Transition rate of the Unruh-DeWitt detector
in curved spacetime
We examine the Unruh-DeWitt particle detector coupled to a scalar field in an arbitrary Hadamard state in four-dimensional curved spacetime. Using smooth switching functions to turn on and off the interaction, we obtain a regulator-free integral formula for the total excitation probability, and we show that an instantaneous transition rate can be recovered in a suitable limit. Previous results in Minkowski space are recovered as a special case. As applications, we consider an inertial detector in the Rindler vacuum and a detector at rest in a static Newtonian gravitational field. Gravitational corrections to decay rates in atomic physics laboratory experiments on the surface of the Earth are estimated to be suppressed by 42 orders of magnitude.
The Unruh-DeWitt model for a particle detector [1, 2] is an important tool for probing the physics of quantum fields wherever noninertial observers or curved backgrounds are present. In such cases there is often no distinguished notion of a “particle,” analogous to the plane-wave modes in Minkowski space, but an operational meaning can be attached to the concept by analysing the transitions induced among the energy levels of a detector coupled to the field. Upwards or downwards transitions can then be interpreted as due to absorption or emission of field quanta, or particles. The best-known applications of this procedure are those for which the spectrum of transitions is thermal, which is the case for uniformly accelerated detectors in Minkowski space , inertial detectors in de Sitter space , and detectors at rest in the exterior Schwarzschild black hole spacetime .
In first-order perturbation theory, the transition probability of the Unruh-DeWitt detector is proportional to a quantity known as the response function, which involves integrating the Wightman distribution of the quantum field over the worldline of the detector. When the quantum state of the field is sufficiently regular and the detector is switched on and off smoothly, the response function is well defined , and the physical interpretation is that the response function is then proportional to the probability of a transition to have occurred by a time at which all interaction has already ceased. If, however, one wishes to address the probability of a transition to have occurred by a time at which the interaction is still ongoing, the response function is no longer well defined because the switching function then has a sharp cut-off at a singularity of the Wightman distribution. In special cases in which the trajectory is stationary, the vacuum state is invariant under the Killing vector generating this stationary motion and the detector has been switched on in the infinite past [1, 2, 3, 4], the issue can be bypassed by formally integrating over the whole trajectory and factoring out the infinite total proper time, because by stationarity the transition rate can then be argued to be time-independent. But in a general setting this is not possible, and seemingly inconspicuous regularisations of the Wightman distribution can lead to unphysical results, even for uniformly accelerated motion in Minkowski space [6, 7].
A way to address this problem is to regard the sharp detector switch-off as a limit of a family of smooth switch-offs and investigate how the results depend on the way in which the limit is taken. In  this issue was investigated for a massless scalar field in four-dimensional Minkowski space, with the quantum field in the Minkowski vacuum. The response function with a smooth switching function was written in a form in which the integrand is no longer a distribution but a genuine function, and it was shown that a well-defined notion of a transition rate emerges when the switching time scale is small compared with the total duration of the coupling. It was also shown that in the appropriate limits this transition rate coincides with that obtained by regularising the sharply switched-off detector by a nonzero spatial size [6, 7]. The key point is that when the Wightman distribution under the integral is represented by an -regularised function, the regulator limit and the limit to sharp switching do not in general commute and the first must be taken before the second.
The aim of this paper is to extend these results to a more general setting. For this, we will start in section 2 with a review of the Unruh-DeWitt detector, with special attention to the procedure introduced in  that allows limits of switching functions to be considered. In section 3 the results of  are generalised to a situation in which Minkowski space is replaced by an arbitrary four-dimensional globally hyperbolic spacetime, the Minkowski vacuum state by an arbitrary Hadamard state and the massless scalar field by a scalar field with arbitrary mass and curvature coupling. We shall in particular obtain a simple and manifestly well-defined expression for the difference in the response of detectors that have the same switching function but move in different quantum states of the field, on different trajectories or even in different spacetimes. The limit of sharp switching is discussed in section 4. In sections 5 and 6 we use these results to obtain the detector transition rate in two examples of interest: an inertial detector in the Rindler vacuum in Minkowski space, and a detector at rest in a static Newtonian gravitational field. The results are summarised and discussed in section 7. Certain technical properties of the detector response in the Rindler vacuum are established in the Appendix.
