Fly at constant velocity and it is vacuum; accelerate and the same vacuum is a heat bath ── temperature depends on the observer's state of motion too
The finale of the main series. Episode 1 wrote that temperature is not a property of matter. Here we push that claim all the way ── temperature depends on the observer's state of motion too. The same empty space that an observer moving at constant velocity calls "vacuum" is felt by an accelerating observer as a heat bath at temperature \(T\) (the Unruh effect, 1976). Carry a thermometer and accelerate, and the reading goes up. And that temperature is written \(T=\hbar a/2\pi ck_B\) ── an expression containing \(\hbar\), \(c\) and \(k_B\) all at once, which in this collection means the one place where quantum (ℏ), relativity (c) and heat (k_B) intersect. The origin of the \(2\pi\) is beautiful. Episode 6's "circle in imaginary time" becomes, for an accelerating observer, literally an angle ── and one lap of an angle is \(2\pi\). That is all it is. Apply the same argument to a black hole and the Hawking temperature falls out.
Vacuum means "a state with not a single particle." But in quantum field theory, how you count "how many particles there are" depends on the observer. A state with zero particles for an observer moving at constant velocity is not zero for an accelerating one ── that is the content of the Unruh effect.
What matters here is that what the accelerating observer sees is not mere noise but a proper thermal distribution. Measure the energy distribution of the particles and it follows Episode 2's Boltzmann factor \(e^{-E/k_BT}\) exactly. A temperature is genuinely definable.
\(a\) is the observer's (proper) acceleration. The telltale sign is that \(\hbar\) (quantum), \(c\) (relativity) and \(k_B\) (heat) are all present ── this is an expression that can only exist where the three fields meet.
| Acceleration | a [m/s²] | Unruh temperature |
|---|---|---|
| Earth's gravity | 9.8 | 4×10⁻²⁰ K |
| a fighter jet at 9 g | 88 | 3.6×10⁻¹⁹ K |
| to match the CMB at 2.7 K | 6.7×10²⁰ | 2.7 K |
| to match room temperature, 300 K | 7.4×10²² | 300 K |
Earth's gravity gives \(10^{-20}\) K ── ten more orders of magnitude below the lowest temperature ever reached (of order \(10^{-10}\) K). Which is why you can never notice it in daily life.
Turned around, the smallness of that number \(4\times10^{-21}\) is built out of \(\hbar\) (small) times \(1/c\) (small) times \(1/k_B\) (large) ── the invisibility of the Unruh effect is precisely why quantum, relativity and heat do not ordinarily meet.
This \(2\pi\) did not show up vaguely as "pi." It is genuinely the \(2\pi\) of a circle.
Write spacetime in the coordinates of an observer flying with constant acceleration \(a\) (Rindler coordinates) and rotate time onto the imaginary axis as in Episode 6: the metric takes the form of polar coordinates in a plane ── the radius is the spatial direction, and the angle is the imaginary-time direction.
The decisive point is that an angle must come back to itself after one lap. One lap is \(2\pi\). Otherwise there is a conical singularity (a sharp point of discontinuity) at the origin. For spacetime to be smooth, the imaginary-time period must be
Set that equal to Episode 6's conclusion, "imaginary-time period \(=\hbar/k_BT\)": \(\hbar/k_BT=2\pi c/a\), i.e. \(T=\hbar a/2\pi ck_B\).
The temperature came out of a demand that spacetime be smooth. That is the most beautiful thing in this episode.
To hold still just outside a black hole's horizon you must keep accelerating in order not to fall. By the equivalence principle (Episode 6 of the sister series "Relativity That Clicks"), acceleration and gravity cannot be told apart ── so there should be an Unruh temperature there too.
Indeed, converted to the temperature seen from far away:
The form is the same as Unruh's, \(T=\hbar\kappa/2\pi ck_B\), with the surface gravity \(\kappa\) playing the role of the acceleration. So is the derivation ── the condition that the Euclideanised spacetime have no conical singularity fixes the imaginary-time period, and with it the temperature.
And since it goes as \(1/M\), the lighter the black hole the hotter. A solar mass gives \(6\times10^{-8}\) K (eight orders colder than the CMB), a lunar mass gives 1.7 K, and lighter still gets hotter and hotter until it ends in explosive evaporation.
The figure lays out the temperatures of the universe on a single logarithmic scale. The upper slider moves acceleration and the lower one black-hole mass, each driving its own cursor.
What is fun is that the two cursors move along the same axis ── acceleration and black-hole mass, two utterly different things, become comparable on one temperature scale. And both sit many orders of magnitude away from everyday temperatures (room temperature, the CMB).
