Static interactions and stability of matter in Rindler space
Dynamical issues associated with quantum fields in Rindler space are addressed in a study of the interaction between two sources at rest generated by the exchange of scalar particles, photons and gravitons. These static interaction energies in Rindler space are shown to be scale invariant, complex quantities. The imaginary part will be seen to have its quantum mechanical origin in the presence of an infinity of zero modes in uniformly accelerated frames which in turn are related to the radiation observed in inertial frames. The impact of a uniform acceleration on the stability of matter and the properties of particles is discussed and estimates are presented of the instability of hydrogen atoms when approaching the horizon.
pacs:04.62.+v, 11.15.-q, 11.10.Kk
The study of quantum fields in Rindler space has played an important role in developing our understanding of quantum fields in non-trivial space-times. The importance of these studies derives to a large extent from the existence of a horizon in Rindler space. With its close connection to Minkowski space via the known local relation between fields in uniformly accelerated and inertial frames, Rindler space provides the simplest possible context to investigate kinematics and dynamics of quantum fields close to a horizon.
The kinematics of non-interacting quantum fields in Rindler space FULL73 (), BOUL75 (), DAVI75 () and their relation to fields in Minkowski space together with the interpretation in terms of quantum fields at finite temperature UNRU76 (), SCCD81 () are well understood. The relation between acceleration and finite temperature remains an important element of the thermodynamics of black holes. Although formally established, the relation between fields in Rindler and Minkowski spaces has posed some intriguing interpretational problems. Here we mention in particular the issue of compatibility of the radiation generated by a uniformly accelerated charge observed in Minkowski space and the apparent absence of radiation in the coaccelerated frame. This problem was finally resolved by the realization that the counterpart of Minkowski space radiation is the emission of zero energy photons in Rindler space HIMS921 (), HIMS922 (), REWE94 (). Dynamical issues of quantum fields have received less attention (cf. CRHM07 ()). Related to topics to be discussed in the following are the study of the relation between interacting quantum fields in Minkowski and Rindler spaces within the path-integral formalism UNRU84 (), the calculation of level shifts in accelerated hydrogen atoms PASS98 () and the investigations of the decay of protons MULL97 (), VAMA00 ().
Dynamical issues of quantum fields in Rindler space are in the center of our work. Subject are static forces and their properties relevant for the structure of matter in Rindler space. We will study the forces acting between static scalar sources, electric charges and massive sources generated by exchange of massless scalar particles, photons and gravitons respectively. The relevant quantities involved are the corresponding static Rindler space propagators. While the time dependent propagators in Rindler and Minkowski spaces are trivially related to each other by a coordinate transformation with corresponding mixing of the components, this is not the case for the static propagators obtained by integration over the corresponding time. The different properties of the static propagators reflect the significant differences of Rindler and Minkowski space Hamiltonians. Though it was realized from the beginning that the interpretation of Rindler and Minkowski particles (cf. e.g. FULL73 ()) is different, the differences in the corresponding Hamiltonians have not received sufficient attention. (The identity of the Rindler and Minkowski space Hamiltonians for the frequently used special case of a massless field in two dimensional space-time may have obscured the issue.) As shown in LOY08 (), the Rindler space Hamiltonian exhibits symmetries which give rise to a highly degenerate spectrum. In particular this degeneracy is also present in the zero energy sector. This 2-dimensional sector of zero modes is the image of the photons generated by the charges uniformly accelerated in Minkowski space. It will be shown to give rise to an unexpected quantum mechanical contribution to the force acting between two charges. Also the classical Coulomb-like contribution to the force will be shown to significantly deviate from the electrostatic force in Minkowski space. The electrostatic force will be determined via Wilson loops and Polyakov loop correlation functions. This method will enable us to separate the contribution of the quantum mechanical transverse photons from that of the classical longitudinal field. It will be the method of choice if one attempts to determine the static force in simulations of non-abelian gauge theories on a Rindler space lattice.
The modifications in the interaction of two charges at rest in Rindler space raise naturally the question of the stability of atomic systems in Rindler space. We shall investigate this issue for an ensemble of hydrogen atoms and show within a non-relativistic reduction that indeed only metastable states exist. We will estimate the probability of ionization as a function of the distance to the horizon and present arguments concerning the stability of other forms of matter.
