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Unruh Effect Tee

\[T = \frac{\hbar a}{2\pi c k_B} \approx 4.06\times10^{-21}\ \mathrm{K}\times\frac{a}{1\ \mathrm{m\,s^{-2}}}\]
$29.99 USD
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Size guide S–5XL · inches

Meet The Accelerant, who feels warmth in empty space only because it never stops speeding up. In 1976 William Unruh showed that an observer with constant proper acceleration a sees the quantum vacuum as a thermal bath at temperature T = ħa/(2πck_B), while an observer at rest sees nothing. The effect is tiny: about 4×10⁻²⁰ kelvin at Earth's surface gravity, and it takes 10²⁰ m/s² to reach 0.4 K. It is a firm prediction of quantum field theory but has not yet been directly observed. The Accelerant on a pixel-art unisex tee; equations, symbols and sources in the physics panel below.

"Empty to you. Warm to me."

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Details

Size & fitS–5XL

Unisex heavy cotton (Gildan 5000), classic fit.

SizeWidthLengthSleeve
S182815.1
M202916.5
L223018
XL243119.5
2XL263221
3XL283322.4
4XL303423.7
5XL323525

For scale: size M is 20 in (50.80 cm) wide, about 3.143 × 1034 Planck lengths (lP = 1.616255 × 10−35 m). A tape measure is quicker.

Measurements in inches, ±1.5 in tolerance.

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The physics

If you accelerate steadily through empty space, quantum field theory predicts you would see it as a warm bath of particles, though someone not accelerating sees nothing. The temperature is tiny: you would need about 10²⁰ m/s² to feel 0.4 K. It is predicted, not yet directly seen.

The Accelerant has never stood still, and so has never seen empty space as empty.

Quantum field theory says the vacuum is the state with no particles, but who counts as having "no particles" depends on the observer. In 1976 William Unruh showed that an observer moving with constant proper acceleration a, the acceleration they would feel themselves, sees the ordinary vacuum as a warm bath of particles with a thermal spectrum at temperature T = ħa/(2πck_B), where ħ is the reduced Planck constant, c the speed of light and k_B the Boltzmann constant. An inertial observer in the same region sees no particles at all. Related work by Stephen Fulling (1973) and Paul Davies (1975) points the same way.

The second equation shows why. A constantly accelerating observer follows a hyperbola in spacetime, x² − c²t² = (c²/a)², drawn on the back of this shirt. The hyperbola approaches the light-cone lines but never crosses them, so there is a horizon: a region of spacetime the observer can never receive signals from. Tracing over the vacuum on the far side of that horizon leaves a thermal state, the same mathematics as Hawking radiation from a black hole.

Worked number: ħ/(2πck_B) ≈ 4.06×10⁻²¹ K per m/s². At Earth's surface gravity, 9.81 m/s², T ≈ 4×10⁻²⁰ K. To feel 0.4 K you would need a ≈ 10²⁰ m/s², about ten million million million times g.

That is why the effect has not yet been directly observed, though it is a firm prediction of quantum field theory, and experiments with accelerated electrons, analogue systems and high-intensity lasers have been proposed to test it.

The Accelerant: "You call it vacuum. Speed up the way I do and you will feel the heat."

Equations

\[T = \frac{\hbar a}{2\pi c k_B} \approx 4.06\times10^{-21}\ \mathrm{K}\times\frac{a}{1\ \mathrm{m\,s^{-2}}}\]
\[x^{2} - c^{2}t^{2} = \left(\frac{c^{2}}{a}\right)^{2}\]
SymbolMeaningUnit
\(T\) Unruh temperature seen by the accelerating observer \(\mathrm{K}\)
\(\hbar\) reduced Planck constant \(\mathrm{J·s}\)
\(a\) constant proper acceleration of the observer \(\mathrm{m/s²}\)
\(c\) speed of light \(\mathrm{m/s}\)
\(k_B\) Boltzmann constant \(\mathrm{J/K}\)
\(x\) position in an inertial frame \(\mathrm{m}\)
\(t\) time in that inertial frame \(\mathrm{s}\)

Sources

  1. Unruh (1976) Notes on black-hole evaporation, Phys. Rev. D 14, 870
  2. Crispino, Higuchi & Matsas (2008) The Unruh effect and its applications, Rev. Mod. Phys. 80, 787
  3. NIST CODATA 2022 — Boltzmann constant
  4. NIST CODATA 2022 — reduced Planck constant

Source Unruh, W. G. (1976). Notes on black-hole evaporation. Physical Review D 14, 870–892.doi.org/10.1103/PhysRevD.14.870

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