Revista Brasileira de Física, Vol. 18, n? 1, 1988
~e~resentations
of the LorentzScrew ~ransforrnation
MARCELO SAMUEL BERMAN
Depto. de Ciências Exatas, F. Fil. Ci. L., fundaçso Regional Joinville, 78920, Joinville, SC, Brasil
and
FERNANDO
DE MELLO GOMIDE
Depto. de Fisica, Instituto Tecnol6gico de Aeronáutica, 12225, Súo JacP dos Campos, SP, Brasil
Recebido em 22 de outubro de 1987
A Lorentz 4-screw i s a p a r t i c u l a r transformation,
d e f i n e d and
studied by J.L.Synge. We advance h i s t h w r y , by presenting two represent a t i o n s o f t h i s transformation, the s p i n o r i a l , and the t r i d i m e n s i o n a l (on
the surface o f the u n i t sphere).
Abstraot
1. INTRODUCTION
~ . ~ . ~ i ni s~ responsi
e '
b l e f o r the name and study o f the Lorentz
4-screw transformation. A Lorentz 4-screw i s a Lorentz t r a n s f o r m a t i o n
consisting of a r o t a t i o n i n a timel ike 2 - f l a t
T,
f o l lowed
by
ceded by) a r o t a t i o n i n a spacel i k e 2 - f l a t ?r*, the two 2 - f l a t
( o r pce-
r
and
r*
being orthogonal t o one another.
I f we c a l 1
(7, 5, 2, 2) a
u n i t orthogonal t e t r a d ,
and p o i n t i n g i n t o the f u t u r e , so t h a t
7 and .? l i e
2 t i me 1 i k e
i n IT* w h i l e (E
,; )
lie
i n r, we can w r i t e f o r the o v e r a l l transformation:
-+
-+
= I
cos 0
-+
-k
J" = - I s i n 8
+
+
-+
J s i n 0,
3
J cos 8,
3
2 cosh x + L
si = 2 s i n h x + L-+
=
x
cosh x
sinh
,
41. I)
We s h a l l choose
cosh
senh
x
x
= sec a
,
= tan a
.
I n our n o t a t i o n , the two angles o f r o t a t i o n a r e 0 and a. We s h a l l study
two types o f r e p r e s e n t a t i o n s :
Revista Brasileira de Flsica, Vol. 18,
n?
1, 1988
a
-
Spinorial representation;
b
-
Representation on the three-dimensional surface, or,
said
in other terms;
Representation by transformation of triads of nu11 rays.
2. SPINORIAL REPRESENTATION OF 4-SCREW
We may represent a spinor by two points 5 and V in the
plane ( ~ ~ n ~ e ' ) .I f 5 and V represent the initiat spinor, let 5"
Argand
and V"
represent the Lorentz 4-s<rrew transformat ion on thern. Synge showed that
the 8 transfarmation is given by
On the other hand,
the
x
rotation is given by
By combining both results, we obtain
v"
=
Relations (2.3) give the b s c r e w spinor representation.
3. REPRESENTATION OF 4-SCREW ON THE TRIDIMENSIONAL SURFACE
Any two triads of nu11 rays (M,N,P) and (M', N' , P') transform
in a manner that can be associated to the transformation of the tetrad
-+ + -+ -+
-b
(I,J,K,L). into
31, K'-+ , L'),
as they were introduced earl ier.
By
(I',
cutting spacetirpe across by the hyperplane X , / i = 1, the section of the
nu11 cone X,Xr = O is the sphere:
x:
+
x; + x ; =
1,
Revista Brasileira de Flsica, Vol. 18, n? 1, 1988
To each nu11 r a y t h e r e corresponds a p o i n t on i t s surface, so t h a t
(M,
of
the
N,
P) a r e p o i n t s , from now on, on t h i s t r i d i m e n s i o n a l surface
sphere.. L e t us choose the f o l l o w i n g p o i n t s
f=
(oI1*oI0)
3=
(-1 IO,O,o)
9
,
-P
K = (0,0,1 ,o)
-P
L = (OIO$O,
The
x
I
.
