Quantum non-equilibrium effects in rigidly-rotating thermal states

Based on known analytic results, the thermal expectation value of the stress-energy tensor (SET) operator for the massless Dirac field is analyzed from a hydrodynamic perspective. Key to this analysis is the Landau decomposition of the SET, with the aid of which we find terms which are not present in the ideal SET predicted by kinetic theory. Moreover, the quantum corrections become dominant in the vicinity of the speed of light surface (SOL). While rigidly-rotating thermal states cannot be constructed for the Klein-Gordon field, we perform a similar analysis at the level of quantum corrections previously reported in the literature and we show that the Landau frame is well-defined only when the system is enclosed inside a boundary located inside or on the SOL. We discuss the relevance of these results for accretion disks around rapidly-rotating pulsars.


Introduction
In relativistic fluid dynamics, global thermal equilibrium can be attained if the product βu µ between the inverse local temperature β and the four-velocity u µ of the flow satisfies the Killing equation [1,2,3,4,5]. A special property of thermal equilibrium is that the stress-energy tensor (SET) T µν eq = (E + P)u µ u ν + Pg µν corresponds to that of an ideal fluid of energy density E and pressure P [2,6,7,8]. 1 In this letter, we will show that a quantum field theory (QFT) computation of the SET for rigidly-rotating thermal states (RRTS) contains non-ideal terms, as well as corrections to E which become important near the speed of light surface (SOL). We discuss the relevance of these corrections in the context of an astrophysical application.

Kinetic theory analysis
In space-times with axial symmetry, RRTS in thermal equilibrium can be described using the Killing vector corresponding to rotations about the z axis, i.e., βu = β 0 (∂ t + Ω∂ ϕ ), where Ω is the angular velocity of the rotating state [7]. On Minkowski space, the particle four-flow N µ eq and stress-energy tensor T µν eq corresponding to RRTS are given by: while β and u = u µ ∂ µ are given by: Email address: victor.ambrus@e-uvt.ro (Victor E. Ambrus , ) 1 We use Planck units with c = = k B = 1, while the metric signature is (−, +, +, +).
where γ is the Lorentz factor of a co-rotating observer at distance ρ from the z axis: The Killing vector βu becomes null on the SOL, where ρΩ → 1 and co-rotating observers travel at the speed of light. From Eq. (2), it can be seen that the temperature β −1 diverges as the SOL is approached. The energy density E for massless particles obeying Fermi-Dirac (F-D) and Bose-Einstein (B-E) statistics is given by [6]: while P = E/3. Since E and P diverge as inverse powers of the distance to the SOL, RRTS are well-defined only up to the SOL. While such divergent states clearly cannot occur in nature, rigid rotation can be induced in astrophysical systems, such as accretion disks around rapidly-rotating neutron stars or magnetars, where the intense magnetic field can lock charged particles into rigid rotation. 2 We investigate the role of quantum corrections in such systems in Sec. 6.

Stress-energy tensor decompositions
Before discussing the quantum analogue of Eqs. (4), the tools necessary to analyse the SET in out of equilibrium states must be introduced. The main difficulty comes due to the equivalence between mass and energy in special relativity, which makes the distinction between the velocity u µ and the heat flux q µ ambiguous. For a general (time-like) choice of u µ , N µ can be decomposed as [10]: where n = −u µ N µ and the flow of particles in the local rest frame (LRF) V µ is given by: In the above, ∆ µν = u µ u ν + g µν is the projector on the hypersurface orthogonal to u µ . The decomposition of the SET reads: where the dynamic pressure ω, flow of energy in the LRF W µ and shear stress π µν , together with V µ , represent nonequilibrium terms. The quantities on the right hand side of Eq. (7) can be obtained through: For a massless fluid, ω = 0. The heat flux q µ is defined as [10]: In the Eckart (particle) frame [2,8,11], u µ e is defined as the unit vector parallel to N µ . Observers in the LRF of the Eckart frame see a flow of energy (W µ e = q µ e ) and no flow of particles (V µ e = 0). Since N µ cannot be obtained using the QFT approach considered in this paper, the Eckart velocity u µ e cannot be defined. Hence, we will not consider the Eckart frame further in this paper.
In the Landau (energy) frame [2,8,12], u µ ≡ u µ L is defined as the eigenvector of T µ ν corresponding to the (real, positive) Landau energy density E L : such that W µ L = 0, which implies that there is no energy flux in the LRF. Since V µ L = − n L E L +P L q µ L is in general non-zero, an observer in the LRF of the Landau frame will detect a nonvanishing particle flux.
Finally, the β-frame (thermometer frame) for the case of rigid rotation is defined with respect to [4]: A special property of the β-frame is that the LRF temperature is highest compared to the temperature measured with respect to any other frame [4]. In general, V µ β and W µ β do not vanish, such that the β-frame is a mixed particle-energy frame [13]. Due to the simplicity of its construction, we will start the analysis of the quantum SET with respect to the β-frame.

