# Unruh effect

The hypothetical Unruh effect (or sometimes Fulling–Davies–Unruh effect) is the prediction that an accelerating observer will observe black-body radiation where an inertial observer would observe none. In other words, the background appears to be warm from an accelerating reference frame; in layman's terms, a thermometer waved around in empty space, subtracting any other contribution to its temperature, will record a non-zero temperature. The ground state for an inertial observer is seen as in thermodynamic equilibrium with a non-zero temperature by the uniformly accelerated observer.

The Unruh effect was first described by Stephen Fulling in 1973, Paul Davies in 1975 and W. G. Unruh in 1976.[1][2][3] It is currently not clear whether the Unruh effect has actually been observed, since the claimed observations are under dispute. There is also some doubt about whether the Unruh effect implies the existence of Unruh radiation.

## The equation

The Unruh temperature, derived by William Unruh in 1976, is the effective temperature experienced by a uniformly accelerating detector in a vacuum field. It is given by[4]

$T = \frac{\hbar a}{2\pi c k_\text{B}},$

where $a$ is the local acceleration, $k_\text{B}$ is the Boltzmann constant, $\hbar$ is the reduced Planck constant, and $c$ is the speed of light. Thus, for example, a proper acceleration of 2.5 × 1020 m·s−2 corresponds approximately to a temperature of 1 K.

The Unruh temperature has the same form as the Hawking temperature $T_\text{H} = \hbar g/(2\pi c k_\text{B})$ of a black hole, which was derived (by Stephen Hawking) independently around the same time. It is, therefore, sometimes called the Hawking–Unruh temperature.[5]

## Explanation

Unruh demonstrated theoretically that the notion of vacuum depends on the path of the observer through spacetime. From the viewpoint of the accelerating observer, the vacuum of the inertial observer will look like a state containing many particles in thermal equilibrium—a warm gas.[6]

Although the Unruh effect would initially be perceived as counter-intuitive, it makes sense if the word vacuum is interpreted in a specific way.

In modern terms, the concept of "vacuum" is not the same as "empty space": space is filled with the quantized fields that make up the universe. Vacuum is simply the lowest possible energy state of these fields.

The energy states of any quantized field are defined by the Hamiltonian, based on local conditions, including the time coordinate. According to special relativity, two observers moving relative to each other must use different time coordinates. If those observers are accelerating, there may be no shared coordinate system. Hence, the observers will see different quantum states and thus different vacua.

In some cases, the vacuum of one observer is not even in the space of quantum states of the other. In technical terms, this comes about because the two vacua lead to unitarily inequivalent representations of the quantum field canonical commutation relations. This is because two mutually accelerating observers may not be able to find a globally defined coordinate transformation relating their coordinate choices.

An accelerating observer will perceive an apparent event horizon forming (see Rindler spacetime). The existence of Unruh radiation could be linked to this apparent event horizon, putting it in the same conceptual framework as Hawking radiation. On the other hand, the theory of the Unruh effect explains that the definition of what constitutes a "particle" depends on the state of motion of the observer.

The free field needs to be decomposed into positive and negative frequency components before defining the creation and annihilation operators. This can only be done in spacetimes with a timelike Killing vector field. This decomposition happens to be different in Cartesian and Rindler coordinates (although the two are related by a Bogoliubov transformation). This explains why the "particle numbers", which are defined in terms of the creation and annihilation operators, are different in both coordinates.

The Rindler spacetime has a horizon, and locally any non-extremal black hole horizon is Rindler. So the Rindler spacetime gives the local properties of black holes and cosmological horizons. The Unruh effect would then be the near-horizon form of the Hawking radiation.

## Calculations

In special relativity, an observer moving with uniform proper acceleration a through Minkowski spacetime is conveniently described with Rindler coordinates. The line element in Rindler coordinates is

$ds^2 = -\rho^2 d\sigma^2 + d\rho^2,$

where $\rho = 1/a$, and where $\sigma$ is related to the observer's proper time $\tau$ by $\sigma = g\tau$ (here c = 1). Rindler coordinates are related to the standard (Cartesian) Minkowski coordinates by

$x= \rho \cosh\sigma$
$t= \rho \sinh\sigma.$

An observer moving with fixed $\rho$ traces out a hyperbola in Minkowski space.

An observer moving along a path of constant $\rho$ is uniformly accelerated, and is coupled to field modes which have a definite steady frequency as a function of $\sigma$. These modes are constantly Doppler shifted relative to ordinary Minkowski time as the detector accelerates, and they change in frequency by enormous factors, even after only a short proper time.

Translation in $\sigma$ is a symmetry of Minkowski space: It is a boost around the origin. For a detector coupled to modes with a definite frequency in $\sigma,$ the boost operator is then the Hamiltonian. In the Euclidean field theory, these boosts analytically continue to rotations, and the rotations close after $2\pi$. So

$e^{2\pi i H} = 1.$

The path integral for this Hamiltonian is closed with period $2\pi$ which guarantees that the H modes are thermally occupied with temperature $\scriptstyle (2\pi)^{-1}$. This is not an actual temperature, because H is dimensionless. It is conjugate to the timelike polar angle $\sigma$ which is also dimensionless. To restore the length dimension, note that a mode of fixed frequency f in $\sigma$ at position $\rho$ has a frequency which is determined by the square root of the (absolute value of the) metric at $\rho$, the redshift factor. From the equation for the line element given above, it is easily seen that this is just $\rho$. The actual inverse temperature at this point is therefore

$\beta= 2\pi \rho.$

Since the acceleration of a trajectory at constant $\rho$ is equal to $1/a$, the actual inverse temperature observed is

$\beta = {2\pi \over a}.$

Restoring units yields

$k_\text{B}T = \frac{\hbar a}{2\pi c}.$

The temperature of the vacuum, seen by an isolated observer accelerated at the Earth's gravitational acceleration of g = 9.81 m·s-2, is only 4×10−20 K. For an experimental test of the Unruh effect it is planned to use accelerations up to 1026 m·s-2, which would give a temperature of about 400,000 K.[7][8]

To put this in perspective, at a vacuum Unruh temperature of 3.978×10−20 K, an electron would have a de Broglie wavelength of h/√(3mekT) = 540.85 m, and a proton at that temperature would have a wavelength of 12.62 m. If electrons and protons were in intimate contact in a very cold vacuum, they would have rather long wavelengths and interaction distances.

At one astronomical unit from the sun, the acceleration is $\frac{GM_{S}}{\mathrm{\left(1~AU\right)}^{2}} = 0.005932~\mathrm{m\cdot s^{-2}}$. This gives an Unruh temperature of 2.41×10−23 K. At that temperature, the electron and proton wavelengths are 21.994 km and 513 m, respectively. Even a uranium atom will have a wavelength of 2.2 m at such a low temperature.

## Other implications

The Unruh effect would also cause the decay rate of accelerated particles to differ from inertial particles. Stable particles like the electron could have nonzero transition rates to higher mass states when accelerated fast enough.[9][10][11]