M-226 W. W . Hansen Laboratories of Physics

Stanford University

S t anford, Cal i fornia

Report No. M-226

October 1960

JACOBIANS AND RELATIVISTIC KINEMATICS

BY

K. G. Dedrick

It i s of ten necessary t o ca lcu la te the energy-angle d i s t r ibu t ion of

pa r t i c l e s i n one Lorentz frame of reference from knowledge of the d i s t r i -

bution i n some other frame. This problem occurs i n the in te rpre ta t ion of

experiments where an energy-angle d i s t r ibu t ion i s measured i n the labora-

t o ry frame of reference and it i s desired t o calculate the d i s t r ibu t ion

i n the center of mass. Problems of t h i s type a re discussed i n Section I.

A d i s t i n c t but r e l a t ed problem i s the subject of Section 11, which

i s concerned w i t h react ions where two react ion product p a r t i c l e s a re

produced and hence spec i f ica t ion of t h e bombarding energy gives r i s e t o

a de f in i t e laboratory energy-angle re la t ionship f o r t he react ion products.

I n t h i s problem we are in te res ted i n calculat ing the laboratory y i e l d

spectrum of pa r t i c l e s produced when the bombarding p a r t i c l e s a re d i s t r i -

, buted over a range of energies. An example of t h i s problem is t h a t of

calculat ing t h e laboratory y i e ld spectrum of s ingle photopions given

knowledge of the incident photon spectrum and the cross sect ion f o r pion

photo-production as a function of photon energy.

In both of these problems, Jacobians of t he Lorentz transformations

occur as mult ipl icat ive fac tors i n transforming p a r t i c l e y ie lds between

coordinate systems. In the descr ipt ion of the second problem mentioned

above, where one i s concerned w i t h a d i s t r ibu t ion of bombarding energies,

it i s useful t o group a l l possible react ions in to classes having Jacobians

and angle transformations of a given form.

the subject of Section 111.

This c l a s s i f i ca t ion scheme i s

- 1 -

1

I. THE FIRST PROBLEM

The t o t a l energy, momentum and angle i n the laboratory frame of ref-

erence w i l l be denoted by the quant i t ies E, p, and 8 respectively.

The s i tua t ion i n the moving frame o f reference, which i n t h i s study i s the

center-oP-mass frame, i s described by the corresponding primed quant i t ies

E', p ' , and 8 ' . The velocity o f the center-of-mass frame with respect t o

the laboratory i n un i t s of t he veloci ty of l i g h t i s denoted by 6, and I' i s wr i t ten for 1 1

(1 - /32)-2. The Lorentz transformation may be wr i t ten

E = r ( E ' + (~P ' co ' )

PW = P(P'cu' + PE') P s i n 8 = P' s i n 8'

E' = P(E - ~ P w )

P 'W' = r(m - BE) 2 2 2 2 E2 - p Mc = E ' - PI

where

w = COS e; (13' = COS el

P = pc; P' = p ' c

W'(E',(u') denotes the energy-angle d i s t r ibu t ion of pa r t i c l e s i n the

moving frame of' reference, and W(E,u) i s t h e d i s t r ibu t ion i n t h e lab-

oratory. Conservation of pa r t i c l e s requires t ha t

W ' ( E ' , W ' ) ~ E ' dw' = W(E,u)dE dw ( 5 1

where ( E ' ,a1) and (E,cu) are r e l a t ed through the Lorentz transformations

(1) or (3). We then w r i t e

where d ( E ' ,a' )/d(E,cu) i s the Jacobian of t he Lorentz transformation.

'See for example : "The Kinematics of High-Energy Collisions, " by K. Dedrick, ML Report No. 574, February 1959, W. W. Hansen Laboratories o f Physics, Stanford University, Stanford, California.

- 2 -

Subst i tut ion of (6) i n t o ( 5 ) y ie lds

W(E,w) = e a ( E l w ' ) W'(E' ,(I)') ( 7 ) E? UJ

Once t h i s Jacobian has been calculated, other usefu l Jacobians may

be obtained from it:

