einstein 1937

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O N

G R A V I T A T I O N A L W A V E S

BY

A . E I N S T E I N a n d N . R O S E N .

ABSTRACT.

T h e r i g o r o u s s o l u t i o n f o r c y l i n d r i c a l g r a v i t a t i o n a l w a v e s is g i v e n . F o r t h e

c o n v e n i e n c e o f t h e r e a d e r t h e t h e o r y o f g r a v i t a t i o n a l w a v e s a n d t h e i r p r o d u c t io n ,

a l r e a d y k n o w n i n p r i n ci p l e , is g i v e n i n t h e f i rs t p a r t o f t h i s p a p e r . A f t e r e n c o u n -

t e r i n g r e l a t i o n s h i p s w h i c h c a s t d o u b t o n t h e e x i s t e n c e o f ri orous s o l u t i o n s f o r

u n d u l a t o r y g r a v i t a t i o n a l f ie ld s , w e i n v e s t i g a t e r i g o r o u s l y t h e c a s e o f c y l i n d ri c a l

g r a v i t a t i o n a l w a v e s . I t t u r n s o u t t h a t r i g o r o u s s o l u t i o n s e x i s t a n d t h a t t h e

p r o b l e m r e d u c e s t o t h e u s u a l c y l i n d r i c a l w a v e s i n e u c l i d e a n s p a c e .

I . A P P RO X IM A T E S O LU T IO N O F T H E P R O B L E M O F P L A N E W A V E S

A N D T H E P R O D U C T I O N O F G R A V I T AT IO N A L W A V E S .

I t is w e ll k n o w n t h a t t h e a p p r o x i m a t e m e t h o d o f i n te -

g r a t i o n o f t h e g r a v i t a t i o n a l e q u a t i o n s o f t h e g e n e r a l r e l a t iv i t y

t h e o r y l ea d s t o t h e ex i s te n c e o f g r a v i t a t i o n a l w a v e s . T h e

m e t h o d u s e d is a s fo l l ow s : W e s t a r t w i t h t h e e q u a t i o n s

- / ~ . v - - 1

~ g , . J ~ = - - T , . . I )

W e c o n s i d e r t h a t t h e g . , a r e re p l a c ed b y t h e e x p r e s s io n s

g . , = a . , + u . , , 2 )

w h e r e

~ , v = I

if /~ = v

= o i f ~ ¢ v ,

p r o v i d e d w e t a k e t h e t i m e c o 6 r d i n a t e i m a g i n a r y , a s w a s d otLo

b y M i n k o w s k i . I t is a s s u m e d t h a t t h e . a r e s m a l , i .e .

t h a t t h e g r a v i t a t i o n a l fie ld is w e a k . I n t h e e q u a t i o r s t h e

3`.. a n d t h e i r d e r i v a t i v e s w i ll o c c u r in v a r i o u s p o w e r s . I f t h e

3`.. a r e e v e r y w h e r e s u f f ic i e n tl y s m a l l c o m p a r e d t o u a i t y o n e

o b t a i n s a f i r s t - a p p r o x i m a t i o n s o l u t io n o f t h e e q u a t io n s b y

n e g l e c t i n g in ( I ) t h e h i g h e r p o w e r s o f t h e 3 `.. ( ~ n d t h e i r

d e r i v a t i v e s ) c o m p a r e d w i t h t h e l o w e r o n e s. I f o n e i n t r o d u c e s

f u r t h e r t h e ~ . , i n s t e a d o f t h e 3`.. b y t h e r e l a t io n s

1

VOL. 2 2 3 , N0 . I 3 3 3 - - 4 4 3

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  A. EI NST EI N AND N. ROSEN. [J. F . I .

then (I) assumes the form

G . . . . - G . . . . - ~ . . . . ~ + v . o ~ = - 2 T . , . 3 )

The specialization contained in (2) is conserved if one

performs an infinitesimal transformation on the coordinates:

X :

x ~ + ~ , 4 )

where the ~ are infinitely small but otherwise arbi tra ry

functions. One can therefore prescribe four of the ~ or

four condit ions which the ~ must satisfy besides the equa-

tions (3); this amounts to a specialization of the coordinate

sys tem chosen to describe the field. We choose the co-

ordinate system in the usual way by demanding that

~ o o = o . 5 )

It is readily verified tha t these four conditions are compatibl e

with the approximate gravitational equations provided the

divergence T~, ~ of T~ vanishes, which must be assumed

according to the special theory of relativity.

