IL I~UOVO CIMENT0 VoT.. 16 B, N. 2 11 Agosto 1973

Perspectives on the Schwarzschild Singularity (*).

:F. I. COOPERSTOCK and G. g. G. Ju~EvIcus (**)

Department o] Physics, University o] Victoria - Vivtoria, British Oolumbia, Oanada

(ricevuto il 21 Settembre 1972)

Summary. - - A survey and discussion of some of the literature per- raining to the Schwarzschild singularity is presented. Scalar indeter- minaeies, with particular reference to the metric of Bach and Weyl, a r e

cited as evidence for the reality of the Schwarzschild singularity. An analogy is drawn with the work of Janis, Newman and Winieour.

l . - I n t r o d u c t i o n .

The Schwarzschild (1) solution has unders tandab ly played the most funda- men ta l role of all solutions of Einstein 's general relativistic field equations. Being the unique (~) spherically symmetr ic e m p t y space solution, the Schwarzschild solution is to general re la t iv i ty a t the ve ry least what the inverse square law force is to Newtonian theory . In fac t i t is much more, for various reasons. On the purelY ter res t r ia l scale, the full general theory is cur ren t ly beyond the level of verification. Einstein 's t heo ry has found almost all of its observa- t ional support , albeit l imited and vigorously contes ted (8), in the domain of the solar sys tem where the Schwarzschild solution describes wha t is v e ry near ly the static, spherically symmetr ic gravi ta t ional field of e m p t y space. The phys- ical field is not exac t ly static, nor exac t ly spherically symmetr ic , nor exac t ly

(*) Supported, in part, by the National Research Council of Canada, Grant A5340. (**) N.R.C. Predoctoral Fellow. (1) K. SCHW~ZSCH~D: Sitzber. Pre~ss. Akad. Wiss. Physik-Math. KI., 189 (1916). (~) G. BIRKHO~F: Relativity and Modern Physics (Cambridge, Mass., 1923), p. 253. (a) See, for example, R. It. DIC~: Remarks on the observational basis o] general rdat~vlty, in Gravitation and Relativity, edited by CHIU and HOFFMAN~ (New York, 1964). (,) Units are chosen in which c ~--G----1.

387

3 ~ F . I . COOP:ERSTOCK &n~ G. J. G. JUNEVICUS

empty . However , the successful h is tory of m a n ' s idealistic m a t h e m a t i c a l descript ion of the physical world encourages h im to pursue the s t u d y of such ideal representa t ions . The ~se of the Schwarzschild solution as an ideal ma the - mat ica l descript ion of the physical ly observed gravi ta t ional p h e n o m e n a in the solar sys t em has been a good example of t h a t successful history. I ndeed it would be safe to say t h a t the accep tance of Eins te in ' s general t heo ry of re la t iv i ty b y the world 's scientific e o m n m n i t y a t large res ts p r imar i ly on the

success of the Schwarzsehild solution. Solutions of Einste in 's highly non- l inear field equat ions are ha rd to come by, and when they are found, more

of ten t h a n not t hey are devoid of physical interest . For these reasons, it is ent i re ly reasonable t h a t the physical ramif icat ions

of the Sehwarzsehild solution should have been pursued as vigorously and nniversa l ly as they have been in recent years. The centre of a t t en t ion has focussed inward with growing intensi ty, par t icu lar ly within the last decade, f rom the ((exterior region ~) r > 2m where the theory draws its p r i m a r y sup- por t , to the ((Schwarzschild surface ~> r = 2m and then into the ((interior

region ~> r < 2m and finally to the origin itself, r ~ 0. Wr i t t en in usual fo rm (4)

(1.1) ds2 ~ ( 1 - - ~ ) d t ~ - - ( 1 - - 2 ~ r ) - ~ d r 2 - - r ~ ( d 0 2 + sin~ 0d92) ,

the Schwarzschild solution displays the fea tures t h a t the metr ic tensor com-