Ii Propagators and interactions of scalar particles in Rindler space
ii.1 The quantized scalar field in Rindler space
The Rindler space metric RIND01 ()
is the (Minkowski) metric seen by a uniformly accelerated observer (acceleration in -direction). Rindler space () and Minkowski space () coordinates are related by
The range of is
while the preimage of the Rindler space covers only part of Minkowski space, the right “Rindler wedge”,
The restriction of the preimage of the Rindler space to the right Rindler wedge gives rise to a horizon, the boundary . With this property the Rindler metric can be identified with other static metrics in the near horizon limit. In particular this is the case for the Schwarzschild metric which can be approximated in the limit that the distance from the horizon is small in comparison to the Schwarzschild radius and if the spherical Schwarzschild horizon is replaced by a tangential plane.
We start our study of propagators and interactions with a discussion of a scalar field in Rindler space (cf. for instance FULL73 (), CRHM07 (). We will use the notation of LOY08 ()). The action of a free massive scalar field is given by
The wave equation
with the Laplacian
is solved by the normal modes (in terms of the McDonald functions)
which form a complete, orthonormal set of functions. The normal mode expansion of the scalar field reads
with the creation and annihilation operators . The stationary states associated with the Hamiltonian (
including the lowest energy () state, are degenerate with respect to the value of the transverse momentum . This degeneracy has its origin in the appearance of the transverse momentum (in combination with the mass) as a coupling constant of the “inertial potential” in the Hamiltonian.
In the Rindler wedge (4), the field operator can be represented in terms of plane waves in Minkowski space or in terms the Rindler space normal modes (9) resulting in the representation of the Rindler creation and annihilation operators in terms of the corresponding Minkowski space operators. This relation yields the important result for the expectation value of the Rindler number operator in the Minkowski ground state
which exhibits a thermal distribution with temperature
It is important to realize that the infinite degeneracy of the Rindler eigenstates leads, at long wavelengths, effectively to 1-dimensional thermal distributions with weight .
ii.2 The scalar Rindler space propagator
The propagators in Minkowski and Rindler spaces are related to each other by the coordinate transformation (2). Written in terms of Rindler and Minkowski coordinates respectively, the propagator of a massless field is given by (cf. TRDA77 (), DOWK78 ())
with the notations
As we will see, it is (or equivalently ) which determines the interaction energies rather than the proper distance in Rindler space
The quantity actually can be viewed as the proper distance in the 4-dimensional AdS space. The appearance of this geometry (of a three-hyperboloid) has been noted SCCD81 () in the context of the unusual density of states in the photon energy momentum tensor in Rindler space. The quantity or equivalently exhibits the remarkable invariance under the transformation
The coordinate transformation (17) together with a rescaling of the fields leaves, for , the Hamiltonian (10) invariant and gives rise to the degeneracy of the spectrum. The invariance under scale transformations can be generalized to massive particles LOY08 ().
In Rindler space coordinates the 2-point function (13) satisfies
After a Wick rotation of the Rindler time,
the propagator is periodic in the imaginary time with periodicity (cf. Eq. (12))
In the context of the electromagnetic field we will compute observables in both the real and imaginary Rindler time formalisms. (For discussions of “Euclidean” Rindler space propagators, in particular of their topological interpretation cf. CHDU78 (), LINE95 (), SVZA08 ().)
is obtained where the propagator defined with respect to the Rindler space vacuum is given by
(For an interpretation of the difference between the two propagators in terms of image charges located in the left Rindler wedge , cf. CARA76 ().) Obviously, the propagator depends parametrically on the acceleration . In order to formulate properly the relation between propagators in Minkowski and Rindler spaces one has to define the Rindler space propagator for a fixed value of the acceleration at finite temperature (given by )
In terms of the propagator , the central result concerning the relation between Rindler and Minkowski spaces is the identity
i.e., the Rindler space propagator defined with respect to the Minkowski ground state coincides with the Rindler space finite temperature propagator with the value of the temperature determined by the acceleration (cf. Eq. (12)). This identity makes also manifest that a change in the acceleration does not correspond to a change in temperature of the accelerated system. The acceleration appears not only as temperature in the Boltzmann factor but also as a parameter in the Hamiltonian (10) of the accelerated system. We will encounter observables which make explicit this twofold role of the acceleration.
