E L E C T R O M A G N E T I C
V.
Considering
product
of
a
agree
dimensional
ordinary
algebraic equations
to
IYENGAR
four
two
F I E L D
we
D E C O M P O S A B L E
AND K .
KUPPUSWAMY
R i e m a n n i a n space
surfaces
obtain
w i t h the results
I N
with constant
e x p r e s s i o n s for
obtained by
with
RAO
signature
curvature, and
the e l e c t r o m a g n e t i c
BERTOTTI
S P A C E - T I M E
i- -j- +
u s i n g the
field.
( 1 9 5 9 ) o n the b a s i s o f
This
purely
as
the
RAINSCH
is
seen
geometric
considerations.
1. I f we consider space-time as topologically equivalent to the product o f t w o o r d i n a r y surfaces 2 + (coordinates x" , x') and 2— (coordinates x , x ) , a tensorfield o f arbitrary
type is said t o be decomposable ['] i f (a) its components w i t h mixed indices are zero and
(6) i f its components relative to 2+ depend only o n x° , x and those relative t o 2^ depend
only o n x , x . T h e fundamental tensor g^
must then be such that
2
3
l
2
B
v
0-1)
= V
where
For
pertains
to 2 +
source-free
f«v
t o 2— -
electromagnetism w i t h
M A X W E L L equations are
tensor
and
+
n o n - n u l l electromagnetic fields, the E I N S T E I N -
replaced b y a set o f conditions
involving the energy
momentum
where
0-2)
< > drf
=
T
m
where
.tit
1
\ia.,
V
*
H-V,
a
'
1
' Ï I I
1
|3«
fl\
and
T h e B I A N C H I identities give
(t.3)
G;
: v
- o
a n d then the R A I N I C H algebraic conditions P I for a real electromagnetic field are :
T h e M A X W E L L stress tensor has zero trace
(1.4)
G = 0.
T h e square o f the stress tensor is p r o p o r t i o n a l to the u n i t matrix
30
40
V.
(1.5)
IYENGAR AND K . KUPPUSWAMY R A O
G°- G\ = Q d
l
=
v
<5"
where
fi' =
i
G^.GJ>0.
The electromagnetic energy density is positive definite
(1.6)
G
> 0
0 0
and
(1.7)
a
„ — a,
= 0
where
where
ei T V ^ i l is taken skew-symmetric
1.
iJUUIIWlV i.n
. . all
« > . pairs
p t l l l J of
U l indices
UlUl^J w
H iilt lh. s
O ^ g g =
0 [ a a
T h e3 m a t r i x
tthe
l
eigenvalues (g, Q, - Q, - Q)
has
with
y a positive scalar defined i n
a locally Galilean frame by
(1.8)
a
2
= (H
2
—E )
-h (2 E • H )
2
a
where E a n d H are the electric and magnetic field strengths respectively.
I f the skew-symmetric tensor
*
dual to /
is defined by
defines
the
electromagnetic
field,
then
a
tensor
[]
8
I n the M I N K O W S K I f r a m e ' / * differs f r o m f b y the interchange of E ->- H and H
I f we n o w define a complex tensor co^
a.io)
E.
by
«v =/„ + </;
v
then the M A X W E L L equations can be w r i t t e n i n the differential f o r m as
(1.11)
< v = °
0-12)
% v ,
or i n the integral f o r m as
P
] = 0
[ ]
3
f
(1.13)
f
co^
</(>>,
x^)
= 0.
A l s o i f An a n d /«, be the n u l l eigen vectors o f the R I C C I tensor,then
(1.14)
where
/
l l V
=2e"
2
(X /
u
v
]
c o s « ^ ( - ^ )
1
/
2
V « P X */
[ t
v l
sina}
is given b y
E L E C T R O M A G N E T I C FIELD I N DECOMPOSABLE
2.
