Relativistic charged particle in a uniform electromagnetic field

Transcription

1 INVESTIGACIÓN REVISTA MEXICANA DE ÍSICA EBRERO 011 Relativistic charged particle in a uniform electromagnetic field G.. Torres del Castillo Departamento de ísica Matemática, Instituto de Ciencias, Universidad Autónoma de Puebla, Puebla, Pue., 7570, México. C. Sosa Sánchez acultad de Ciencias ísico Matemáticas, Universidad Autónoma de Puebla, Apartado postal 115 Puebla, Pue.,7001, México. Recibido el 14 de mayo de 010; aceptado el 16 de noviembre de 010 The equations of motion for a relativistic charged particle in a uniform electromagnetic field are solved in a covariant form by calculating the onential of the matrix corresponding to the electromagnetic field tensor. It is shown that owing to the antisymmetry of the electromagnetic field tensor, the onential mentioned above can be easily calculated. Some results are then applied to study the algebraic properties of the energy-momentum tensor of the electromagnetic field the orthogonal transformations in spaces of dimension 4. Keywords: Electromagnetic field; matrix onentials; dominant energy condition; orthogonal transformations. Se resuelven las ecuaciones de movimiento de una partícula relativista cargada en un campo electromagnético uniforme en forma covariante, calculo la onencial de la matriz correspondiente al tensor del campo electromagnético. Se muestra que debido a la antisimetría del tensor del campo electromagnético, dicha onencial se puede calcular fácilmente. Algunos resultados se aplican para estudiar las propiedades algebraicas del tensor de energía-momento del campo electromagnético y las transformaciones ortogonales en espacios de dimensión 4. Descriptores: Campo electromagnético; onenciales de matrices; condición de energía dominante; transformaciones ortogonales. PACS: p; De; 0.0.Qs 1. Introduction The motion of a relativistic charged particle in a constant uniform electromagnetic field has been studied in a large number of papers textbooks see, e.g., Refs. 1 to 6 the references cited therein, making use of various procedures. The corresponding equations of motion can be solved in a manifestly covariant form, ressing the four-velocity or the space-time coordinates of the particle as a function of the proper time of the particle, or making use of a convenient reference frame in which the electromagnetic field takes some simplified form; for instance, when E B = 0 E B is different from zero, there exists an inertial frame where one of the fields, E or B, vanishes the equations of motion can be easily solved see, e.g., Ref. 6. The equations of motion of a charged particle in a uniform electromagnetic field constitute a system of four homogeneous linear first-order ordinary differential equations with constant coefficients for the four-velocity of the particle as a function of the proper time, whose solution can be ressed in terms of the onential of a 4 4 matrix. The aim of this paper is to show that, owing to the antisymmetry of the electromagnetic field tensor, this onential can be calculated directly by appropriately decomposing the argument of the onential into two matrices corresponding to the selfdual the anti-self-dual parts of the electromagnetic field tensor see also Ref. 1. Some of the basic relations thus established are then used to study the algebraic properties of the energy-momentum tensor of the electromagnetic field the orthogonal transformations of any four-dimensional space. In Sec. the four-velocity of a relativistic charged particle in a uniform electromagnetic field is obtained in a covariant form by directly calculating the onential of a matrix formed by the components of the electromagnetic field. The procedure followed here essentially coincides with that employed in Taub s paper [1]; however, here we present the mechanism behind the algebraic relations that simplify the calculation of the onential mentioned above. We also demonstrate, in Sec. 3, that some of the relations obtained in Sec. can be employed