Throughout this paper we will assume a Lorentzian metric of signature , using the sign convention of Misner, Thorne and Wheeler . We use units in which , while keeping . Spacetime points are denoted by sans-serif letters. The symbol denotes a quantity for which is bounded as . denotes a quantity that is bounded in the limit under consideration.
2 Particle detectors and their regularisation
We consider a detector consisting of an idealised atom with two energy levels, and , with associated energy eigenvalues and . The detector is following a timelike trajectory , parametrised by its proper time , in a four-dimensional Lorentzian globally hyperbolic manifold . The coupling of the detector to a real scalar field of mass and curvature coupling is given by the interaction Hamiltonian
where is a coupling constant, is the detector’s monopole moment operator and is a smooth non-negative function of compact support. is called the switching function: the interaction takes place only when is nonvanishing, and because has compact support the interaction has a finite duration. If is not defined for all , we assume the support of to be in the open interval in which is defined.
We take the initial state of the joint system before the interaction to be , where the field state is an arbitrary Hadamard state [10, 11]. We are interested in the probability for the detector to be observed at state after the interaction has been switched off. Treating the coupling constant as a small parameter, working to first order in perturbation theory in , and summing over the unobserved final state of the field, this probability reads [5, 12, 13]
where the response function is given by
and the distributional correlation function is the pull-back of the Wightman distribution to the detector worldline,111We denote both the spacetime Wightman distribution and its pull-back to the detector worldline by the same letter, writing out the arguments explicitly in places where ambiguity could arise.
The response function thus encodes the properties that depend on the state and the detector trajectory, while the prefactor in (2.2) is a constant that only depends on the detector’s internal properties. We shall from now on suppress the prefactor and refer to the response function simply as the probability.
To summarise, (2.3) gives an unambiguous answer to the question “What is the probability of the detector being observed in the state after the interaction has ceased?”
The meaning of the distributional correlation function under the integral in (2.3) is somewhat subtle. Recall that the Wightman distribution in a Hadamard state can be represented by a family of functions [10, 11]
where is a positive parameter, is the squared geodesic distance between and , and is any globally-defined future-increasing function. The logarithm denotes the branch that is real-valued on the positive real axis and has the cut on the negative real axis. is the Van Vleck determinant, which is smooth for sufficiently near-by and , the function is a polynomial in , and the function can be chosen for arbitrarily large by taking the degree of the polynomial sufficiently high. The -prescription in (2.5) defines the singular part of : the action of the Wightman distribution is obtained by integrating against test functions and taking the limit , and this limit can be shown to be independent of the choice of the global time function . Now, the distributional correlation function (2.4) is the pull-back of to the detector’s worldline, which is a submanifold. It follows that the action of the distribution is obtained by pulling back the function to the function , integrating against test functions and taking the limit [5, 14, 15]. Formula (2.3) must thus be understood as
where the integrand is now an ordinary function and the singular part of has been encoded in the prescription. As , where the oveline denotes complex conjugation, we have , and it follows that (2.6) can be written in the equivalent form [6, 8]
Although formulas (2.6) and (2.7) are suitable for computing the detector’s response, these formulas do not display a clear separation between those properties of the response that depend on on the trajectory and the quantum state and those properties that only depend on the choice of the switching function. Neither do these formulas exhibit how the response depends on the proper time along a given trajectory. Several authors [6, 16, 17, 18] have therefore addressed the question: “If the detector is turned on at proper time and read at proper time , while the interaction is still on, what is the probability that the transition has taken place?” If issues of regularisation could be ignored, this would amount to adopting in (2.3) the switching function
where is the Heaviside function. The transition probability then becomes a function of the reading time and can be written as
and we can define the instantaneous transition rate as its derivative with respect to ,
where . thus represents the number of transitions per unit time in an ensemble of identical detectors. It is this instantaneous transition rate that one expects to have the -independent Planckian spectrum in the Unruh effect in Minkowski spacetime and in its generalisations to curved spacetimes, once is taken to to avoid transient effects. A comprehensive recent review of the Unruh effect can be found in .
