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Alexei A. Deriglazov, Walberto Guzmán Ramírez, "Ultrarelativistic Spinning Particle and a Rotating Body in External Fields", Advances in High Energy Physics, vol. 2016, Article ID 1376016, 27 pages, 2016. https://doi.org/10.1155/2016/1376016
Ultrarelativistic Spinning Particle and a Rotating Body in External Fields
We use the vector model of spinning particle to analyze the influence of spin-field coupling on the particle’s trajectory in ultrarelativistic regime. The Lagrangian with minimal spin-gravity interaction yields the equations equivalent to the Mathisson-Papapetrou-Tulczyjew-Dixon (MPTD) equations of a rotating body. We show that they have unsatisfactory behavior in the ultrarelativistic limit. In particular, three-dimensional acceleration of the particle becomes infinite in the limit. Therefore, we examine the nonminimal interaction through the gravimagnetic moment and show that the theory with is free of the problems detected in MPTD equations. Hence, the nonminimally interacting theory seems a more promising candidate for description of a relativistic rotating body in general relativity. Vector model in an arbitrary electromagnetic field leads to generalized Frenkel and BMT equations. If we use the usual special-relativity notions for time and distance, the maximum speed of the particle with anomalous magnetic moment in an electromagnetic field is different from the speed of light. This can be corrected assuming that the three-dimensional geometry should be defined with respect to an effective metric induced by spin-field interaction.
The problem of a covariant description of rotational degrees of freedom has a long and fascinating history [1–13]. Equations of motion of a rotating body in curved background were formulated usually in the multipole approach to description of the body; see  for the review. The first results were reported by Mathisson  and Papapetrou . They assumed that the structure of test body can be described by a set of multipoles and have taken the approximation which involves only first two terms (the pole-dipole approximation). The equations are then derived by integration of conservation law for the energy-momentum tensor, . Manifestly covariant equations were formulated by Tulczyjew  and Dixon [5, 6]. In the current literature, they usually appear in the form given by Dixon (the equations (6.31)–(6.33) in ); we will refer to them as Mathisson-Papapetrou-Tulczyjew-Dixon (MPTD) equations. They are widely used now to account for spin effects in compact binaries and rotating black holes; see [14–16] and references therein.
Concerning the equations of spinning particle in electromagnetic field, maybe the best candidates are those of Frenkel [9, 10] and Bargmann, Michel, and Telegdi (BMT) . Here, the strong restriction on possible form of semiclassical equations is that the reasonable model should be in correspondence with the Dirac equation. In this regard, the vector model of spin (see below) is of interest because it yields the Frenkel equations at the classical level and implies the Dirac equation after canonical quantization .
In this work, we study behavior of a particle governed by these equations (as well as by some of their generalizations) in the ultrarelativistic regime. To avoid the ambiguities in the passage from Lagrangian to Hamiltonian description and vice versa, and in the choice of possible form of interaction, we start in each case from an appropriate variational problem. The vector models of spin provide one possible way to achieve this (for early attempts to build a vector model, see review ). In these models, the basic variables in spin sector are and , where is non-Grassmann vector and represents its conjugated momentum. The spin-tensor is a composite quantity constructed from these variables; . To have a theory with right number of physical degrees of freedom for the spin, certain constraints on the eight basic variables should follow from the variational problem. It should be noted that, even for the free theory in flat space, search for the variational problem represents rather nontrivial task (for the earlier attempts, see  and the review ).
To explain in a few words the problem which will be under discussion, we recall that typical relativistic equations of motion have singularity at some value of a particle speed. The singularity determines behavior of the particle in ultrarelativistic limit. For instance, the standard equations of spinless particle interacting with electromagnetic field in the physical-time parametrization ,become singular as the relativistic-contraction factor vanishes, . Rewriting the equations in the form of second law of Newton, we find an acceleration. For the case, the longitudinal acceleration reads ; that is, the factor, elevated in some degree, appears on the right hand side of the equation and thus determines the value of velocity at which the longitudinal acceleration vanishes, . For the present case, the singularity implies that, during its evolution in the external field, the spinless particle can not exceed the speed of light .