( 8 1

The Jacobian d ( E ' ,w ' ) /d (E7w)

given by evaluating t h e determinant

has a pa r t i cu la r ly simple form and i s

From Eq. (3) we f ind

p p ( l - B o f \

so t h a t t he Jacobian of Eq. (11) i s

- 3 -

1

This las t form is obtained using Eq. (2). ( g ) , and (10) gives

Substi tution of (12) in to ( 8 ) , 2

a(P ' w') E' s i n 8 ' = P ' s i n e

. s i n 0 ' s i n 0

It i s important t o note that i n Eq. ( 5 ) w e wrote dE' du' and

dE dw ra ther than IdE'du' I and IdE dw I as suggested by the usual

p a r t i c l e conservation statement. I f the Lorentz tranformation reveals

t h a t more than one point i n the plane corresponds t o a given

point i n t h e E - cu plane, the pa r t i c l e conservation statement properly

reads

E'-w'

where the summation is over a l l points ( E i , mi ) i n t h e E'-cu' plane which correspond t o the point (E,w) i n the E-cu plane. It is possible

t o make meaningful measurements only i f the re la t ionship between (E,u) and (E' ,a' ) i s single-valued. Consequently, c r i t e r i a f o r t he existence

of a multivalued re la t ionship must be developed.

The Lorentz tranformation equations may be used t o construct curves

i n the E-w plane of constant E' f o r several values of E ' . If any of

these curves touch or cross one another, then it i s c l ea r t ha t more than

one point i n the E'-w' plane corresponds t o a point i n the E-w plane. The same i s t r u e f o r curves of constant w'. In f ac t , i f these constant

E ' (or a') curves are t o cross a t a l l , the curves E' = constant and

E ' + dE' = constant will cross a t some point i n the diagram. Let

E(E ' ,w ' ) and w(E',w') be determined by the Lorentz transformation.

Arbitrary s m a l l changes i n E ' and w' are then made and we ask under

what circumstances w e a re l e d t o the same values of E and w.

2 . Eq. (12) was previously obtained by Panofsky (pr iva te communication) and Eq. (14) i s given by Behr and Hagdorn, CERN Report No. 60-20, May 1960.

- 4 -

On s e t t i n g the lefi-hand sides of these expressions equal t o

and

E(E',w') o(E' ,of ) , respectively, we obtain

A solut ion t o these equations e x i s t s f o r a rb i t r a ry dE' and do3' only

i f the determinant of the coefr ic ients vanishes. This determinant i s the

Jacobian a ( E Y ~ ) / a ( E ' , u ' ) . Similarly, more than one point i n the E-o

plane corresponds t o a point i n the

vanishes.

w i l l be ca l led c r i t i c a l points, and imply a multivalued re la t ionship

between (E,w) and (E',w').

E'-o' plane i f a ( E ' , o ' ) / a ( E , w )

The points a t which a Jacobian e i t h e r vanishes or i s i n f i n i t e

As an example, t he Lorentz transformation f o r $ = @ i s p lo t ted

i n Fig. 1 and shows how points i n the E'-8' plane transform t o the E-8

plane.

t he r e s t energy Me . A t the point (Ekc/Mc = 1, 8 = O), the Jacobian

a(EY~)/a(E',o') vanishes.

spons t o E i = 0 and a l l values of 8 ' . According t o Eq. (12), a (E ,u ) / a<E ' ,a1)

i s zero since P' i s zero a t t h i s point. Other c r i t i c a l points form a

locus and are located a t Ek/Mc2 = 0 for a l l values of 8 .

a(E',w')/a(E,w) i s zero i n tha t P vanishes.

Kinetic energies ra ther than t o t a l energies are shown i n un i t s of 2 2

As may be seen from Fig. 1, t h i s point corre-

In t h i s case,

This discussion shows t h a t Eq. (7) i s va l id i n a l l regions of t he

E-o plane except a t the c r i t i c a l points ment,ioned i n the preceding para-

graph, and t h a t except a t these points t h e mapping of the

onto the

of t h i s r e s u l t i s t h a t p a r t i c l e y ie lds measured i n the laboratory a t some

spec i f ic values of (E,cu) correspond t o the y ie lds a t spec i f ic values of

(E',o') i n the center-of-mass system.

E'-w' plane

E-o plane i s a one-to-one mapping. The p rac t i ca l significance

- 5 -

10

9

a

7

6

5

4

3

2

1

0

,g ---e.

FIG. l--Lorentz transformation f o r p =$j-'/2

- 6 -

I n the next section, considerations of t h i s nature w i l l be seen t o have more serious consequences i n ce r t a in cases.

11. THE SECOND PROBLEM

Here we consider react ions of the type

1 + p 2 + p 3 + p 4

where i s the bombarding p a r t i c l e with r e s t energy Ml, P2 i s the

ld&ei?k p a r t i c l e with r e s t energy M2, and P and P4 a re the react ion

products having r e s t energies M and M4, respect ively. The k ine t i c

energy of t he bombarding p a r t i c l e

I; we are concerned with a d i s t r ibu t ion of bombarding energies.

