On physics of a highly relativistic spinning particle in the gravitational field
PPrepared for submission to JHEP
On physics of a highly relativisticspinning particle in the gravitational field
R. Plyatsko and M. Fenyk
Pidstryhach Institute for Applied Problems in Mechanics and Mathematics,3-b Naukova Street, Lviv 79060, Ukraine
E-mail: [email protected] , [email protected] Abstract:
The Mathisson-Papapetrou equations are used for investigations of influence ofthe spin-gravity coupling on a highly relativistic spinning particle in Schwarzschild’s field.It is established that interaction of the particle spin with the gravitomagnetic componentsof the field, estimated in the proper frame of the particle, causes the large accelerationof the spinning particle relative to geodesic free fall. As a result the accelerated chargedspinning particle can generate intensive electromagnetic radiation when its velocity is highlyrelativistic. The significant contribution of the highly relativistic spin-gravity coupling tothe energy of the spinning particle is analyzed.Keywords: Black Holes, Classical Theories of Gravity a r X i v : . [ g r- q c ] M a y ontents Known important properties of gravitational interaction in general relativity were discov-ered through investigations of motions of small test bodies (particles) in gravitational fieldof a massive body. For example, physics of black holes was studied by consideration ofworld lines and trajectories of simple test particles which follow the geodesic lines in theSchwarzschild and Kerr metrics [1, 2]. Here "simple" means that the particle does notpossess inner structure, with inner rotation or higher multipoles. In classical picture of thegravitational collapse of a massive object quantum properties of the particles do not takeinto account.Electrons, protons and other particles with nonzero spin which in some classical approx-imation can be considered as particles with inner rotation do not follow geodesic trajectoriesexactly. However, as it is emphasized in [1], in usual situations deviations of motions of thespinning test body (particles) from the corresponding geodesic motions are very small: thisconclusion follows from the Mathisson-Papapetrou equations [3, 4]. Just these equations,which are the generalization of the geodesic equations for description of motions of a testrotating body in general relativity, were derived for the first time in [3]. (This paper isabsent in refs. of [1], in contrast to paper [4] which was published much later than [3]).However, unusual situations with spinning particles arise when their orbital velocity in theSchwarzschild or Kerr field becomes very high, close to the speed of light. Then the influenceof the spin-gravity coupling on the particles orbits can be significant [5–16]. The physicalreason of this situation is connected with the fact that in a frame which moves relative toSchwarzschild’s or Kerr’s source with the very high velocity the values of components ofthe gravitational field are much greater than in frames with low velocities. For example,as a result of strong spin-gravity action on the particle, the space regions of existence ofthe highly relativistic circular orbits for spinning particles in the Schwarzschild and Kerr– 1 –ackgrounds are much wider than for spinless particles [7, 8, 10, 11, 15, 16]. This fact isinteresting for analysis of possible mechanism of generation of synchrotron radiation forcharged spinning particles near compact astrophysical objects.The purpose of this paper is to investigate the contribution of the spin-gravity couplingto the energy of a spinning particle moving with high velocity in Schwarzschild’s field, andto obtain some estimation for electromagnetic radiation of a highly relativistic chargedspinning particle. This investigations are based on the analysis of solutions of the exactMathisson-Papapetrou equations.The paper is organized in the following way. In sect. 2 the Mathisson-Papapetrouequations and their physical meaning are discussed. Section 3 is devoted to the analysisof the relations following from these equations in the comoving tetrads representation forSchwarzschild’s metric. The dependence of the spinning particle 3-acceleration relativeto geodesic free fall as measured by the comoving observer on the particle velocity inSchwarzschild’s field is evaluated. For a charged spinning particle the expression for theintensity of its electromagnetic radiation caused by the acceleration is evaluated. In sect. 4we investigate the difference in the values of energies of the spinning and spinless particlesat their high velocities in Schwarzschild’s field. We conclude in sect. 5.