In the equations for spinning particle, instead of the original metric ( in flat and in curved space), emerges the effective metric , with spin- and field-dependent contribution . This turns out to be true for both MPTD and Frenkel equations. This leads to (drastic in some cases) changes [20, 21] in behavior of spinning particle as compared with (1). The present work is devoted to detailed analysis of the behavior in ultrarelativistic regime.
We will use the following terminology. The speed that a particle can not exceed during its evolution in an external field is called critical speed (we prefer the term critical speed instead of maximum speed since generally is spin- and field-dependent quantity; see below). The observer-independent scale of special relativity is called, as usual, the speed of light.
The work is organized as follows. In Section 2, we define three-dimensional acceleration (28) of a particle in an arbitrary gravitational field. The definition guarantees that massive spinless particle propagating along four-dimensional geodesic can not exceed the speed of light. Then, we obtain expressions (46) and (47) for the acceleration implied by equation of a general form (45). They will be repeatedly used in the subsequent sections. In Section 3, we shortly review the vector model of spin and present three equivalent Lagrangians of the free theory. In Section 4.1, we obtain equations of the particle minimally interacting with gravity starting from the Lagrangian action without auxiliary variables. The variational problem leads to the theory with fixed value of spin. In Section 4.2, we present the Lagrangian which leads to the model of Hanson-Regge type , with unfixed spin and with a mass-spin trajectory constraint. In Section 4.3, we present the MPTD equations in the form convenient for our analysis and show their equivalence with those obtained in Section 4.1. In Section 4.4, we discuss the problems arising in ultrarelativistic limit of MPTD equations. The first problem is the discrepancy between the critical speed and the speed of light. We should note that similar observations were mentioned in a number of works. The appearance of trajectories with space-like four-velocity was remarked by Hanson and Regge in their model of spherical top in electromagnetic field . Space-like trajectories of this model in gravitational fields were studied in [23, 24]. The second problem is that the transversal acceleration increases with velocity and blows up in the ultrarelativistic limit.
In , Khriplovich proposed nonminimal interaction of a rotating body through the gravimagnetic moment . In Section 5.1, we construct the nonminimal interaction starting from the Hamiltonian variational problem and show (Section 5.2) that the model with has reasonable behavior in ultrarelativistic limit. The Lagrangian with one auxiliary variable for the particle with gravimagnetic moment is constructed in Section 5.3. In Section 6 we construct two toy models of spinless particle with critical speed different from the speed of light. In Section 7.1 we analyze generalization of the Frenkel equations to the case of a particle with magnetic moment in an arbitrary electromagnetic field in Minkowski space. Here, we start from the Lagrangian action with one auxiliary variable. In Section 7.2 we show that critical speed of the particle with anomalous magnetic moment is different from the speed of light, if we use the standard special-relativity notions for time and distance. In Section 7.3 we show that the equality between the two speeds can be preserved assuming that three-dimensional geometry should be defined with respect to effective metric arisen due to interaction of spin with electromagnetic field. We point out that a possibility of deformed relation between proper and laboratory time in the presence of electromagnetic field was discussed before by van Holten in his model of spin .
Notation. Our variables are taken in arbitrary parametrization , and then . Covariant derivative is and curvature is . The square brackets mean antisymmetrization, . For the four-dimensional quantities, we suppress the contracted indexes and use the notations , , and , . Notations for the scalar functions constructed from second-rank tensors are and .
When we work in four-dimensional Minkowski space with coordinates , we use the metric , then , and so on. Suppressing the indexes of three-dimensional quantities, we use bold letters: , , , and so on.
