IL NUOVO CIMENT0
VoL. 2B, N. 1
11 Marzo 1971
Gravitational Red-Shift (*). D. K. l~oss Iowa State University - Ames, Ia.
(ricevuto il 17 Luglio 1970)
Summary. - - The usual expression for the gravitational red-shift z is calculated exactly in both the standard and isotropie forms of the Schwarzsehild metric. The differing results are found to be meaningless and metric dependent until they are put in terms of quantities directly observable from Earth. The importance of expressing physical quantities in terms of observables is stressed.
1. -
Introduction.
l~ecent w o r k b y ARP (1) on t h e pairing of quasi-stellar radiosources (QSR's) a n d on their a p p a r e n t association with galaxies strongly suggests t h a t the large red-shifts of these objects do n o t arise f r o m t h e i r g r e a t distance as would be the case if t h e y obeyed t h e t t u b b l e r e d - s h i f t - d i s t a n c e relation. O t h e r indications t h a t the Q S R ' s m a y be n o n r a n d o m l y distributed and relatively local objects are found in previous p a p e r s b y ARv (2), SHARP {3), a n d WAGO~E~ (4). STRITTMATTER, FAULKNEI~ and WALMSLEY (5) found t h a t QSR's with red-shift z ~ 1 . 5 are c o n c e n t r a t e d t o w a r d the Galactic poles in a w a y t h a t is difficult to u n d e r s t a n d if t h e y are at cosmological distances. L a r g e red-shifts f r o m r e l a t i v e l y local objects can arise f r o m relativistic (*) To speed up pubheation, the author of this paper has agreed to not receive the proofs for correction. (1) H. ARI~: Astron. Journ., 75, 1 (1970). (2) H. Am': Science, 151, 1214 (1966); Astrophys. Journ., 148, 321 (1967); Astro]izika, 4, 59 (1968). (a) J. R. SHAY: M.S. thesis, San Diego State College (1967). (4) R. V. WAGONER.: Nature, 214, 766 (1967). (5) •. STRITTI~IATTER,J. FAULKNER a n d M. WALMSLEY: Nature, 212, 1441 (1966). 55
56
1). K. ROSS
velocities and f r o m strong g r a v i t a t i o n a l fields. I n the first case we m i g h t also e x p e c t blue-shifts, which h a v e not b e e n o b s e r v e d or at least recognized. The lack of a blue-shift could be e x p l a i n e d if quasars were ejected f r o m our own g a l a x y (6) or f r o m n e a r b y radiogalaxies (7). E v i d e n c e t h a t g r a v i t a t i o n a l fields m a y p l a y a role is suggested b y the four QSR's associated with 1NGC 520. ARP (1) found t h a t t h e four Q S R ' s e x t e n d on a s t r a i g h t line f r o m NGC 520 a n d h a v e red-shifts of 0.77, 0.67, 2.11 a n d 0.72 f r o m the G a l a x y outward. The large red-shift of 2.11 lying b e t w e e n two r e l a t i v e l y low red-shifts m a y r e p r e s e n t a strong g r a v i t a t i o n a l contribution s u p e r i m p o s e d on a general velocity background. I n view of the possibility t h a t we are seeing v e r y large gravitational red-shifts in astronomical bodies, the p r e s e n t p a p e r examines the expressions being used for the g r a v i t a t i o n a l red-shift. This analysis is i n d e p e n d e n t of w h e t h e r or not quasars h a v e g r a v i t a t i o n a l red-shifts.
2. - Calculation o f the gravitational red-shift.
L e t us derive the usual expression for t h e g r a v i t a t i o n a l red-shift. We will assume in w h a t follows t h a t t h e quasar is not m o v i n g with respect to the observer. The s t a n d a r d f o r m for the e x t e r i o r Schwarzschild line e l e m e n t (8) in spherical co-ordinates is (1)
ds 2
(1 --2m/rs)C ~ dt ~ - (1 --2m/rs)-ldr~--r2s(dO ~ + sin20 d ~ ) ,
where m _ K M / c ~. M is the mass of t h e central b o d y a n d K is t h e g r a v i t a t i o n a l constant. The subscript S on various quantities denotes the Schwarzschild line e l e m e n t in all t h a t follows. P r o p e r - t i m e intervals at the surface of the b o d y are r e l a t e d to co-ordinate t i m e intervals b y (2)
c dv R -- (1 - - 2m/Rs)89 dt,
where R s is t h e radius of the quasar. (3)
cdvr
F o r a v e r y d i s t a n t observer we h a v e : e dr,
where we h a v e neglected 2m/ds, with d s the distance to t h e quasar, c o m p a r e d to 1. This is a v e r y good a p p r o x i m a t i o n since 2m/d s ~ 10 -20 for 1 solar m a s s at a distance of 10 Mpc. T h r o u g h o u t this p a p e r we will neglect this q u a n t i t y . We shall c a r r y out t h e calculations e x a c t l y however, in t e r m s of m / R s. N o w the n u m b e r of waves e m i t t e d in t h e i n t e r v a l Av R b y the quasar is y e a r R.