It turns out however that these conditions do not com-

plete ly fix the coordinate system. If , are solutions of (2)

and (5), then the / a f t e r a transformation of the type (4)

are also solutions, provided the ~ satisfy the conditions

F ~ + ~'~ - }a . (~ .o + ;~ o)],~ = o,

o r

e , ~ = o . 7 )

TI a ~:field can be made to vanish by the addition of terms

like chose in (6), i.e., by means of an infinitesimal transfor-

mation, then the gravitational field being described is only

an apparent field.

With reference to (2), the gravi tational equat ions for

empty sp~ee can be written in the form

° }

One obtains plane gravitational waves which move in the

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J a n ., I9 3 ~ f O N G R A V I T A T I O N A L W A V E S . 4 5

d i r e c t i o n o f t h e p o s i t i v e x l - a x i s b y t a k i n g t h e ~ ¢ , o f t h e f o r m

~ ( x l +

i x 4 ) = ~ x l -

t ) ) , w h e r e t h e s e ~ , m u s t f u r t h e r s a t i s f y

t h e c o n d i t i o n s

~11 + i ' ~ 4 = o , ]

% 1 + i~ 4 4 = 0 , [

~ : 1 J r - i ~ 2 4 = O , r ( 9 )

% 1 + i % 4 = o .

O n e C an a c c o r d i n g l y s u b d i v i d e t h e m o s t g e n e r a l ( p r o g r e s s i n g )

p l a n e g r a v i t a t i o n a l w a v e s i n t o t h r e e t y p e s :

( a ) p u r e l o n g i t u d i n a l w a v e s ,

o n l y 7 n , 7 ~ , ~y44 i m t r e n t f r o m z e r o ,

(b ) h a l f l o n g i t u d i n a l , h a l f t r a n s v e r s e w a v e s ,

o n l y ~21 a n d ~ 24 , o r o n l y ~ al a n d ~34 d i f f e r e n t f r o m z e r o ,

(c ) p u r e t r a n s v e r s e w a v e s ,

o n l y ~ 2 2, ~ '23, ~33 a r e d i f f e r e n t f r o m z e r o .

O n t h e b a s i s o f t h e p r e v i o u s r e m a r k s i t c a n n e x t b e s h o w n

t h a t e v e r y w a v e o f t y p e ( a ) o r o f t y p e (b ) i s a n a p p a r e n t f ie ld ,

t h a t i s , i t c a n b e o b t a i n e d b y a n i n f i n i t e s i m a l t r a n s f o r m a t i o n

f r o m t h e e u c l i d e a n f i e ld (~ ,~ = v , , = 0 ) .

W e c a r r y o u t t h e p r o o f in t h e e x a m p l e o f a w a v e o f t y p e

( a ) . A c c o r d i n g t o ( 9 ) o n e m u s t s e t , i f ~ i s a s u i t a b l e f u n c t i o n

o f t h e a r g u m e n t

x~ +

i X 4

h e n c e a l s o

Yll - -- -- (P , } /14 = i ( ~ , Y44 = - - (~ .

I f o n e n o w c h o o s e s ~ ' a n d ~4 ( w i t h ~'~ = ~3 = o ) s o t h a t

~ 1 = X ( X l _ [_ i X 4 ) , ~4 =

i x x l +

i : ~ 4 ) ,

t h e n o n e h a s

~1.1 + } ' ,1 = 2 x ' , ~1,4 + }4 ,1 = 2 ix ' , ~4 ,4 + }4 ,4 = - 2 x ' .

T h e s e a g r e e w i t h t h e v a l u e s g i v e n a b o v e f o r 7 . , 7 1 4 , ~44 i f o n e

c h o o s e s x ' - - ½ -~ . H e n c e i t i s s h o w n t h a t t h e s e w a v e s a r e

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  6

A . E I N S T E I N A N D N . R O S E N . [ J. F . I .

a p p a r e n t . A n a n a l o g o u s p r o o f c a n b e c a r r i e d o u t f o r t h e

w a v e s o f t y p e (b ).