The basic quantity in the following investigation of the properties and consequences of interactions generated by exchange of scalar particles and photons is the static propagator
Alternatively, this result can be obtained by a contour integration of Eq. (13) in the complex plane. The first term is due to the pole infinitesimally close to the real axis while the second, imaginary contribution is the result of the summation of the residues of the poles at . The result for the static propagator and its decomposition into the real non-thermal and the imaginary thermal contributions read
The imaginary part of the static propagator arises since the propagator (LABEL:Dur) is defined with respect to the Minkowski rather than to the Rindler ground state.
ii.3 The interaction energy of scalar sources
Given the propagator, the interaction energy between two scalar sources is obtained by adding to the action (5) the scalar particle-source vertex
where stands for either Minkowski () or Rindler coordinates (). The effective action (the generating functional of connected diagrams) associated with the source is given by
For two point like sources moving along the trajectories which are parametrized in terms of their proper times , we find
We assume the sources to be at rest in Rindler space and evaluate by expressing the proper times by the coordinate times and obtain for sources positioned at
We have introduced the effective coupling constant
which, due to the difference between proper and coordinate times,“runs” with the coordinates of the charges. With the sources at rest, depends only on the differences of the times . After carrying out the integrations, up to a factor , the size of the interval in the integration over the sum of the times, is determined by the static propagator and yields for the interaction energy of two scalar sources
where (cf. Eq. (15)). For two sources at rest in Minkowski space, this procedure yields the interaction energy .
The interaction energy (33) constitutes an explicit example of the bivalent role of the acceleration . The dependence of the real part of the interaction is exclusively due to the dependence of the Hamiltonian (10) on the parameter while the imaginary part depends in addition on the acceleration via the temperature (cf. Eq. (27)). The appearance of a non-trivial imaginary contribution to the “static interaction” generated by exchange of scalar particles and, as we will see also by photons or gravitons, is a novel phenomenon not encountered in the static interactions in Minkowski space. Here we will analyze this phenomenon. Other properties of the interaction (33) will be discussed later in the comparison with the “electrostatic” interaction.
As follows from Eq. (18) the static propagator (27) satisfies the Poisson equation for a point-like source in Rindler space. Since the source is real, the imaginary part of the propagator satisfies the corresponding (homogeneous) Laplace equation. In turn, this implies that the imaginary part of propagator or the interaction energy can be represented by a linear superposition of zero modes of the Laplace operator (7). From Eq. (LABEL:Dur) we read off
It is instructive to compare the Rindler space propagator with the finite temperature propagator in Minkowski space. As above we decompose the propagator of a non interacting scalar field into thermal and non-thermal contributions
carry out the Fourier transform of the thermal part
and obtain for massless particles
The Fourier transformed thermal propagators in Rindler (27) and in Minkowski space (37) become identical to order apart from the factor and differ from the corresponding ground state contribution only by the constant . The convergence to this limit is not uniform in Rindler space, since it requires . Due to the twofold role of the acceleration the finite temperature contribution to the propagator in Minkowski space is only part of the leading order correction to the propagator (cf. Eq. (27)) in Rindler space. The structure of the Minkowski space static propagator suggests that the imaginary part is due to on-shell propagation of zero-energy massless particles. The difference between Rindler and Minkowski space propagators is due to the different dimensions (0 and 2 respectively) of the space of zero modes. Furthermore, while the Minkowski space zero mode is constant, the zero modes in Rindler space exhibit a non-trivial dependence on all the three coordinates. Finally for massive particles no zero mode exists in Minkowski space (Eq. (36)) while in Rindler space together with the degeneracy in the spectrum also the zero modes persist. In this case the spectral representation in Eq. (34) remains valid provided we replace in the arguments of the McDonald functions.
The imaginary part of the propagator determines the particle creation and annihilation rates. To leading order in the coupling constant , the probability for a change in the initial state in the time interval is given by (cf. Eqs. ((28)-(32))
The rate for a change in the initial state within an arbitrarily large time interval is easily obtained to be
The total response (not observable in the accelerated frame) of the field to external sources determines the imaginary part of the static propagator. Furthermore by defining the reaction rate in terms of the proper time of the external sources (cf. HIMS921 (), HIMS922 (), CRHM07 (), REWE94 ()) it is seen that the imaginary part is determined by the total rate for Bremsstrahlung of uniformly accelerated sources observable in Minkowski space.