Over seven episodes, temperature changed its face three times.
| Episode | Temperature is | The ratio that matters |
|---|---|---|
| 1 | another name for energy; \(k_B\) is a conversion factor | \(E/k_BT\) |
| 2 | the slope of the logarithm of a count, \(\beta=d\ln\Omega/dE\) | \(E/k_BT\) |
| 3 | the gauge of which degrees of freedom can be woken | \(\hbar\omega/k_BT\) |
| 4 | what fixes the odds of violating the second law | \(\sigma/k_B\) |
| 5 | the reciprocal \(1/T\) is the real quantity; with a ceiling it goes negative | \(\Delta E/k_BT\) |
| 6 | the reciprocal of a period in imaginary time | \(E\beta\hbar/\hbar\) |
| 7 | a geometric quantity that depends on the observer's motion too | \(\hbar a/ck_BT\) |
Temperature is not a property of matter. Nor is it "hotness."
It is the slope with which energy is shared out, the reciprocal of a circle's circumference in imaginary time, and a quantity that depends on the observer's state of motion. The one thing that never changed ── only the dimensionless ratio, divided by \(k_B\), ever matters. \(k_B\) itself, as Episode 1 showed, is a mere exchange rate, and it disappears the moment you measure temperature in energy.
"Dimensionful quantities are stage machinery; what matters are dimensionless ratios" ── the view that runs through the whole collection held for temperature as well. The Physics Cube (\(c\), \(\hbar\), \(G\)) now gains a fourth axis, \(k_B\). And this episode's \(T=\hbar a/2\pi ck_B\) was a rare expression in which three of those four show their faces at once.
Established (as theory): the Unruh effect \(T=\hbar a/2\pi ck_B\) (Fulling 1973, Davies 1975, Unruh 1976); Hawking radiation and the Hawking temperature \(T=\hbar c^3/8\pi GMk_B\) (1974); that Euclideanised Rindler coordinates become polar coordinates and the condition of avoiding a conical singularity fixes the imaginary-time period at \(2\pi c/a\); that the same works for a black hole via the surface gravity; and the numbers \(4.05\times10^{-21}\) K per (m/s²) and \(6.17\times10^{-8}\ \mathrm{K}\times M_\odot/M\). All standard as theory, with several independent derivations.
But: (1) Neither the Unruh effect nor Hawking radiation has been directly observed. The required acceleration (of order \(10^{20}\ \mathrm{m/s^2}\)) and the temperature of astrophysical black holes (\(10^{-8}\) K, eight orders below the CMB) are far outside current technology. Analogue systems in the laboratory (sonic black holes, BECs) have reported the corresponding phenomena, but those are different systems obeying the same equations, not a test in real spacetime. (2) The statement "particle number depends on the observer" rests on the field-theoretic fact that the definition of a particle (which modes count as positive-frequency) is observer-dependent. The accurate framing is that the vacuum itself does not change; the way it is read does. (3) The Unruh effect is derived for the idealisation of constant acceleration continuing forever; finite-duration acceleration gives departures from a thermal spectrum. (4) The step that uses the equivalence principle to read "gravity = acceleration" holds only locally (the same caveat as Episodes 6 and 7 of "Relativity That Clicks"). (5) The figure is a schematic laying out orders of magnitude; the scale is logarithmic.
What is vacuum to an observer moving at constant velocity looks to an accelerating observer like a heat bath at \(T=\hbar a/2\pi ck_B\) (the Unruh effect). Earth's gravity gives \(4\times10^{-20}\) K ── too small to see, and that smallness is the numerical statement of why quantum, relativity and heat do not ordinarily meet.
The \(2\pi\) came from Episode 6. Euclideanise an accelerating observer's spacetime and it becomes polar coordinates, with imaginary time in the role of an angle. An angle must have period \(2\pi\) or a conical singularity appears, so the imaginary-time period is fixed at \(2\pi c/a\); set that equal to Episode 6's "period \(=\hbar/k_BT\)" and out comes the temperature. The temperature fell out of a demand that spacetime be smooth. The same argument at a black hole gives \(T=\hbar c^3/8\pi GMk_B\) ── the lighter, the hotter.
And the series' conclusion. Temperature is neither a property of matter nor "hotness"; it is the slope with which energy is shared out, the reciprocal of a circle's circumference in imaginary time, and a geometric quantity that depends on the observer's state of motion. One thing held across all seven episodes ── only the dimensionless ratio divided by \(k_B\) ever matters. \(k_B\) itself is an exchange rate, and it disappears if you measure temperature in energy. The Physics Cube has gained its fourth axis.
Print / make a PDF: ⌘+P (Ctrl+P on Windows). On screen the two sliders ── acceleration and black-hole mass ── drive cursors along the same temperature scale. "See the answer" opens each solution.