The initial form of the Mathisson-Papapetrou equations is [3]
Dds (cid:18) mu λ + u µ DS λµ ds (cid:19) = − u π S ρσ R λπρσ , (2.1) DS µν ds + u µ u σ DS νσ ds − u ν u σ DS µσ ds = 0 , (2.2) S λν u ν = 0 (2.3)where u λ ≡ dx λ /ds is the particle’s 4-velocity, S µν is the antisymmetric tensor of spin, m and D/ds are the mass and the covariant derivative along u λ , respectively. Here, and inthe following, greek indices run through 1, 2, 3, 4 and latin indices run through 1, 2, 3; thesignature of the metric (–, –, –, +) and the unites c = G = 1 are chosen.Note that eqs. (2.1), (2.2) and (2.3) have an important unusual feature as compareto equations in classical (nonrelativistic) mechanics which describe the propagation of thecenter of mass of a rotating body and possible changes of its angular velocity. Namely, inclassical mechanics the motion of such a body is fully determined by the given initial valuesof the coordinates and velocity of the center of mass and the value of the angular velocity.Another situation takes place with eqs. (2.1), (2.2) and (2.3). Indeed, because the left-handside of eq. (2.1) contains the term D S µν ds , (2.4)which is proportional to the second derivative of the angular velocity, the fixed initial valuesof the coordinates, linear velocity and only angular velocity without given initial value of– 2 –he angular acceleration, in general, are insufficient for determination of a single solutionof eqs. (2.1), (2.2) and (2.3). This situation cannot be changed if after using relation (2.3)in eq. (2.1) instead of (2.4) to write − S λµ D u µ ds − DS λµ ds Du µ ds , (2.5)because expression (2.5) contains the second derivative of the linear velocity and thento determine some unique solution it is not sufficient to point out only initial values ofcoordinates and linear velocity, without acceleration.More simple case, without the second derivatives in the Mathisson-Papapetrou equa-tions, takes place when one consider the deviation of the particle motions from geodesicsin the linear spin approximation.Then it is sufficient instead of eqs. (2.1) and (2.2) to dealwith equations m Du λ ds = − u π S ρσ R λπρσ , (2.6) DS µν ds = 0 (2.7)(at relation (2.3), it follows from (2.1) and (2.2) that m = const ).To avoid the terms with too high derivatives in the exact Mathisson-Papapetrou equa-tions it was proposed to consider instead of (2.1), (2.2) and (2.3) some modified equations[17, 18] DP λ ds = − u π S ρσ R λπρσ , (2.8) DS µν ds = 2 P [ µ u ν ] , (2.9) S λν P ν = 0 , (2.10)where P ν = mu ν + u λ DS νλ ds (2.11)is the particle 4-momentum. An important difference in eqs. (2.1), (2.2), (2.3) and (2.8),(2.9), (2.10) consists in the form of relations (2.3) and (2.10): because of the second term inthe right-hand side of expression (2.11), the vector P ν , in general, is not parallel to u ν andrelation (2.3) does not follow from (2.10). (By the way, from eqs. (2.8), (2.9) and (2.10)some explicit expression for the components of u λ through P µ are obtained [19]).Often relations (2.3) and (2.10) are treated as supplementary conditions for the Mathis-son-Papapetrou equations. Without any supplementary condition, these equations describesome wide range of the representative points [4] which can be in different connection witha rotating body. To describe just the inner rotation of the body it is necessary to fix theconcrete corresponding representative point. In Newtonian mechanics, the inner angularmomentum of a rotating body is defined relative to its center of mass and just the motionof this center represents the propagation of the body in the space. In relativity, the positionof the center of mass of a rotating body depends on the frame [20, 21]. Then condition(2.3),which follows from the usual definition of the center of mass position[22], is common for the– 3 –o-called proper and nonproper centers of mass. (We use the terminology when the properframe for a spinning body is determined as a