had not been anticipated.

What the detector finds is not a rough approximation to a thermal spectrum, and the distinction matters more here than anywhere else in the essay. It is a Planck distribution — the same shape as a furnace’s, the same shape as the microwave background’s — with the right factor in front and the right dependence on frequency, and the only thing put in was a worldline. That is what makes “temperature” more than a way of speaking: a quantity with the dimensions of temperature could be defined out of any spectrum, and only a Planck one earns the name.

The number, and why nobody has felt it #

The constant is small. One kelvin requires an acceleration of 2.5 × 10²⁰ metres per second squared.

At one gravity the temperature is 4 × 10⁻²⁰ K, which is twenty orders of magnitude below the cosmic microwave background and eighteen below the lowest temperature ever produced in a laboratory. A centrifuge at 100,000 g reaches 4 × 10⁻¹⁵ K.

The accelerations that would do it exist only in two places. An electron in the field of a focused petawatt laser experiences around 10²² m/s², which corresponds to about 40 K — and that is the basis of most proposals to detect the effect. And the surface gravity of a black hole, where the same formula gives the Hawking temperature.

The identity with Hawking’s result #

The equality is not an analogy and is worth stating as an identity.

An observer hovering at fixed distance outside a black hole is accelerating — that is what it takes not to fall in — and the acceleration required, measured locally, is the surface gravity redshifted appropriately. The radiation they detect is the Unruh radiation of their own acceleration. An observer falling freely past the same point accelerates not at all, and detects nothing.

A floor that cannot be told from gravity is the equivalence principle in its original form, and applied to a quantum field it does the work directly. A static observer near a horizon and an accelerated observer in flat space cannot tell their situations apart by any local experiment, and a detector is a local experiment. So whatever one of them measures the other must measure too — which is why the two temperature formulae are not merely analogous but the same formula, with the local acceleration in both.

So the awkward question of where Hawking radiation comes from has a companion: whether a freely falling observer sees any is exactly the question of whether an inertial observer sees Unruh radiation, and the answer to that is no. The radiation is real for the observer who is accelerating and absent for the one who is not, and both descriptions are of the same state of the same field.

That is a strong statement about what “real” means for a particle. A detector that clicks is not ambiguous; what is observer-dependent is the description of the state that made it click.

The Rindler wedge, and what is missing from it #

It is worth being concrete about the horizon, because the whole argument turns on it and it is easy to picture wrongly.

An observer accelerating uniformly for ever traces a hyperbola in spacetime whose asymptotes are light rays. Everything on the far side of one of those asymptotes is permanently out of reach: a signal sent from there never catches up. The region the observer can ever see or influence is a wedge, and it is called the Rindler wedge.

Two observers who hold a fixed distance apart while accelerating have to accelerate differently, and that is the wedge’s geometry made concrete rather than an oddity of rigid bodies. A rigidly accelerated ruler has a larger proper acceleration at its trailing end, nearer the horizon, than at its leading one — the two ends sit at different distances from the same asymptote — and the difference is fixed by the geometry rather than by anything the ruler is made of.

So the Unruh temperature is not uniform across an extended object, and the variation follows exactly the Tolman law for a temperature in a gravitational field, which is the same equivalence-principle statement again: a clock lower in a field runs slow, and a temperature deeper in a field is correspondingly higher.

Where the energy comes from #

The obvious objection is that the accelerated detector absorbs energy from somewhere, and the vacuum has none to give.

The answer is that the agent doing the accelerating supplies it. Keeping a detector on a hyperbolic worldline requires a force, and that force does work at a rate that exceeds what is needed for the kinematics alone — the excess going into the radiation the detector emits back into the field. The bookkeeping is consistent, and it was worked out carefully because it looked as though it might not be.

That is the same structure as the force a radiating charge exerts on itself: the accelerating agent pays, and the field is the intermediary. The Unruh case is more delicate because what the detector “sees” and what it emits are different questions, and separating them is where most of the technical work has gone.

The classical statement in the same situation is Larmor’s. A charge accelerating in vacuum radiates, into a pattern set by its own velocity, and feels a reaction force for doing so. A detector accelerating in vacuum finds a thermal bath. The two are not the same effect — one is a field emitted, the other a field absorbed — and they are paid for by the same agent, which is whatever is maintaining the acceleration.

What a detector actually does #

The word “detector” has been doing a lot of work, and pinning it down is where the calculation becomes physics rather than a statement about mode decompositions.

The standard object is an Unruh–DeWitt detector: a two-level system with an energy gap, carried along the accelerated worldline and coupled weakly to the field. Its transition rate is computed from the field’s correlation function evaluated along that worldline — and along a hyperbola, that correlation function is periodic in imaginary proper time, which is exactly the condition for a thermal response.

What is computed is the behaviour of a system with discrete levels, because that is what a detector is. The Unruh calculation gives the rate at which such a system is excited by the field it is moving through, and the number that matters is the ratio of the excitation rate to the de-excitation rate. That ratio comes out as $e^{-\Delta E/kT}$ — which is the definition of being in contact with a bath at temperature $T$, and is a stronger statement than any single rate would be.