F o l l o w i n g B E R T O T T I [ ' ] w e use a polar frame o f reference and choose
(2.1)
dsi
= -
(2.2)
dst
= ( 1 - ^ )
(1 +
where ds+ an d /• refer to 2+ ; and ds(2.1)
41
SPACE-TIME
df
+ (l
+ ~ \
dy* +
1
dx*
* &
a
a n d s refer to 2— - If r
s
and s 4 -
M
we see that
a n d (2.2) reduce t o flat space. D e n o t i n g b y R j and 5^- the R I C C I tensors correspon{
d i n g t o 2+ and 2— respectively and the R I C C I tensor f o r the whole space b y G - we obtain
{
(2-4)
R„
(2.5)
R,
= 7i
n
3 8
= 0 = 5
1
(2-6)
5
(2.7)
»
i
0 0
/
z
^
-
1
= S
t
l
,
e
-4-11
^ « = —
I t follows that
(2.8)
G
i
= R
j
i
j
+S
i
j
and
(2.9)
G
= R + S .
, 3. F r o m (1.4) we have
(3.1)
G° + G\ + G\ + G\ = 0
a n d f r o m (1.5)
(3.2)
(GlY = (G\y ={G\y
= (G\f
a n d f r o m (3.1), (3.2) a n d (2.3) t o (2.7) i t follows that
(3.3)
G l - G \ I G8«<?1 ! Go =
W e n o w calculate the n u l l eigenvectors o f the R I C C I tensor. F o r ^ > 0, we have
(3.4)
(3.5)
B* k
v
= — AS:'*
£1**11 = 0
,
P
,
ii£ / = — A P ,
v
/u -
0
;
^
= — 1 .
F r o m (3.4) a n d (3.5) m a k i n g use o f (3.3) we get for k" ^ 0
=
K
n
(—g !g )
00
ll
= K, = 0
1/2
k
0
42
V . IYENGAR AND K . KUPPUSWAMY
RAO
and
I,
= /, = 0 .
Thus
(3.6)
K =[k
a
(3.7)
/«
4.
0
f
(-g
0
0
ig
[-'f2g00
=
L
1
)
k0,
U
2
/c ,0
,
0
(-iT
0 0
/^ )"
1
0],
'U^K,
2
0 , 0 ] .
F r o m (1.14) the nonvanishing components o f / are f o u n d to be
(4.1)
f
(4.2)
/
ai
=
Q'I
-
e V » sin a ,
2 3
cos a ,
2
which are constant.
F r o m (1.9) the nonvanishing components o f f*
are
v/
(4.3)
f*
(4.4)
'
= y*
oi
e
f*
sin « ,
= — e
1 / s
c
os«.
F r o m (1.10) we get
(4.5)
« 0 1 = e 1 /*
(4.6)
a ) „ = — i e 1 ' 3 e''" .
Since f r o m
(1.5)
(4.5) and (4.6) n o w become
(4.7)
'
«
0 1
= (eg)'/* i «
e
(4.8)
/(Gg)'/'ef
a
I t can easily be verified that these values o f OJ^ satisfy the M A X W E L L equations (1.11)
an d (1.12) as well as (1.13).
T h e electromagnetic
BERTOTTI
['].
field
given
by
(4.1)
and
(4.2)
is
the
same as
obtained
by
ELECTROMAGNETIC
FIELD
IN DECOMPOSABLE
SPACE-TIME
4
3
R E F E R E N C E S
[']
[ ]
B
BERTOTTI, B .
:
P h y s . R e v . , 116,
(1959).
RAINICH, G. V.
:
Trans.
WÎTTEN,
:
G r a v i t a t i o n : a n i n t r o d u c t i o n to c u r r e n t r e s e a r c h , J . W I L E Y
L .
A m . Math. S o c ,
27,
(1925).
OSMANIA U N I V E R S I T Y
(Manuscript
A N D SONS,
received
(1962).
29 th August
HYDERABAD (INDIA),
Ö Z E T
İmzası
h + 4-
şeklinde
olan dört
yüzeyin çarpımı gibi d ü ş ü n ü l e r e k ,
elektromanyetik
tarafından,
alan
sâdece
için
boyutlu
RATNICH'İU
ifadeler
elde
geometrik m ü l â h a z a l a r a
RIEMANN
cebirsel
ediliyor.
Bunların,
dayanarak
görülüyor.
uzayı,
sâbit
bağıntılarım
1959
elde e d i l e n
eğriliği hâiz
uygulamak
iki
suretiyle
yılında,
BERTOTTI
iradelere
uydukları
1968)