ponents g00 and gll become zero and infinite respect ive ly a t r = 2m. I n the ear ly years following tile discovery of the metr ic , this was regarded as evidence for the existence of a physical s ingulari ty on the Schwarzschild surface unti l 1933 when LE~[.~iTI~E (5) and others (~-i0) la ter devised co-ordinate t ransfor - mat ions which rendered a regular, nonsingular appearance to the metr ic tensor a t r = 2m. Wi th the Sehwarzschild radii of the Sun and E a r t h being only app rox ima te ly th ree k i lometers and one cen t imete r respect ively, some au- thors (~) voiced the opinio~ t h a t since the Sehwarzschild surface, and hence also the interior region, lies well within ord inary as t ronomica l bodies where the m a t t e r tensor does not vanish, eq. (1.]) for r ~ 2 m is beyond the range of concern for the physical world. I n spite of this, Su (~) a rgued t h a t the iute- rior region is still wo r t hy of theore t ica l invest igat ion for the light which i t m a y shed on the na t u r e of Eins te in ' s theory .

(5) G. LEZ~IAITttE: Ann. Soc. Sci. Bruxelles, A53, 51 (1933). (6) J . L . SYz~G]~: Proc. Roy. Irish Acad., A53, 83 (1950). (7) D. FINKELSTEIN: Phys. Rev., 110, 965 (1958). See Mso; A. S. EDDINGTOZr Nature, 113, 192 (1924). (s) 3~. D. ]~RUSKAL: Phys. leer., 119, 1743 (1960). (9) Yu. A. RYLOV: Sov. Phys. JETP , 13, 1235 (1961). (lo) W. ISRAEL: Phys. •ev., 143, 1016 (1966). (11) See, for example, J. L. SYNGE: Relativity: The General Theory (Amsterdam, 1960), p. 283.

PJ~RSP~--CTIV'ES ON THE SCB-WARZSCHILD SIlqGULARITY ~

With the lid pried open, numerous investigators, encouraged to a certain ex t en t by the work of Oppv.~H~n~R and S ~ Y D ~ (12) on gravi tat ional collapse, rushed to explore the bewildering interior region r < 2m (13). At the present t ime, the na tu re of bodies which have collapsed into the interior region an d thei r a t t endan t phenomena appear familiar and substant ive to these investi- gators.

Through the years , a small coterie of critics have quest ioned what is now the convent ional wisdom on the subject. ~r (~4) argued on the basis of energet ics t ha t the Schwarzschild radius was the min imum radius which a body could have. SALZ~A~ and S ~ z ~ A ~ (15) p resen ted evidence t h a t a t e s t part icle would ac tual ly a t t a in the speed of light a t r ~ 2m in violation of a p r imary t ene t of relat ivi ty. J ~ I s , !qEW-M~N and WrNICO~a (16) consid- ered the limiting case of zero coupling of a spherically symmetr ic solution of coupled gravi ta t ional and zero-res t -mass scalar fields. Their solution ren- dered the r----2m surface intrinsically singular and also pointlike, obviating the difficulties of the Sehwarzschfld interior. BEL (17) also presented evidence for the intrinsically singular pointlike charac ter of the Schwarzschfld singu- lari ty. RosE~ (~8) showed t ha t the singulari ty in the line e lement cannot be r emoved if one restr icts oneself to t ransformat ions which mainta in the stat ic fo rm of the Schwarzschfld solution. He also argued tha t since r and t switch thei r spacelike and timelike characters respect ively in the t ransi t ion to the inter ior region, a contradict ion results because r has been defined to be space- like f rom the outset .

I n this paper we side with some of the conclusions of the critics, if not always with the manners by which these conclusions are a r r ived at.

I n Sect. 2 we discuss and analyse some of the afore ment ioned work. I n Sect. 3 we discuss the axial ly symmetr ic solutions of BACH and WEYL (~)

and demons t ra te the inde terminacy which arises for the special case of the Schwarzschild solution. We postula te this inde te rminacy as a plausible indi- ca tor of an intrinsic singulari ty with par t icular reference to the ve r t ex of an ordinary cone in :Euclidean space. The parallel to the work of J ~ I S , :NEW~'~ and WINICOVR is also discussed.

I n Sect. 4 we offer some concluding remarks .