F u r t h e r m o r e w e w i s h to s h o w t h a t a ls o t y p e (c) c o n t a i n s

a p p a r e n t f i e ld s , n a m e l y , t h o s e in w h i c h ~22 = ~aa ~ o , ~2a = O

T h e c o r r e s p o n d i n g % ~ a r e y u -- 744 o , a l l o t h e r s v a n i s h i n g .

S u c h a w a v e c a n b e o b t a i n e d b y t a k i n g } ' = x , ~4 = _ i x

i .e . b y a n in f in i te s i m a l t r a n s f o r m a t i o n f r o m t h e e u c l i d e a n

s p a ce . A c c o r d i n g l y t h e r e r e m a i n a s r ea l w a v e s o n l y t h e t w o

p u r e t r a n s v e r s e t y p e s , th e n o n - v a n i s h i n g c o m p o n e n t s o f

w h i c h a r e

72 2 ----- - - "~ /33 , ( C l )

o r

7 3 . c 2 )

I t f o ll ow s h o w e v e r f r o m t h e t r a n s f o r m a t i o n l aw f o r t e n s o r s

t h a t t h e s e t w o t y p e s c an b e t r a n s f o r m e d i n t o e a c h o t h e r b y

a s p a t i a l r o t a t i o n o f t h e c o 6 r d i n a t e s y s t e m a b o u t t h e x l- ax is

t h r o u g h t h e a n g le rr/4. T h e y r e p r e s e n t m e r e l y t h e d e c o m p o -

s i t io n i n t o c o m p o n e n t s o f t h e p u r e t r a n s v e r s e w a v e ( t h e o n l y

o n e w h i c h h a s a r e a l s i g n if i c a n c e ). T y p e ct is c h a r a c t e r i z e d

b y t h e f a c t t h a t i t s c o m p o n e n t s d o n o t c h a n g e u n d e r t h e

t r a n s f o r m a t i o n s

X 2

: X 2, ~'1 ~ ~'1 , A~3 : X 3 , X 4 ~ X 4 ,

O F

X 3 f : ~ 3 , X l = W 1, ~3~2 = ~ 2 ~ K 4 = 0C4,

i n c o n t r a s t t o c2, i .e . c l is s y m m e t r i c a l w i t h r e s p e c t t o t h e

x , - x 2 - p l a n e a n d t h e & - x a - p l a n e .

W e n o w i n v e s t i g a t e t h e g e n e r a t i o n o f w a v e s , a s i t f o l l o w s

f r o m t h e a p p r o x i m a t e ( li n ea r iz e d ) g r a v i t a t i o n a l e q u a t i o n s ,

T h e s y s t e m o f t h e e q u a t i o n s t o b e in t e g r a t e d is

. . . .

= - m )

L e t u s s u p p o s e t h a t a p h y s i c a l s y s t e m d e s c r i b e d b y T ~ is

f o u n d in t h e n e i g h b o r h o o d o f t h e o r ig i n o f c o 6 r d i n a t e s . T h e

~ ,-fie ld is t h e n d e t e r m i n e d m a t h e m a t i c a l l y in a s i m i l a r w a y t o

t h a t in w h i c h a n e l e c t r o m a g n e t i c f i e ld is d e t e r m i n e d t h r o u g h

a n e le c t r i c a l c u r r e n t s y s t e m . T h e u s u a l s o l u t i o n is t h e o n e

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Jan. , I937.] O N GRA VITATIONAL WA VES. 47

given by retarded potentials

- = _ _ '

( I i )

~' P 271- J f

Here r signifies the spatial distance of the point in question

from a volume-e lement, t =

x 4 / i ,

the time in question.

If one considers the material system as being in a volume

having dimensions small compared to r0, the distance of our

point from the origin, and also small compared to the wave-

lengths of the radiation produced, then r can be replaced by

r0, and one obtains

_ t t ~ T - ] ( t - ° ) d v '

r

~ - I - f T , , d y e ( t - r ° ) . (I2)

271 10

The ~,, are more and more closely approximated by a plane

wave the greate r one takes r0. If one chooses the point in

question in the neighborhood of the xl-axis, the wave normal

is parallel to the xt direction and only the components ~22,

N23, 733 correspond to an actual gravitational wave according

to the preceding. Th e corresponding integrals (I2) for a

system producing the wave and consisting of masses in motion

relative to one another have directly no simple significance.

We notice however tha t / 44 denotes the (negatively taken)

ene rgy densi ty which in the case of slow motion is practically

equal to the mass density in the sense of ordinary mechanics.