2. Three-Dimensional Acceleration of Spinless Particle in General Relativity
By construction of Lorentz transformations, the speed of light in special relativity is an observer-independent quantity. As we have mentioned in Introduction, the invariant scale is closely related with the critical speed in an external field. In a curved space, we need to be more careful since the three-dimensional geometry should respect the coordinate independence of the speed of light. To achieve this, we use below the Landau and Lifshitz procedure  to define time interval, three-dimensional distance, and velocity. Then, we introduce the notion of three-dimensional acceleration which guarantees that massive spinless particle propagating along four-dimensional geodesic can not exceed the speed of light. Expression (47) for longitudinal acceleration implied by equation of the form in (45) will be repeatedly used in subsequent sections.
Consider an observer that labels the events by the coordinates of pseudo Riemann space [27, 28]to describe the motion of a particle in gravitational field with metric . Formal definitions of three-dimensional quantities subject to the discussion can be obtained representing interval in block-diagonal form This prompts introducing infinitesimal time interval, distance, and speed as follows:Therefore, the conversion factor between intervals of the world time and the time measured by laboratory clock is
Introduce also the three-velocity vector with componentsor, symbolically, . We stress that, contrary to , the set is nonholonomic basis of tangent space (let be a basis of tangent space and , where , be the dual basis for ; i.e., ; is the holonomic basis (i.e., is tangent to some coordinate lines ) if ; for the matrix , which determines the dual basis , this condition reduces to the simple equation ; for the matrix which determines our decomposition we have , , and ; then, for instance, ; so the set generally does not represent a holonomic basis). This does not represent any special problem for our discussion since we are interested in the differential quantities such as velocity and acceleration.
Equation (8) is consistent with the above definition of : . In the result, the interval acquires the form similar to special relativity (but now we have ):This equality holds in any coordinate system . Hence, a particle with the propagation law has the speed , and this is a coordinate-independent statement.
These rather formal tricks are based  on the notion of simultaneity in general relativity and on the analysis of flat limit. Four-interval of special relativity has direct physical interpretation in two cases. First, for two events which occur at the same point, the four-interval is proportional to time interval; Second, for simultaneous events, the four-interval coincides with distance; . Assuming that the same holds in general relativity, let us analyze infinitesimal time interval and distance between two events with coordinates and . The world line is associated with laboratory clock placed at the spacial point . So, the time interval between the events and measured by the clock isConsider the event infinitesimally closed to the world line . To find the event on the world line which is simultaneous with , we first look for the events and which have null-interval with , . The equation with has two solutions: ; then and . Second, we compute the middle pointBy definition, the event with the null coordinate (13) is simultaneous with the event (in the flat limit, the sequence , , of events can be associated with emission, reflection, and absorption of a photon with the propagation law ; then the middle point in (13) should be considered simultaneous with ). By this way, we synchronized clocks at the spacial points and . According to (13), the simultaneous events have different null-coordinates, and the difference obeys the equationConsider a particle which propagated from to . Let us compute time interval and distance between these two events. According to (13), the eventat the spacial point is simultaneous with .
According to (12) and (13), the time interval between the events and (15) isSince the events and (15) are simultaneous, this equation gives also the time interval between and . Further, the difference of coordinates between the events and (15) is . As they are simultaneous, the distance between them isSince (15) occurs at the same spacial point as , this equation gives also the distance between and . Equations (16) and (17) coincide with the formal definitions presented above, in (5) and (6).
We now turn to the definition of three-acceleration. The spinless particle in general relativity follows a geodesic line. If we take the proper time to be the parameter, geodesics obey the systemwhereDue to this definition, system (18) obeys the identity .
The system in this parametrization has no sense of the case we are interested in, . So, we rewrite it in arbitrary parametrization .this yields the equation of geodesic line in reparametrization-invariant form
Formalism (5)–(9) remains manifestly covariant under subgroup of spacial transformations , , and . Under these transformations, is a scalar function and is a vector while and are tensors. Since , the inverse metric of turns out to be . Introduce the covariant derivatives of a vector field :The three-dimensional Christoffel symbols are constructed with help of three-dimensional metric written in (6), where is considered as a parameter:As a consequence, the metric is covariantly constant, .
