Ультрарелятивистская спиновая частица во внешнем гравитационном и электромагнитном полях Текст научной статьи по специальности «Физика»

UDC 530.1; 539.1

FAST SPINNING PARTICLE IN EXTERNAL ELECTROMAGNETIC AND GRAVITATIONAL FIELDS

A. A. Deriglazov

Depto. de Matemática, ICE, Universidade Federal de Juiz de Fora, MG, Brasil. E-mail: alexei.deriglazov@ufjf.edu.br

Three-dimensional acceleration in general relativity can be defined so that spinless particle in the process of evolution in an external gravitational field can not exceed the speed of light. We ask what happens if a spinless particle is replaced by test-particle with spin one-half. Inclusion of spin-gravitational interaction gives generalized Papapetrou equations with modified metric along the world-line. The modified metric should be used to calculate the acceleration of spinning particles. Inclusion of spin-electromagnetic interaction in flat space gives generalized Frenkel equations, where an effective metric along world-line arises for the particle with anomalous magnetic moment. This implies that intervals of time (and distance) probed by such particle in the presence of electromagnetic field slightly differ from those in empty space.

Keywords: ultra-relativistic spinning particle, Frenkel equation, BMT equation, Papapetrou equation

1 Introduction

Basic notions of Special and General Relativity have been formulated before the discovery of spin, so they describe space-time properties as they are seen by spinless test-particle. In this work we raise the question whether these notions remain the same if the spinless particle is replaced by more realistic spinning test-particle.

To account spin, we need a systematically constructed classical model for relativistic description of rotational degrees of freedom. In recent works [1-4], we have obtained equations of motion of spinning particle which generalize those of Frenkel-BMT (Papapetrou) to the case of an arbitrary electromagnetic (gravitational) fields. This allows us to start the detailed analysis of old suggestions [5-7] on possible modifications of metric properties of spacetime as they are seen by spinning particle in the presence of electromagnetic (gravitational) fields.

Our starting observation is that speed of light does not represent special point of the complete equations of motion. As a consequence, if we preserve the usual special-relativity definitions of time and distance, the critical speed that the particle with anomalous magnetic moment can not overcome during its evolution in electromagnetic field slightly differs from the speed of light. To improve this point, we notice that interaction of spin with electromagnetic field induces the matrix (6), which can be considered as effective metric along the world-line. So, we can follow the general-relativity prescriptions to analyze the spinning particle in an external electromagnetic field.

Therefore we turn to the analysis of ultra-relativistic behavior of spinless particle moving along geodesic in gravitational field. We propose three-

dimensional acceleration (which reduces to that of Landau-Lifshitz for the constant gravitational field), and show that this guarantees impossibility for spinless particle to exceed the speed of light. Then we apply the formalism to the effective metric which arose for spinning particle in Minkowski space in the presence of electromagnetic field. With our definition of acceleration, the critical speed coincides with the speed of light. The price to pay is that intervals of time (and distance) probed by such particle in the presence of electromagnetic field slightly differ from those in empty space. At last, we show that interaction of spin with gravitational background also implies slight deformation (24) of initial metric. The deformed metric need to be used for definition of the acceleration.

2 Spinless particle on electromagnetic background

Spinning particle in arbitrary electromagnetic and gravitational backgrounds obeys rather complicated equations of motion [2-4]. So it is instructive to begin our discussion with more simple case of spinless particle. Typical relativistic equations of motion (EM) have singularity at some value of a particle velocity. For instance, the standard Lagrangian of spinless particle in electromagnetic field

L = -1 x2 - ^m2c2 + , (1)

2A 2 c

implies the manifestly relativistic and reparametrization

invariant equations DDxM = mc2Dxv, where

D = r1^ dr' They became singular B8 X2 —0. Using

reparametrization invariance of EM, we take physical time as the parameter, t = t, then xM = (ct, x(t)), xM = (c, v(t)) and * = } __,. For the acceleration

X 2 V c2 —v

dt, the EM imply va = [c2 - v2]3

e(vE)

that

a

3 J

mc

is the longitudinal acceleration vanishes as |v| ^ c. Hence the singularity implies that the particles speed

c

We will use the following terminology. The speed vcr that a particle can not exceed during its evolution in an external field is called critical speed. The observer

c

usual, the speed of light. According to the expression for a above, critical speed of a spinless particle coincides with the speed of light.