That last statement is the operational content. A detector at rest is excited never and de-excited at a rate given by spontaneous emission; an accelerated one is excited at a rate whose ratio to its de-excitation rate is the Boltzmann factor at the Unruh temperature. Detailed balance holds, which is what allows the word “temperature” to be used without qualification.

The experiments #

Nothing has yet measured the Unruh temperature directly, and the attempts are instructive about what would count.

Electron storage rings. Electrons circulating in a storage ring reach accelerations around 10²² m/s², and their spins polarise — the Sokolov–Ternov effect — to a degree that comes out at about 92 per cent rather than the naive 100. Bell and Leinaas argued in 1983 that the shortfall is an Unruh temperature acting on the spin, and the calculation reproduces the number. The argument is not universally accepted, because circular motion is not the uniform acceleration the effect is derived for, and the equilibrium is not obviously thermal.

Laser-driven electrons. A free electron in the field of an intense laser has the required acceleration, and the proposed signature is a modification of the emitted radiation spectrum. The experiments are difficult because the classical radiation from the same acceleration is enormous and must be subtracted.

And analogue systems. Bose–Einstein condensates, water waves and optical fibres can be arranged to have effective horizons, and thermal emission at the corresponding temperature has been reported in several of them. What such experiments test is the kinematics — that a horizon plus a quantum field gives a thermal spectrum — and not the gravitational or relativistic content.

What every argument here is about is the light-cone structure of spacetime rather than any object in it. A horizon is a boundary in that structure: a surface beyond which no signal can reach the observer, defined entirely by where their future light cones can go. The whole content of the effect is that such a boundary turns a pure state into a mixed one for the observer it belongs to, because the degrees of freedom on the far side have been traced out rather than measured.

What the effect is really about #

Stripped of the numbers, the statement is about information rather than heat.

An accelerated observer cannot see beyond their horizon. The field’s vacuum state is correlated across that horizon — pairs of modes on either side are entangled — so an observer restricted to one side has access to half of an entangled pair and no access to its partner. Tracing out what cannot be seen turns a pure state into a mixed one, and the mixed state that results is exactly thermal.

That is why the temperature is universal: it does not depend on the field, on the detector, or on anything but the geometry of the horizon. And it is why the same argument produces an entropy proportional to the horizon’s area: the entropy is the entanglement across it, and entanglement lives on the boundary.

The entanglement, written down #

The claim that the temperature is entanglement seen from one side can be made concrete without any heavy machinery, and doing so is the shortest route to why the spectrum is exactly Planckian.

The horizon divides spacetime into two wedges — the one the accelerated observer inhabits and its mirror image, from which no signal reaches them. Each wedge has its own set of modes, and the vacuum an inertial observer calls empty can be written out in terms of both sets at once.

What it looks like, written that way, is a sum over states with equal numbers of quanta in the two wedges: one quantum on each side, paired; two on each side, paired; and so on, with each term weighted by a factor that falls exponentially with the number. Nothing is present on one side without a partner on the other, and the weighting is the only free structure in it.

That is a maximally entangled state between the two wedges, and it is the whole content of the effect. An observer confined to one wedge has no access to the partners, so their description is obtained by summing over everything the other wedge might be doing — and summing over an entangled partner turns a pure state into a mixed one.

The mixture that results is exactly a thermal state, because the exponential weighting in the sum is exactly a Boltzmann factor. The exponent contains the mode’s frequency divided by the acceleration, and matching it against $e^{-E/kT}$ reads off the temperature with the $2\pi$ in place.

Which explains three things at once. Why the spectrum is Planckian rather than merely thermal-looking: the weighting is a geometric series in the occupation number, which is precisely what a bosonic thermal state is. Why the temperature does not depend on the field: the weighting comes from the transformation between the two mode sets, which is a property of the geometry. And why the effect and the horizon’s entropy are the same fact: the entropy of the mixed state is the entanglement entropy across the horizon, counted in one case and heated in the other.

Analogues, and what they are evidence for #

The analogue experiments deserve a careful sentence, because they are frequently reported as having detected Hawking radiation and the claim needs qualifying in a specific way.

The idea is that a horizon does not require gravity. Any medium in which a wave has a finite speed, flowing faster than that speed in some region, has a surface past which waves cannot travel upstream — which is kinematically a horizon. Sound in a flowing fluid does it, ripples on a stream do it, and light in a suitably engineered nonlinear medium does it.

The prediction is that such a horizon should emit correlated pairs of waves — one escaping upstream, one carried downstream — with a thermal spectrum at a temperature set by the flow’s gradient at the horizon. That is Hawking’s calculation with the sound speed in place of the light speed, and the derivation transfers line by line.

