General Relativity Derivation of Beam Rest-Frame Hamiltonian

Home , Rest frame

Proceedings of the 2001 Particle Accelerator Conference, Chicago GENERAL RELATIVITY DERIVATION OF BEAM REST-FRAME

HAMILTONIAN £

J. Wei Ý , Brookhaven National Laboratory, New York, USA

Ù = dÜ Abstract where =dØ are the contravariant components of the

spatial velocity, the Greek letter indicates the three spatial

Analysis of particle interaction in the laboratory frame of

= d c components, dØ , is the speed of light, and

storage rings is often complicated by the fact that particle

½ µ

motion is relativistic, and that reference particle trajectory ´

½ g Ù Ù 4

is curved. Rest frame of the reference particle is a conve-

= ´ g µ

44 (1)

c g c nient coordinate system to work with, within which particle 44 motion is non-relativistic. We have derived the equations

of motion in the beam rest frame from the general relativ- is the generalized Lorentz factor. The contravariant com-

i i

È = Ñ Ñ Í ¼ ity formalism, and have successfully applied them to the ponents of the four-momentum are ¼ , where

analysis of crystalline beams [1]. is the proper mass of the particle.

fF g

Acted upon by a non-gravitational four-force i , the 1 DERIVATION equations of motion of the particle can be written as

The motion of charged particles under Coulomb interac- i

DÈ

ik i

= g F ; F Í ¼; i tion and external electromagnetic (EM) forces can be most k (2)

conveniently described in the rotating rest frame of the ref- d

4 ¢ 4 fg g fg g erence particle of which the orientation of the axes are con- where the matrix is the inverse of ij , and

stantly aligned to the radial, tangential, and vertical direc- the covariant differentiation is deﬁned as

tion of the motion. In this frame, particle motion within the

i ÐÑ i

dÈ @g @g g DÈ @g

Ñk ik Ñi

Ð k Ð Ð

· Í · È ;

beam bunch is non-relativistic. =

ik ik

k i Ñ

d ¾ @Ü @Ü @Ü We derive the equations of motion using the general rel- d (3) ativity formalism. First, we express the equations of mo- Among the non-gravitational forces, the external EM force tion in a general tensor formalism. The Lorentz force ex- acting upon the particles is expressed by means of the EM

perienced by the particle is constructed as a product of the

fF g

ﬁeld tensor ij as

EM ﬁeld tensor and the four-velocity. Starting from the e

laboratory frame, the EM ﬁeld tensor is written by means k

F = F Í ; i ik (4) of the components of the EM ﬁelds. Then, tensor algebra c

is used to transform this ﬁeld tensor into the rest frame.

e fF g

where is the electric charge, and ij is anti-symmetric. With a similar transformation, the metric tensor of the rest The Maxwell’s equations are given by

frame is also obtained. The equations of motion can thus

@F @F

be constructed in the rest frame which include centrifugal @F

ik kÐ Ði

· · =¼;

i k

force, Coriolis force, time-dependent external EM forces, Ð

@Ü @Ü @Ü i

Ô (5)

@ ½ Í ¼

and electrostatic Coulomb forces. Finally, these equations ik

´ jg jF ; µ=

i c

are re-scaled in terms of dimensionless quantities for the @Ü

jg j

convenience of computer simulation and analysis.

ij iÐ jÑ

F = g g F jg j

where ÐÑ , is the absolute value of the de-

fg g ¼ 1.1 Tensor formalism terminant of the metric ij , and is the charge density measured in the system of inertia. Adopting the formalism used by M6 oller,[2] we consider

the motion of a particle in an arbitrary system of coordi-

Ü µ i = ½; ¾; ¿; 4

nates ´ , where indicate the space-time 1.2 Laboratory frame

fg g

components. The metric tensor ij is deﬁned in terms

X µ

In the Cartesian coordinate system of inertia ´ , the

¾ i j

d× d× = g dÜ dÜ of the differential line element as ij , so-called laboratory frame, Eq. 2 can be written in the con-

where the summation is performed over the four indices. ventional vector form. In this frame, the only non-zero

i i

Ü = Ü µ Let ´ be the equation of the time track of the components of the metric tensor are

motion, being the proper time of the particle. The con-

g = g = g =½; aÒd g ½: travariant components of the four-velocity are =

½½ ¾¾ ¿¿

44 (6)

dÜ

i In terms of the conventionally deﬁned electric and mag-

Í =´ Ù ; cµ;

d netic ﬁelds

£ Work performed under the auspices of the US Department of Energy

E =´E ;E ;E µ; B =´B ;B ;B µ; Ý

¾ ¿ ½ ¾ ¿ [email protected], on joint appointment at ORNL and BNL ½ (7)

