Pis'ma v ZhETF, vol.86, iss.9, pp. 702-706
© 2007 November 10
On the physical meaning of the Unruh effect
E. T. Akhmedov1), D.Singleton1') + ITEP, 117218 Moscow, Russia + Physics Department, CSU Fresno, Fresno, CA 93740-8031, USA Submitted 20 September 2007
We present simple arguments that detectors moving with constant acceleration (even acceleration for a finite time) should detect particles. The effect is seen to be universal. Moreover, detectors undergoing linear acceleration and uniform, circular motion both detect particles for the same physical reason. We show that if one uses a circularly orbiting electron in a constant external magnetic field as the Unruh-DeWitt detector, then the Unruh effect physically coincides with the experimentally verified Sokolov-Ternov effect.
PACS: 04.62.+v
Hawking radiation [1] and the closely related Unruh [10] radiation are often seen as first steps toward combining general relativity and quantum mechanics. Under achievable conditions for gravitational system these effects are too small to be experimentally testable. In this letter we examine the physical meaning of the Unruh effect and argue that for uniform, circular acceleration the Unruh effect has already been observed. Given the close connection between the Hawking and Unruh effects this experimental evidence for the latter gives strong support for the former.
It has been shown [10] that a detector moving eternally with constant, linear acceleration a should detect particles with Planckian distribution of temperature T = a/2ir. The non-inertial reference frame which is co-moving with the detector has an event horizon. Even massless particles radiated a distance 1 ¡a behind the detector would never catch up with an eternally accelerating detector. It is the reference frame co-moving with the eternally accelerating detector which "sees" the Rindler metric. Thus it seems that the Unruh effect is strongly related to the existence of the horizon. However, if the effect only exists for an eternally accelerating observer/detector then it can be discarded as unphysi-cal since one can never have a detector that undergoes constant acceleration from infinite past time to infinite future time. Due to the Hawking radiation [1] black-holes do not exist eternally. As well a positive cosmo-logical constant (giving a de-Sitter space-time) should eventually be radiated away to zero.
The real question is whether or not a detector which moves with linear, constant acceleration for a finite time will see particles (e.g. a detector which is initially stationary, accelerates for a finite time and then continues
e-mail: akhmedov0itep.ru, dougs0csufresno.edu
with constant velocity). We are interested whether the detector gets excited or not during the period when it moves homogeneously. We are not interested in the detector's reaction during the periods when the acceleration is turned on or off. The reaction of the detector which we are interested in does not come from internal forces where one part of the detector can move with respect to another (like the arrow of an ammeter which moves with respect to its box if it is shaken), but is due to the existence of a universal radiation in the detector's non-inertial reference frame. We consider two kinds of homogeneous accelerations: (i) from a force that is constant in magnitude and direction resulting in linear accelerated motion; (ii) from a force that is constant only in magnitude resulting in circular motion. We take as our definition of a particle that thing which causes a detector to click, i.e. jump from one of its internal energy levels to a higher one. We do not know any other invariant definition of a particle.
If detectors do click during homogeneous, accelerated motion occurring for a finite time, then the Unruh effect does not depend on the existence of a horizon2), since for finite time acceleration the co-moving frame "sees" a metric different from Rindler and does not have a horizon: a massless particle with light speed velocity following the detector will eventually catch up with it if the detector accelerates for finite time.
Once this idea is accepted, we can go further and state that there is no significant physical difference between detectors in homogeneous, linear acceleration versus uniform circular motion. Note, the reference frame co-moving with the detector performing eternal homo-
2'Here we understand notion of the horizon as the eternally existing surface from inside of which classically nothing can ever escape.
geneous, circular motion does not have a horizon (only a light-surface). A particle can eventually catch up with a circularly moving detector.
In this letter we show (following other authors) that detectors performing homogenous linear and circular accelerations (or any other homogeneous non-inertial motion in the empty Minkowski space) do detect particles, and they do this for the same physical reason. Moreover, we show that the circular Unruh effect has been well known for a long time under a different name and has even been experimentally observed.