to study the algebraic properties of the energy-momentum tensor of the electromagnetic field to find the orthogonal transformations of the fourdimensional spaces.. Solution of the equations of motion In the framework of special relativity, the equations of motion of a charged particle with rest mass m electric charge q in an arbitrary electromagnetic field can be written in the covariant form du α = q dτ β α U β, 1 where U α α = 0, 1,, 3 is the four-velocity of the particle, τ is its proper time, c is the speed of light in vacuum, αβ is the electromagnetic field tensor, with sum over repeated indices see, e.g., Ref. 6. Equivalently, this system of equations can be ressed in the more elementary matrix form du dτ = q U,

2 54 G.. TORRES DEL CASTILLO AND C. SOSA SÁNCHEZ where U is the column matrix with entries U 0, U 1, U, U 3, is the 4 4 matrix 0 E x E y E z E x 0 B z B y E y B z 0 B x, 3 E z B y B x 0 where E x, E y, E z B x, B y, B z are the Cartesian components of the electric magnetic fields, respectively, with respect to some inertial reference frame. If one assumes that the electromagnetic field is uniform, the entries of are all constant, the solution of Eq. is given by see, e.g., Refs. 7 8 Uτ = U0, 4 with the onential of a matrix being defined by means of the series ansion 1 A = n! An. n=0 In spite of the fact that is a 4 4 matrix, this onential can be very easily computed if one writes the matrix as the sum of two complex matrices cf. Ref. 1 with S A = 1 S + A, 5 0 x y z x 0 i z i y y i z 0 i x z i y i x 0 0 x y z x 0 i z i y y i z 0 i x z i y i x 0, 6, 7 where x, y, z are the Cartesian components of the complex vector field E + ib denotes complex conjugation note that A is the complex conjugate of S. The usefulness of this decomposition comes from the fact that S A commute with each other, their squares are proportional to the identity matrix, SA = AS, S = I, A = I, 8 where I is the 4 4 identity matrix, x + y + z =E + ib =E B +ie B see the discussion in Sec.., below. Hence, the powers of S are given by S n = n I, S n+1 = n S, 9 for n = 0, 1,,.... It may be remarked that the two Lorentz invariants [6] of the electromagnetic field, E B E B, arise here in a natural way. Since S A commute with each other, from Eq. we have [ Uτ = qτ ] S + A U0 = qτ S qτ A U0, with A being the complex conjugate of Then, with the definition S. = E B + ie B 10 so that =, making use of Eq. 9 we obtain qτ S= n=0 1 n! n q τ I+ 1 n=0 1 n + 1! n+1 q τ S= cosh q τ I+ 1 sinh q τ S, 11 provided that = 0. In the case where = 0, only the first two terms of the series contribute [see Eqs. 8 9] we have qτ S = I + qτ S. 1 Note that is a square root of the complex number E B + ie B, that the final ression in Eq. 11 is an even function of ; therefore, since the two square roots of a complex number differ by a 1 factor, both square roots produce the same result. Note also that Eq. 1 can be obtained from Eq. 11 in the limit 0, therefore it is enough to consider Eq. 11 only. Rev. Mex. ís

3 rom Eq. 11 we then obtain RELATIVISTIC CHARGED PARTICLE IN A UNIORM ELECTROMAGNETIC IELD 55 = cosh q τ + 1 sinh q τ cosh q τ cosh q τ I + 1 A + sinh q τ 1 cosh q τ sinh q τ S sinh q τ SA, 13 which only involves the matrices I, S, A, SA, according to Eq. 4 gives the solution to Eq. for any initial condition..1. Some particular cases As shown in Eqs. 11 1, the ression for S depends essentially on, this ression is simplified if E B = 0, in which case is real or pure imaginary [see Eq. 10]. As is well known, if E B = 0 then there exist inertial frames in which B or E is equal to zero, provided that E B is greater than zero or less than zero, respectively see, e.g., Ref. 6. If E B = 0 E > B, then is real Eq. 13 reduces to = cosh q τ sinh q τ I 1 cosh q τ SA, 14 with = E B. Note that, since E B is invariant under the Lorentz transformations, in this case is the magnitude of the electric field in a reference frame where B = 0. On the other h, from Eqs. 5 8 we find that = 1 4 S + SA + A = 1 I + SA, 15 hence, eliminating SA