We note in passing that the transition rate (2.10) is not directly related to transition rates that could be measured with a single ensemble of detectors. Given an ensemble of identical detectors on a given trajectory, gives the fraction of detectors that have undergone a transition when observed at time , but as an observation alters the dynamics of the system, no longer has this interpretation after a first observation has been made. To measure , one therefore needs a set of identical ensembles, such that each ensemble is used to measure at just a single value of . Note in particular that may well be negative at some values of [20, 21]. may thus be difficult to measure operationally, but it is nevertheless of interest as a nonstationary generalisation of the transition rate that naturally arises in stationary situations.
Returning to formulas (2.9) and (2.10), the difficulty with them as written is that the ‘switching function’ (2.8) is not smooth. We are no longer guaranteed that replacing by in (2.9) and (2.10) and taking the limit would give a result that is independent of the choice of the global time function in (2.5). Case studies have shown that the result depends on the choice of the time function for Minkowski vacuum in Minkowski space [6, 7, 20, 21] and the Euclidean vacuum in de Sitter space [20, 21], and the methods of Appendix A of  can be adapted to show that the same holds for arbitrary Hadamard states in an arbitrary spacetime. Formula (2.10) does therefore not provide a well-defined notion of an instantaneous transition rate.
One way to address this problem was introduced in  and further developed in [7, 20, 21]. The idea is to replace the correlation function in (2.10) by a correlation function in which the field operator has been smeared over a spacelike hypersurface orthogonal to the trajectory. The weight function in the smearing is characterised by a positive length parameter , which acts as a regulator and corresponds physically to the spatial size of the detector in its instantaneous rest frame. At the end the pointlike detector limit is taken. This scheme does not rely on the choice of a time function to regularise the Wightman distribution, and in Minkowski space the introduction of the spatial hypersurfaces is straightforward  and there are partial results regarding independence of the choice of the weight function . An implementation of the scheme in de Sitter space was given in [20, 21, 22]. However, the spacelike surfaces introduced in Minkowski space in  are not easily generalisable to spacetimes without a high degree of symmetry, and it would seem desirable to attach a meaning to the instantaneous transition rate within the framework of the conventional regularisation of the Wightman distribution (2.5).
A way that stays fully within the conventional regularisation of (2.5), (2.6) and (2.7) was introduced in  in the special case of Minkowski spacetime, massless scalar field and the Minkowski vacuum state. Adopting a Lorentz frame with global Minkowski coordinates and choosing as the global time function, formula (2.7) for the transition probability becomes
where is the squared geodesic distance between and and . The limit can be computed explicitly, with the result 
When the switching function equals 1 over an interval of length , and the switch-on and switch-off each take place within an interval of length with a profile that scales with but whose shape is otherwise fixed, the leading behaviour of the transition rate (defined as the derivative of (2) with respect to ) at is
where now is the squared geodesic distance between and . In the limit , the transition rate (2.13) agrees with that obtained from spatial smearing in , and it reproduces the expected Planckian spectrum when the trajectory is uniformly linearly accelerated and . Further properties of this transition rate are discussed in [7, 8].
In this paper we generalise the Minkowski vacuum results (2) and (2.13) to a general Hadamard vacuum state in four-dimensional spacetime, for a field with arbitrary values of the mass and the curvature coupling. We shall show that most of the arguments in  carry over to this situation, and we shall find the expressions that generalise (2) and (2.13). These expressions will then be applied to two examples.
