Electromagnetic Field in the Newman-Unti-Tamburino Spacetime

Nonlinear Phenomena in Complex Systems, vol. 28, no. 3 (2025), pp. 248 - 260 Electromagnetic Field in the Newman-Unti-Tamburino Spacetime N. G. Krylova∗and V. M. Red’kov† B. I. Stepanov Institute of Physics of NAS of Belarus, 68 Nezavisimosti Ave., 220072 Minsk, BELARUS‡ (Received 23 May, 2025) Using the conventional tetrad method by Tetrode-Weyl-Fock-Ivanenko, we specify the Maxwell equations for Newman-Unti-Tamburino (NUT) spacetime. We apply the covariant Majorana–Oppenheimer matrix presentation of the Maxwell theory. Separation of the variables is performed, and the equations for angular and radial components are solved in terms of hypergeometric and confluent Heun functions respectively. We find the NUTcharge dependent quantization rule for the angular separation constant. Behavior of the radial components with structure of outgoing and ingoing waves is studied near the outer event horizon, and we demonstrate that the probability of particle-antiparticle production on the outer event horizon decreases with the increase of the NUT charge; the expression of temperature for the Hawking radiation of the photons coincides with that for the fermions production on the horizon. The effective constitutive relations, generated by metric structure of NUT spacetime, are derived; it is shown that the existence of the NUT charge leads to entanglement of electric and magnetic field components in these relations. PACS numbers: 02.30.Gp, 02.40.Ky, 03.65Ge, 04.62.+v Keywords: quantum mechanics, Maxwell theory, Majorana–Oppenheimer formalism, Riemannian geometry, Newman-Unti-Tamburino spacetime, Hawking radiation, constitutive relations DOI: https://doi.org/10.5281/zenodo.17237014 1. Introduction The Newman-Unti-Tamburino (NUT) metric is an axially symmetric vacuum solution of Einstein equations with two parameters, the black hole mass and the NUT parameter. The NUT spacetime is generalization of the Schwarzschild one, due to the presence the NUT parameter (or NUT charge) [1–3]. NUT parameter is understood as a gravito-magnetic charge, or as gravito-magnetic monopole, or magnetic (gravitomagnetic) mass [3, 4]. For NUT spacetime the singularities of the Misner string type arise. This leads to the difficulties in thermodynamical analysis and, as a consequence, in physical interpretation ∗E-mail: nina-[email protected] †E-mail: [email protected]net.by ‡Also at Belarusian State Agrarian Technical University, 99 Nezavisimosti Ave., Minsk, BELARUS of the NUT parameter [5]. Mostly, the NUT parameter is interpreted as a linear source of a pure angular momentum [6] or the twist parameter of the surrounding vacuum spacetime or electromagnetic (EM) universe in the presence of the EM field [7]. The existence of the Misner string as well as the non-vanishing gtφ components in metric tensor, lead to the spacetime areas where T-symmetry is broken and circular timelike (null) geodesics exist. By this reason, sometimes NUT spacetime is considered as nonphysical one. However, in [8] it was shown that geodesics of the freely falling observer are not closed time-like ones, so the NUT spacetime can be geodesically complete, without causal pathologies. Currently, the black holes with NUT parameter are considered as one of the most intriguing cosmological objects. As shown in [9], the black hole with NUT charge has the smaller Hawking temperature, 248 Electromagnetic Field in Background of the NUT Spacetime 249 and pure gravitomagnetic monopoles without ordinary mass (if exist) may not be decayed due to the Hawking radiation by now. In [8] it was demonstrated that the supercritically charged black holes with NUT parameter belong to traversable wormhole solutions. Besides, the NUT black holes may exhibit a twist in the lensing pattern [10], and an asymmetry of black hole shadow or the Lense-Thirring effect [3, 11]. Classical equations of motion in NUT spacetimes were extensively studied [12–14]. However, the papers on the quantum-mechanical problems of the particles in the background of NUT spacetimes are few. In [15], within the Newman-Penrose formalism, the Maxwell equations have been studied in Taub-NUT background which has singularity at whole axes θ= 0, π. After separating the variables, solutions of the angular equations were constructed in terms of Jacobi polynomials. The radial equations (see equations (28a) and (44a) in [15]) were transformed respectively to equations with hypergeometric and Heun’s structure on the lefthand side, while the right-hand sides include terms of different order in frequency ω. Only approximate solutions with the zeroth order in frequency ωin the right-hand side have been found in terms of the hypergeometric and Heun functions. The goal of the present paper is to study the electromagnetic field in background of original NUT spacetime which has singularity only at semiaxes θ=π. We apply the conventional tetrad method developed by Tetrode-Weyl-FockIvanenko in [16–19], and covariant Majorana– Oppenheimer matrix presentation of the Maxwell theory [20–22], in the background of the original NUT spacetime. We have solved the angular and radial equations in terms of hypergeometric and Heun functions, respectively. 