As will be shown, the above integrals can be expressed

thro ugh this quanti ty. This can be done because of the

existence of the ene rgy -mo men tum equations of the physical

syst em :

T,~, ~ = o. (13)

If one multiplies the second of these with x2 and the fourth

with ½x22 and integrates over the whole sys tem, one obtains

two integral relations, which on being combined yield

fr , l O ' f

2 0 x 4 2 x 2 ~ T 4 4 d v (13a)

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48 A. EINSTEIN AND N. ROSEN. [J. F. I.

Analogously one obtains

f T33d. I a2 f

0 x 4 2 x 3 2 T 4 4 d v

I 2

f T2 d 2o f xm T d. .

One sees from this that the time-derivatives of the moments

of inertia determine the emission of the gravitational waves

provided the whole method of application of the approxi-

mation-equat ions is really justified. In particu lar one also

sees that the case of waves symmetrical with respect to the

xl x2

and

x1 x~

planes could be realized by means of elastic

oscillations of a material system which has the same sym-

me tr y properties. For example one might have two equal

masses which are joined by an elastic spring and oscillate

toward each other in a direction parallel to the x3-axis.

From consideration of energy relationship it has been

concluded th at such a system in sending out gravitational

waves mus t send out energy which reacts by damping the

motion. Nevertheless one can think of the case of vibrat ion

free from damping if one imagines that besides the waves

emi tted by the system there is present a second Concentric

wave-field which is propagated inward and brings to the

syst em as much energy as the outgoing waves remove. This

leads to an undamped mechanical process which is imbedded

in a system of standing waves.

Mathematically this is connected with the following

considerations clearly pointed out in past years by Ritz and

Tetrode. The integration of the wave-equation

[B~ = - 4~rp

by the

retarded

potential

f

~ = r

is mat hem ati cal ly not the only possibility. One can also do

it with

g~ r

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J an ., 1937.] O N G R A V I T A T I O N A L W A V E S . 9

i.e. by means of the adva nced potential , or by a mixtur e

of the two, for example,

I ~ E P ] ( t + r ) - Ir - E P ] ( t - - r )

d r

g~ -= 2 J r

The last possibili ty corresponds to the case without damping,

in which a standing wave is present.

It is to be remarked th at one can thin k of waves generated

as described above which approximate plane waves as closely

as desired. One can obt ain the m, for example, thr ou gh a

limit-process by considering the wave-source to be removed

further and further from the point in quest ion and at the

same time the oscillating moment of inertia of the former

increased in proportion.

I I . R IGOROUS SO LUTION FOR CYLINDRICAL WA VES.

We choose the coordinates Xl, x2 in the meridian plane in

such a wa y t ha t Xx = o is the axis of ro tat io n an d x2 runs fro m

o to infinit y. Le t xa be an angle coordin ate specifyin g the

position of the me rid ian plane. Also, let the field be sym-

metrical ab ou t ever y plane x2 = const, and ab out ever y

meridian plane. The required sy mm et ry leads to the vanish-

ing of all components g,, which contain one and only one

index z; the same holds for the index 3- In such a grav ita-

tional field only

g l l g 2 2 g 3 3 g 4 4 g14

can be differen t from zero. For convenience we now take all

the cobrd inate s real. One can fur the r tr ans for m the coOrdi-

nat es xl, x4 so th at two condi tio ns are satisfied. As such we

take

g14 = O , [

g l l = - - g 44 . [ I 4 )

It can be easily shown that this can be done without intro-

ducing a ny singularities.

We now write

- - g l l = g 44 = A , ]

- g : ~ = C j

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5o A . EI N ST EI N AND N . ROSEN . [J. F . I .

w h e r e A , B , C > o .

l a t e s t h a t

I n t e r m s o f t h e s e q u a n t i t i e s o n e c a l c u -

( I ) B44 C44 I [ B 4 2 C42

2 .Rl l 2 g l ,R = ~ - - [- ~ 2 ~ - [-

2 A R 2 2 I ) A 44

B 2 g 22 R = - £

2 A ( i ) A44

C R a a 2 g a aR = A

- - B - - ~ + A k B +

B1C1 A I B ,

c l h ]

+ N K + A , B + - ~ ] J '

C44 A 11 C ll

-t- _ _

C A C

i [ C12 C42

+ ~ [ c~ - c - ;

2A19 , 2A 4 2 ]

+ A 2 ~ ; ]

B44 A n B n

- - +

B A B

i [ 2A12 2A 4 2

+ 2 [ A 2 A 2

B I 2 B42 ]

+ B z B 2

I [

B12 C12

2 [ - ~ - + ~

A I I

- - - A , B - d ]

< ]

+ A k B + C ] .]