3 Regulator-free response function in a general Hadamard state
In this section we obtain a regulator-free expression for the response function by computing explicitly the limit in (2.7). Following the procedure used in , we split the -integral into the subintervals and , with , estimate the integrand in each subinterval and finally combine the results.
We shall make use of the small expansions
where the Ricci scalar , the squared (covariant) acceleration and are evaluated at the point and the dots indicate proper time derivatives.
Consider in (2.7) the subinterval , and let denote the pointwise limit of as . Replacing by creates under the -integral an error that equals times the quantity
where the functions , , and are each evaluated at the pair and . We shall show that this error term does not contribute to (2.7) after the limit is taken.
Consider first in (3.1) the contribution from and . We split this contribution into its odd and even parts in . The part that is odd in can be written as
where . As the switching function makes the upper limit of the -integral finite, it follows from (3.1a) and (3.1d) that the quantities and are bounded by constants that are independent of . We can therefore write (3.3) as
where the estimate holds uniformly in . As the functions and are at small , the integrand in (3.4) is bounded by a constant times , and (3.4) is thus of order . Similarly, the part that is even in can be written as
Consider then in (3.1) the contribution from the logarithmic terms. Keeping track of the branches of the logarithms, we can write this contribution as
We now turn to the subinterval in (2.7). The singularity of at implies that we cannot directly replace by , and we will need to examine the small behaviour of more closely.
The imaginary part of the logarithm is bounded and its contribution in (3.7) is therefore of order . To estimate the real part of the logarithm, we write , with , and use the expansions (3.1) to obtain
where the term holds uniformly in . Hence
where again the term holds uniformly in . The contribution in (3.7) is therefore of order . As this estimate holds uniformly in , by virtue of the compact support of , the logarithmic term does thus not contribute in (2.7).
Finally, consider the term in (2.5). From (3.8) we see that is bounded, and hence . It follows from (3.1b) that we may replace by , and we may similarly replace the factor by . What remains is to analyse the small behaviour of the expression
where and are evaluated at . In the special case in which the spacetime is Minkowski space and the global time function is the Minkowski time coordinate in a given Lorentz frame, this analysis was carried out in , and the techniques used therein generalise to (3.10) in a straightforward way. Splitting into its even and odd parts in as , and writing with , we find222Our formula (3.11b) corrects a typographical error in formula (3.4b) of .
In (3.11a) the integral of the first term is elementary, and multiplying the second and third term by yields a total -derivative that can be taken outside the integral. The result is
The integral in (3.11b) is elementary, and multiplying the result by we obtain
3.3 Joining the subintervals
Following , we take the term proportional to under the -integral, add and subtract under the -integral the term and group the terms in the form
where in the last term the interchange of the -integral and the -integral is justified by absolute convergence of the double integral. The limit can now be taken by simply setting . In the last term the reason is that the -integral, when regarded as a function of , has a Taylor expansion that starts with . In the term involving , the reason is that the real part of has the small behaviour of plus an integrable function of , by virtue of (2.5) and (3.1a). The final result for the response function is thus
The first two terms in (3.16) depend only on the switching function but neither on the quantum state, the spacetime or the trajectory. If we compare two detectors in different quantum states of the field, on different trajectories or even in different spacetimes, but having the same switching function, the difference of the responses is given by
where and are the pull-backs of the unregularised Wightman distributions in the two situations. The representation (2.5) of the Wightman distribution in a Hadamard state guarantees that the divergences in (3.17) cancel and the integral is well defined. This is particularly convenient for numerical calculations.
4 Sharp switching limit
In this section we discuss the response function (3.16) in the limit of sharp switch-on and switch-off. As in the case of Minkowski vacuum , we shall isolate the divergence due to the sharp switching from a finite remainder and show that a well-defined notion of instantaneous transition rate can be defined in an appropriate limit.
where , and are parameters satisfying and , and , , are non-negative functions satisfying for and for . This means that the detector is turned on smoothly during the interval , with a profile determined by the function , it then remains turned on at constant coupling strength for the time , and it is finally turned off smoothly during the interval , with a profile determined by the function . The functions are regarded as fixed. We initially regard and as fixed but will eventually allow to vary.