Let us point out two possible modifications which could yield non vanishing acceleration as v ^ c.

First, this would be for the sufficiently singular external force. For example, for a particle in electromagnetic background, this could be Dxv [1 + n(DxFdaFDx)Dxa], The right dimension can be supplied by the constant k proportional to

e8 ft4

Or •

Second, in the presence of external fields, we can construct additional reparametrization invariants. For instance, we can use the derivative D' = where

\J—xgx dr :

9ßv

c2 - v2 - k(xFFx),

(2)

reparametrization in variance, there are only only three independent equations in the system. This allows to interpret them as equations of motion of relativistic particle. C) The equations make sense for any velocity. D) If f* ~ (D'x)2 or less, acceleration of the particle may be non vanishing at |v| = c, but vanishes at the critical velocity vcr. If f* ~ (D'x)4, Eq. (3) has no critical points. So the particle could exceed the speed of light and then continues accelerate. Below, we repeat this analysis for more realistic case of a particle with spin.

3 Spinning particle in electromagnetic background within the standard special-relativity notions

Consider spinning particle with mass m, electric charge e and magnetic moment ^ interacting with an arbitrary electromagnetic field = d*Av —

dv AThe manifestly Poincare and reparametrization invariant Lagrangian on configuration space with coordinates x*(t), w* (t) and A(t) reads [4]

L = — [xNx + ADwNDw 4A

with k equal to mkc? 01 ^he surface xgx = 0

is slightly different from the sphere c2 — v2 = 0. For the particle with equations D'D'x* = mec?D'xv, acceleration vanishes at the critical velocity different from the speed of light. To see this, we compute

—xFFx = c2Ei (Sij — ^Vf) Ej + v2BiNijBj.

-d [xNx + XDwNDw]2 - 4X(iNDw)2

X

--m~

2

22

+ 2W + cAx.

(4)

^t-, this

Here N = 1 — is projection operator on the plane

orthogonal to the vector v, so we can write BNB = denoted = ^

(NB)2 = B\. Then the factor (2) reads

This depends on one free parameter a particular value corresponds to spin one-half particle. Similarly to Eq. (1), the only auxiliary variable is A, this provides the mass- shell condition. It has been

-xgx = c - v + k

vv

2E(1 - v_)E + v

2Bi

Besides, when |v|

1 - ™

c-

N

|v|

then = 0. To

x

electromagnetic field, we have added the minimal interaction e A^. As for basic variables of spin they couple with AM through the term DwM = w^ — A et fwv. The Frenkel spin-tensor [8] is a composite

^^nce —xgx kc2\E\]. If E and B are not mutually parallel in the laboratory system, this expression does not vanish for any orientation v | v| = c

The particle could exceed the speed of light and then approaches to the critical velocity.

Combining these observations, let us write relativistic equation of the form

[(xgx)ntv — xt(gx)v] xv = —(xgx)2 f*(F, D'x), (3)

with f* being polynomial on (D'i)2^ to n = 2. The system has the following properties. A) It is manifestly relativistic covariant, that is speed of light represents the invariant scale of the model. B) Due to

quantity constructed from momentum as follows1 :

and its conjugated

= 2(W

') = (J

i0

Dl, Jj

: 2eijkSk ). and Di

is

Here Si is three-dimensional spin-vector dipole electric moment.