frame where the axis of the body rotation isat rest; correspondingly, the proper center of mass is calculated in the proper frame.) Theusual solutions of the Mathisson-Papapetrou equations at condition (2.3) in the Minkowskispacetime describe the motion of the proper center of mass of a spinning body, whereasthe helical solutions describe the motions of the family of the nonproper centers of mass[20, 21]. Detailed analysis of different centers of mass is presented in [23] where it is shownthat helical motions are fully physical in the context of the kinematical interpretation.In contrast to condition (2.3), relation (2.10) picks out a unique world line of a spinningparticle in the gravitational field. However, from physical point of view eq. (2.10) hasexplicit restriction for its applications in the region of the highly relativistic motions of aspinning particle relative to the source of the gravitational field [9, 24]. Different situationsthat arise with condition (2.10) for a fast spinning particle are considered in [25–29].Both at condition (2.3) and (2.10), the Mathisson-Papapetrou equations have the con-stant of motion S = 12 S µν S µν , (2.12)where | S | is the absolute value of spin. When dealing with the Mathisson-Papapetrouequations the condition for a spinning test particle | S | mr ≡ ε (cid:28) (2.13)must be taken into account [30], where r is the characteristic length scale of the backgroundspace-time (in particular, for the Schwarzschild metric r is the radial coordinate).Equations (2.1) and (2.2) at condition (2.3) can be presented through the 3-componentvalue S i [12], where by definition S i = 12 u √− gε ikl S kl , (2.14)here ε ikl is the spatial Levi-Civita symbol. Then eq. (2.2) takes the form [12] u ˙ S i + 2( ˙ u [4 u i ] − u π u ρ Γ ρπ [4 u i ] ) S k u k + 2 S n Γ nπ [4 u i ] u π = 0 , (2.15)where a dot denotes differentiation with respect to the proper time s , and square bracketsdenote antisymmetrization of indices; Γ nπ are the Christoffel symbols. Let us consider eqs. (2.1), (2.2) and (2.3) in the linear spin approximation when accordingto (2.6) the deviation of the spinning particle world line from the geodesic line, for which Du λ /ds = 0 , is determined by the term − u π S ρσ m R λπρσ . (3.1)– 4 –ecause the components S ρσ are proportional to S , according to (2.13) expression (3.1) isproportional to the small value ε . It means that when the particle velocity is not very high,i.e. when the relation | u π | (cid:29) is not satisfied, it is possible to search the solutions of theMathisson-Papapetrou equations in the form of some small corrections to the correspondingsolutions of the geodesic equations (at the condition that the values of the Riemann tensorcomponents are not very high). Concerning the case | u π | (cid:29) more detailed analysis isnecessary. Indeed, at first glance, even when the relation | u π | (cid:29) is satisfied and theabsolute value of the expression (3.1) becomes much grater than at the low velocity, onecan suppose that this situation is a result of the kinematic effect only, when the value ofthe proper time of the highly relativistic particle is much less than for a slow particle. Toverify this supposition, it is appropriate to consider the value of expression (3.1) in theframe comoving with the particle.For description of the comoving frame of reference we use the set of orthogonal tetrads λ µ ( ν ) , where λ µ (4) = u µ and the relations λ µ ( ν ) λ π ( ρ ) g µπ = η ( ν )( ρ ) , g µν = λ ( π ) µ λ ( ρ ) ν η ( π )( ρ ) (3.2)takes place (here, in contrast to the indices of the global coordinates, the local indices areplaced in the parenthesis; g µν and η ( ν )( ρ ) are the metric tensor of the curved spacetime andthe Minkowski tensor, respectively). Without loss in generality, we direct the first spacelocal vector (1) along the direction of spin. Then from eq. (2.6) we have [12] a ( i ) = − S (1) m R ( i )(4)(2)(3) , (3.3)where a ( i ) are