Proceedings of the 2001 Particle Accelerator Conference, Chicago

fg fg g g

the EM ﬁeld tensor is of the rest frame is related to ij of the laboratory

¼ Ð Ñ

¼ ½

g = « « g :

frame by the relation ÐÑ Using the transfor-

ij i j

¼ B B E

¾ ½

¿ mation matrix, the contravariant components of a vector

B C

B ¼ B E

¿ ½ ¾

B C

fa g

fF g = :

ij (8) in the laboratory frame are transformed into the rest

@ A

B B ¼ E

¾ ½ ¿

i i k

a = « a

frame as : On the other hand, the covariant com-

E E E ¼

¾ ¿

½ ponents of a tensor of rank 2, e.g. the EM ﬁeld tensor, are

¼ Ð Ñ

F = « « F

It is straightforward with these expressions to verify that transformed as ÐÑ . With these relations, the

ij i j Eq. 2 is equivalent to the equations equation of motion in the beam rest frame of ﬁxed orienta-

tion can be obtained from Eq. 2 by an explicit evaluation.

e d d´ Ùµ

Ñ = eE · Ù ¢ B; Ñ c = eÙ ¡ E; ¼

¼ (9)

c dØ

dØ 1.4 Rotating beam rest frame

½=¾

¾ ¾

dØ = d = ½ Ù where , and =c is in this case The equations of motion can be greatly simpliﬁed in the Lorentz factor. terms of the variables tangential and normal to the direc- tion of the motion of the reference particle. We thus seek 1.3 Beam rest frame of ﬁxed orientation for another transformation into the rotating beam rest frame

Consider a reference particle circulating around the ver- of which the orientation of the spatial axes are constantly

Ü Þ Ý !

tical axis X with a uniform angular velocity at a radius aligned to the radial ( ), tangential ( ), and vertical ( ) di-

ff g Ê . Its time track may be described in the laboratory rection of the motion of the reference particle.

frame in terms of the proper time Deﬁne the rotating beam frame as the rest frame of Þ

which the orientations of the spatial axes Ü and rotate

½ ¿

f ´ µ=´Ê cÓ× ; ¼;Ê×iÒ ; µ ; Ü

(10) relative to those of the ﬁxed orientation Ü and

¼ ½ ¼ ½ ¼ ½

½=¾

~ ~

Ü cÓ× ¼ ×iÒ ¼ Ü

= ! = ¬

where is the revolution angle, ½ ,

B C B C B C

Ý ¼ ½ ¼ ¼ Ü

= Ê!

and ¬c is the velocity. Introduce a rigid system

B C B C B C =

: (14)

@ A @ A @ A

~ ~

Þ Ü

×iÒ ¼ cÓ× ¼

Ü µ

of coordinates ´ which follows the reference particle in

¼ ¼ ½ its motion, so that the particle is constantly situated at the ¼ origin of this frame of reference, and that the spatial axes Express the electric and magnetic ﬁelds in the laboratory

have constant orientations. The transformation connecting

Þ i

i frame in terms of the tangential component in and the

X µ ´Ü µ the variables ´ and are

normal component in Ü

i i 4

X = f ´ µ·« Ü ; aÒd Ü = ;

i (11)