The velocity in (8) behaves as a vector, , so below we use also the covariant derivative
We associated with the one-parameter family of three-dimensional spaces . Note that velocity has been defined above as a tangent vector to the curve which crosses the family and is parameterized by this parameter, .
To define an acceleration of a particle in the three-dimensional geometry, we need the notion of a constant vector field (or, equivalently, the parallel-transport equation). In the case of stationary field, , we can identify the curve of with that of any one of . So, we have the usual three-dimensional Riemann geometry, and an analog of constant vector field of Euclidean geometry is the covariantly constant field along the line , . For the field of velocity, its deviation from the covariant constancy is the acceleration 
To define an acceleration in general case, , we need to adopt some notion of a constant vector field along the trajectory that crosses the family . We propose the definition which preserves one of basic properties of constant fields in differential geometry. In Euclidean and Minkowski spaces, the canonical scalar product of two constant fields does not depend on the point where it was computed. In (pseudo) Riemann space, constant vector field is defined in such a way that the same property holds . In particular, taking the scalar product along a line , we have . For the constant fields in the three-dimensional geometry resulting after Landau-Lifshitz decomposition, we demand the same (necessary) condition: . Taking into account that , this condition can be written as follows:This equation is satisfied, if we take the parallel-transport equation to beDeviation from the constant field is an acceleration. So we define acceleration with respect to physical time as follows:For the special case of stationary field, , definition (28) reduces to (25) and to that of Landau and Lifshitz; see page 251 in .
The extra term that appeared in this equation plays an essential role in providing that for the geodesic motion we have . As a consequence, geodesic particle in gravitational field can not exceed the speed of light. To show this, we compute the longitudinal acceleration implied by geodesic equation (21). Take ; then , and spacial part of (21) iswhereis nonsingular function as . Computing derivative on the l.h.s. of (29), we complete up to covariant derivative :For the derivative contained in the last term we find, using covariant constancy of ,Then, (29) acquires the formwhereWe apply the inverse matrixand use the identityand thenNext, we complete up to acceleration (28). Then, (37) yieldsContracting this with , we use and obtain longitudinal accelerationThis implies as .
The last term in (28) yields the important factor in (39). As equations of motion (38) and (39) do not contain the square root , they have sense even for . Without this factor, we would have as , so the particle in gravitational field could exceed and then continue to accelerate. The same happens if we try to define acceleration using usual derivative instead of the covariant one. Indeed, instead of (28), let us define an acceleration according to the expression . Then for the geodesic particle we obtain, instead of (39), the longitudinal acceleration r.h.s. of (39)r.h.s. of (39). The extra term does not involve the factor and so does not vanish at .
Let us confirm that is the only special point of function (39) representing the longitudinal acceleration. Using (19), (6)–(10), (23), and the identitieswe can present the right hand side of (39) in terms of initial metric as follows:The quantity has been defined in (10). Excluding according to this expression, we obtainFor the stationary metric, , (42) acquires a specially simple form:This shows that the longitudinal acceleration has only one special point in the stationary gravitational field; as . Then, the same is true in general case (41), at least for the metric which is sufficiently slowly varied in time.
While we have discussed the geodesic equation, the computation which leads to formula (39) can be repeated for a more general equation. Let us formulate the result which will be repeatedly used below. Using the factor , we construct the reparametrization-invariant derivativeConsider the reparametrization-invariant equation of the formand suppose that the three-dimensional geometry is defined by . Then, (45) implies the three-accelerationand the longitudinal accelerationThe spacial part of the force is , where is given by (10) and the connection is constructed with help of the three-dimensional metric according to (23). For the geodesic equation in this notation, we have . With this , (46) and (47) coincide with (38) and (39).