(6) J. TERRELL: Science, 145, 918 (1964); Astrophys. Journ., 147, 827 (1967). (7) F. HOYLE and G. BURBIDGE: Astrophys. Journ., 144, 534 (1966). (0) K. SCHWARZSCIIILD:Sitzber. Preuss. Akad. Wiss. Berlin, 189 (1916).
GRAVITATIONAL RED-SHIFT
57
E q u a t i n g this with the n u m b e r of waves received at the observer, ~ A ~ , t h e n gives (4)
Vo/V = h v . / h v ~ = (1 - - 2 m / R s ) -'~ ,
where we s u b s t i t u t e d (2) a n d (3). I n t e r m s of w a v e l e n g t h (9)
(5)
).I,L
=
(1 -
2 m / i t ~ ) -~-
and
(6)
zs
(]L
-
2o)/~,~ = (1 - - 2 m / R s ) -~- - -
-
1.
The Schwarzschild line e l e m e n t can also be p u t into the isotropic f o r m (lo)
(7)
ds'=
(
)(
)
m"~ -~ c2dt 2 1 - - ~ rm ~ '2 1-]-~r~r
(
1 - ] - - ~mr ~
( d r ~ + r ~ d 0 2 + r~ sin2Od~ ~)
The subscript i on various quantities denotes the isotropic line element. Carrying out the red-shift calculation a b o v e t h e n gives the result
(8)
zi
=
(mtt~i
1--
These results in the two metrics are quite different for large re~It. The reason for this discrepancy is t h a t b o t h (6) and (8) contain R which is not a measurable q u a n t i t y a n d which is co-ordinate s y s t e m dependent. N e i t h e r (6) nor (8) is meaningful as it stands. N e i t h e r result is the (~correct,> expression for z. P u t t i n g in the co-ordinate t r a n s f o r m a t i o n which relates r s and r~ will c o n v e r t (6) into (8) or vice versa b u t accomplishes nothing since neither (6) nor (8) is a useful c o - o r d i n a t e - s y s t e m - i n d e p e n d e n t result. B o t h results m u s t be p u t in t e r m s of quantities which are directly measurable f r o m t h e o b s e r v e r ' s position (~).
3. - Results in t e r m s o f observable quantities.
We need an expression for the radius of the quasar R in t e r m s of measurable quantities. The m o s t obvious q u a n t i t y to use is the observed angular diameter. The actual m e a s u r e m e n t of the angular d i a m e t e r m a y t u r n out to
(9) G. BURBIDGE and M. Bt:RBmCE: Quasi-Stellar Objects (San Francisco, 1967). (lo) ~. ADLER, M. BAZIN •nd ~{. SCHIFFER: Introduction to General Relativity (New York, 1965). (11) D. K. Ross and L. I. SCHIFF: Phys. Rev., 141, 1215 (1966).
58
D.K.
ROSS
be impossible if t~/d is too small, b u t t h e e x p e r i m e n t is a t least c o n c e p t u a l l y possible. W e will first derive an expression r e l a t i n g R s a n d t h e a n g u l a r radius in t h e s t a n d a r d f o r m of t h e m e t r i c a n d t h e n t u r n to t h e isotropie form. I n o r d e r to h a v e a well-defind s i t u a t i o n to analyse, we will consider a r a y of light w h i c h leaves t h e q u a s a r at r a d i u s R s e x a c t l y h o r i z o n t a l l y (see Fig. 1). P
quasar
d
0
Fig. 1. The path of a light ray from the edge of the quasar of radius R to the observer at point 0 is shown. The light ray leaves point P parallel to the y-axis. The distance d is very greatly shortened in the diagram. The co-ordinates (r, ~) of an arbitrary point Q on the trajectory of the light ray are also shown. This trajectory PQO is not a straight line because space is not flat near the quasar. R, r s and d can have either S or i subscripts denoting either Schwarzschild or isotropic co-ordinates.