( o r a) r o t a t i o n i s then given by
This corresponds t o M and N f i x e d , w h i l e P runs around the meridiangreat
c i r c l e throuph
x,l) t e t r a d
the angle o . This corresponds t o the f o l i o w i n g (
f, 3.
transformations
k 1 = 1 ,
311= 3 ,
21 = s e c d +
= tan cd +
211
tan
sec
a2 ,
.
So t h i s i s a t i m e l i k e r o t a t i o n through an angle a.
+ +
Now, c a l c u l a t i n g a spacelike r o t a t i o n 0 , where t h e , t r i a d (M, N,
+
P) i s t r a n s f o n e d i n t o
Revista Brasileira d e Ffsica, Vol. 18, no 1, 1988
,
(O,O,-I,<)
$1
si E ( c o s e ,
s e n 8 , O,
i)
.
This corresponds t o a r o t a t i o n o f P through the angle 8 on
++++
XiX2.
The t e t r a d ( I , J , K , L )
the
plane
transforms i n t o
11= T c o s e
-?sino
?r
+?sino
+
,
+
J cose,
The obvious r e s u l t (we s h a l l demonstrate i t
ir!
next section) i s
t h a t a Lorentz 4-screw i s represented by the p o i n t s (M,N,P)
on the f o l -
lowing transformat ion:
a)
M
b)
and N do n o t move
moves on the ( ~ 1 x 3 )plane along an angle a, and thennwves
along the
(XiX2)
plane an a n g l e 0 ( t h e r e v e r s e o r d e r i s
a1 so possi b l e ) . The movement i s always on t h e
t r i d i men-
s i o n a l surface o f the u n i t sphere.
4. DEMONSTRATION OF THE RESULT STATED IN LAST SECTION
L e t us begin w i t h the 0 r o t a t i o n (spacelike).
nu1 1 v e c t o r s
3, 3,
The normal i zed
and
3 are
3
3' , 31, P1 nu 11
vectors are
The
normalized
Revista Brasileira de Flsica, Vol. 18, n? 1, 1988
-+
Now, l e t us obta i n the a r o t a t i o n (time1 i k e ) , The normal ized M1,
$li,
3"
nu1 l v e c t o r s are
=
$11
J;COS
The
(9"31,21,ifi')
(COSa cos 8, cos a, s i n a, i ) .
a
t e t r a d can be obtained by
the
following
r e l a t i o n s due t o syngel
whkre cPmmz i s the well-known permutation symbol. Taking eq.'(4.3)
(4.4)
we g e t a t l a s t
7
1
1
= (-sino,
c o s e , 0, 0)
5"
= (-cos 8 , - s i n
e,
,
0, O),
,
31=
(O,
21,=
(O, 0, t a n a , isec a) ,
O, s e c a , i t a n a)
into
Revista Brasileira de Ffsica, Vol. 18, n(>1, 1988
I f we compare r e l a t i o n s (1 , i ) and (3.2)
with
(4.5)
f i n d t h a t they a r e e q u i v a l e n t . So, what we constructed
was
we
a
shalt
4-screw
Lorentz t r a n s f o r m a t i o n indeed!
5. CONCLUDING REMARKS
As syngel remarked,
the s i x parameter group o f p r o j e c t i v e trans-
formations o f Euctidean 3-space i n t o i t s e l f
i s e q u i v a l e n t t o the general
Lorentz t r a n s f o r m a t i o n o f spacet ime. By c o n s i d e r i ng the t h r e e p o i n t s ( M,
N,
P)
on the t r i d i m e n s i o n a l surface o f the u n i t sphere, we were
able to
generate the Lorentz 4-screw, which i s then easi 1y represented. The spi nor
r e p r e s e n t a t i o n a l s o gives us another i n s i g h t o f a Lorentz 4-screw.
REFERENCE
I. J.L.
synge,
R e k t i v i t y : The SpeciaZ Theory, 2nd, ed.,
N o r t h Hol land,
1964, pages. 69-110.
Um parafuso- 4 de Lorentz é una tracsformação de Lorentz p a r t i c u l a r que f o i d e f i n i d a e estudada por J.L.Synge. Desenvolvemos sua t e o r i a
e x i b i n d o duas representações dessa transformação, a e s p i n o r i a l e a t r i d i
mensional (na s u p e r f í c i e da e s f e r a u n i t á r i a ) .
-