Klein-Gordon field
We now analyse the construction of RRTS from a QFT perspective. A first surprise comes from the analysis of the RRTS of the Klein-Gordon field: in the unbounded Minkowski space, there exist modes which have a non-vanishing Minkowski energy ω (i.e., with respect to the static Hamiltonian H s = i∂ t ), while their co-rotating energy ω = ω − Ωm, measured with respect to the rotating Hamiltonian H r = i(∂ t + Ω∂ ϕ ), vanishes. For such modes, the Bose-Einstein density of states factor (e β ω − 1) −1 diverges, yielding divergent thermal expectation values (t.e.v.s) at every point in the space-time [14,15]. The kinetic theory result (4) is clearly unaffected by this vanishing co-rotating energy modes catastrophy. Indeed, the problematic modes are no longer present in the QFT formulation if the system is enclosed within a boundary placed inside or on the SOL [15,16]. Furthermore, a recent perturbative QFT analysis allows the computation of quantum corrections to the kinetic theory SET [17], which we will analyse in detail in what follows. For completeness, we present an analysis of the connection between these perturbative results and the non-perturbative QFT approach in Appendix A.
We are again forced to regard the RRTS of the Klein-Gordon field as ill-defined. The natural question to ask is whether the problem with defining the Landau frame persists when the system is enclosed inside a boundary. Following Ref. [15], we construct the Landau frame for the case when the system is enclosed inside a cylinder of radius R = Ω −1 (i.e. placed on the SOL), on which Dirichlet boundary conditions are imposed. Fig. 1 shows that the Landau frame is well defined arbitrarily close to the boundary, where the Landau velocity v L = ρu ϕ L /u 0 L decreases to 0 due to the boundary conditions. It can also be seen in Fig. 1 that both v L and E L increase monotonically as β 0 is increased. Figure 1(b) also shows E L for the unbounded Minkowski space (13) for the case when β 0 Ω = 1. The curve is interrupted at ρΩ ≃ 0.942, where E L becomes complex.

Dirac field
The QFT analysis of the RRTS of the Dirac field is presented in Ref. [14]. The β-frame decomposition can be performed using u β (2) for the components of the SET given in Eqs. (25c)-(25f) in Ref. [14], yielding: while P β = E β /3 and π µν β = 0. It is remarkable that W µ β for the Dirac field (16b) has the same expression as that for the Klein-Gordon field (12). As in the case of the Klein-Gordon field, the first term in Eq. (16a) corresponds to E F−D (4), while the second term represents a quantum correction which dominates in the vicinity of the SOL. Figure 2(a) demonstrates this behaviour and it can be seen that the correction increases when either Ω or β are increased.
The eigenvalue equation (10) can be solved analytically in terms of the Landau energy and velocity: In contrast to the case of the Klein-Gordon field, the Landau frame is well-defined everywhere inside the SOL, since 4E 2 β /9W 2 β > 1 when ρΩ < 1. The ratio E L /E β decreases from 1 on the rotation axis down to 1 3 + 1 √ 3 as the SOL is approached, where W β → 1 3 E β . At fixed ρΩ < 1, E L approaches E β as either Ω or β are decreased, as confirmed in Fig. 2(b).
The Landau velocity v L = ρu ϕ L /u 0 L ≥ ρΩ is compared to ρΩ in Fig. 2(c). The difference 1 − ρΩ/v L decreases to zero as the SOL is approached, while its value at the origin increases monotonically as β 0 Ω is increased.
For completeness, we list below π µν L :