W e t h e n w a n t to calculate t h e angle a at w h i c h t h e light r a y crosses t h e y-axis. This angle ~ "will be a c o n c e p t u a l m e a s u r e of t h e a n g u l a r radius. Since we act u a l l y o b s e r v e light or radio w a v e s c o m i n g f r o m t h e edge of a q u a s a r in m a k i n g a n a n g u l a r - d i a m e t e r m e a s u r e m e n t , it is m o r e sensible to use light r a y s t h a n to do p u r e g e o m e t r y . W e can derive t h e p a t h followed b y t h e light r a y b y l e t t i n g t h e light follow a null geodesic. T h e n ds 2 ~ 0 in (1) a n d dividing t h r o u g h b y dq: gives
(9)
"2
(1 - - 2 m / r s ) i ~ - - (1 - - 2 m / r s ) - ' ~ - - r s ~
= O,
w h e r e we let i . : dt/dq etc., a n d sin0 = 1. q is an a r b i t r a r y p a r a m e t e r . To derive t h e e q u a t i o n s of m o t i o n for t a n d % we consider t h e v a r i a t i o n a l p r o b l e m (lo)
(10)
~"
T h e i n t e g r a n d is g i v e n b y (9). (11)
dx ~ dx~
T h e t e q u a t i o n is
d [(1 - - 2 m / r s ) 2 t ] = O, d--q
a n d t h e ~ e q u a t i o n is
(lz)
d
dq [2~r~] = 0 .
GRAVITATIONAL R E D - S H I F T
59
W e will use (9) i n s t e a d of t h e r e q u a t i o n since it is m o r e c o n v e n i e n t . :Now (11) a n d (12) i m p l y (13)
(1
2m/rs)t
-- A
and (14)
~r~ = B ,
w h e r e A a n d B are c o n s t a n t s . arrangement,
(15)
U s i n g (13) a n d (14) in (9) t h e n gives, a f t e r re-
A 2 _ rs.3__ ( 1 - - 2 m / r s ) ( B 2 y s ) = 0
N o w let o~s ~ m / r s be a dimensionless variable. Also let r's _ drs/d~, etc. The angle ~ is defined in Fig. 1. I n t e r m s of o~s a n d ~, (15) can be w r i t t e n as (16)
- - 2O9s) = 0 .
A S - - Ors 2 B 2 / m 2 - - (CO~sB 2 / m 2 ) ( 1
T a k i n g t h e d e r i v a t i v e w i t h r e s p e c t to ~0 a n d d i v i d i n g o u t u n w a n t e d q u a n t i t i e s t h e n gives (17)
g
w s + eos : 3w~.
:Now far f r o m t h e source w h e r e t h e space a s y m p t o t i c a l l y a p p r o a c h e s a flat space, we h a v e t h a t (18)
drs i
ds ctg ~ .
d ~ I~ - 90~
T h u s in o r d e r to find a we n e e d o n l y t h e slope of t h e light r a y t r a j e c t o r y at t h e o b s e r v e r a n d n o t t h e full details of t h e t r a j e c t o r y . T h u s we n e e d o n l y a first i n t e g r a l of (17). To i n t e g r a t e (17) let us i n t r o d u c e
(19)
Ps
do) s
|
T h e n (17) b e c o m e s
(20)
Ps dPs + d~ S
3 ~s
~
2 ~s,
which i n t e g r a t e s to (21) w h e r e C~ is a c o n s t a n t .
To find C~ we n o t e t h a t t h e slope d r s / d ~ l ~ _ o = 0 w h e n
60
D.K.
e) s - = m / R s
Thus P s = O
at this point, a n d we h a v e
C1
(22)
Ross
- - 2 ( m / ~ s ) 3 -~- ( m / R s )
2.
A t the position of the o b s e r v e r we h a v e t h e n f r o m (21) (23)
P ~ I r -90 ~ ~ - - [ ( m / R ~ ) 2 - -
where we neglect t e r m s we h a v e
(m/ds)~-~(m/l~s)~.
2(m/Rs)~] ~ ,
Also since P s
(--m/r~)(drs/d~),
drSd~ ~= 9oo : (ds/m)[(m/Rs)2--2(m/Rs)3]
(24)
and using (18) t h e n gives finally (25)
tg a = (Rs/ds)(1 - - 2m/Rs)-~.