7d1-" e7 p1

3 3

P1 i s not chosen f ixed as i n Section

The center-of-mass motion depends on the energy of the bombarding

pa r t i c l e , and for a given value of the bombarding energy, t h e energy of

a react ion product p a r t i c l e i n the center of mass i s uniquely determined

by some function of the bornbarding energy.

energies and angles of the react ion products a re determined by a function

of the bombarding energy Eo and the center-of-mass angle 8 ' . For a

d i s t r ibu t ion of bombarding energies, there e x i s t s a d i s t r ibu t ion function

U(EO, cos e ' ) which i s the number of p a r t i c l e s produced i n the center of

m a s s i n the d i f f e r e n t i a l so l id angle

Eo and E + dEo are causing the react ion. The spec- e ne r g i e s between

trum of e i t h e r of the two react ion products i n the laboratory may be

described by a d i s t r ibu t ion function W(E, cos e ) . Par t i c l e conservation

re quires*

In t h i s way, t he laboratory

s i n 8 ' d@' de' when bombarding

0

U ( E o , ~ ' ) d E o d ~ ' = W(E,w)dE dm (19)

where :

E = bombarding energy 0

E = laboratory energy of react ion product

LU' = COS e l ; LU = COS e .

The Jacobian most readi ly calculated i s

dE h = dEo dw'

* Eq. (19) i s va l id i f the mapping from (E,cu) t o (EO,w') i s a

- 7 - one-to-one mapping. See the discussion of Eq. (16).

- .

so t h a t (19) becomes

I n t h i s problem, f3 and E’ may be w r i t t e n as functions of Eo alone. The Jacobian i s

This Jacobian can a l so be wr i t ten

w T&$= (R)

This l a t t e r expression i s perhaps eas i e r t o understand physically since

(aw/aw’ )

i s taken of the d i s t r ibu t ion of Eo through the fac tor

Evaluation of t he Jacobian of Eq. (22) yields*

i s the familiar solid-angle transformation fac tor and account EO

(dE/dEO)cu.

This expression i s va l id f o r both react ion product pa r t i c l e s provided

proper subscripts a re employed.

with rest energy M4 i s then

Equation (24) f o r the react ion product

~~

Jc Variations of t h i s expression may be constructed i n the manner used

i n deriving (8), (9), and (10).

- 8 -

Y

or

In addition t o t h e c r i t i c a l points a t

the poss ib i l i t y that the quantity i n the square brackets i n Eq. (26) i s

zero f o r ce r t a in values of LU' W e now w r i t e

P& = 0 and P4 = 0, there e x i s t s

4'

If the magnitude of f i s less than unity, there e x i s t s a locus of

c r i t i c a l points f o r the Jacobian corresponding t o a range of

important, then, t o know how f var ies w i t h bombarding energy i n a l l ul;. It i s

possible react ions.

of t h i s section.

The quantity f

function of E ' the

terms of E;, f i s 0'

where* :

This i s the subject f o r discussion i n the remainder

i s most conveniently discussed when expressed as a

t o t a l energy avai lable i n the center of m a s s . In

given by

E ' = E' + E& = t o t a l energy avai lable i n the center of mass. 0 3

- % E = M + M2; 61 = M1 1 1

- M4 E = M3 + M4; 82 = M3 2

Now E; i s never l e s s than E and i n endo-ergic reactions i s grea te r

than E so t h a t a react ion can proceed only i f E' > E ~ . Hence we are 1'

1' 0

* This expression i s not va l id f o r M = 0, i n which case f is iden- 2

t i c a l l y zero since El; i s independent of p and hence (dE/+/dp) vanishes. See a l so Eq. (37).

- 9 -

in te res ted i n the behavior of (28) for values of

E o r E depending on which i s the grea te r . Further, since 6 1 = c1

and ( f j 2 1 5 c2? we see that f as given by (28) i s posi t ive f o r values

of EA i n the region of i n t e re s t , and becomes unity as EA approaches

i n f i n i t y . The values of E ' leading t o f2 - 1 = 0 a re readi ly calcu-

l a t e d and may be examined t o see i f indeed they fall i n the region of

i n t e r e s t .