Astrophysical application
Let us now apply our results in the context of the millisecond pulsar PSR J1748-2446ad reported in Ref. [18]. Its pulse frequency is ν ≃ 716 Hz, such that the SOL is located at a distance ρ SOL = c/2πν ≃ 66.685km from the rotation axis. The typical surface temperature for a neutron star with characteristic age τ c ≥ 2.5 × 10 7 years is T s ≃ 10 5 K [19]. Its radius is r s 16 km [18], such that the temperature on the rotation axis can be extrapolated as T 0 = T s /γ s ≃ 9.7 × 10 4 K. Let us now investigate the magnitude of the quantum corrections for massless Dirac fermions dragged into rigid rotation by the pulsars magnetic field (B surf ≤ 1.1 × 10 9 G [18]) by considering the folowing quantity: where the appropriate units were reinserted. As pointed out in Ref. [17], the quantum correction is very small due to the presence of the Planck constant h. The value of γ at which E β = 2E F−D is γ ≃ 7.4 × 10 12 , which would correspond for an electron to an energy of mγc 2 ≃ 3.8 × 10 18 eV, comparable to cosmic rays energies. At such high values of γ, the distance to the SOL is of order ∼ 6 × 10 −22 m, where the rotation of the accretion disk is most likely no longer rigid. Since our analysis was performed at the level of massless fermions, it is worth mentioning that in the case of the pulsar PSR J1748-2446ad, the relativistic coldness [8] has the value ζ 0 = mc 2 /k B T 0 ≃ 6.1×10 4 in the case of electrons, while the ratio mc 2 /hν ≃ 1.7 × 10 17 also has a large value. These numbers indicate that the massless limit results presented in this paper may be inaccurate close to the rotation axis, where the properties of RRTS are heavily influenced by the value of m in both the kinetic theory [6] and in the QFT [14] approaches. Also in these latter references, it can be seen that the mass dependence dissapears in the vicinity of the SOL, such that at γ ∼ 7.4×10 12 , the particle constituents behave as though they were massless.

Conclusion
In summary, we investigated rigidly-rotating thermal states of massless Klein-Gordon and Dirac particles. In comparison to relativistic kinetic theory results, the QFT approach yields a non-ideal SET. An analysis of the quantum SET reveals the presence of quantum corrections to the energy density, as well as non-equilibrium terms such as the shear pressure tensor. These quantum terms become dominant as the speed of light surface (SOL) is approached. While for the Dirac field, the Landau frame can be defined everywhere up to the SOL, this is not so for the Klein-Gordon field, which we analysed based on the quantum corrections calculated in Ref. [17]. The Landau frame becomes everywhere well defined when the system is enclosed inside a boundary placed inside or on the SOL.
An evaluation of the order of magnitude of the quantum corrections in a realistic astrophysical system (i.e. for a millisecond pulsar) shows that for such systems, quantum corrections become important only at cosmic ray energies, in which case the rigid rotation must be mantained up to subnuclear distances from the SOL. Since the left hand side of the above expression has a pole at ω = Ωm, the above expansion is not well defined when ω < Ωm. It is worth mentioning that the modes for which ω < 0 are no longer allowed when the system is enclosed inside a boundary placed inside or on the SOL [15,16]. Despite the fact that the modes with ω < 0 cannot be excluded from the mode sum in Eq. (A.7), we will show that the above procedure can still be used to recover the results in Ref. [17]. The sum over m can be performed using the following formula: Γ( j + 1 2 ) j! √ π a n, j z 2 j , (A. 13) where the coefficients a n, j can be determined as follows: a n, j = 1 (2 j)! lim such that a n, j vanishes when j > n. The following particular