These results imply, that under the transformation from the inertial to the accelerated system comoving with the uniformly accelerated charge, the particles generated in Minkowski space by Bremsstrahlung are mapped into zero energy excitations in Rindler space. In this way the conflict between particle production in Minkowski space and the conservation of energy in Rindler space in the presence of static sources is resolved. The on-shell zero modes describe the radiation field in Rindler space without any change in the energy in Rindler space and may be viewed as a “polarization” cloud of on-shell particles induced by external sources or by classical detectors (cf. GROV86 (), MAPB93 (), UNRU92 (), HURA00 ()). Similar remarks apply for the acceleration induced decay of protons MULL97 (), VAMA00 (). Essential for this important role of the zero modes is, as indicated above, the peculiar symmetry of the Rindler Hamiltonian which gives rise to the extensive degeneracy.
Iii Wilson and Polyakov loops of the Maxwell field
iii.1 Wilson loops in Minkowski and Rindler space
In this section we consider the Maxwell field coupled to external charges given by the action
With changing emphasis we will describe various methods for evaluating the interaction energy of static sources. The computation of Wilson loops BAMU94 () and Polyakov loop correlation functions JASM02 () constitute the preferred techniques in analytical and numerical studies of interaction energies of static sources in gauge theories. In the comparison of these two methods the emphasis will be on the consequences of the Wick rotation to imaginary time for the interaction energy which will be seen to be of relevance also for static interactions in Yang-Mills theories. Evaluation of the Wilson loops in different gauges will enable us to identify the origin of real and imaginary parts respectively of the electrostatic interaction.
Wilson loops are defined as integrals over the gauge field along a closed curve in space-time
The invariance of the Wilson loop under gauge and (general) coordinate transformations and reparameterization which is explicit in Eq. (41) makes the Wilson loop a particularly useful tool for our purpose. Up to self energy contributions, the interaction energy of two oppositely charged sources is given by the expectation value (e.g. in the Minkowski space ground state) of a rectangular Wilson loop in a time-space plane with side lengths and
with the ground state expectation value
The gauge fields along the loop can be interpreted as resulting from two opposite charges which are separated in an initial phase from distance to (for a rectangular loop this initial phase is reduced to one point in time), remain separated at this distance for the time and recombine in a final phase. In order to make the contributions from the turning-on period negligible, the interaction energy of static charges is defined by the limit. In terms of the photon propagator
the Wilson loop is given by (cf. BAMU94 ())
In Lorenz gauge,
the Minkowski space photon propagator is expressed in terms of the scalar propagator (cf. Eq. (13)) as
with the Minkowski space metric . The Rindler space photon propagator is obtained by the change in coordinates (2)
where we have used the notation
Under the coordinate transformation, the Lorenz gauge condition becomes
with the covariant derivative .
iii.2 Interaction energy of static charges in Rindler space
iii.2.1 Wilson loops of gauge fields in Lorenz gauge
Invariance of the Wilson loop under coordinate transformations does not imply invariance of the interaction energy. Under the coordinate transformation (2) the shape of a loop changes, as is illustrated in Fig. 1 for the case of a rectangular loop in Rindler space. With the change in shape also the value of the interaction energy changes which is defined with respect to two different limits ( or ).
We compute the interaction energy for a rectangular loop with 2 of the 4 segments of the loop varying in time and kept fixed while the other two segments are computed at fixed . The sum of the two contributions from the integration along the axis can be carried out without specifying the segments in the spatial coordinates. Inserting (cf. Eq. (47))
into Eq. (45) we find
Here is given by (15) with the spatial coordinates of the vertices of the rectangle. Introducing as integration variables can be rewritten as
The last step can be verified by differentiation. The contribution to the interaction energy of two oppositely charged sources can be extracted in the limit from the two - independent terms in (52) (the integrals of the - dependent terms converge).