in favor of, Eq. 14 can also be written in the form = I + 1 sinh q τ 1 1 cosh q τ. 16 Both invariants of the electromagnetic field are equal to zero if only if = 0; in that case from Eq. 16 we obtain = I + qτ + 1, 17 which correspond to the first three terms of the series ansion of the onential on the left-h side of the equation; in fact, making use of Eq. 15 that both S A are equal to zero one finds that 3 = 0 therefore n = 0 for n 3. As we shall prove in Sec. 3., in all cases for each value of τ is a Lorentz transformation; the transformations of the form 17 are known as null rotations or parabolic Lorentz transformations. In a similar manner one finds that if E B = 0 B > E, then is pure imaginary writing = ib 0, with B 0 real positive the magnitude of B in a reference frame where E = 0, Eq. 13 reduces to = I + 1 sin qb 0τ B cos qb 0τ B 0 18 which also leads to Eq. 17 in the limit as B 0 tends to zero. Equation 18 is a periodic function of τ with angular frequency ω = qb 0 /, which produces a rotation of the particle s four-velocity with this angular velocity see also Ref Origin of the algebraic simplifications As we have seen, the properties 8 simplify the calculation of the onential of the matrix ; however, the derivation presented above, or that given in Ref. 1, does not elucidate the origin of these key relations. However, as we shall presently show, by using an appropriate basis, relations 8 become trivial. Indeed, making use of the unitary 4 4 matrix M = i i which represents a change of basis, with the new basis containing two complex vectors; the new basis, called a null tetrad, is formed by four null four-vectors, one finds that Rev. Mex. ís

4 56 G.. TORRES DEL CASTILLO AND C. SOSA SÁNCHEZ K MSM 1 = z x i y 0 0 x + i y z z x i y 0 0 x + i y z 0 L MAM 1 = z 0 x + i y 0 0 z 0 x + i y x i y 0 z 0 0 x i y 0 z. 1 Making use of the tensor product or direct product of two matrices, defined by see, e.g., Ref. 10 α β α β a b aα aβ bα bβ a b α β γ δ γ δ c d γ δ α β α β = aγ aδ bγ bδ cα cβ dα dβ, c d γ δ γ δ cγ cδ dγ dδ we see that the matrices K L can be ressed in the form K = z x i y 0 1 x + i y z z L = x iy x + i y z 0 1, then, using the fact that A BC D = AC BD, it trivially follows that K L commute with each other. Similarly, noting that z x + i y x i y z = 0 1, we readily obtain K = I, similarly, L = I, which are equivalent to the last two Eqs. 8. It may be also noticed that z x i y = σ, x + i y z where σ is the vector formed with the stard Pauli matrices, therefore, K L can be conveniently ressed in terms of the Pauli matrices. The matrices S A or, equivalently, K L correspond to the self-dual anti-self-dual parts of αβ, respectively. If the dual of an antisymmetric tensor t αβ is defined by t αβ 1 ε αβγδt γδ, 3 where ε αβγδ is totally antisymmetric with ε the tensor indices are raised or lowered with the aid of the metric tensor g αβ = diag 1, 1, 1, 1 4 its inverse, it follows that t αβ = t αβ. Hence the eigenvalues of the duality operator are ±i. The entries of the matrices 6 7 are S α β A α β, respectively, with S αβ = αβ i αβ, A αβ = αβ + i αβ, so that S αβ = i S αβ, A αβ = i A αβ. Similar results hold for all possible metrics in four-dimensional spaces see Sec. 3..1, below as can readily be shown making use of the spinor formalism [9]..3. Comparison with other covariant approaches In Ref. 3, the series corresponding to is calculated by showing first that all powers of are linear combinations of the four matrices I,,, the matrix corresponding to the dual of αβ [equivalent to 1/iS A], which implies that the desired onential is also a linear combination of these matrices, then finding the coefficients by calculating traces. In the approach followed in Ref., is calculated by establishing some algebraic relations between the matrix formed with the dual of αβ, then translating the problem into that of solving a homogeneous fourth-order linear ordinary differential equation with constant coefficients. By contrast, the calculation presented in Ref. 1 in this section is much shorter direct, by virtue of Eqs. 8, has some other useful applications, as we shall show in Sec. 3. The algebraic relations employed in Refs. 3 follow at once from Eqs. 8. On the other h, in Ref. 5, the solution of the matrix equation 1 is obtained making use of the onential of a differential operator a decomposition analogous to Eq. 5 see also Ref. 4. Rev. Mex. ís

