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in a recent investigation of the rotating quantum vacuum , davies et .
@xcite uncovered a remarkable issue involving the speed - of - light cylinder of flat space - time . a particle detector rotating at the same angular velocity as this vacuum fails to remain inert in unbounded space - time and remains inert when the space - time is bounded by an infinite cylindrical mirror only if the mirror has a radius less than that of the speed - of - light cylinder .
it is only in this case that the concept of a rotating vacuum is unambiguously defined . in the present article ,
we attempt to extend this discussion of rotation to the problem of defining a rigidly rotating quantum thermal distribution .
whereas we might have expected that this distribution would be pathologically only on and outside the speed - of - light cylinder , in fact we find that it is pathological almost everywhere on the unbounded space - time and likewise on the space - time bounded by a cylindrical mirror except when the mirror has a radius less than that of the speed - of - light cylinder .
this problem is closely related to the definition of a hartle - hawking vacuum on kerr space - time . in particular , on the asymptotically flat region of the space - time for a black hole which is rotating arbitrarily slowly ,
the contributions made to the field by the upgoing modes become negligible and when these are discarded the anti - commutator function associated with the hartle - hawking vacuum of  @xcite coincides with the thermal distribution considered here .
we first introduce a cylindrical co - ordinate system @xmath0 rigidly rotating at a fixed angular velocity @xmath1 ; this is related to the usual cylindrical minkowski co - ordinate system @xmath2 by the transformation @xmath3 these co - ordinates are appropriate to the discussion of rigidly rotating observers ( rros ) in flat space - time .
these are analogous to observers co - rotating with the horizon in kerr space - time .
the scalar wave equation is separable in these co - ordinates . indeed
the corresponding positive klein - gordon norm modes , @xmath4 may be obtained from the standard modes of minkowski space - time in cylindrical co - ordinates , @xmath5 by the identification @xmath6 since it is the norms not the frequencies of the rro modes which determine the commutation relations of the associated creation and annihilation operators , the rotating vacuum naively coincides with the conventional vacuum of minkowski space - time .
the thermal distribution at inverse temperature @xmath7 and rigidly rotating at angular velocity @xmath1 will be described by the density operator @xmath8 with the hamiltonian @xmath9 the thermal anti - commutator function @xmath10 associated with this distribution is therefore characterized by the condition  @xcite @xmath11 it can be checked that at least formally this condition is satisfied by the anti - commutator function defined by the mode sum @xmath12\times\\      j_m\left(\sqrt{\omega^2-k^2}r\right)j_m\left(\sqrt{\omega^2-k^2}r'\right ) .
\label{eq : twopointpathological}\end{gathered}\ ] ] as the energy @xmath13 as measured by @xmath14 tends towards zero , the density of states factor @xmath15 becomes infinite although the modes themselves remain non - zero whenever @xmath16 .
these modes clearly make divergent contributions to the mode sum  ( [ eq : twopointpathological ] ) except when either @xmath17 or @xmath18 is zero .
the anti - commutator function @xmath10 is thus pathological except when at least one of the two points is on the @xmath19-axis . in  @xcite
, a similar pathology was noted in the mode sum for the anti - commutator function of the hartle - hawking state considered in  @xcite .
we now introduce a cylindrical mirror of arbitrary radius @xmath20 . for brevity
, we only treat the case of a field which satisfies dirichlet conditions on this cylinder . introducing a non - dimensional radial variable @xmath21 ,
a complete set of orthonormal solutions to the field equation subject to the boundary conditions is @xmath22 where @xmath23 and @xmath24 denotes the @xmath25 positive zero of @xmath26 .
the normalization factor has been calculated by making use of an identity for the bessel functions  ( * ? ? ?
* see page 765 ) .
the modes which have positive norm are precisely those which have positive frequency @xmath27 .
the vacuum state associated with the field when it is expanded in terms of this set of modes has a anti - commutator function which is given by the mode sum @xmath28 .
\label{eq : vacuumtwopoint}\end{gathered}\ ] ] the anti - commutator function @xmath29 \label{eq : thermaltwopoint}\end{gathered}\ ] ] is associated with a thermal distribution described by the density operator  ( [ eq : densityoperator ] ) with hamiltonian  ( [ eq : hamiltonian ] ) . this mode sum suffers from a similar pathology to that on the unbounded space - time unless there are no positive frequency modes for which @xmath30 is zero .
it is a well known property of the zeros of @xmath26 that @xmath31  ( * ? ? ?
* see  15.3 ) and so we see from  ( [ eq : omegadiscrete ] ) that if @xmath32 , no such modes exist . on the other hand ,
the asymptotic behaviour of this first zero is  ( * ? ?
* see page xviii ) @xmath33 from which we see that if @xmath34 , there are modes of this type for all sufficiently large @xmath35 .
it follows that  ( [ eq : thermaltwopoint ] ) is well behaved if and only if the mirror lies within the speed - of - light cylinder .
when the mirror lies inside the speed - of - light cylinder , static observers and rros both make measurements with respect to the vacuum state whose anti - commutator function is given in  ( [ eq : vacuumtwopoint ] ) .
these measurements can be calculated from @xmath36 .