I [ B I B , C i C 4

- - ~ c--

A 4 ( B 1

A I B4

c l

+ A , B + - g ] ] '

2 ( . R 4 4 - - -

~ B n C n

e . R } = ~

B1C~

B C

B 4 G

N K

B,4 G 4

2 R,4 = B - + C - -

( I 6 )

w h e r e s u b s c r i p t s i n t h e r i g h t - h a n d m e m b e r s d e n o t e l i f t e r-

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J a n . , 1 9 3 7 . ] O N G R A V I T A T I O N A L W A V E S .

51

e u t i a t i o n . I f w e t a k e a s f i e ld e q u a t i o n s t h e s e e x p r e s s i o n s s e t

e q u a l t o z er o , r ep l a c e th e s e c o n d a n d t h i r d b y t h e i r s u m a n d

d i ff e re n c e , a n d i n t r o d u c e a s n e w v a r i a b l e s

a = l og A , ]

½ og B/C),p (15a)

3 ½1og(BC) , J

w e g e t

1 - - 2 o 3 - 1 = o , ( 1 7 )3 " 4 4 A I - ~ [ ~ 4 - ~ - 3 3 " 4 2 " @ ~ t 2 3 "1 2 - - 2 0L 1 3" 1

2 (O ~ 1 1 - - O 4 4 ) - IT . 2 3 "1 1 - - 2 3 "4 4 - [ - [ ~ 1 2 - ~ - 3 " 1 2 - - ~ 4 2 - - 3 "4 2 = O , ( 1 8 )

~ 1 1

- - ~ 4 4 - t - [ ~ 1 3 1 - - ~ 4 ' ~ 4 - ] = O , ( I 9 )

2 3 " l l - [ - 1 [ ~ 1 2 - t - 3 3 "1 2 - t - ~ 4 2 - - 3 " 42 - - 2 G~ l'Y 1 - - 2 0 / 4 3 " 4 ] = O , ( 2 0 )

2 3 "1 4 - t - [ ~ i ~ 4 - t - 3 " 13 " 4 - - 2 o~ 13 "4 - - 2 0 ~ 4 3 " 1 ] = O . ( 2 1 )

T h e f ir s t a n d f o u r t h e q u a t i o n s o f t h is g r o u p g i v e

3 "1 I - - 3 " 4 4 - J r - ( 3" 12 - - 3 "4 2) = O . ( 2 2 )

T h e s u b s t i t u t i o n

3 = lo g 0 , 0 = BC)~, 2 3 )

l e a d s t o t h e w a v e e q u a t i o n

0 "1 1 - - 0 "4 4 = O , ( 2 4 )

w h i c h h a s t h e s o l u t i o n

0 = f ( X l A t- X 4 ) -[ - g ( X l - -

X 4 ) , ( 2 5 )

w h e r e f a n d g a r e a r b i t r a r y f u n c t io n s . E q . (1 8) r e d u c e s t o

O g l l _ _ O ~ 4 4 . 71 - 1 ( ~ 1 2 _ _ ~ 4 2 . .. [ - 3 " 4 2 _ _ 3 " 1 2 ) = O .

I8a)

E q u a t i o n ( I 7 ) t h e n s h o w s t h a t 3 c a n n o t v a n i s h e v e r y w h e r e .

W e m u s t n o w s e e w h e t h e r t h e r e e x i s t u n d u l a t o r y p r o c e s s e s

f o r w h i c h 3 d o e s n o t v a n is h . W e n o t e t h a t s u c h a n u n d u l a -

t o r y p r o c e s s is r e p r e s e n t e d , in t h e f i r s t a p p r o x i m a t i o n , b y a n

u n d u l a t o r y fl, t h a t is b y a 5 - f u n c t i o n w h i c h , s o f a r a s i t s

d e p e n d e n c e o n x , a n d a l so i ts d e p e n d e n c e o n x 4 is c o n c e r n e d ,

p o s s es s es m a x i m a a n d m i n i m a ; w e m u s t e x p e c t t h i s a ls o fo r

a r i g o r o u s s o l u t i o n . W e k n o w a b o u t 3 t h a t e~ = ~ s a t is f ie s

t h e w a v e e q u a t i o n (2 4) a n d t h e r e f o r e t a k e s t h e f o rm ( 25 ).