(12) j . R. (~PPENHEIMER and I-I. SNYDER: Phys. l~ev., 56, 455 (1939). (1*) See, W. ISRAEL: Phys. ~ett., 21, 47 (1966), and references contained therein. (14) W. H. McC~EA: Astro~hys. ~orvegica, 9, 89 (1964). (15) 1 ~. SALZMAN and G. SALZ~AN: ~ett. •uovo Cimento, 1, 859 (1969). (1~) A. I. JANIS, ~. T. NEWMAN and J. WINICOUR: P~y8. Re~). ~et$., 20, 878 (1968). See also, Phys. Rev., 176, 1507 (1968). (17) L. BEL: Journ. Math. Phys., 1@, 1501 (1969). (is) N. ROSY.N: The Sehwarzsehild singularity, in Relativity, edited by CARM~LI, FICKLER and WITTEN (New York, 1970), p. 229. (19) R. BACH and H. W~.YL: Math. Zeits., 13, 134 (1922).

390 F.I. COOPEKSTOCK and G. J. G. JuNEVICUS

2. - D i s c u s s i o n o f s o m e p r e v i o u s w o r k .

I t is well known (2o) t h a t the veloci ty of tes t particles, as measured locally

b y s t a t ionary observers approaches the ~-elocity of light as r - ~ 2 m . ]~ow- ever, SYxaE (6) demons t r a t ed t h a t the world-line of a t es t part icle will no t be t angen t to the null cone a t r = 2m and hence, p resumably , the part icle will no t a t t a in the speed of light. SXLZ_~r~" and S~LZ_~An" (15) a rgued t h a t the par - ticle can bo th a t t a in the speed of l ight and not be on the null cone a t r = 2m

b y v i r tue of the fac t t h a t (( the t r ans fo rma t ion f rom the r, t to the u~ v (Krus- kal co-ordinates, see ref. (s)) co-ordinates carries us outside the range of our experience with local Loren tz t ransformations,>. Their a rgumen t leads us to the following considerat ions. I f r--~ 2m is a physical ly una t ta inab le si tua- tion, then the f u n d a m e n t a l principle t h a t v = c is una t t a inab le b y f ini te-mass part icles is left iuCact. However , if the s i tuat ion is a t t a inab le and the par t ic le can reach and cross this bounda ry continuously, as has been assumed in recent years , t hen the demands of cont inui ty appea r to force one to accept t h a t the part icle does a t t a in the speed of light a t r - - 2 m . Wi th the physics t h e r e b y abused, one would be ill advised, we feel, to a t t e m p t justif ications for ma the - mat ica l conclusions which derive f rom wha t we regard as unacceptab le t rans - format ions .

R0SE~ (zs) defined the Schwarzschild solution as the stat ic, spherical ly symmet r i c empty - space solution for which space- t ime is fiat a t spat ial infini ty and then defined ((static)) b y the conditions (~)

c~gs~ (2.1) ~ x ~ = 0 , ao~ = 0

on the metr ic tensor. He acknowledged t h a t the t r ans format ions b y FINKEL- STEIn" (7), KRTYSKAL (s), etc. do r em ove the singulari ty a t r ~ - 2 m bu t re jec ted the metr ic forms which result . ROSEn" a rgued t h a t they do not represen t the Schwarzschild solution b y v i r tue of thei r violat ing the stat ic conditions.

We feel t h a t it is only reasonable to call a space- t ime stat ic if the met r ic can be expressed in a fo rm which satisfies eqs. (2.1). Clearly, for example , f reely falling observer in the Schwarzschild field would deduce a t ime-dependen t charac te r for the metr ic , ye t the object ive charac te r of the field remains intact . We accept the legi t imacy of a nonsta t ic appea rance for the Schwarzschild solution rendered b y the t r ans fo rmat ions in ref. (6-~0), but only /or r > 2m. We re jec t the notion t h a t these transform%tions r emove the r = 2m singulari ty. The t r ans format ions themselves are, with Yarious combinat ions, singular, complex

(20) L. D. LANDAV and E. M. LIFSHITZ: The Classical Theory o/ Fields 3rd revised English ed. (Reading, Mass., 1971), p. 298. See also, ref. (is). (21) Latin indices range 0-3, Greek indices range 1-3.

P~RSPECTIVES 0 2 T H ~ SCH~rARZSCHILD SINGULARITY 3 9 1

or undefined on r = 2m or t he interior ragion. Unfor tuna te ly , a lmost unl imited f reedom vis d vis allowed co-ordinate t ransformat ions has been read into the principle of general eovariance. Complex t ransformat ions , for example, readi ly a l te r the physical charac te r of a space-t ime in spite of the fac t t h a t f rom a mathemat ica l point of view, t hey m a y be eminent ly logical and reasonable. When a t ransformat ion becomes singular, it is no t a t ransformat ion a t all and when it is undefined, it fits into the same category.