\label{eq : ddiff}\end{gathered}\ ] ] we can derive from this a set of expressions for the non - zero components of the energy - momentum stress tensor corresponding to the the conformally invariant field , the details of which can be found in  @xcite .
the mode by mode cancellation of the high frequency divergences which afflict both anti - commutator functions in the coincident limit makes these expressions amenable to numerical analysis and the results are shown in figure  [ fig : thermalexpct ] .
they are compared with the planckian forms corresponding to a rigidly rotating thermal distribution at temperature @xmath37 which are @xmath38 where @xmath39 and @xmath40 are given by @xmath41 and are the speed and lorentz factor of an rro at the appropriate space - time point .
we find that they are in close agreement everywhere except , as expected , close to the mirror .
a renormalized expectation value differs from that an rro measures by a term due to polarization of the vacuum by the mirror .
this term can be calculated by making use of the relationship between that feynman propagator and the euclidean green function ; on the euclidean section of the manifold , the analysis becomes essentially identical to that of a uniformly accelerating infinite flat mirror on the euclidean section of the rindler manifold and we can proceed along the lines of  @xcite . we find that the euclidean green function which vanishes on the mirror is @xmath42 where @xmath43 , @xmath44 and @xmath45 .
the second term in the braces is absent in the case of the euclidean green function on the unbounded manifold and so this is the term that remains after renormalization .
now , closing the points and making a transformation to polar variables @xmath46 and @xmath40 defined by @xmath47 we find that we can perform the integral over @xmath40 to obtain @xmath48 a similar thing can be done for the components of the energy - momentum stress tensor and the resulting expressions together with  ( [ eq : phicyl ] ) are amenable to numerical analysis . once again , the details can be found in  @xcite .
the results are presented in figure  [ fig : renexpct ] .
they are compared with the casimir divergence close to the mirror which can be calculated by an asymptotic analysis following  @xcite .
the relevant expressions are @xmath49 , &      \qquad(\bar{r}&\to1).\label{eq : phicas}\\ { \left\langle\hat t^r_r\right\rangle^{}_{\mathrm{ren}}}&\sim-\frac{1}{720\pi^2r_{0}^4(1-\bar r)^2 } , &      \qquad(\bar{r}&\to1),\label{eq : trrcas}\\ { \left\langle\hat t^t_t\right\rangle^{}_{\mathrm{ren}}}&\sim\frac{1}{720\pi^2r_{0}^4(1-\bar r)^3 }      \left[1+\frac{19(1-\bar r)}{14}\right ] , &      \qquad(\bar{r}&\to1),\label{eq : tttcas}\\ { \left\langle\hat t^\varphi_\varphi\right\rangle^{\mathrm{}}_{\mathrm{ren}}}&\sim-\frac{1}{360\pi^2r_{0}^4(1-\bar r)^3 }      \left[1+\frac{6(1-\bar r)}{7}\right ] , &      \qquad(\bar{r}&\to1).\label{eq : tphiphicas}\end{aligned}\ ] ] and are in agreement with the general expressions of  @xcite .
we have used these to calculate renormalized expectation values in the thermal distribution when @xmath50 and checked that they are well approximated by the general expressions given in  @xcite .
we found that the anti - commutator function associated with the rigidly rotating thermal distribution on unbounded minkowski space - time is pathological almost everywhere .
the pathology is caused by the existence of non - zero modes which have zero energy as measured by the hamiltonian @xmath14 relevant to rros . in  @xcite
, a similar pathology was noted in the anti - commutator function of the hartle - hawking state considered in  @xcite . in this case
@xmath14 is the hamiltonian relevant to observers rigidly rotating with the horizon .
the corresponding modes are thus at the critical point of superradiant scattering .
when minkowski space - time is bounded by an infinite cylinder of radius larger than the speed - of - light cylinder we found that the anti - commutator function is once again pathological almost everywhere because of the existence of these modes for all sufficiently high @xmath35 . in a future article we will show that when the kerr black hole is enclosed within a mirror of constant boyer - lindquist radius larger than the minimum radius of the speed - of - light surface , for all sufficiently high @xmath35 there are complex frequency modes whose real parts lie in the regime we associate with superradiance in the absence of the mirror .
this set of modes has the critical point of superradiant scattering as an accumulation point .    , the radius of the cylinder , is unity .
the temperature is @xmath51 and the angular velocity is @xmath52 .
the dashed line is a plot of the value for a rigidly rotating thermal distribution  ( [ eq : phirot][eq : tzzrot ] ) . ]
, the radius of the cylinder , is unity . in the right hand graphs
, the dashed line gives the analytically calculated casimir divergence  ( [ eq : phicas][eq : tphiphicas ] ) while the solid line in the second graph and the points in the others give the numerically calculated values . ]
we show that the rigidly rotating quantum thermal distribution on flat space - time suffers from a global pathology which can be cured by introducing a cylindrical mirror if and only if it has a radius smaller than that of the speed - of - light cylinder . 
 when this condition is met , we demonstrate numerically that the renormalized expectation value of the energy - momentum stress tensor corresponds to a rigidly rotating thermal bath up to a finite correction except on the mirror where there are the usual casimir divergences .