F r o m th is , h o w e v e r , t h e u n d u l a t o r y n a t u r e o f t h is q u a n t i t y

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5 2 A E I N S T E I N A N D N R O S E N [J F I

d o e s n o t n e c e s s a r i l y f o l l o w . W e s h al l i n f a c t s h o w t h a t q,

c a n h a v e n o m i n i m a .

S u c h a m i n i m u m w o u l d i m p l y t h a t t h e f u n c t i o n s f a n d g

i n ( 25 ) h a v e m i n i m a . A t a p o i n t (X l , x4) w h e r e t h i s w e r e t h e

c a s e w e s h o u l d h a v e ~' = ~'4 = o , ~ 'n = > o , ~'44 ~ o . B u t

b y ( I 7 ) a n d ( 2 o) t h i s is i m p o s s i b l e . T h e r e f o r e ~, h a s n o

m i n i m a , t h a t i s i t is n o t u n d u l a t o r y b u t b e h a v e s , a t l e a s t in a

r e g io n o f s p a c e a r b i t r a r i l y e x t e n d e d in o n e d i r e c t io n , m o n o -

t o n i c a l ly . W e s h a ll n o w c o n s i d e r s u c h a r e g io n o f s p a c e .

I t i s u s e f u l to s e e w h a t s o r t o f t r a n s f o r m a t i o n s o f x l a n d x4

l e a v e o u r s y s t e m o f e q u a t i o n s ( I4 ) i n v a r i a n t . F o r t h is

i n v a r i a n c e i t i s n e c e s s a r y a n d s u f f i c ie n t t h a t t h e t r a n s f o r m a -

t io n s a t i s f y t h e e q u a t i o n s

021 024 ]

09;1 OX4

02, 024

Ox4 Oxl

(26 )

T h u s w e m a y a r b i t r a r i l y c h o o s e 2 1(xl, x4) t o s a t i s f y t h e

e q u a t i o n

0 2 2 1 0 2 2 1

- - o (26 a )

Oxl 2 Ox4 ~-

a n d t h e n ( 26 ) w i ll d e t e r m i n e t h e c o r r e s p o n d i n g 24. S i n c e ev is

i n v a r i a n t u n d e r t h i s t r a n s f o r m a t i o n a n d a l so s a ti s f ie s t h e w a v e

e q u a t i o n , t h e r e e x i s t s a t r a n s f o r m a t i o n w h e r e 21 is r e s p e c t i v e l y

e q u a l o r p r o p o r t i o n a l to e~. I n t h e

n e w

c o 6 r d i n a te s y s t e m w e

h a v e

e~ = axl

o r ~, = lo g a -}- lo g Xl ( 2 7 )

I f w e i n s e r t t h i s e x p r e s s i o n f o r ~, i n ( I 7 ) - ( 2 7 ) t h e e q u a t i o n s

r e d u c e t o t h e e q u i v a l e n t s y s t e m

~ 1 1 ~ 4 4 + i =

1 o , 2 s )

X l

a n d

I

B 3

al = ½x1(¢tl2 + /342) 2 x l (2 9)

~ 4 = x 1 ~ 1 ~ 4 . 3 0 )

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J an ., I937.1 O N G R A V I T A T I O N A L W A V E S . 5 3

E q u a t i o n (2 8) is t h e e q u a t i o n f o r c y l i n d r ic a l w a v e s i n a t h r e e -

d i m e n s i o n a l s p a c e , i f x l d e n o t e s t h e d i s t a n c e f r o m t h e a x i s

o f r o t a t i o n . T h e e q u a t i o n s (2 9) a n d ( 3o ) d e t e r m i n e , f o r

g i v e n /3 , t h e f u n c t i o n a u p t o a n ( a r b i t r a r y ) a d d i t i v e c o n s t a n t ,

w h i l e , b y ( 2 7 ), r i s a l r e a d y d e t e r m i n e d .