3. Vector Model of Relativistic Spin
The variational problem for vector model of spin interacting with electromagnetic and gravitational fields can be formulated with various sets of auxiliary variables [17, 30–33]. For the free theory in flat space, there is Lagrangian action without auxiliary variables. Configuration space consists of the position and non-Grassmann vector attached to the point . The action reads [20, 32]The matrix is the projector on the plane orthogonal to :The double square-root structure in expression (48) seems to be typical for the vector models of spin [22, 34]. This yields the primary constraint in (62) and, at the end, supplementary spin condition (77). The parameter is mass, while determines the value of spin. The value corresponds to an elementary spin one-half particle. The model is invariant under reparametrizations and local spin-plane symmetries  (the reparametrizations are , , and ; i.e., both and are scalar functions; the local spin-plane transformations act in the plane determined by the vectors and ).
The spin is described by Frenkel spin-tensor . In our model, this is a composite quantity constructed from and its conjugated momentum as follows:and then . Here, is three-dimensional spin-vector and is dipole electric moment . In contrast to its constituents and , the spin-tensor is invariant under local spin-plane symmetry and thus represents an observable quantity. Canonical quantization of the model yields one-particle sector of the Dirac equation .
In formulation (48), the model admits minimal interaction with electromagnetic field and with gravity. This does not spoil the number and the algebraic structure of constraints presented in the free theory. To describe the spinning particle with magnetic and gravimagnetic moments, we will need the following two reformulations.
In the spinless limit, and , functional (48) reduces to the standard expression, . The latter can be written in equivalent form using the auxiliary variable as follows: . Similarly to this, (48) can be presented in the equivalent formIn this formulation, our model admits interaction of spin with an arbitrary electromagnetic field through the magnetic moment; see Section 7.1. Another form of the Lagrangian isIts advantage is that the expression under the square root represents quadratic form with respect to the velocities and . To relate Lagrangians (48) and (52), we exclude from the latter. Computing variation of (52) with respect to , we obtain the equationwhich determines :We substitute into (52) and use , and then (52) turns into (48). In formulation (52), our model admits interaction of spin with gravity through the gravimagnetic moment; see Section 5.3.
4. Minimal Interaction with an Arbitrary Gravitational Field
4.1. Lagrangian and Hamiltonian Formulations
The minimal interaction with gravitational field can be achieved by covariantization of the formulation without auxiliary variables. In expressions (48) and (49), we replace and usual derivative by the covariant one; . Thus, our Lagrangian in a curved background reads Velocities , and projector transform like contravariant vectors and covariant tensor, so the action is manifestly invariant under general-coordinate transformations.
Let us construct Hamiltonian formulation of model (56). Conjugate momenta for and are and , respectively. Due to the presence of Christoffel symbols in , the conjugated momentum does not transform as a vector, so it is convenient to introduce the canonical momentumthe latter transforms as a vector under general transformations of coordinates. Manifest form of the momenta is as follows:with These vectors obey the following remarkable identities:Using (49), we conclude that and ; that is, we found two primary constraints. Using the relations in (60), we find one more primary constraint, . At last, computing given by (58); we see that all the terms with derivatives vanish, and we obtain the primary constraintIn the result, action (56) implies four primary constraints, andThe Hamiltonian is constructed excluding velocities from the expressionwhere is the Lagrangian multipliers associated with the primary constraints. From (58), we observe the equalities and . Together with (60), they imply . Using this in (63), we conclude that the Hamiltonian is composed of the primary constraintsThe full set of phase-space coordinates consists of the pairs and . They fulfill the fundamental Poisson brackets and and then , , and . For the quantities , , and , these brackets imply the typical relations used by people for spinning particles in Hamiltonian formalism.To reveal the higher-stage constraints and the Lagrangian multipliers, we study the equation . implies the secondary constraintthen can be replaced on . Preservation in time of and gives the Lagrangian multipliers and :where we have denotedPreservation in time of gives the equation which is identically satisfied by virtue of (67). No more constraints are generated after this step. We summarize the algebra of Poisson brackets between the constraints in Table 1. and represent a pair of second-class constraints, while , , and the combinationare the first-class constraints. Taking into account that each second-class constraint rules out one phase-space variable, whereas each first-class constraint rules out two variables, we have the right number of spin degrees of freedom, .