5 RELATIVISTIC CHARGED PARTICLE IN A UNIORM ELECTROMAGNETIC IELD urther applications In this section we show that the results derived above can be employed in the study of the algebraic properties of the energy-momentum tensor of the electromagnetic field, to find the orthogonal transformations of a four-dimensional space The energy-momentum tensor of the electromagnetic field The components of the energy-momentum tensor of the electromagnetic field are given by [6] T α β = 1 4π α γ γ β 1 4 ρ γ γ ρδ α β, 5 which amounts to saying that, in terms of the matrix, the 4 4 matrix T = T α β is ressed as [see Eqs. 5 8] T = 1 [ 1 4π 4 tr I ] = 1 [ S + A 1 4π 4 tr I ] = 1 [ SA π 4 tr I ], 6 where tr denotes the trace. As is well known, the trace of the energy-momentum tensor 5 vanishes, T α α = 0; on the other h, tr SA = tr M 1 KMM 1 LM = tr KL = 0, since KL is the direct product of two traceless matrices. Hence, from Eq. 6 we see that tr = 4 + = 8E B 7 [which also follows directly from Eq. 3] T = 1 SA. 8 π An important, nontrivial consequence of this last equation is that, by virtue of Eq. 8, T = 1 4π S A = 1 4π [E B + 4E B ] I or, equivalently, T α γt γ β = 1 4π [E B + 4E B ] δ α β. 9 With the aid of Eq. 9 one can prove that the electromagnetic field satisfies the so-called dominant energy condition. or an observer whose four-velocity is V α, the energy flux four-vector of the electromagnetic field is T α βv β. This four-vector is nonspacelike since, using the symmetry of T αβ Eq. 9, T α βv β T αγ V γ = T β αt α γv β V γ = 1 4π [E B + 4E B ]V β V β 0. As another application of Eq. 9 we can prove that the energy-momentum tensor of the electromagnetic field is of the form T αβ = k α k β, for some four-vector k α only if both invariants of the electromagnetic field vanish. In effect, the fact that the trace of T αβ is equal to zero amounts to k α k α = 0; hence, T α γt γ β = k α k γ k γ k β = 0 from Eq. 9 it follows that E B E B must vanish simultaneously. 3.. Orthogonal transformations As pointed out in Refs. 1 to 5, the onential appearing in Eq. 4 is analogous to the ression of a Lorentz transformation. As we show licitly below, the onential in Eq. 4 is a Lorentz transformation. In fact, the four-velocity U α of a particle must satisfy the condition U α U α = c the solution U α τ to the system of linear equations 1 must depend linearly of the initial condition U α 0 even if the αβ are not constant. Since the Lorentz transformations are the only linear transformations of the Minkowski space into itself that leave invariant the products A α A α, U α τ must be related to U α 0 by means of a Lorentz transformation. Let V be a real four-dimensional vector space, with a non-singular metric tensor, g αβ that is, detg αβ 0. In the case of the Minkowski space-time, with respect to an appropriate basis, g αβ = diag 1, 1, 1, 1 30 or its negative, depending on the conventions adopted, but we can also consider, e.g., g αβ = diag1, 1, 1, 1, corresponding to the Euclidean four-dimensional space, g αβ = diag 1, 1, 1, 1. The orthogonal transformations are those linear transformations from V onto itself that leave the metric tensor invariant; in the case of the metric tensor 30 the orthogonal transformations are just the homogeneous Lorentz transformations. If the 4 4 matrix a α β is an orthogonal transformation, then a α µa β νg αβ = g µν, 31 from which one can prove that the orthogonal transformations form a group under composition. If As is an orthogonal matrix depending on a real parameter, s, in such a way that At + s = AtAs, 3 then, differentiating with respect to t at t = 0 we obtain A s = A 0As, or, defining A 0 A s = As. 33 rom Eq. 3 it follows that A0 = I, therefore the solution to Eq. 33 is given by As = s A0 = s. 34 Rev. Mex. ís