The action implies EM which generalize those of Frenkel and Bargmann-Michel-Telegdi to the case of an arbitrary electromagnetic field. They have been studied in [2,4]. For the present discussion we need only-symbolic form of the equation for position variable, this reads

d dr

a/-xgx

= fß ,

(5)

The basic variables do not represent observable quantities since they are not invariant under spin-plane local symmetry presented in the model [1].

v

x

c

vv

n — w n

where /M is the polynomial /M = V—xgxa1 + a2 + , a3 . + with the coefficients ai that are finite as

V — xgx xgx ' 1

xgx ^ 0. The effective metric arises for the particle with anomalous magnetic moment ^ =1

= [n + — 1)(JF + FJ) +62(M — 1)2FJJF ]a

JMVI

To achieve this, we represent interval in 1 + 3 blockdiagonal form [10]

-ds2

/ dx dx — c

v^Z^00 (dx0 + dx4) c goo

(6)

+ gij —

dx1 dxj.

goo /

where b = 4m2e3— ) • The variational problem (4) yields also the value-of-spin and Frenkel conditions JMV JuV = 6ft2, JMV= 0, where the canonical momentum is = dXV — f They provide consistent quantization which yields one-particle (positive energy) sector of the Dirac equation, see [1].

Inclusion of these constraints into a variational problem, as well as the search for an interaction consistent with them turn out to be rather non trivial tasks [5,9], and the expression (4) is probably the only solution of the problem. So, the appearance of effective metric (6) in Eq. (5) seems to be unavoidable in a systematically constructed model of spinning particle.

The speed of light does not represent special point of the manifestly relativistic equation (5). Singularity occurs at the critical velocity ver determined by the xgx = 0

homogeneous fields there are configurations admitting ver > c. In general case we expect that ver is both field and spin-dependent quantity. Using spacial part

ver

Explicit computation gives va ~ a/—xgx, that is the acceleration along the direction of velocity vanishes as |v| ^ Ver > c.

In resume, if we insist to preserve the usual special-relativity definitions of time and distance, critical speed of spinning differs from the speed of light. To see,

ver = c

g

space-time metric. Then we can follow the general-relativity prescription to define time and distance in the presence of electromagnetic field. So, let us stop for a moment to discuss the definitions of velocity and acceleration in general relativity.

Three-dimensional acceleration of spinless particle in general relativity

Consider pseudo Riemann space M(1'3) = {x^, gMv(xp), goo < 0}.

This prompts to introduce infinitesimal time interval, distance and speed as follows:

dt = v^ (dxo + ^ dx1),

c goo

dl2 = (gij — goig°j )dx®dxj = Yijdx®dxj, v = —. (8) goo dt

xo

the physical time t is

dl

dt = a/—goo (i + goi dx1) dxo c goo dxo

Introduce also the three-velocity vector

dt

dx«

i

dxi dxo,

(9)

(10)

or, symbolically v1 = dir- This is consistent with the above definition of v: v2 = (^)2 = v2 = v^jvj. In the result, the interval acquires the form similar to special relativity (but now we have v2 = vyv)

—ds2 = —c2dt2 + dl2 = —c2dt

22

1 — —

1 c2

(11)

This equality holds in any coordinate system xM. Hence

ds2 = 0 v2 = c2

The formalism remains manifestly covariant under subgroup of spacial transformations xo = x/o, x1 = xi(x'j), aj (x/) = dfXj- Under these transformations goo is a scalar function, goi is a vector while g.j and Y.j ^e tensors. Since gijYjk = 5lk, the inverse metric of Yj turns out to be (y — 1)ij = gij- The velocity (10) behaves as a vector vi(xo) = aj(x/k(xo))v/j(xo), so it

Dk

Do

Dk v1 = dk v1 + f ikj (Y )v

dxk

Dov1 = dxo Dkv1. (12)

(7)

The Christofltel symbols fijk(y) are constructed with

help of three-dimensional metric Yj (xo,xk) written in

xo

Three-dimensional velocity and acceleration can be defined in such a way that speed of light represents coordinate independent quantity and, more over, a particle during its evolution in curved background can not exceed the speed of light.

rjk = 2 Yia(dj Yak + dk Yaj — daYjk ).

(13)

As a consequence, the metric y is covariantly constant, Dk Yij = 0.

Hence we associated with the one-

parameter family of three-dimensional spaces MXo =

2

v

[xk, jj, DkYj = 0}. Note that velocity has been defined above with help of the curve in parameterized by this parameter, xl(x0).