E = E cÓ× · E ×iÒ ; B = B cÓ× · B ×iÒ

Ü ½ ¿ Ü ½ ¿

= E ; B = B

where the check ( ) denotes the inverse operation, the sum- E

Ý ¾ Ý ¾

½; ¾; ¿

= E ×iÒ · E cÓ× ; B = B ×iÒ · B cÓ×

mation over is on spatial components . The coef- E

Þ ½ ¿ Þ ½ ¿ «

ﬁcients ij are obtained by means of successive inﬁnitesi- (15) mal Lorentz transformations without rotation of the spatial The equations of motion can be obtained from Eqs. 2 and

axes,[2] 3 by using Eq. 8 and the transformation relations as

¼ ½

Ñ ~

« ¼ « ¬ ×iÒ ¾ 4 ¾ ¾ ¾

½½ ½¿

Ü ¾ ! Þ_ ! Ü Ñ ! Ê ´½ µ=

B C

¼ ½ ¼ ¼

B C

¢ £

@Î e

f« g = ;

¼ ¼ ¾ ¼

@ A

; B Ý_ B ´_Þ · !Üµ

= e´½ µE ·

« ¼ « ¬ cÓ×

¿½ ¿¿

Þ Ý Ü

c @Ü

¬ ×iÒ ¼ ¬ cÓ×

Ý = e´½ µE ·

< Ý

(12)

¢ £

@Î e

~ ~ ~

¼ ¾ ¼ ¾

« = cÓ× cÓ× · ×iÒ ×iÒ « = ×iÒ cÓ×

> ½¿

where ½½ ,

B ´_Þ · !Ü B ´_Ü !Þ µ ; ·

Ü Þ

~ ~ ~ >

c @Ý

cÓ× ×iÒ « = cÓ× ×iÒ ×iÒ cÓ× « =

> ¿¿

, ¿½ ,

¾ 4 ¾

~ ~ ~

Þ ·¾ ! Ü_ ! Þ =

×iÒ ×iÒ · cÓ× cÓ× = ! = :

, and The fact >

~ >

¢ £

> @Î

that differs from the revolution angle by the Lorentz e

¾ ¼ ¼ ¼

; Ý_ ´_Ü !Þ µ B B

· = e´½ µE

Ü Ý Þ

c @Þ factor is the consequence of the Thomas precession. To obtain the metric tensor of the rest frame, we need (16) to derive the relation between the line elements of the two where

coordinate systems. Differentiation of Eq. 11 gives

¼ ¼

E = ´E ¬B µ; B = ´B · ¬E µ

Ü Ý Ü Ý

Ü Ü

¼ ¼

E = ´E · ¬B µ; B = ´B ¬E µ

Ý Ü Ü

Ý (17)

Ý Ý

« ´ µ f ´ µ

i i j

¼ ¼

dX · Ü =« dÜ · d « dÜ ;

E = E ; B = B

Þ Þ

d d

(13)

i i ½

f« g = f«

where g is the transformation matrix be- are the electric and magnetic ﬁelds after a Lorentz transfor-

j j

tween the line elements of the laboratory and the rest frame. mation without rotation, and Î is the electric potential. For Î

Let the prime (’) denote the rest frame. The metric tensor particles of the same bunch, the potential C describing the

1679 Proceedings of the 2001 Particle Accelerator Conference, Chicago

Coulomb interaction is Here, the dots denote differentiations with respect to the

¾ X

normalized time Ø. The normalized Coulomb potential and

Î ´Ü; Ý ; Þ µ=

C rf potential satisfy

¾ ¾ ¾

´Ü Üµ ·´Ý Ý µ ·´Þ Þ µ

j j j

j X

(18) ½

Î ´Ü; Ý ; Þ µ=

¾ ¾

where the summation is performed over all the other parti- ¾

´Ü Üµ ·´Ý Ý µ ·´Þ Þ µ

j j j

j Î cles and their image charges. The potential Öf describ-

(25) E

ing a radio-frequency (rf) electric ﬁeld × in the laboratory and

eE ´Ü; Ý ; Þ µ Ê

frame satisﬁes @Î

× Öf

= (26)

@Î ´Ü; Ý ; Þ µ

@Þ Ñ c ¬

Öf

= eE

× (19)

@Þ In the normalized units, the revolution period of the refer- Since the particle motion in the rest frame is non- ence particle in the storage ring is ¾ .

relativistic, the Lorentz factor is simpliﬁed as

½ ¾ ¾

2.2 Hamiltonians

´½ µ ; ÛiØh ¬ :

(20)