6 58 G.. TORRES DEL CASTILLO AND C. SOSA SÁNCHEZ On the other h, denoting by a α βs α β the entries of the matrices As, respectively, from Eq. 31 we have a α µsa β νsg αβ = g µν. Differentiating with respect to s at s = 0, taking into account that A0 = I A 0 =, we get α µg αν + β νg µβ = 0 or, equivalently, with the stard definition αβ g αγ γ β, νµ + µν = 0, 35 i.e., µν is antisymmetric, as in the case of the electromagnetic field tensor. Thus, the onential appearing in Eq. 4 is an orthogonal transformation i.e., a Lorentz transformation from Eq. 34, by suitably choosing the value of s, one can obtain Lorentz transformations. In fact, any proper orthochronous Lorentz transformation is of this form see, e.g., Ref Orthogonal transformations of the Euclidean space In the case where the metric tensor has the form g αβ = diag 1, 1, 1, 1, we have α β = αβ by virtue of Eq. 35 the matrix = α β is of the form 0 e 1 e e 3 e b 3 b e b 3 0 b 1, e 3 b b where e 1, e, e 3 b 1, b, b 3 are six arbitrary real numbers [cf. Eq. 3]. Now t αβ = t αβ, therefore the eigenvalues of the duality operator are ±1, the self-dual antiself-dual parts of αβ are defined by S αβ αβ + αβ, A αβ αβ αβ, respectively, so that S αβ = S αβ, A αβ = A αβ. Thus where, S = A = with the definitions = 1 S + A, 36 0 g 1 g g 3 g g 3 g g g 3 0 g 1 g 3 g g 0 g 1 g g 3 g g 3 g g g 3 0 g 1 g 3 g g g e + b, g e b.,, By means of a straightforward computation one finds that, as in the previous case, S commutes with A, the squares of S A are proportional to the identity matrix SA = AS, S = g I, A = g I 37 [cf. Eqs. 8], but now S A are two independent real antisymmetric matrices, g g are always real nonnegative see the discussion below. rom Eqs. 37 we have, e.g., S n = 1 n g n I, S n+1 = 1 n g n S, for n = 0, 1,,.... Thus, t = 1 ts 1 ta 1 ts = cos t g I + 1 sin t g g S [cf. Eq. 11], with an analogous ression for 1/tA, replacing g by g. Hence t = cos t g cos t g I + 1 sin t cos g g t g S + 1 g + 1 g g sin t g cos t g A sin t g sin t g SA. Making use of Eqs , this last ression can be written in terms of I, S, A,. As in the case where the metric tensor is given by Eq. 30, one can readily see that relations 37 trivially follow from the ression of S A in an appropriate basis. In fact, with the aid of the unitary 4 4 matrix M = 1 one obtains M M 1 = i i i 0 i ig 3 ig 1 + g ig 1 g ig 3 i g 3 i g 1 g i g 1 + g i g The first term corresponds to S the second one to A. Rev. Mex. ís

7 RELATIVISTIC CHARGED PARTICLE IN A UNIORM ELECTROMAGNETIC IELD Conclusions By contrast with the approaches followed in other works, the procedure employed in this paper allows us to solve the equations of motion in a straightforward manner to derive certain algebraic relations [notably Eq. 9] that do not seem obtainable with other elementary methods in such a simple way. As we have shown, the fundamental relations established in order to solve Eq. have a wider applicability analogs that enable us to readily find the onentials of the generators of all the orthogonal groups in four dimensions. Acknowledgment One of the authors C.S.S. wishes to thank the Vicerrectoría de Investigación y Estudios de Posgrado of the Universidad Autónoma de Puebla for financial support through the programme Jóvenes Investigadores. The authors would like to thank the referee José Luis López Bonilla for helpful comments for pointing out Ref. 1 to them. 1. A.H. Taub, Phys. Rev A.T. Hyman, Am. J. Phys G. Muñoz, Am. J. Phys J. redsted, J. Math. Phys S.A. Chin, J. Math. Phys J.D. Jackson, Classical Electrodymanics, nd ed., Wiley, New York, M. Hirsch S. Smale, Differential Equations, Dynamical Systems, Linear Algebra, Academic Press, New York, K.B. Wolf, Rev. Mex. ís G.. Torres del Castillo, Spinors in our-dimensional Spaces, Birkhäuser, Boston, G.B. Arfken H.J. Weber, Mathematical Methods for Physicists, 6th ed., Elsevier Academic Press, Amsterdam, 005, Chap. 3. Rev. Mex. ís