In all cases we consider Minkowski space-time, and take % = 1 and c = 1. For simplicity we consider a linear interaction of the detector with a free scalar field. We consider the following two processes: (i) the detector is originally in its ground state and then gets excited because of its non-inertial motion; (ii) the detector is originally in its excited state and then relaxes to its ground state. In both cases the background QFT is originally in its ground state. We want to find the probability rates for these two processes. As a result of these processes the background QFT will become excited, i.e. the detector will radiate quanta of the background QFT when performing the above two processes.
To leading order in perturbation theory the probability rate per unit time is [3]:
Wzr OC
/+oo
dTe^iA£TG \x(t-T/2) , x(t + t/2) , (1)
-oo
up
where t is the detector's proper time; A£ = £, — ¿"down > 0 is the discrete change of the detector's internal energy level; the "—" sign, both in the LHS and in the exponent, corresponds to the first process, while the "+" sign corresponds to the second process mentioned above; G[x(t — r/2), x(t + r/2)] = = (o (f>[x(t - t/2)] </>[x(t + t/2)] is the Wightman function of the scalar field <f>. This function measures the correlation between fluctuations of the scalar field at two points in the space-time in the vacuum of the scalar QFT. In our case these two points are on the same trajectory x(t). Because of this these points are causally connected to each other even for the eternally, linearly accelerating detector. However, as we will see below the important contribution to wT in all cases comes from the imaginary t.
The reason why we consider the detector approach to the Unruh effect is that then all our considerations can be made completely generally covariant [4]. This allows us to address the question as to whether or not a detector making a particular motion in Minkowski space-time sees/detects particles.
Eq. (1) shows that the probability rates wT are Fourier images of the Wightman function. The Wight-man function is a universal characteristic of the field, and its features universally characterize the reaction of a detector moving along the trajectory x(t). Of course the spectrum of the detected particles depends on the detector's trajectory.
Note that eq. (1) is written for the simplest linear type of interaction of the detector with <j> [3, 4]. In cases with a more complicated interaction, say non-linear or via derivatives of the field, one would get probability rates that are Fourier images of powers or derivatives of the Wightman function. It will be clear from the discussion below that this would not change the spectrum of the detected particles, but would only alter the time necessary to reach the equilibrium distribution over the detector's energy levels under the homogeneous background radiation.
Thus, the question is reduced to the study of the characteristic features of the Wightman function of free massless particles:
G(x,y) =
•y|
2 '
(2)
with various homogeneous trajectories - x(ti) = x and x{t2) = y - plugged into it. Below we are going to consider three different trajectories. All poles of the two-point correlation functions (both in coordinate and momentum spaces) have physical meanings based on intuition from condensed matter physics.
In the case of motion with constant velocity one can show that (see e.g. [4]): w_ = 0, and w+ oc AS. The physical meaning of this result is as follows: If the detector moves with constant velocity in the vacuum of a QFT there is zero probability for it to get excited, w- = 0. However, if the detector was originally in the excited state, there is a non-zero probability for it to radiate spontaneously, w+ ^ 0.
For the case of eternal, constant, linear acceleration -x(t) = sinh [at], i cosh [at], 0, 0) with t the detector's proper time and a its acceleration - the Wightman function is:
G [«(i-r/2) , x(t + t/2)J
oc
sinh2 [f (r-ic)]
(3)
The integral in eq. (1) is taken using contour integration in the complex t plane. Since A£ > 0, the integral w_ in eq. (1) uses a contour which is closed with a large, clockwise semi-circle in the lower complex half-plane. This contour is denoted by C_. For w+ the contour is closed with a large, counterclockwise semi-circle in the
x
a
upper complex half-plane, and is denoted by C+. This choice of contours for wT is used everywhere below.
Unlike the constant velocity case, the Wightman function now has non-trivial poles encircled by the C_ contour, hence, w_ ^ 0. The positions of the poles are easy to find, so the integral in eq. (1) can be calculated exactly with the result:
AS
w _ oc
exp
2 7r AS
w+ oc AS
exp
2 7r AS
(4)
Therefore a detector moving with constant acceleration in the vacuum of the background QFT does detect particles. The detecte
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