I n o r d e r t h a t t h e w a v e s m a y b e r e g a r d e d a s w a v e s i n a

e u c l id e a n s p a c e t h e se e q u a t i o n s m u s t b e sa t i s f i ed b y t h e

e u c l i d e a n s p a c e w h e n t h e f ie ld is i n d e p e n d e n t o f x4. T h i s

f ie ld is r e p r e s e n t e d b y

A = I ; B = I ; C = x l 2,

if w e d e n o t e t h e a n g l e a b o u t t h e ax i s o f r o t a t i o n b y x3. T h e s e

r e l a t i o n s c o r r e s p o n d t o

a = o , 3 = - - l o g x l , ~, = l og x l ,

a n d f r o m t h i s w e se e t h a t t h e e q u a t i o n s ( 2 7 ) - ( 3 o ) a r e in f a c t

s a t i s f i e d .

W e h a v e s t i l l t o i n v e s t i g a t e w h e t h e r

st tion ry

w a v e s

e x is t , t h a t i s w a v e s w h i c h a r e p u r e l y p e r i o d ic in t h e t i m e .

F o r 3 i t is a t o n c e c le a r t h a t s u c h s o l u t i o n s ex i s t. A l -

t h o u g h i t i s n o t e s s e n t ia l , w e sh a ll n o w c o n s i d e r t h e c a s e w h e r e

t h e v a r i a t i o n o f 3 w i t h t i m e is s i n u s o i d a l . H e r e 3 h a s t h e

f o r m

3 = X0 + X I s i n ~x4 + X2 co s ~x4 ,

w h e r e X0 , X 1, X 2 a r e f u n c t i o n s o f x l a l o n e . F r o m ( 30 ) i t t h e n

f o l lo w s t h a t a i s p e r i o d i c i f a n d o n l y i f t h e i n t e g r a l

f / 3 1 / ~ 4 d x 4

t a k e n o v e r a w h o l e n u m b e r o f p e r io d s v a n is h e s .

I n t h e c a s e o f a s t a t i o n a r y o s c i l l a t i o n , w h i c h i s r e p r e s e n t e d

b y

= X 0 + X I

sin ~x4,

t h i s c o n d i t i o n is a c t u a l l y f u lf il le d s i n c e

f 3 1 ~ 4 d x 4 = f ( X o t A v X I s i n

¢ o x 4 c o X 1 c o s w x 4 d x 4 ~-- o .

O n t h e o t h e r h a n d , in t h e g e n e r a l c as e , w h i c h i n c l u d e s t h e c a s e

o f p r o g r e s s i v e w a v e s , w e o b t a i n f o r t h i s i n t e g r a l t h e v a l u e

l r X - ~ p

\ 1-~-2 - - X 2 X I ) o ~ T ,

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5 A E I N S T E I N A N D N R O S E N [ J F I

where T is the interval of t ime over which the integral is

take n. Thi s does not vanish, in general. At distan ces xl

from xi = o great compared with the wave-lengths, a pro-

gressive wave can be represented with good approximation in

a domain containing many waves by

= X0 -t- a sin ~0 x4 - - X l ,

where a is a co ns ta nt which, to be sure, is a su bs ti tu te for a

fu nct io n de pe nd in g we ak ly on xl). In this case XI = a cos ~0xi,

X2 = -- a sin

~OXl

so th at the integral can be approxim ately)

represented by -- ½a~o2T and thus cannot vanish and always

has the same sign. Progressive wave s therefo re produc e a

secular change in the metric.

This is re la ted to the fact that the waves t ra nsp or t energy,

which is bound up with a systematic change in t ime of a

gravitating mass localized in the axis x = o.

N o t e . T h e s e c o n d p a r t o f t h i s p a p e r w a s c o n s i d e r a b l y a l t e r e d b y m e a f t e r

t h e d e p a r t u r e o f M r R o s e n f o r R u s s i a si n c e w e h a d o r i g i n a l l y i n t e r p r e t e d o u r

f o r m u l a r e s u l t s e r r o n e o u s ly I w i s h t o t h a n k m y c o l le a g u e P r o f e ss o r R o b e r t s o n

f o r h i s f r i e n d l y a s s i s t a n c e i n t h e c l a r i f i c a t i o n o f t h e o r i g i n a l e r r o r I t h a n k a ls o

M r H o f f m a n n f o r k i n d a s s i s t a n c e i n t r a n s l a t i o n

A EINSTEIN