the local components of the particle 3-acceleration relative to geodesic free fallas measured by the comoving observer; S (1) is the single nonzero component of the particlespin. Note that the right-hand side of eq. (3.3) is the direct consequence of expression(3.1).Taking into account the definition of the gravitomagnetic components B ( i )( k ) of the grav-itational field in general relativity according to [31] B ( i )( k ) = − R ( i )(4)( m )( n ) ε ( m )( n )( k ) , (3.4)eq. (3.3) can be written in the form a ( i ) = − S (1) m B ( i )(1) . (3.5)Let us analyze eq. (3.5) in the specific case when the spinning particle is moving in thegravitational field of Schwarzschild’s mass. We use the standard Schwarzschild coordinates x = r, x = θ, x = ϕ, x = t when the nonzero components of the metric tensor g µν are g = − (cid:18) − Mr (cid:19) − , g = − r , g = − r sin θ, g = 1 − Mr , (3.6)– 5 –here M is the mass of Schwarzschild’s source of the gravitational field. We consider thecase when the particle moves in the plane θ = π/ and its spin (as well as the first spacelocal axis (1)) is orthogonal to this plane. It is convenient to orient the second space axisalong the direction of the particle’s motion. Then by direct calculation according to (3.2),(3.4) and (3.6) we obtain B (1)(2) = B (2)(1) = 3 Mr u (cid:107) u ⊥ (cid:112) γ − (cid:18) − Mr (cid:19) − / , (3.7) B (1)(3) = B (3)(1) = 3 Mr u ⊥ γ (cid:112) γ − , (3.8)where γ is the relativistic Lorentz factor of the moving particle as estimated by an observerwhich is at rest relative to the source of the gravitational field. Let us compare the valuesfrom (3.7) and (3.8) at low and high velocities. When the velocity is low with u (cid:107) = δ , u ⊥ = δ , | δ | (cid:28) , | δ | (cid:28) , and γ − (cid:28) , where by (2.9) ∆ = (cid:18) − Mr (cid:19) − δ + δ , (3.9)it follows from (3.7) and (3.8) that B (1)(2) = B (2)(1) ≈ Mr δ δ ∆ (cid:18) − Mr (cid:19) − / , (3.10) B (1)(3) = B (3)(1) ≈ Mr δ ∆ . (3.11)That is, at low velocity the common term M/r in the expressions for the gravitomagneticcomponents (3.10) and (3.11) is multiplied by corresponding small factors: (cid:12)(cid:12)(cid:12)(cid:12) δ δ ∆ (cid:12)(cid:12)(cid:12)(cid:12) (cid:28) , (cid:12)(cid:12)(cid:12)(cid:12) δ ∆ (cid:12)(cid:12)(cid:12)(cid:12) (cid:28) . In the highly relativistic region, when γ (cid:29) and both u (cid:107) and u ⊥ have order γ , it followsfrom (3.7) and (3.8) that B (1)(2) = B (2)(1) ∼ Mr (cid:18) − Mr (cid:19) − / γ, (3.12) B (1)(3) = B (3)(1) ∼ Mr γ . (3.13)When only u ⊥ (cid:29) , with u (cid:107) (cid:28) u ⊥ , the values from (3.12) are proportional to u (cid:107) , and thevalues from (3.13) are proportional to γ . In the case, when u (cid:107) (cid:29) and u ⊥ (cid:28) u (cid:107) , thevalues from (3.12) and (3.13) are proportional to u ⊥ and u ⊥ , respectively. So, according to(3.5), (3.12), (3.13) the absolute values of a ( i ) become much grater at the highly relativisticvelocities of the spinning particle. It means that the smallness of ε from (2.13) does notlead to the conclusion about the small influence of the particle spin on its acceleration as– 6 –stimated by the comoving observer. (Note that in the above considered partial case of theparticle motion in Schwarzschild’s field the relation | S (1) | = | S | takes place.)It follows from (3.5), (3.7), (3.8) that the absolute value of the spinning particle accel-eration | (cid:126)a | = (cid:113) a + a + a is determined by the expression | (cid:126)a | = 3 Mr | S | mr | u ⊥ | (cid:113) u ⊥ , (3.14)and the vector (cid:126)a is oriented along the radial direction. According to (3.14) | (cid:126)a | does notdepend on the radial component of the particle velocity and essentially depends on itstangential velocity. In the case of the highly relativistic motion with u ⊥ (cid:29) by (3.14) wehave | (cid:126)a | = 3 Mr εγ , (3.15)where γ is the Lorentz factor calculated by the tangential velocity u ⊥ , and ε is