It has been observed. A condensate of cold atoms, made to flow supersonically through a constriction, emits correlated phonon pairs across its acoustic horizon, and the correlations and the spectrum have been measured and found consistent with the prediction for that system.

What that is evidence for is worth being exact about. It confirms the kinematics: that a horizon plus a quantum field gives thermal emission, that the spectrum has the predicted shape, and that the pairs are correlated as the calculation says. It does not confirm anything about gravity, because there is no gravity in it; and it does not settle any question about what happens to information at a black hole’s horizon, because the analogue system’s underlying description is entirely known and unitary by construction.

Which is still a substantial thing to have. The step in Hawking’s argument that has always attracted suspicion is the one where modes are traced back through the horizon to arbitrarily short wavelengths, and the analogue systems have a natural short-distance cutoff — the atomic spacing — where a black hole’s calculation has none. Finding the thermal spectrum anyway is evidence that the result does not depend on that troubling extrapolation, which is exactly the reassurance the original calculation could not supply for itself.

The shortest derivation #

There is a route to the temperature that is three lines long and explains why the answer is so clean, and it is worth including because it makes the result look inevitable rather than surprising.

In quantum statistical mechanics, a thermal state at temperature $T$ is what comes of making imaginary time periodic with period $\hbar/k_BT$. That is not a trick; it is the standard equivalence between the Boltzmann factor $e^{-H/kT}$ and the evolution operator $e^{-iHt/\hbar}$ at imaginary time.

Now take flat spacetime and continue it to imaginary time. The accelerated observer’s coordinates become polar coordinates in the resulting Euclidean space, with their proper time as the angle. An angle is periodic, and requiring the geometry to be smooth at the origin — where the horizon was — fixes its period at exactly $2\pi$.

Converting that period back gives $T = \hbar a/2\pi c k_B$, with the $2\pi$ arriving from the geometry of a plane rather than from any calculation about fields. The same argument applied to a black hole’s exterior gives the Hawking temperature by fixing the periodicity of imaginary time at the horizon, and the two derivations are one derivation.

The Euclidean argument is a clock’s rate pushed one step further than relativity usually pushes it. A moving clock’s rate depends on the frame it is read in; continue the time coordinate to imaginary values and a frame’s time stops being a length and becomes an angle, whose periodicity is a temperature. Nothing about fields is used in the step, which is why the same argument produces the Hawking temperature from the black hole’s exterior geometry with no calculation about radiation in it at all.

What the pictures cannot show #

The derivation assumes eternal uniform acceleration. A detector that starts and stops does not see a clean thermal spectrum, and the transients last for a time of order the inverse temperature — which for any achievable acceleration is longer than any experiment. This is the main practical objection to every proposed measurement.

The detector is idealised. The standard calculation uses a two-level system with a specified coupling, and what a real detector reports depends on its response function. Different detectors accelerating identically report different things, all consistent with the same thermal state.

Circular acceleration is not this. A detector in a circular path has a time-dependent acceleration direction and no single Rindler horizon, and its response is thermal-like rather than thermal. Every storage-ring argument has to confront that difference.

And nothing here is gravitational. The entire calculation takes place in flat spacetime with no curvature anywhere. The connection to black holes runs through the equivalence principle, and the fact that a flat-space calculation predicts a black hole’s temperature is the reason the result is taken as seriously as it is.

Why it matters, given that it cannot be felt #

An effect twenty orders of magnitude below anything measurable needs a defence, and it has three.

It is the cleanest statement that the vacuum is not empty. Quantum field theory’s ground state has structure — correlations across every surface, fluctuations at every scale — and most demonstrations of that fact are indirect: the Lamb shift, the Casimir force, the anomalous magnetic moment. The Unruh effect says the structure is visible directly to anybody who accelerates, which converts a technical statement about renormalisation into a statement about observation.

It is where black-hole thermodynamics stops being a formal analogy. The area law, the surface gravity, the temperature and the entropy of a horizon were first written down as an analogy with thermodynamics that was too good to be accidental. The Unruh calculation, done in flat space with no gravity in it, produces the same temperature — which is why the analogy is now regarded as an identity and why the information problem is a problem rather than a curiosity.

And it fixes what a particle is not. The number of particles is not an invariant. That is a stronger statement than the familiar ones about length and duration, because a particle count is an integer and integers feel as though they should be agreed on. Whether two events are simultaneous is a question with an observer in it, and so, it turns out, is whether there is anything there.

The ladder from here #

Later rungs on this anchor: the Bogoliubov transformation between inertial and Rindler modes, which is where the Planck factor actually comes from; the Unruh–DeWitt detector and its response function; the thermodynamics of the Rindler wedge, including the entropy of the horizon; and the relation to the Euclidean approach, in which the temperature is the periodicity of imaginary time and the whole derivation is three lines.

The neighbouring ladders are the wall of silence behind a rocket, which is the horizon this temperature belongs to, the horizon that nothing marks, which is the gravitational case, and the entropy that lives on a surface, which is the same entanglement counted rather than heated.