EA greater than e i t h e r <

1 2' I 1

0

The d e t a i l s of t h i s examination a re too lengthy t o be included

here, but they show t h a t i n only two cases a re there values of

the region of i n t e r e s t s a t i s o i n g EA i n

f2 - 1 = 0. These cases a re :

5 + M2 > M + M4 and M > 5 ( implies M2 > M4) 3 3 Case 1:

M + M4 > % + % and % > M (implies M4 > M2) 3 3

Case 2:

That is, unless the c r i t e r i a for these cases a re sa t i s f i ed , e i t h e r there

a re no c r i t i c a l points i n the Jacobian due t o t h e f ac to r given by (27) or the Jacobian i s c r i t i c a l for ce r t a in "1; for a l l E6 i n the region

of i n t e r e s t . This l a t t e r condition obtains i f f2 - 1 < 0 for a l l EA i n the region of i n t e re s t . The value of El2 sa t i s fy ing f2 - 1 = 0

for both Cases 1 and 2 i s 0

?LM3 + M2M4 b3 [?) - Y (?) 1 M3 - Y

and the bombarding energy Eo giving r i s e t o the above value of EA2 i s

2 For values of Eo greater than 5' we f ind t h a t f - 1 is posi t ive

%? it for Case 1 and negative f o r Case 2. Similarly, for Eo l e s s than

may be seen t h a t

Thus the Jacobian has c r i t i c a l points for Case 1 when the bombarding e n e r a

i s l e s s than the value given by ( 3 0 ) , and fo r Case 2 when the bombarding

energy i s grea te r than the value given by (30).

f2 - 1 i s negative f o r Case 1 and posi t ive for Case 2.

- LO -

1

111. GENERAL FEATURES OF THE LORENTZ TBANSFOFMATIONS

In problems of the type discussed in Section I1 the characteristics of the energy-angle plots for product particles can be separated into cer- tain classes.

transformations, viz:

and the laboratory angle of a reaction product particle; of the Jacobian as given by (22) or (26).

This classification is based on two characteristics of the (a) the relation between the center-of-mass angle

(b) the behavior

The Jacobian has been discussed in the previous section in terms of

f(E6) given by Eq. (28). laboratory angle transformation can be characterized by a number

may be expressed as a functior, of

In a similar manner, the center-of-mass to y which

E;. The angle transformation is

J- sin 0;

yi + cos 8' tan Bi =

i

and is obtained by dividing Eq. (2) by Eq. (lb) and using the definition

The subscript

sidered. It is shown in reference 1 that. for y < 1 the angle trans- formation gives a single-valued relation between

'i value of ei.

i denotes which of the product particles is being con-

i and Of, and for 'i

> 1 there are usually two values of 0; corresponding to a given

By using energy and momentum conservation in the center-of-mass

system it can be shown that

where (33) refers to E are defined below Eq. (28); and as before Eo is the total energy

available in the center-of-mass.

to those of f(E6)

y4 (i.e., i -T: 4); the quantities 6*, E ~ , and

2 The general features of (33) are similar

of Eq. (28) and are described immediately below Eq. (28),

- 11 -

1

2 with the exception t h a t

following spec ia l cases:

7 - 1 is zero i n the region of i n t e re s t i n t he

Case 3: 5 + M2 > M + M4 and M4 > M2 (implies 5 > M3) 3

M3 ’ M1) Case 4: M + M4 > 5 + M2 and M2 > M4 (implies 3

The value of E l 2 sa t i s fy ing y2 - 1 = 0 f o r both Cases 3 and 4 i s c

M M M2M4 + [ M4 - M2 (:)I

M4 - M2

The bombarding energy Eo giving rise t o the above value of EA2 i s

1 E = a m 4 - M2,

(34)

( 3 5 )

For values of E

f o r Case 3 and negative f o r Case 4. it may be seen t h a t

grea te r than Ea, we f ind t h a t y2 - 1 i s posi t ive 0

Similarly, f o r Eo less than Ea,

y2 - 1 i s negative f o r Case 3 and pos i t ive f o r Case 4. The general features of these Lorentz tranformations f o r M2 # 0 are

shown i n Fig. 2-4. The behavior of 7 and f i s shown as a function of

t he bombarding energy

of the transformations a l so shown i n these figures.

on these l a t t e r p lo t s correspond t o la rge values of E 0’ l i n e s correspondingly denote s m a l l values of Eo. In endo-ergic reactions

(See Fig. 4) t h e bombarding-energy threshold i s given by

Eo, and from these graphs one may sketch the mapping

The heavy s o l i d l i n e s

and the l i g h t

The energies E and % re fer red t o i n Figs. 3 and 4 are given by

( 3 5 ) and ( 3 0 ) , respectively.