One might be inclined to c i rcumvent Rosen's objection to the logic of regard- ing r as bo th a t imelike and a spacelike co-ordinate b y not ing t h a t the Schwarz- sehild solution could have been der ived jus t as well for example in the manne r of L E ~ i T ~ E (~). I n Lemaitre% synchronous form, the r -co-ordinate is always spacelike and the t -co-ordinate is always timelike. However , the fac t t h a t the r and t co-ordinates mus t abrup t ly switch thei r spaeelike and t imelike charac- ters when the metr ic is expressed in a manifest ly s tat ic fo rm might be inter-

p re ted as a reasonable danger signal.

3. - S c a l a r i n d e t e r m i n a c i e s .

I n suppor t of the content ion t ha t the r = 2m surface is no t t ru ly singu- lar, various authors (22) have der ived considerable encouragement f rom an examinat ion of the scalar S ~R~kzR ~jkz where R ~ is the Riemann tensor. They have no ted t ha t f rom the metr ic tensor components of eq. (1.1)

~8m ~ ( 3 . 1 ) ~ _

7, 6

The finiteness of S at r----2m, in contradis t inct ion to its infinite value a t r ---- 0, is cited as evidence for the co-ordinate na tu re of the Sehwarzschild sin- gularity. 1V[oreover, with varying degrees of firmness, the infinite value of is proclaimed to be a necessary and sufficient condition for the existence of an intrinsic singularity. Few would quarre l with the sufficiency b u t Bo~No~ and SACKFIEL]) (23) cited the intrinsiealiy singular v e r t ex of an ordinary cone, where S, t hey claimed, is no t infinite bu t zero, as a eonnterexample to the neces- si ty condition. I n this Section we will examine the na tu re of this scalar in a n o w m a n n e r .

Consider the axially symmetr ic , static family of metrics genera ted b y the l~ewtonlan potent ia l of a line mass (2~) in the Weyl-Levi-Civi ta f ramework

(z2) See, for example, J. L. ANDERSON: Principles o/ l~elativity Physics (New York 1967), p. 386. (~8) W. B. BONNER and A. SXOKFI~LD: Commun. Math. Phys., 8, 338 (1968). (~) Ref. (1,). See also, H. P. ROBV.RTSON and T. W. NOONAN: Relativity and Cosmology, Chap. 10 (Philadelphia, 1968).

392 ~'. I . COOPERSTOCK and G. J. ~. JUN]~VICUS

in cy l indr ica l po l a r co -o rd ina t e s :

(3.2)

w h e r e

d s 2 = exp [2).] dt 2 - exp [ 2 ( v - ).)] (d@~+ dz 2) --@2 exp [ - -2) . ] dq~ 2 ,

(R~ -}-/~2 - - 21) ~v 2 In (R1 + R , ) ~ - - 4l ~ (3.3) 2~ = ~v In i]R~ + R2 + 2 1 ) ' 2v = 4R, R2 '

(3.4) ~1~ = (z - 1)~ + q~ , m

/ ~ = (z + t)~ + e ~ , ~ = T '

21 a n d m a re t he l eng th a n d m a s s of the line r e s p e c t i v e l y in the N e w t o n i a n p ic tu re . F o r t he special case ~ 1 (m = 1), t he r egu la r co -o rd ina te t r a n s f o r - m a t i o n

(3.5)

or e q u i v a l e n t l y

(3.6)

z = ( r - - m ) cosO ,

Q= V r - - 2 m r s i n O ,

R1 q-/~2 q- 2m R 2 - - R 1 2 , cos 0 = 2m -

p u t s t he m e t r i c in to t he u sua l Schwarzsch i ld fo rm, eq. (1.1). Since Rx q-R2 has a m i n i m u m va lue of 2m (co r re spond ing to t he (~ line m a s s )) itself), eq. (3.6) impl ies t h a t r ~ 2 m . This condi t ion can also be seen f r o m eq. (3.5) w h e r e t he va lues r < 2m r e n d e r t he cy l indr ica l po la r eo-ord in~te Q i m a g i n a r y . I n f a c t t h e line - - m < z < m , ~ = 0 m a p s in to t h e Schwarzsch i ld su r face r = 2m.