It should be noted that and turn out to be space-like vectors. Indeed, in flat limit and in the frame where , the constraints imply . This implies and . Combining this with constraint (66), we conclude and .
We point out that the first-class constraint can be replaced on the pairthis gives an equivalent formulation of the model. The Lagrangian which implies constraints (62) and (71) has been studied in [17, 30, 31]. Hamiltonian and Lagrangian equations for physical variables of the two formulations coincide , which proves their equivalence.
The dynamics of basic variables is governed by Hamiltonian equations , where , and the Hamiltonian is given in (72). Equivalently, we can use the first-order variational problem equivalent to (56):Variation with respect to gives constraints (61) and (62), while variation with respect to , , , and gives the dynamical equations. By construction of , the variational equation is equivalent to and so on. The equations can be written in a manifestly covariant form as follows:
According to general theory [29, 36, 37], neither constraints nor equations of motion determine the functions and . Their presence in the equations of motion implies that evolution of our basic variables is ambiguous. This is in correspondence with two local symmetries presented in the model. The variables with ambiguous dynamics do not represent observable quantities, so we need to search for variables that can be candidates for observables. Consider antisymmetric tensor (50). As a consequence of and , this obeys the Pirani supplementary condition [4, 5, 7]Besides, the constraints and fix the value of squareso we identify with the Frenkel spin-tensor . Equations (77) and (78) imply that only two components of spin-tensor are independent, as it should be for spin one-half particle. Equations of motion for follow from (76). Besides, we express (74) and (75) in terms of the spin-tensor. This gives the systemwhere has been defined in (68). Equation (81), contrary to (76) for and , does not depend on . This proves that the spin-tensor is invariant under local spin-plane symmetry. The remaining ambiguity due to is related with reparametrization invariance and disappears when we work with physical dynamical variables . Equations (79)–(81), together with (77) and (78), form a closed system which determines evolution of a spinning particle.
To obtain the Hamiltonian equations, we can equally use the Dirac bracket constructed with help of second-class constraints:Since the Dirac bracket of a second-class constraint with any quantity vanishes, we can now omit and from (72); this yields the HamiltonianThen, (74)–(76) can be obtained according to the rule . The quantities , , and , being invariant under spin-plane symmetry, have vanishing brackets with the corresponding first-class constraints and . So, obtaining equations for these quantities, we can omit the last two terms in , arriving at the familiar relativistic HamiltonianEquations (79)–(81) can be obtained according to the rule . From (84), we conclude that our model describes spinning particle without gravimagnetic moment. The Hamiltonian with gravimagnetic moment has been proposed by Khriplovich  adding nonminimal interaction to the expression for . The corresponding Lagrangian formulation will be constructed in Section 5.1.
Let us exclude momenta and the auxiliary variable from the Hamiltonian equations. This yields second-order equation for the particle’s position . To achieve this, we observe that (79) is linear on .Using the identitywe find inverse of the matrix :so (85) can be solved with respect to , . We substitute into the constraint ; this gives expression for :We have introduced the effective metricThe matrix is composed of the original metric plus (spin- and field-dependent) contribution; . So, we call the effective metric produced along the world line by interaction of spin with gravity. The effective metric will play the central role in our discussion of ultrarelativistic limit.
These equations, together with conditions (91) and (78), form closed system for the set (). The consistency of constraints (91) and (78) with the dynamical equations is guaranteed by Dirac procedure for singular systems.
4.2. Lagrangian Action of Spinning Particle with Unfixed Value of Spin
Lagrangians (48) and (56) yield the fixed value of spin (78); that is, they correspond to an elementary particle. Let us present the modification which leads to the theory with unfixed spin and similarly to Hanson-Regge approach , with a mass-spin trajectory constraint. Consider the following Lagrangian in curved background:where