2. Ricci rotation coefficients for NUT space NUT-metric is determined by the line element ds2= Φ dt + 4asin2(θ/2)dφ2−dr2 Φ −a2+r2dθ2+ sin2θdφ2, (2.1) Φ=1−rgr+ 2a2 r2+a2=∆ ρ2, where tis a time coordinate, t, θ, φ are spherical coordinates, rg= 2Mis a Schwarzschild horizon of black hole with mass M,ais a NUT parameter, ρ=r2+a2,∆ = r2+rgr−a2. The corresponding metric tensor is nondiagonal gαβ = Φ 0 0 2aΦ (1 −cos θ) 0−1 Φ0 0 0 0 −a2−r20 2aΦ (1 −cos θ) 0 0 4a2Φ (1 −cos θ)2−a2+r2sin2θ  . We chose the following tetrad e(a)α(x) =  √Φ 0 0 2a√Φ (1 −cos θ) 01 √Φ0 0 0 0 √a2+r20 0 0 0 √a2+r2sin θ  . (2.2) Applying the known formulas [23] γabc =1 2(λabc +λbca −λcab), λabc =∂e(a)α ∂xβ−∂e(a)β ∂xαeα (b)eβ (c), Nonlinear Phenomena in Complex Systems Vol. 28, no. 3, 2025 250 N. G. Krylova and V. M. Red’kov we find the relevant Ricci rotation coefficients γab0=  0Φ0 2√Φ0 0 −Φ0 2√Φ0 0 0 0 0 0 a√Φ a2+r2 0 0 −a√Φ a2+r20  , γab1= 0000 0000 0000 0000  , γab2=  0 0 0 a√Φ a2+r2 0 0 r√Φ a2+r20 0−r√Φ a2+r20 0 −a√Φ a2+r20 0 0  ,(2.3) γab3=  0 0 −a√Φ a2+r20 0 0 0 r√Φ a2+r2 a√Φ a2+r20 0 1 tan θ√a2+r2 0−r√Φ a2+r2−1 tan θ√a2+r20  . 3. Maxwell equations, separating the variables It is convenient to apply the matrix complex Silberstein – Majorana – Oppenheimer formalism, so the covariant matrix Maxwell equation reads (for more detail see [20–22, 24]) αaeβ (a) ∂ ∂xβ+1 2jmnγmnaΨ = 0, Ψ =  0 E+icB , (3.1) where Eand Bare electric and magnetic fields vectors, generators jmn of the complex vector representation of orthogonal group SO(3.C)equal j23 =s1, j01 =is1, j31 =s2, j02 =is2, j12 =s3, j03 =is3. In the cyclic basis, the matrix sread s1= 0 0 0 0 0−i0 0 0 0 0 0 000i  , s2=1 √2  0 0 0 0 0 0 −i0 0−i0−i 0 0 −i0  , s3=1 √2  0 0 0 0 0 0 −1 0 0 1 0 −1 0 0 1 0  . Taking into account expression for the tetrad (2.2) and the Ricci rotation coefficients (2.3), the Maxwell matrix equation (3.1) is obtained in the following form hα0ρ √∆+α32a ρr1−cos θ 1 + cos θ∂ ∂t −α1√∆ ρ ∂ ∂r −i α0s1∆0 2√∆ρ−(r+ia)√∆ ρ3+α3s2−α2s3(r+ia)√∆ ρ3−1 ρΣθ,φiΨ=0, (3.2) Σθ,φ =α2∂ ∂θ +α31 sin θ ∂ φ+s1 1 tan θ, α0=−i, α1= 0 0 1 0 0−i0 0 −1 0 0 0 0 0 0 i  , Нелинейные явления в сложных системах Т. 28, № 3, 2025 Electromagnetic Field in Background of the NUT Spacetime 251 α2=1 √2  0−1 0 1 1 0 −i0 0−i0−i −1 0 −i0  , α3=1 √2  0−i0−i −i0−1 0 0 1 0 −1 −i0 1 0  . As the NUT-metric does not depend on the time and angle φ, we should search wave functions in the form Ψ = e−iωteimφ  0 R1(r)T1(θ) R2(r)T2(θ) R3(r)T3(θ)  .(3.3) Substituting the last in the equation (3.2), we get R2T22aω tan θ 2+mcsc θ) + R1T1∆0 √2√∆+i√2ρ2ω √∆+√2√∆T1R0 1+R2T0 2= 0,(3.4) R1T12aω tan θ 2−cot θ+mcsc θ+R3T32aω tan θ 2+ cot θ+mcsc θ +2√2√∆R2T2(r+ia) ρ2+√2√∆T2R0 2−R1T0 1+R3T0 3= 0, (3.5) R2T22aω tan θ 2+mcsc θ+R3T3∆0 √2√∆−i√2ρ2ω √∆+√2√∆T3R0 3−R2T0 2= 0,(3.6) −R1T12aω tan θ 2−cot θ+mcsc θ+R3T32aω tan θ 2+ cot θ+mcsc(θ) +i√2ρ2R2T2ω √∆+R1T0 1+R3T0 3= 0. (3.7) With the use of simple algebraic calculation we separate the variables. The radial equations read (we introduce the separation constants ξ1, ξ2, ξ3, ξ4) √2√∆R0 1+R1∆0 √2√∆+i√2ρ2ω √∆+ξ1R2= 0,(3.8) √2√∆R0 3+R3∆0 √2√∆−i√2ρ2ω √∆+ξ2R2= 0,(3.9) √2 2√∆R0 2+√2R2√∆(r+ia) ρ2+iρ2ω 2√∆+ξ3R3= 0,(3.10) √2 2√∆R0 2+√2R2√∆(r+ia) ρ2−iρ2ω 2√∆−ξ4R1= 0; (3.11) Nonlinear Phenomena in Complex Systems Vol. 28, no. 3, 2025 252 N. G. Krylova and V. M. Red’kov and for the angular components: T0 2+T22aω tan θ 2+mcsc θ−ξ1T1= 0, T0 3+T32aω tan θ 2+ cot θ+mcsc θ−ξ3T2= 0,(3.12) T0 1−T12aω tan θ 2−cot θ+mcsc θ−ξ4T2= 0, T0 2−T22aω tan θ 2+mcsc(θ)Big)+ξ2T3= 0.(3.13) Without loss of generality won can take ξ1= −ξ4= Λ1,ξ2=ξ3= Λ2.Then the last equations take the form √2√∆R0 1+R1∆0 √2√∆+i√2ρ2ω √∆+ Λ1R2= 0, √2 2√∆R0 2+√2R2√∆(r+ia) ρ2−iρ2ω 2√∆+Λ1R1= 0, (3.14) √2√∆R0 3+R3∆0 √2√∆−i√2ρ2ω √∆+ Λ2R2= 0, √2 2√∆R0 2+√2R2√∆(r+ia) ρ2+iρ2ω 2√∆+Λ2R3= 0; (3.15) and T0 2+T22aω tan θ 2+mcsc θ−Λ1T1= 0, T0 1−T12aω tan θ 2−cot θ+mcsc θ+ Λ1T2= 0, (3.16) T0 3+T32aω tan θ 2+ cot θ+mcsc θ−Λ2T2= 0, T0 2−T22aω tan θ 2+mcsc θ+ Λ2T3= 0.(3.17) Three of four equations in both these systems are independent. The fourth equation can be expressed as a combination of the last three if the following condition performed: 4aω −Λ2 2+ Λ2 1= 0.(3.18) 4. Angular equations solution Expressing T2from the first equation in the system (3.16) and substituting it into the second one, we get the second-order equation for the function T1: T00 1+ cot θT0 1 +Λ2 1+ 4a2ω2−2aω(1 + 2m+ 4aω) 1 + cos θ −(1 + m2) sin2θ+ 2 cot θm sin θ+aω tan θ 2T1= 0. (4.1) In the same way, one get the equation for the function T2: T00 2+ cot θT0 2+Λ2 1+ 2aω + 4a2ω2 −4aω(m+ 2aω) 1 + cos θ−m2 sin2θT2= 0. (4.2) Introducing the new variable z= sin2θ 2, one transform the equations (4.1)-(4.2) to the form (1 −z)zT00 1+ (1 −2z)T0 1+Λ2 1+ 2aω(2aω + 1) −(4aω +m+ 1)2 4z+(m−1)2 4(z−1) T1= 0, (4.3) (1 −z)zT00 2+ (1 −2z)T0 2+Λ2 1+ +2aω(2aω + 1) −2aω(2aω +m) z+m2 4(z−1)zT2= 0. (4.4) We search the solution with the structure T1=zA(z−1)BG1, T2=zC(z−1)DG2; substituting the last in the equations (4.3)-(4.4), one get (1 −z)zG00 1+ (1 + 2A−2z(A+B+ 1))G0 1 +2aω(2aω + 1) −(A+B)(A+B+ 1) + Λ2 1G1= 0, Нелинейные явления в сложных системах Т. 28, № 3, 2025 Electromagnetic Field in Background of the NUT Spacetime 253 (1 −z)zG00 2+ (1 + 2C−2z(C+D+ 1))G0 2 +2aω(2aω + 1) −(C+D)(C+D+ 1) + Λ2 1G2= 0, here A=±1 2(1 + m+ 4aω), B =±1−m 2, C=±1 2(m+ 4aω), D =±m 2. The equations for G1,G2have the structure of hypergeometric type z(1 −z)G00 + [c−(a+b+ 1)z]G0−abG = 0, G=2F1(a, b, c;z); then the general form of solutions is (K1, K2 stand for some numerical constants): T1=K1zA(z−1)BG(a1, b1, c1;z), T2=K2zC(z−1)DG(a2, b2, c2;z), a1, b1=1 2+A+B±1 2q(1 + 4aω)2+ 4Λ2 1, c1= 1 + 2A; a2, b2=1 2+C+D±1 2q(1 + 4aω)2+ 4Λ2 1, c2= 1 + 2C. The cases of positive and negative values of number mshould be considered separately: m > 0, A =1 21 + m+ 4aω, B =m−1 2, C=1 2m+ 4aω, D =m 2, c1= 2 + m+ 4aω, a1=1 21+2m+ 4aω −1 2q(1 + 4aω)2+ 4Λ2 1, b1=1 21+2m+ 4aω +1 2q(1 + 4aω)2+ 4Λ2 1, c2=c1−1, a2=a1, b2=b1; m < 0, A =−1 21 + m+ 4aω, B =−m−1 2, C=−1 2m+ 4aω, D =−m 2, c1= 1 −m−4aω, a1=1 21−2m−4aω −1 2q(1 + 4aω)2+ 4Λ2 1, b1=1 21−2m−4aω +1 2q(1 + 4aω)2+ 4Λ2 1, c2=c1−1, a2=a1, b2=b1. We introduce the quantization rule in usual way by imposing the condition that the power series for the hypergeometric function are terminated, namely (btakes on non-positive integer values): m > 0, b1=1 21+2m+ 4aω +1 2q(1 + 4aω)2+ 4Λ2 1=−n1⇒ Λ2 1= (m+n1)(1 + m+n1+ 4aω) = N1(N1+ 1 + 4aω); (4.5) Nonlinear Phenomena in Complex Systems Vol. 28, no. 3, 2025 254 N. G. Krylova and V. M. Red’kov m < 0, b1=1 21−2m−4aω +1 2q(1 + 4aω)2+ 4Λ2 1=−n2⇒ Λ2 1= (1 −m+n2)(−m+n2−4aω) = N2(N2−1−4aω); (4.6) in the second case, the constraint n2−m−4aω − m > 0has to performed; so that in both cases Λ2>0. 5. Solving the radial equations In the system (3.14) we apply the following substitutions R1=F/√∆, R2=√2G/√∆, so we obtain F0+iρ2ω ∆F+Λ1 √∆G= 0, G0−iρ2ω ∆+∆0 2∆ −2(r+ia) ρ2G+Λ1 √∆F= 0. (5.1) Eliminating the function G, one derives the second-order equation for the function F(taking into account the explicit expressions for ∆,ρand introducing r1and r2as the roots of equation ∆ = 0:r1= 1/2(rg−qr2 g+ 4a2),r2= 1/2(rg+ qr2 g+ 4a2)): F00 +2 r−iaF0+ω2+−Λ2 1−2aω + 2ir1ω+ 2r2 1ω2 (r−r1)(r1−r2) −−Λ2 1−2aω + 2ir2ω+ 2r2 2ω2 (r−r2)(r1−r2)−ir1ω−r2 1ω2 (r−r1)2−ir2ω−r2 2ω2 (r−r2)2F= 0. (5.2) In the same way, we get the second-order equation for the function G: G00 +2 r−iaG0+ω2+−Λ2 1−2aω + 2r2 1ω2 (r−r1)(r1−r2)+2ia +r1 2r1(r−r1)(r1−r2)−−Λ2 1−2aω + 2r2 2ω2 (r−r2)(r1−r2) −2ia +r2 2r2(r−r2)(r1−r2)+1+4r2 1ω2 4(r−r1)2+1+4r2 2ω2 4(r−r2)2−i a(r−ia)−2 (r−ia)2G= 0. (5.3) Solutions of the equation (5.2) are searched in the form F=1 (r−ia)(r−r1)α(r−r2)βe−γrf; substituting the last into (5.2) leads to f00 −2γ−2α r−r1−2β r−r2f0+−−Λ2 1+ 2αβ + 2αγ (r2−r1)+2ir1ω+ 2r2 1ω2−2aω (r−r1) (r2−r1) +−Λ2 1+ 2αβ −2βγ (r2−r1)+2ir2ω+ 2r2 2ω2−2aω (r−r2) (r2−r1)f= 0 (5.4) at γ=±iω;α=−ir1ω, 1 + ir1ω;β= −ir2ω, 1 + ir2ω. In a new variable v=r−r1 r2−r1 the equation (5.4) is transformed to the confluent Нелинейные явления в сложных системах Т. 28, № 3, 2025 Electromagnetic Field in Background of the NUT Spacetime 255 Heun equation: f00 +−2γ(r2−r1) + 2α v+2β v−1f0+−A v+B v−1f= 0,(5.5) A=−Λ2 1+ 2αβ + 2αγ (r2−r1)+2ir1ω+ 2r2 1ω2−2aω, B=−Λ2 1+ 2αβ −2βγ (r2−r1)+2ir2ω+ 2r2 2ω2−2aω. The solution of the equation (5.5) can be written as follows f=C1HeunC [−A, B −A, 2α, 2β, 2γ(r1−r2), v] +C2v1−2αHeunC [−A1, B −A1−2(2α−1)β, 2−2α, 2β, 2γ(r1−r2), v], here A1=A−2(2α−1) (β+γ(r2−r1)) .Then, the original function R1has the form R1=(r−r1)α−1/2(r−r2)β−1/2 (r−ia)e−γrf. In the same way, we solve the system (3.15). Applying the change R3=W/√∆, R2=√2G/√∆, we get W0−iρ2ω ∆W+Λ2 √∆G= 0, G0−−iρ2ω ∆+∆0 2∆ −2(r+ia) ρ2G+Λ2 √∆W= 0. (5.6) Expressing the function Gfrom the first equation and substituting into the second, we get W00 +2 r−iaW0+ω2−Λ2 2−2aω + 2ir1ω−2r2 1ω2 (r−r1)(r1−r2) +Λ2 2−2aω + 2ir2ω−2r2 2ω2 (r−r2)(r1−r2)+ir1ω+r2 1ω2 (r−r1)2+ir2ω+r2 2ω2 (r−r2)2W= 0. (5.7) The solution of (5.7) is searched in the form W=1 (r−ia)(r−r1)χ(r−r2)ξe−γrw, we get w00 −2γ−2χ r−r1−2ξ r−r2w0+Λ2 2−2χξ −2χγ (r2−r1)+2ir1ω−2r2 1ω2−2aω (r−r1) (r2−r1) −Λ2 2−2χξ + 2ξγ (r2−r1)+2ir2ω−2r2 2ω2−2aω (r−r2) (r2−r1)w= 0 (5.8) Nonlinear Phenomena in Complex Systems Vol. 28, no. 3, 2025 256 N. G. Krylova and V. M. Red’kov at γ=±iω;χ=ir1ω, 1−ir1ω;ξ=ir2ω, 1−ir2ω. Taking into account the condition (3.18), the equation (5.8) reads w00 −2γ−2χ r−r1−2ξ r−r2w0+−−Λ2 1+ 2χξ + 2χγ (r2−r1)−2ir1ω+ 2r2 1ω2−2aω (r−r1) (r2−r1) +−Λ2 1+ 2χξ −2ξγ (r2−r1)−2ir2ω+ 2r2 2ω2−2aω (r−r2) (r2−r1)w= 0. (5.9) Comparing the equations (5.4) and (5.9), one can see that they are complex conjugated ones, so we have w=f∗(a symbol ∗denotes complex conjugation). The function R3is determined by the expression R3=(r+ia) (r−ia)R∗ 1. Algebraic equation to find the function R2is derived from equations (3.14-3.15): Λ2R3−Λ1R1+i√2ρ2ω √∆R2= 0. 6. Behavior near horizon To estimate the behavior of the functions R1, R2in the vicinity of the outer horizon, one should consider the equations (5.2), (5.3) at r→ r2. Preserving only the largest terms in these equations, one gets F00 +2 r−iaF0+r2ω(−i+r2ω) (r−r2)2F= 0, G00 +2 r−iaG0+1+4r2 2ω2 4(r−r2)2G= 0. So, the solutions of the last equations in the vicinity of the horizon have the form F∼(r−r2)−ir2ω,(r−r2)1+ir2ω; G∼(r−r2)1 2−ir2ω,(r−r2)1 2+ir2ω. Then, for the original functions R1=F/√∆, R2=√2G/√∆, the solutions represent the incident and reflected waves R1∼(r−r2)1/2+ir2ω,(r−r2)−1/2−ir2ω; R2∼(r−r2)ir2ω,(r−r2)−ir2ω. According the procedure proposed in [25–27], the scattering probability Γ =  Ψout(x>x2) Ψout(x<x2) 2 (6.1) is the probability of creating an outgoing particle outside the outer horizon and an ingoing antiparticle of negative energy inside the horizon. Then substituting the outgoing wave solutions into the formula (6.1), the probability of particleantiparticle pair creating is Γ = e−4πωr2.(6.2) the mean number ¯ Nωof photons emitted with a given frequency is determined by relation (ignoring the backscattering effect): ¯ N=Γ 1−Γ=1 e4πωr2−1.(6.3) We get the Bose-Einstein distribution ¯ N=1 1 + eω/T , T=1 4πr2 =1 2π(rg+qr2 g+ 4a2) ,(6.4) where Tdetermines the Hawking temperature. This expression for Hawking temperature coincides with the result obtained previously for the fermions production on the horizon. Нелинейные явления в сложных системах Т. 28, № 3, 2025