This expresses the observable R s a n d d s. I f we c a r r y out this same t h a t t h e differential equation now in isotropie co-ordinates, (26)
w~ ~ ~oi
angle ~ in t e r m s of the co-ordinate distances calculation using the isotropie metric we find analogous to (17) for (o~ __ m/r~, where r~ is is
(02i( 2 --~i~2)( / 1-
w~/4)-1 -~ e)i'2 (2 - - w,/2)(1-- w~/4) -1
The details are o m i t t e d for b r e v i t y . gives the final result (27)
Solving (26) b y introducing P as before
(1+
;.)
Now the distance of the quasar d a p p e a r i n g in (25) a n d (27) can be p u t into t e r m s of (( m e a s u r a b l e ~>p a r a m e t e r s v e r y easily. Assume t h a t the quasar has a given l u m i n o s i t y L. This is m e a s u r a b l e or a t least calculable conceptually. I t is certainly co-ordinate s y s t e m i n d e p e n d e n t . L e t us t h e n m e a s u r e the flux F f r o m the quasar at a distance d away, where d is v e r y large a n d corresponds to the o b s e r v e r ' s position on E a r t h . Then (28)
F =
L/As,
where A~ is the a r e a of a sphere of radius d. I n the s t a n d a r d m e t r i c we h a v e (29)
A d = 4z~d~,
61
GRAVITATIONAL RED-SHIFT
b u t in the isotropic metric 4z d~ since we assume mid << 1 as before. Thus d is the same in b o t h metrics a n d is g i v e n b y (28) with (29) s u b s t i t u t e d . We can consider d to be m e a s u r a b l e conceptually a n d d ~ = d s. Referring to (25) a n d (27), we see t h e n t h a t these implicitly give ~ and R s in t e r m s of the m e a s u r a b l e quantities d a n d a in the two different metrics. The quantities d a n d a are the same in b o t h metrics whereas the ~ ' s are clearly different. W e would now like to solve (25) and (27) f o r / ~ a n d t h e n s u b s t i t u t e these values of R into t h e respective equations for the red-shift (6) and (8). U n f o r t u n a t e l y this is not easily done for (25) a n d (27) are b o t h cubic algebraic equations for R. Since we are not i n t e r e s t e d in the detailed answer, let us instead consider t h e e q u a t i o n satisfied b y the q u a n t i t y y _ ( 1 - - 2 m / R s ) -89 in the s t a n d a r d metric. This is (31)
73
q
~
q
f r o m (25), where q ~ d tg ~ a n d is a m e a s u r a b l e q u a n t i t y . I n the isotropic m e t r i c f r o m (27), we find t h a t the q u a n t i t y fl ~ (1 +m/2R~)(1--m/2R~) -1 satisfies t h e equation (32)
q f12
q
which is identical with (31). Now, however, f r o m (6) we notice t h a t in the s t a n d a r d metric we can write (33)
z s -- y - - 1 .
Also f r o m (8) we find in the isotropic m e t r i c t h a t (34)
z~ -- fl - - 1 .
Since we h a v e shown t h a t fl and y obey identical equations w h e n w r i t t e n in t e r m s of m e a s u r a b l e quantities, we h a v e t h u s shown t h a t the expressions for the red-shift z in the two metrics (6) and (8) also reduce to the same result when the radius of the quasar R is e x p r e s s e d in t e r m s of m e a s u r a b l e quantities. I t is this unique result for z expressed in t e r m s of observables in (31) and (33) t h a t is physically significant. N e i t h e r of the original equations (6) or (8) for z with R p r e s e n t has a n y meaning, for t h e y are b o t h m e t r i c - d e p e n d e n t .
62
9
D . K . ROSS
RIASSUNTO
(*)
Si caleola e s a t t a m e n t e la c o n s u e t a espressione del r e d - s h i f t g r a v i t a z i o n a l e nella f o r m a s t a n d a r d e in quella i s o t r o p a della m e t r i c a di S c h w a r z s e h i l d . I r i s u l t a t i differenti risult a n o essere p r i v i di signifieato e d i p e n d e n t i d a l t a m e t r i c a a m e n o ehe n o n li si p o n g a i n t e r m i n i di q u n n t i t s d i r e t t a m e n t e o s s e r v a b i l i dalla t e r r a . Si p o n e i n e v i d e n z a l ' i m p o r t a n z a di e s p r i m e r e le q u a n t i t ~ fisiche i n t e r m i n i di osservabilit~t.
(*)
Traduzione a eura della Redazione.
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