For large , the integration along the spatial segments of the rectangle (the horizontal segments of the loop in Fig. 1) yields a -independent term and does therefore not contribute to the interaction energy. The non-diagonal element gives rise to two space-time contributions to the Wilson loop which in the large limit become independent of the spatial coordinates and cancel each other. Thus we obtain the asymptotic value of the Wilson loop expressed in terms (15)
with the interaction energy
The integration “constant” arises from the first term in (50) and represents the self-energy of the static charges. Regularizing the divergent integrals by point splitting, is given in terms of the proper distance in AdS (cf. Eq. (15))
Before discussing the properties of the “electrostatic” interaction and the comparison with the static scalar interaction we briefly describe a simple alternative method for calculating the interaction energy of two static charges. In analogy with the corresponding calculation for scalar fields (cf. Eqs. (28)-(32)) the current in Eq. (40) generated by two charges moving along the trajectories is parametrized as
resulting in the photon-charge vertex
where the propagators in different coordinates are obtained from each other by the corresponding coordinate transformations (cf. Eq. (47)). Unlike in the scalar case, for exchange of photons no factor renormalizes the coupling constants when changing from the proper to coordinate time . We define the Fourier transform in time of the -component of the propagator (Eq. (49)) by the limit
and disregarding the (divergent) constant, for opposite charges , the result (54)
In Fig. 2 are plotted real and imaginary parts of the interaction energy of two static charges in comparison with the interaction energy of two scalar sources. The distance which determines both scalar (33) and vector (54) interaction energies is the proper distance of the AdS space and not of Rindler space, i.e., the static interaction energies generated by scalar particle and photon exchange are invariant under the scale transformation (17). At small distances where the effect of the inertial force is negligible both interaction energies display the familiar linear divergence in the real part and as in Minkowski space, vector and scalar exchange give rise to the same behavior. For small distances the imaginary part is constant and agrees with the constant in the finite temperature propagator (37) in Minkowski space
With increasing distance the inertial forces become important. They weaken both scalar and vector interaction energies and change completely the asymptotic behavior
Significant differences in the asymptotics for scalar and vector exchange are obtained. Qualitative differences are observed for the imaginary part which vanishes asymptotically for scalar exchange and diverges logarithmically with for photon exchange. Given the highly relativistic motion of the sources in Minkowski space, spin effects are expected to be important. Indeed the comparison of Eqs. (33) and (53) shows that the differences between scalar and vector exchange are due to the additional factor in (53) arising from the transformation of the propagator from Minkowski to Rindler space. In this context we also note the stronger asymptotic suppression of the “electrostatic” force in comparison to the force generated by scalar exchange. From the point of view of a Minkowski space observer (cf. Eq. (58)), cancellation between magnetic and electric forces generated by the spatial components of the current and by the charges respectively is to be expected.
Of particular interest is the behavior of the interaction energy if one of the sources approaches the horizon while the position of the other source is kept fixed. For
the interaction energies are given by
In both cases the interaction is dominated by the imaginary radiative contribution and no bound states are expected in this regime. We also find in analogy with the “no hair” theorem BEKE72 (), BEKE98 (), TEIT72 () for Schwarzschild black holes that a scalar source close to the horizon cannot be observed asymptotically while vector sources are visible. Formally the difference arises from the running of the scalar coupling (32) to zero when approaching the horizon while the electromagnetic coupling remains constant. In detail the results for Rindler and Schwarzschild metrics are different. A significant improvement can be obtained by modifying the Rindler metric
The dynamics in this space is very close to the dynamics in Rindler space unless the difference between the and matters. This is the case when deriving the “no hair” theorem where only the waves matter COWA71 ().
iii.2.2 Wilson loops of gauge fields in Weyl gauge
In the (covariant) Lorenz gauge only the -component of the propagator contributes (cf. Eqs. (54) and (60)) which simplifies significantly the evaluation of the electrostatic interaction energy and by the same token hides the difference in origin of its real and imaginary parts. Separation of constrained and dynamical variables becomes manifest in Weyl gauge
In this gauge one works with two unconstrained dynamical fields describing the photons and a longitudinal field constraint by the Gauß law. The gauge field is decomposed accordingly
into the longitudinal component given by the scalar field and the transverse field satisfying
and the Laplacian
Longitudinal and transverse fields are not coupled to each other. Their actions are given by