In the case of constant gravitational field (xk), we have usual three-dimensional Riemann geometry M3 = [xl, Yj(xk)}, and the standard notion of a parallel transport. So an analog of constant vector field of Euclidean geometry is the covariantly-constant field, D0^i = 0, and the acceleration with respect to physical time is defined by

50 r D»V

dT

dx0

-l

dvi dx0

Dof + 2(ZdoYY-1y = 0.

(15)

Then we define the acceleration with respect to physical time as follows:

dt dx0

Dovi + ±(vdoYY-1y

(16)

d2x*

aft'

dxa dxft ds ds

dA \ d~xgx

+ (g)xc

xfi

d~xgx

0.

A = x0

part of geodesic equation (18) reads

dT \

dx0)

-l

d

Gi

dx0

-| VYV

1--tT-

(19)

where

dT

Gi(g*v, v) = — d^l rioo — rijkvjvk

2

) ^o +rijkvj vk .(14)

00 — r jk -1

k

dx0

P 0k vk

(20)

To define an acceleration in general case, we need to adopt some notion of a constant vector field (parallel transport equation) along the trajectory xi(x0) that cross the family M^o- In Euclidean space the scalar product of two constant fields has the same value at any point. In particular, taking the scalar product along a line xi(x0), we ^ave dXo (^, n) = 0. For the constant fields in our case it is natural to demand the same (necessary) condition: dXo [^i(x0)Yij(x0,xi(x0))ni(x0)] = 0. Taking Dk Yij = 0

written as follows

(D0£ + 1(Zd0YY-1), n) + (£, D0n + \(y-1d0Yn)) = 0. So we take the parallel-transport equation to be

is non-singular function as v ^ c. We compute derivative on l.h.s. of the equation (19)

dT\

dx0

M ij D0vj +

(vd0Yv)

= Gi + r ijkvj vk,

2(c2 — vyv)

jk

(21)

where M j = Sij + c2_v)-)!V- We apply the inverse matrix

Mij = Sij —

i(v7)j

D0vi

acceleration (16). Then (21) reads

_ 1 f dr \-

2 V dx0 J

(vd0YY-1 )i —

ui (vd0yv) i

+ —

vi(vY^j^ [rjk(Y)vkv1 + Gj].

Then the acceleration along the velocity is

(22)

1

(vd0Y v) — (vd0Yv)

1 dT

vY a = dx )

+ (l — ^) (vY)i[^iki(Y)vkvl + Gi].

(vY v)

(23)

For the special case of constant gravitational field, (xi), the definition (16) reduces to that of Landau and Lifshitz, see page 251 in [10].

Let us estimate the acceleration as v ^ c. Particle in general relativity propagates along geodesies of M(1'3). If we take the proper time to be the parameter, geodesies obey the system

_ dx * dx , .

The system has no sense for the case we are interested in, ds2 ^ 0. So we rewrite it in arbitrary parametrization A (here we denote x* = d^)

(18)

To see, which equation for a1 implied by (18), we

vi

This implies vYa ^ 0 as vyv ^ c2. That is acceleration along the direction of velocity vanishes as the speed approximates the speed of light. With these definitions the spinless particle in an external gravitational field can not overcome the speed of light.

The last term in the definition (16) yields the important factor (vd0YY-1 )i in Eq. (23). As EM (22) and (23) do not contain \Jc2 — vyv, they have sense even for v > c. Without this factor, the particle in

c

accelerate.