I n the event t h a t

a

% = 0, such as OCCLWS i n decay-in-flight reactions,

the formulae fo r f and y of (28) and ( 3 3 ) a re meaningless and special

r e l a t ions must be developed. These are:

- 12 -

Y

In the l i m i t

equal t o unity, so t h a t t he angle re la t ionship becomes double-valued for

Eo > Ec, where

Eo -+ 00, it i s easy t o show t h a t y i s greater than or

The mapping of the transformation and the quant i t ies f and y are

sketched i n Fig. 5 . The graphs i n Figs. 2-5 are sketches which have been drawn t o point

out various features of the transformation equations. In some cases the

d e t a i l s shown in the mappings of the transformations may be found t o be

of l i t t l e consequence i n the design of experiments. A s an example, con-

s ide r the react ion

y + p + n + E+

where we choose

M~ = 938.20 Mev

M = 535.45 Mev 3

M4 = 139.63 MeV

In t h i s reaction, M1 + M2 < M3 C Mb and b$ > M4 so t h a t conditions

f o r the L a s t case shown i n Fig. 4 are s a t i s f i e d . Equations (35) and (36) y i e ld i n t h i s example

Ea = 153.35 MeV

Eth :- 151.50 MeV

- 13 -

1

This shows t h a t the angle r e l a t i o n for the pi-meson i s double-valued only for incident photon energies between these values of E and Eth. a

- 14 -

C A S E

MI -M, = M,-M,

IMPLIES M I =M, AND M,=M4

M 3- M, > M I - M

IMPLIES M,>M, AND M2>M,

M I - M, > M,-M,

IMPLIES MI > M, AND M4>M,

C H A R A C T E R I S T I C S OF THE L O R E N T Z

TRANSFORMATIONS FOR

M,+M, = M,+M,

ANGLE RELATIONSHIP AND JACOBIAN

I--+----

E o - Angle rel. sinyle-volueo for 0 1 1 E, . ;YO particles appear ot .

lab

Jacobian regular for 0 1 1 E, .

Eo - -

Angle rel. single-valued for a l l E, . Jacobian regular for all Eo .

Angle rel. for a1 I Eo

Jacobian c r certain 8'

Eo ~

doub

tico or a

e-va I ued

for I E o .

TR A NSFO R MAT I ON M A PP I NG

I1 I i i I 1

I

90" I800 @LAB

I I I \

Eo = CONSTANT

4 K

0 I I

O0 9oo I 80" @ L A B

Figure 2

- 15 -

.I .

CHARACTERISTICS OF THE LORENTZ

TRANSFORMATIONS FOR M , +M, > M, +M,

NGLE RELATIONSHIP AND JACOBIAN TRANSFORMATION MAPPING I

CASE

1 I

M, ' M 4

M I ' M 3

AND

M 3 > M l

I M P L I E S M P > M 4

A n g l e r e l . s i n q l e - v a l u e d f o r 0 1 I E, . J a c o b i a n c r i t i c a l f o r c e r t a i n a' f o r 0 1 1 Eo .

A n y l e r e l . s i n q l e - v a l u e d f o r a l l E, . J a c o b i a n c e r t a i n and r e p

I Y

c r i t i c a l f o r Q' i f Eb>E,>O o r i f Eo> Eb .

A n g l e r e l . s i n g l e - v a l u e d i f Eo>Eo>O an0 d o u b l e - v a l u e d i f EO>LO .

J o c o b i a n c r i t i c a l f o r c e r t a i n 8' f o r a l l Eo

I J

9 oo I 80" 0 O0

@ L A B _-

I

- 16 -

CHARACTERISTICS OF THE LORENTZ

TRANSFORMAT IONS FOR M , +M2 <M, +Me

C A S E ANGLE RELATIONSHIP AND JACOBIAN

I

TRANSFORMATION MAPPING

A n q l e r e l . d o u b l e - v a l u e d f o r a l l E$E,.

J a c o b i a n r e q u l a r e v e r y w h e r e f o r E,-E,.

I E o

O 0 Eth

A n g l e r e l . d o u b l e - v a l u e d f o r a l l E,j=Eth.

J a c o b i a n c r i t i c a l f o r c e r t a i n a' i f Eo?'Eb .

0 O0 L 9 oo I eoo

Figure 4

- 17 -

C H A R A C T E R I S T I C S O F T H E L O R E N T Z

T R A N S F O R M A T IONS FOR

M, = 0

C A S E

( DECAY IN

F L I G H T )

A N G L E R E L A T I O N S H I P AND JACOBIAN

I I Y Y

0 0 E,

E o - Angle rel. single-valued i f Ec’Eo=-O and double- valued i f E d E C .

Jacobian critical f o r # = g o o at Q I I values o f E o *

TRANSFORMATION M A P P I N G

Figure f3

- 18 -

ACKNOWLEDGEMENT

The author would l i k e t o thank W.K.H. Panofsky for a discussion

leading t o t h i s work, H. Clark f o r checking many of' t he formulae, and

R. Harris for his excel lent work with the figures.