determinedin (2.13).We use expression (3.15) to estimate the electromagnetic radiation of a spinning particlewhich posses the electric charge q . Indeed, according to the known result of the classicalelectrodynamic the intensity I of the electromagnetic radiation in the frame where thevelocity of the charge particle is equal to 0 with nonzero acceleration w is given by theexpression [32] I = 2 q w c , (3.16)where c is the speed of light. Inserting into (3.16) expression (3.15) as w in units where c = 1 we get I = 6 q M r ε γ . (3.17)Equation (3.17) shows that due the term γ the value I can be significant for some hightangential velocities even for small values of ε and far from Schwarzschild’s horizon ( r (cid:29) M ).The results presented in this section describe the properties of the spin-gravity cou-pling in the proper frame of the spinning particle. In this context the question arises: canthe highly relativistic spin-gravity coupling significantly deviate trajectories of the spin-ning particle from the geodesic trajectories by their description in the terms of the globalSchwarzschild coordinates? Different cases of the essentially nongeodesic orbits of the highlyrelativistic spinning particle in Schwarzschild’s field are investigated in [7, 9, 10]. According to the geodesic equations there is expression for the energy of a spinless particlewith mass m in Schwarzschild’s field: E = mu = m (cid:18) − Mr (cid:19) / γ, (4.1)– 7 –here γ = √ u u is the relativistic Lorentz factor calculated by the particle velocity relativeto the source of the Schwarzschild field, r is the standard radial coordinate. That is thisenergy is proportional to the γ factor, similar as in the case of the free particle motionin special relativity. Other situations arise in the case of the spinning particle motions inSchwarzschild’s field. Then by the Mathisson-Papapetrou equations the expression for aspinning particle can be written as [19] E = mu + g u λ DS λ ds + 12 g µ,ν S νµ . (4.2)In contrast to (4.1) the value of energy (4.2) depends not only on the initial velocity of aspinning particle and on r , but on the spin-gravity coupling as well. In the partial case ofthe radial motion of the spinning particle in Schwarzschild’s field the value of its energydoes not depend on the absolute value and orientation of the spin and coincides exactlywith the value of energy of the spinless particle, as well as in this case the world line of thespinning particle coincides with the corresponding geodesic world line (it is easy to obtainthis result after writing eqs. (2.1)–(2.3) at condition θ = const , ϕ = const ). However, anynonzero value of the particle tangential velocity leads to some deviation of the value of thespinning particle energy from the value of the energy of the spinless particle. Naturally,when | u ⊥ | (cid:28) , i.e. for low values of the tangential velocity, this deviation is small. It isinteresting to investigate the dependence of the spinning particle energy on the tangentialvelocity in the highly relativistic region when | u ⊥ | (cid:29) . For this purpose it is convenient todeal with the exact Mathisson-Papapetrou equations in the form of the first order differentialequations for the 11 dimensionless quantities y i where by definition y = rM , y = θ, y = ϕ, y = tM ,y = u , y = M u , y = M u , y = u ,y = S mM , y = S mM , y = S mM . (4.3)These equations are presented in the explicit form in [12] as ˙ y = y , ˙ y = y , ˙ y = y , ˙ y = y , ˙ y = A , ˙ y = A , ˙ y = A , ˙ y = A , ˙ y = A , ˙ y = A , ˙ y = A , (4.4)where A i are the corresponding functions of y i and contain the two constants of motion:the energy and angular momentum (a dot denotes the usual derivative with respect to thedimensionless value x = s/M ). At the fixed initial values of y i different values of theseconstants correspond to motions of different centers of mass of the spinning particle.Let us consider eqs. (4.4) in the partial case of