**Quantum field theory **predicts that the temperature of empty space should depend on the observer’s motion, increasing proportionally with acceleration. Here I attempt an accessible introduction to this striking effect, related to **Hawking radiation **and discovered independently by **Fulling, Davies, **and** Unruh**, assuming only sophomore-level physics (including hyperbolic functions) with some assistance from **Mathematica**.

## Hyperbolic Motion

Constant acceleration in Newtonian mechanics is **parabolic**, while constant acceleration in Einsteinian mechanics is **hyperbolic** and asymptotic to light speed c = 1 in natural units. For 1+1-dimensional **Minkowski spacetime**, the *difference* in squared space and time displacements is the square of **proper time** displacement,

d\tau^2 = dt^2 - dx^2.
For *constant* **proper acceleration**, this has the solution

dt = d\tau \cosh a\tau,\\dx = d\tau \sinh a\tau,
with velocity

v = \frac{dx}{dt} = \tanh a\tau \le 1,
where for small times v \sim a \tau \sim a t. If t = 0 and a x = 1 at \tau = 0, then integration gives

a t = \sinh a \tau,\\a x = \cosh a\tau.
Hyperbolic identities then imply

a^2 x^2 - a^2 t^2 = 1
and

a x + a t = e^{a\tau}.

## Quantum Vacuum

Due to **Heisenberg indeterminacy**, electromagnetic fluctuations fill the vacuum. Consider a single such sinusoidal wave of angular frequency \omega_0 = k_0 in natural units. If you move at constant velocity, you observe the wave **doppler-shifted** to a different frequency. But if you move at constant acceleration, you observe the wave doppler-shifted to *a range* of frequencies corresponding to your range of velocities. For an accelerated observer at time \tau,

\phi[x,t] = \exp\left[i(k_0 x + \omega_0 t)\right] = \exp\left[i\omega_0(x + t)\right] = \exp\left[{i \frac{\omega_0}{a} e^{a \tau}} \right] = \phi[\tau].

Expand this waveform as a sum of **harmonics**

\phi[\tau] = \int_{-\infty}^{\infty}\,\frac{d\omega}{2\pi} \Phi[\omega] \exp[-i \omega \tau]
where the **Fourier components**

\Phi[\omega] = \int_{-\infty}^{\infty}d\tau\, \phi[\tau] \exp[+i \omega \tau].
To** regularize** this divergent integral, subtract a tiny imaginary part i \epsilon from the angular frequency \omega to incorporate a decaying exponential factor e^{-\epsilon \tau} in the integrand, and zero it after integrating. Find

\Phi[\omega] = \lim_{\epsilon\rightarrow 0} \int_{-\infty}^{\infty}d\tau\, \exp\left[ i \frac{\omega_0}{a} e^{a\tau} \right] \exp[+i (\omega - i \epsilon) \tau]\\~ \\= \lim_{\epsilon\rightarrow 0}\, \exp\left[ i\frac{\pi}{2} \left(\frac{\epsilon + i\omega}{a}\right) \right]\left(\frac{a}{\omega_0}\right)^{(\epsilon + i \omega)/a}\frac{1}{a} \Gamma\left[\frac{\epsilon + i \omega}{a} \right] = \exp\left[-\frac{\pi}{2} \frac{\omega}{a}\right]\left(\frac{a}{\omega_0}\right)^{i \omega / a}\frac{1}{a} \Gamma\left[i \frac{\omega}{a} \right],
where \Gamma[n+1] = n! analytically continues the factorial function to the complex plane. The spectrum is the absolute square of the **Fourier transform**,

S[\omega] = \left|\Phi[\omega]\right|^2 = \frac{\pi}{a\omega} \left(\coth\left[ \pi \frac{\omega}{a} \right]-1 \right) = \frac{2\pi}{\omega a} \frac{1}{e^{2\pi \omega/a} - 1} \propto \frac{1}{e^{\hbar \omega/k T}-1},
where the **Planck factor **suggests **Bose-Einstein statistics** and a thermal **photon** bath of **temperature** kT = a \hbar / 2\pi. In SI units,

T = \frac{a \hbar}{2\pi k c} \sim 40~\text{zK}~\left(\frac{a}{g_E} \right),
where g_E is** Earth’s** **surface gravity**, and a **zeptokelvin** is very cool.

## Hawking-Unruh Temperature

Just prior to the 1970s work of Fulling, Davies, and Unruh, **Stephen Hawking** famously predicted that despite their reputations **black holes **should radiate with an effective temperature

T = \frac{\kappa \hbar}{2\pi k c},
where \kappa is the black hole’s surface gravity (observed at infinity). The Unruh and Hawking results may be linked by the **equivalence principle**, which equates acceleration and gravity, and by **event horizons**. The black hole horizon is a boundary that **causally** disconnects the interior from the exterior. Similarly, when you accelerate, a **Rindler horizon** appears a distance c^2/a \sim 1~\text{ly} \left(g_E / a \right) behind you, causally disconnecting you from a region of spacetime whose photons you can outrun (so long as your acceleration continues).