The first term in (3.16) is equal to , where is a positive constant. The second term in (3.16) was analysed in , with the result that it only depends on and through the combination and has at small the asymptotic form
where is a constant and the full expansion of the -term proceeds in positive powers of . The last term in (3.16) can be analysed by breaking the integrations into the various subintervals as in , with the result
The qualitatively new feature compared with  is the logarithmic singularity in , but the contribution from this singularity can be verified to be of order and hence subleading in (4.3). Collecting, we find
By the smoothness of the spacetime and the trajectory, we may differentiate the trajectory-dependent contribution (4.3) with respect to termwise, and the same holds for the trajectory-independent contributions by their explicit structure. We may therefore take the -derivative in (4.4) termwise, with the result
Equations (4.4) and (4.5) are our main result. The transition probability (4.4) diverges as , but the divergence has been isolated into an explicit logarithmic term that is independent of the trajectory or the quantum state of the field. The -derivative of the transition probability is given by (4.5) and remains finite as . Equation (4.5) provides a definition of what is meant by the detector’s transition rate, without the need to introduce spatial profiles or other regulators. In the special case of a massless field in Minkowski spacetime, in the Minkowski vacuum, the limit in (4.5) recovers the transition rate obtained in  via a spatial profile regularisation.
We end the section with two comments. First, if we interpret the naive expression (2.10) for the transition rate as
and apply the methods of section 3, we find that (4.6) is equal to the limit of our transition rate (4.5) plus an additional term proportional to . The additional term vanishes if the pull-back of to the trajectory is an affine function of . The transition rate of a sharply-switched detector can thus be calculated equivalently from the limit in (4.5), where the singularity in the Wightman distribution is cancelled by an explicit counterterm, or from the limit in (4.6), provided the time time function used to regularise (4.6) is an affine function of on the trajectory.
Second, if we compare two detectors in different quantum states of the field, on different trajectories or even in different spacetimes, but having been in operation for the same length of proper time, we recover for the difference of the transition rates in the limit the formula
The difference in the transition rates in two given situations can thus be written as a Fourier transform of a function that requires no regularisation. Formula (4.7) is useful for both analytical and numerical calculations, especially in cases where the transition rate in one of the two situations is already known. In the next two sections we shall apply this formula to two such examples.
5 Inertial detector in the Rindler vacuum
In this section we consider a detector moving inertially through the Rindler wedge in Minkowski space, coupled to a massless scalar field in its Rindler vacuum state. This is the state in which the uniformly accelerated detectors associated with the Rindler wedge do not get excited. A naive application of the equivalence principle could be argued to imply that as an accelerated detector moving through the ‘unaccelerated’ (Minkowski) vacuum state gets excited thermally, an unaccelerated detector moving through the ‘accelerated’ (Rindler) vacuum should also get excited thermally. Working in the limit , we shall show that this does not hold: the detector does have a nontrivial transition rate, but the rate is neither thermal nor constant in the detector’s proper time, and it diverges as the detector approaches the Rindler horizon.
Let be a set of standard Minkowski coordinates in Minkowski space. We take the detector to move in the ‘right-hand-side’ Rindler wedge, , denoted by . The Rindler vacuum Wightman distribution in reads 
where is the Minkowski vacuum Wightman distribution, the points and are in and . The difference of and consists thus of the integral of over an orbit of the associated Killing vector in the opposite Rindler wedge. Note that since the two Rindler wedges are spacelike separated, the difference term is a nonsingular function, and the distributional character of comes entirely from the first term on the right-hand side in (5.1). As we shall be using formula (4.7), we are suppressing the distributional issues in (5.1).
We take the trajectory of the detector to be
where is a positive constant and is the proper time. The trajectory stays in during the proper time interval . We must therefore consider the detector response in a finite proper time interval.