Us ing d i f fe rent ia l f o r m s in con junc t ion wi th co-ord ina tes $ a n d ~ def ined b y

(3.7) R~ -- R2 21 cosh ~ , R~ - - R~ = 2l cos r/,

wi th basis 1 - % r m s (zs)

(3.s)

01 = ( t g h ~ ) ~ d t ,

O* = l ( e tgh ~)~ (s inh ~)v' (sinh~ ~ + sin* ~)(1-v')~2 d ~ ,

( 7 0 s = 1 c tgh ~ (sinh ~)v' (s inh ~ ~ + sins ~1)(1-r d ~ ,

( 7 0 4 = 1 c t g h ~ s i n h ~ s i n ~ d ~ ,

(35) W. ISRAEL: Di]]erential ]orms in general relativity, Comm. Dublin Inst . Adv. Stud. Set. A, No. 19 (1970).

P~RSPECTIV'ES OXq T ~ SCKWA~RZSCHLLD S I N G U L A R I T Y ~ 9 ~

the Bach and Weyl line element becomes

(3.97 ds~ = (01)~-- (Oq ~ - - ( 0 , )~ - - (0') ~ .

The nonvanishing frame components of the metric tensor are thus

(3.1o)

and

(3.11)

g . = - - - g 2 2 = - - g 8 3 : - - g 4 4 = 1

gAB ---- gAB "

Taking the exterior differentials of eq. (3.8), expressing them in terms of the basis 2-forms 0 ~/k 0 B and comparing the expressions with the first equations of structure

(3.12) dO "~ = --'w~B A 0 B ,

yields the unique set of connection 1-forms w~B. The wedge products w~oAw~ and the exterior differentials dud B of the

connection 1-forms yield the curvature 2-forms ~ B from the second equations of structure

(3.13) ~'~B = dw~s § w~~ A w ~ .

Finally, the flume components of the Riemann tensor R~Bo~ are read off from

(3.14) ~C2~: �89 A 0 ~

after the ~2~B are expressed in terms of the basis 0~ 0 ~. From eqs. (3.10) and (3.11), the scalar S is seen to be composed of a sum

squares of frame components

(3.15)

= 4[(Rl~lW § (R~IW + ( R ~ ) ~ + (R~I~) ~ § (Rh,W + (R%W +

+ (R%4) ~ + (R~,~,) ~ + (R%,)~ + ( R ~ ) ~] ,

where the nonvanishing components which derive from the basis 1-forms of eq. (3.8) have been explicitly expressed. These components are listed in the Appendix. Because of this convenient structural form for S, there can be no cancellation among the individual terms. Consider the frame component

~p(~-- l) sin~ cosy (3.16) R i m : 22r ~/2) 2r (sinh ~/2) 2~'-~+~ (sinh 2 ~ -k sin 2 ~)3-v."

394 r . i . COOPERSTOCK a n d G. g. G. JUI~]~VICUS

F r o m eq. (3.7), ~ - - 0 c o r r e s p o n d s to t h e l ine R ~ + R 2 = 21 w h e r e s inh~ /2

van i shes . Thus , a p a r t f r o m t h e e n d - p o i n t s w h e r e ~] = 0 or ~z, R~2~ c l e a r l y

b e c o m e s in f in i te for F # 1 a n d ~ - - 0. I n o t h e r words , t h e l ine is i n d i s p u t e d l y

s i n g u l a r for l ine m a s s dens i t i e s o t h e r t h a n t h o s e wh ich c o r r e s p o n d to t h e

S c h w a r z s c h i l d so lu t ion . F o r ~o = 1 a n d ~ ~ 0, t h e S c h w a r z s c h i l d so lu t ion in r eg ions

wh ich do n o t i nc lude t h e r = 2m su r face , /~1231 = 0 a n d w h e n t h e sca l a r S is

c o m p u t e d f r o m t h e n o n v a n i s h i n g f r a m e c o m p o n e n t s us ing eq. (3.15), i t is f o u n d

to ag ree , a s i t should , w i t h eq. (3.1). H o w e v e r , for ~o = 1 a n d ~ = 0, wh ich

c o r r e s p o n d s to t h e S c h w a r z s c h i l d sur face , Rt2a~ is i n d e t e r m i n a t e . F o r t h e sca l a r S (~s)

l i m S(yJ ~---~0

l i r a S(~

l i r a S(~

3( 9 = 1,

= 1 ) = f ini te , c o r r e s p o n d i n g to t h e u s u a l S c h w a r z s c h i l d sca la r ,

# 1 ) = c ~ ,

# 0 ) - f ini te , c o r r e s p o n d i n g to t h e u s u a l Sc hw a rz sc h f ld sca la r ,

= 0 ) = c ~ ,

= 0) = i n d e t e r m i n a t e .