5 World-line geometry probed by spinning particle

The construction can be applied, without modifications, to the effective metric (6) appeared in EM of spinning particle (5). The metric depends on xi via the field strength F(x0,xi), and on x0 via the field strength as well as via the spin-tensor J(x0). The geodesic equation is replaced now by Eq. (5). Adopting that velocity and acceleration of spinning particle in

v

1 VYV 1--2"

c

a

1

2

c

1

2

c

2

c

1

a=

d

electromagnetic field are given by Eqs. (10) and (16), widely assumed as the reasonable equations of spinning

we obtain the theory with critical speed equal to the particle in gravitational fields. In the EM (5) now

speed of light. The intervals of time (and distance) appears the effective metric [3]

probed by such particle are given now by Eq. (8). _ _

They slightly differ from those in empty space. gMv = \9 + e( + J) + e ^ , (^4)

Due to ^ — 1-factor in Eq. (6), the deformation

of world-line geometry in electromagnetic field will where e = 8m2c2- j-0' = J"and (c¡)

be seen only by a particle with anomalous magnetic the curvature tensor. Hence, to guarantee the

moment. In a gravitational field the deformed geometry observer-independence of с and the equality «сг = с,

could be probed by any spinning particle. To see we need to deiine velocity (10) and acceleration (16)

this, let us consider the Frenkel electron in a curved usinS the deformed metric g instead of Í?. background with the metric gMV. We use the model

constructed in [3]. The Lagrangian can be obtained Acknowledgement from (4) (with = 0) replacing by iW and

usual derivative of by the covariant derivative, This work has been supported by the

^ = + ГаеThis leads to EM Brazilian foundation CNPq (Conselho Nacional de

consistent with those of Papapetrou [11], the latter are Desenvolvimento Cienthfico e Tecnolygico - Brasil).

References

[1] Deriglazov A. A. and Pupasov-Maksimov A. M. 2014 Eur. Phys. J. С 74 3101.

[2] Deriglazov A. A. and Pupasov-Maksimov A. M. 2014 Nucl. Phys. В 885 1.

[3] Ramirez W. G., Deriglazov A. A. and Pupasov-Maksimov A. M. 2014 JEEP 03 109.

[4] Deriglazov A. A. 2014 Phys. Lett. В 736 278.

[5] Hanson A. J. and Regge T. 1974 Annals of Phys. 87(2) 498.

[6] Holten J. W. van. 1991 Nucl. Phys. В 356 3.

[7] Hojman S. A. and Asenjo F. A. 2013 Class. Quant. Grav. 30 025008.

[8] Frenkel J. 1926 Nature 117 (1926) 653.

[9] Mukunda M. Dam H. van and Biedenharn L. C. 1982. Lecture Notes in Phys. 165 (Springer-Verlag)

[10] Landau L. D. and Lifshitz E. M. 2011. Classical Theory of Fields, Course of Theoretical Physics vol 2 (Pergamon)

[11] Papapetrou A. 1951 Proc. Roy. Soc. bond. A 209 248.

Received 24-10.20Ц

А. А. Дериглазов

УЛЬТРАРЕЛЯТИВИСТСКАЯ СПИНОВАЯ ЧАСТИЦА ВО ВНЕШНЕМ ГРАВИТАЦИОННОМ И ЭЛЕКТРОМАГНИТНОМ ПОЛЯХ

Трехмерное ускорение в общей теории относительности можно определить так, что бесспиновая частица в процессе эволюции во внешнем гравитационном поле не сможет превысить скорость света. Мы выясняем, что происходит если бесспиновую частицу заменить на частицу со спином 2. Учет спин-гравитационного взаимодействия приводит к обобщенным уравнениям Папапетроу с модифицированной вдоль мировой линии метрикой. Именно модифицированная метрика должна использоваться для вычисления трехмерного ускорения спиновой частицы. Взаимодействие спина с электромагнитным полем в плоском пространстве приводит к обобщенным уравнениям Френкеля, в которых возникает эффективная метрика вдоль мировой линии частицы с аномальным магнитным моментом. Для такой частицы интервалы времени (расстояния) в электромагнитном поле и в пустоте отличаются друг от друга.

Ключевые слова: ультра-релятивистская спиновая частица, уравнения Френкеля, уравнения Папапетроу.

Дериглазов А. А., кандидат физико-математических наук, доцент. Федеральный университет Жуиз де Фора. Depto. de Matemática, ICE, MG, Бразилия. E-mail: alexei.deriglazov@ufjf.edu.br