the spinning particle motion in theplane θ = π/ with the spin orthogonal to this plane. It means that in notation (4.3) weput y = π/ , y = 0 , y = 0 , y = 0 and others nonzero functions y i ( x ) can be findby the numerical integration of eqs. (4.4). The important point in this procedure is finding– 8 – ⊥ (0) E spin /E geod Table 1 . On comparison of the energies of the spinning and spinless particles at different orbitalvelocity for u ⊥ (0) > . u ⊥ (0) E spin /E geod -2.35 0.99-4.70 0.96-11.75 0.76-17.62 0.44-21.15 0.20 Table 2 . On comparison of the energies of the spinning and spinless particles at different orbitalvelocity for u ⊥ (0) < . values of the energy and angular momentum which correspond just to the proper center ofmass of the particle for the fixed initial values of y i , not to the helical solutions. For thispurpose we use computer searching. As a typical case here we present the results for thespinning particle with ε ≡ S mM = 10 − (note that in contract to ε from (2.13), the value of ε does not depend on r ) whichbegins motion from the position r = 2 . M with the initial value of the radial velocity u (cid:107) = − − with different initial values of the tangential velocity u ⊥ . Table 1 describe thesituations when the sign of the u ⊥ is positive with the orientation of the particle spin when S ≡ S θ > . Table 2 corresponds to the cases with u ⊥ (0) < and the same value of the S as in table 1. Both tables 1 and 2 show the ratio of the energy of the spinning particle E spin to the value of the energy of the spinless particle E geod which moves along the geodesiclines and start with the same initial velocity as the spinning particle. For u ⊥ (0) = 0 wehave E spin /E geod = 1 , exactly. When | u ⊥ (0) | (cid:28) the value E spin is almost equal to E geod with high accuracy. Other situations arise when | u ⊥ | is growing up to the highly relativisticmotions with u ⊥ (cid:29) . According to tables 1 and 2 at the highly relativistic regime thedifference between E spin and E geod is growing significantly with growing | u ⊥ | . There areessential difference of the data in tables 1 and 2: in the first case E spin /E geod > whereasin the second case E spin /E geod > . This property corresponds to the known result thatat S > and u ⊥ (0) > the spin-gravity coupling acts on the particle as some attractiveforce whereas at S > and u ⊥ (0) > this action is repulsive [10, 12].So, the contribution of the spin-gravity coupling to the energy of a spinning particle inScwarzschild’s field becomes large when its velocity is highly relativistic.– 9 – Conclusions
In addition to the known results concerning the influence of the spin-gravity coupling onworld lines and trajectories of the highly relativistic spinning particle in the Scwarzschildfield [10, 12], in this paper we present the results about the effect of the highly relativisticspin-gravity coupling on the particle’s energy. Depending on the correlation of the spinorientation and the particle orbital velocity the values of the spinning particle energy canbe much larger or less than the corresponding values for the spinless particle.In the case of highly relativistic motions of the charged spinning particle in Scwarz-schild’s field, in sec. 3 we considered the intensity of the energy of its electromagneticradiation as estimated in the proper frame of the particle. It is important that this valueis proportional to the γ , i.e. becomes very large for highly relativistic orbital velocities ofthe particle.In further investigations it is interesting to apply the results of this paper to the analysisof possible role of the highly relativistic spin-gravity coupling in the astrophysical processeswith fast spinning particles in strong gravitational fields. Acknowledgments
This work was supported by the budget program of Ukraine "Support for the developmentof priority research areas" (CPCEC 6451230).
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