T h e i n d e t e r m i n a c y in S does n o t show up in t h e u s u a l e v a l u a t i o n s b e c a u s e

one, in effect , se ts W 1 f r o m t h e ou t s e t . T h e r e is n o t h i n g los t in d o i n g so

for a l l p o i n t s o t h e r t h a n t h o s e on t h e S c h w a r z s c h i l d sphere . F o r r # 2m, no

i n d e t e r m i n a c y ar i ses . This s i t u a t i o n is s o m e w h a t a n a l o g o u s to t h a t p r e s e n t e d

b y J~kNIs, ~NEW3r~XN a n d W~xICO~-n (~6). I n t h e i r case , for ze ro c o u p l i n g to a

s ca l a r field, t h e y h a v e a n i n d e t e r m i n a c y in t h e S c h w a r z s c h i l d r a d i u s w h i c h

t h e y a t t e m p t to r e so lve b y a l i m i t i n g p r o c e d u r e . W i t h t h e i r choice of l imi t ,

b a s e d on a <, 1 ) e r t u rba t i on p o i n t of v i ew ~>, t h e i r space <~ s u d d e n l y co l l apses

f r o m a r a d i u s s l i gh t ly g r e a t e r t h a n r = 2m to zero ~> a n d i t <( co inc ides w i t h

t h e e x t e r i o r S c h w a r z s c h i l d space b u t w h e r e t h e S c h w a r z s c h i l d " s p h e r e "

r = 2m now b e c o m e s a singular point ~.

I n our v i ew th i s is a t legist a n as e q u a l l y b i z a r r e d e s c r i p t i o n as t h a t wh ich

is n o w g e n e r a l l y a s c r i b e d to t h e u s u a l S c h w a r z s c h i l d so lu t ion . W e feel t h a t

t h e i r e x a m p l e , l i ke ours , is m o r e r e a s o n a b l y i n d i c a t i v e of a b r e a k - d o w n of p h y s -

i ca l ly m e a n i n g f u l a n a l y s i s a t r - 2m, a n i n d i c a t i o n of a n i n t r i n s i c a l l y sin-

g u l a r r eg ion a t r = 2m. W e fee l t h a t a t t e m p t s to choose l im i t s s e rve to ob- scure th i s b r e a k - d o w n .

B y w a y of a n a l o g y , c o n s i d e r t h e i n t r i n s i c a l l y s i ngu la r ~ e r t e x of a n o r d i n a r y

cone in E u c l i d e a n space . T h e su r f ace is i n t r i n s i c a l l y fiat . T h e R i e m a n n t e n s o r

(2r Another peculiar feature of the Bach and Weyl metric which was pointed out to us by ISI~AEL concerns the circumferences C of circles around the line in the l imit of vanishing radius. Away from the end points C ~ 2=Ql-~[4(12--z~)] ~12, C ~ O for

< 1, C ~ ~ for ~ > 1, C-~fini te for ~ = 1.

PI~RSP~CTIVES ON T H E SCHWARZSCHILD SINGULA.RITY 395

and hence the scalar S vanish. However , this is t rue for points o ther t h an the ve r t ex point where S is easily seen to be indeterminate . The limit of S as the ve r t ex is approached, is zero (~) bu t a t the singular v e r t ex itself, S is indeterminate .

We feel t ha t such an inde te rminacy is a reasonable indicator of a physical singularity.

4. - Conc lud ing remarks .

In spite of the fac t t ha t the physical field of the solar sys tem is not precisely the exter ior Schwarzschild field, the Sehwarzsehild solution has served admi- rab ly in providing post -Newtonian corrections. Indeed, such tolerance of ap- proximat ion on the pa r t of I~ature is the basis for all successful mathe- mat ica l idealizations of the real world. However , the r ~ 2m surface of the Sehwarzschild solution is, according to Israel 's theorem (~s), the unique nonsin- gular even t horizon among all static vacuum solutions which are asymptot ical ly flat. Static in ternal per turba t ions introduce real singularities thus changing the qual i ta t ive charac te r of the problem.