´È ;È ;È µ

Ý Þ Using the canonical momentum Ü , the parti-

Typically, the dimensionless quantity is much smaller cle system of Eq. 24 can be described by the Hamiltonian

than 1. The terms on the left hand side of Eq. 16 in-

¡ £ ¢

½ ½

¾ ¾ ¾ ¾

clude the centrifugal and Coriolis forces. Those on the right ¾

ÜÈ · Î À = È · È · È ´½ ÒµÜ · ÒÝ ·

Ü Ý Þ ¾ hand side are external EM forces and electrostatic Coulomb ¾

forces. Again, since the particle motion is non-relativistic, (27)

ÜÈ the magnetic force produced by the motion of the particles where the cross term Þ describes the coupling be- is negligible compared with the electrostatic force. tween the tangential and normal motion.

2 STORAGE RING EXAMPLE 2.3 General case The formalism presented in the previous sections can be We consider the case that the beam is guided by a bend-

easily generalized to a storage ring that consists of both B ing ﬁeld ¼ satisfying

bending and straight sections. Denote the guiding ﬁeld and

eB Ê = Ñ c ¬ ;

B Ê

¼ ¼

(21) bending radius in the bending section as ¼ and , respec- B and focused by a quadrupole ﬁeld of gradient ½ tively. The equations of motion in the bending sections are

given by Eq. 24, while the system Hamiltonian is given by

B = B Ý; B = B · B Ü; B =¼;

Ü ½ Ý ¼ ½ Þ (22)

Eq. 27. In the straight sections where the guiding ﬁeld is B where ½ may vary for different piece of magnets, and the

zero, the strength Ò of the focusing ﬁeld can be deﬁned by

ﬁeld variation at the magnet end is neglected. The equa-

B B ¼ normalizing the gradient ½ in the straight section to in

tions of motion is simpliﬁed by linearizing Eq. 16 the bending section. The Hamiltonian becomes

@Î

¾ ¾ ¾

¢ £ ¡

½ ½

Ñ ; Ü ! Þ_ Ñ ! Ê = e¬ B Ü

¼ ¼ ½

¾ ¾ ¾ ¾ ¾

À = È ÒÜ · ÒÝ · È · Î · È ·

@Ü (28)

< Ü Ý Þ

¾ ¾

@Î

Ñ Ý = e¬ B Ý ;

¼ ½

@Ý C

> If the circumference of the ring is , the revolution period

@Î

¾ :

of the reference particle is C=Ê in the normalized unit.

Ñ Þ · ! Ü_ = : ¼ @Þ (23) 3 DISCUSSIONS

Here, has been assumed small compared with 1. The equations of motion derived for the beam rest frame 2.1 Dimensionless variables made possible direct utilization in storage ring analysis of Eq. 23 can be simpliﬁed in form when it is expressed techniques like the molecular dynamics methods developed

typically for non-relativistic systems. These equations can

Ò B Ê=B ¼ in terms of dimensionless variables. Let ½

represent the strength of the focusing magnetic ﬁeld, and be easily generalized for storage rings containing multipole

½=¿

¾ ¾

¾ magnets and multi-species of ions. [3]

Ö Ê =¬

¼ be a characterization of the inter-

particle distance in the presence of Coulomb interaction in Derivations presented in this paper are by-products of ¾

¾ a study on crystalline beams in collaboration with A.M.

Ö = e =Ñ c ¼ the storage ring, where ¼ is the classical ra-

Sessler and X-P. Li. Ê=¬ c

dius of the particle. Express the time Ø in unit of , the

Ý Þ

spatial coordinates Ü, , and in unit of , and the energy

¾ ¾

¾ 4 REFERENCES

e =

in unit of ¬ . Eq. 23 becomes in these units 8

@Î [1] J. Wei, X-P. Li, A.M. Sessler, Phys. Rev. Lett., 73, (1994)

; Ü Þ_ ·´ ·½ ÒµÜ = >

> 3089.

@Ü

< @Î

[2] C. M6 oller, The Theory of Relativity, Oxford, 1952.

; Ý · ÒÝ =

(24)

@Ý >

> [3] J. Wei, H. Okamoto, A.M. Sessler, Phys. Rev. Lett., 80,

@Î

Þ · Ü_ =

: (1998) 2606. @Þ

1680