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Unruh effect

The Unruh effect, discovered in 1976 by Bill Unruh of the University of British Columbia, 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. The quantum state which is seen as ground state for observers in inertial systems is seen as a thermodynamic equilibrium for the uniformly accelerated observer.

Explanation

Unruh demonstrated 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.

Although the Unruh effect is initially counter intuitive, it makes intuitive sense if the word vacuum is interpreted appropriately, as below.

Vacuum interpretation

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

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 can be linked to this apparent event horizon, putting it in the same conceptual framework as Hawking radiation. On the other hand, the Unruh effect shows 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 is then the near-horizon form of the Hawking radiation.

Calculations

The Unruh effect involves the Rindler coordinates $rho$ and $tau$, which have metric


ds^2 = -rho^2 dtau^2 + drho^2 ,

This is just ordinary Minkowski space in relativistic polar coordinates:

$x= rho cosh\left(tau\right),$
$t= rho sinh\left(tau\right),$

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


e^{2pi i H} = 1 ,

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


beta= 2pi rho ,

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


beta = {2pi over a}

The temperature observed by a uniformly accelerating particle is (in engineering units):

$kT = frac\left\{hbar a\right\}\left\{2pi c\right\}$

The Unruh effect can only be seen when the Rindler horizon is visible. If a refrigerated accelerating wall is placed between the particle and the horizon, at fixed Rindler coordinate $rho_0$, the thermal boundary condition for the field theory at $rho_0$ is the temperature of the wall. By making the positive $rho$ side of the wall colder, the extension of the wall's state to $rho>rho_0$ is also cold. In particular, there is no thermal radiation from the acceleration of the surface of the Earth, nor for a detector accelerating in a circle, because under these circumstances there is no Rindler horizon in the field of view.

The temperature of the vacuum, seen by an isolated observer accelerated at the Earth's gravitational acceleration of g = 9.81 m/s², is only 4×10−20 K. For an experimental test of the Unruh effect it is planned to use accelerations up to 1026 m/s², which would give a temperature of about 400,000 K.

Other implications

The Unruh effect also causes 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.

Although Unruh's prediction that an accelerating detector sees a thermal bath is not controversial, the interpretation of the transitions in the detector in the non-accelerating frame are. It is believed that each transition in the detector is accompanied by the emission of a particle, and that this particle will propagate to infinity and be seen as Unruh radiation. The existence of Unruh radiation is not universally accepted. Some claim that it has already been observed, while others claims that it is not emitted at all. While the skeptics accept that an accelerating object thermalises at the Unruh temperature, they do not believe that this leads to the emission of photons, arguing that the emission and absorption rates of the accelerating particle are balanced.

Experimental Observation of the Unruh effect

Under experimentally achievable conditions for gravitational systems this effect is too small to be observed. In 2005 it was shown that if one takes an accelerated observer to be an electron circularly orbiting in a constant external magnetic field, then the experimentally verified Sokolov-Ternov effect coincides with the Unruh effect.