We shall compute the transition rate from formula (4.7), using the Minkowski vacuum as reference state. Unlike the infinite case, in which the excitation rate is zero, a detector moving inertially through Minkowski vacuum over a finite proper time does have transient excitations due to the switching. This transition rate has been found in several papers (see e.g. ) and equals
where is the sine integral function. diverges as (which is an artifact of omitting the terms in (4.5)) and approaches for large .
The -integral in (5.5) can be done by contour integration. Closing the contour in the upper half-plane, there are two infinite series of contributing poles, at respectively and with , and a single contributing pole at . Summing over the residues, we find
Note that the integrand in (5.6) remains finite as , and the integral is well defined. Regarding fixed and as variable, we see that tends to zero as , but it diverges to as , which is the limit in which the trajectory approaches the Rindler horizon. We shall show in the Appendix that the asymptotic form of at is
and the divergence is thus logarithmic in .
The response of inertial detectors in the Rindler vacuum was previously studied by Candelas and Sciama , but in a somewhat different framework. Candelas and Sciama investigate the whole family of trajectories (5.2), and they compute the transition rate on each trajectory at the point , where is a fixed positive constant. This means that the inertial trajectories are being compared along a Rindler trajectory of acceleration . In the limit , it is found that the response approaches the Minkowski vacuum value . Candelas and Sciama’s interpretation of this result is that “for the case of a charge moving inertially in Minkowski space-time through an accelerated vacuum the spectrum of field fluctuations perceived by the charge is the same as if the vacuum were unaccelerated” (, p. 1717). They view the limit of large as a means to eliminate the transient effects due to the detector starting its inertial motion at .
In our view these transient effects are already contained in the Minkowski vacuum part (5.3) of the response, and the additional Rindler vacuum contribution (5.6) shows that the inertial detector in the Rindler vacuum responds genuinely differently than in the Minkowski vacuum. In particular, the additional Rindler vacuum contribution (5.6) diverges as the trajectory approaches the Rindler horizon. This divergence was found previously by Davies and Ottewill , both by a numerical evaluation of the response and by a comparison of the response with the expectation value of . The divergence can indeed be expected on the grounds that in the Rindler vacuum the expectation values of both and the stress-energy tensor diverge on the horizon [23, 25, 26].
6 Detector at rest in Newtonian gravitational field
In this section we shall find the response of an Unruh-DeWitt detector at rest in a static, Newtonian gravitational field: a static, asymptotically flat spacetime that satisfies the linearised Einstein equations with pressureless matter as source. The quantum field is assumed massless, but with arbitrary curvature coupling, and its vacuum state is taken to be the Boulware-like vacuum defined in terms of the global timelike Killing vector. We shall see that in this situation the detector’s excitation rate is always zero, but the de-excitation rate (response for negative ) in general differs from that of an inertial detector in the Minkowski vacuum in Minkowski space, and the gravitational correction depends on the details of the mass distribution even when the detector is far from the source. We also provide an order-of-magnitude estimate for this gravitational correction in atomic physics decay rates on the Earth’s surface.
6.1 General matter distribution
The metric takes the form , where is the Minkowski metric and is the linearised correction. We use a system of Minkowski coordinates in which and the correction components are independent of . We assume to be small enough for validity of linearised Einstein’s equations, and at large we assume to have the asymptotically flat falloff . We further take to be in the Lorentz gauge, , where .
We need the Wightman distribution in this spacetime, in the Boulware-like vacuum that reduces to the Minkowski vacuum at the asymptotically flat infinity. Working in perturbation theory to the order that is consistent with linearised Einstein’s equations, this Wightman distribution must be the sum of the Minkowski vacuum contribution and a correction that is first-order in and dies off at infinity. Restricting the attention to a detector that remains at constant and was switched on in the infinite past, equation (4.7) shows that the transition rate reads
To find , we first calculate the Feynman Green’s function to first order in , adapting the procedure that was introduced in  in the context of vacuum polarisation. We then find from the relation