In our view, this uniqueness is a fu r the r indicator of a pure ly mathemat ica l r a the r t han a physical phenomenon. We feel t ha t the theorem is a manifes- ta t ion of the high degree of s y m m e t r y which admits a nonphysical ly meaningful

mathemat ica l extension to the interior region. I n this regard it is d propos to ment ion the problem of singularities in cos-

mology. Singularities were not given serious a t t en t ion when LxFsm~rz and KHALATNIKOV (9.9) claimed t ha t t hey arose only b y v i r tue of the special sym- metr ies of the cosmological models and t h a t per turba t ions would p reven t thei r occurrence. I t was only af te r H~.WKr~G (80) p roved t h a t per turba t ions to these models would not p reven t the growth of singularities t h a t t hey were regarded as physically re levant . Wi th the co-ordinate singulari ty problem, it is known t ha t the special s y m m e t r y of the Schwarzschild solution in the static case (31) is responsible for the alleged unique propert ies of the Schwarzschild surface. E v e n if one would grant t ha t these propert ies are what the major i ty of investi-

(2~) W. RINDLER: Essential l~elativity (New York, 1969), p. 185. RII~DLER correctly states that the curvature components are zero near the vertex of a cone whereas BON~OR and SACKFIV.LD (ref. (~a)) state that they vanish at the vertex itself. (2s) W. ISRAEL: Phys. l~ev., 164, 1776 (1967); General Relativity and Gravitation, 2, 53 (1971). (2a) E. 1VI. LIFSHITZ and I. N. KHAT.ATNIKOV: Adv. in Phys., 12, 185 (1965). (a0) S. W. HAWKING: Phys. tCev. Left., 15, 689 (1965). (al) I t has been conjectured that the Kerr solution plays the same role in the stationary case (see ref. (~s)).

3 9 6 F . I . COOPERSTOCK and G. J . G. JUN]~VICUS

gators believe they are, theiz' dependence on the special symmet ry would, in

our view, render their physical significance highly tenuous.

As a final note, it is perhaps wor th recalling tha t the ul t imate arb i t ra tor is :Nature herself.

We express our grat i tude to Profs. W. I S a ~ L , G. 5IILLEIr and J. L. Su

and Dr. P. YODZIS for interesting discussions.

~kPP~ENDIX

The nonvanishing frame components of the l~iemann tensor are

/ ~ 1 ~o s e c h ~ / 2 - - B coseeh$ + C eosech ~ - - 1 2/) '

2D sinh $ /71321 = R1231 = ~o(~o2_1) s in2 U '

~oB sech ~ ~/2

/~131~ = - - 2D sinh ~- '

R1,14 ----_ ~fA C sech ~ ~/2 2D sinh ~ '

sech 2 6/2 [A (yj2_ 1) sin 2 U + 2(A C + B) ctg U], /~3424 : R 2 ~ 3 4 - - 4 D "

.R34 = [AB sech 4 6/2 + 4(1 - - yj2) cos ~ U + 4C] 4D

[ 1 1 (~ 2 - 1 ) Ccos2~_~s in22u + R2323 -- CD

{2 cosh ~ - - C cosech 2 ~ + ~ sin 2 U cosech ~ ~ cosh $} - - +N

~ sin S U + [2 cosh 2 ~ + sinh 2 ~ - - cosh ~]

D

B sech 2 $/2 [cosech ~ - - ctgh ~ - - sinh 25/C] 2D

A sech ' ~/2 [(1 - - ~) C ctgh 6/2 - - B ] Rz~24 = 4D -{-

4C [cosech' $(cosh $ - - yJ) - - 1] + 4(y~' ~ 1) cos ' ~/ + 4D

P~RSPECTI~ES ON THE $CHWA.RZSCHILD SINGULARITY ~

where

A = ~ eosech I s inh ~ - - e t g h ~ eosh ~ ,

B ( - - ~ ' ~ ~ ~) C q- (1 - - ~2) e tgh ~ s i n h ' ~ c~ , = coseeh ~ s i n h ~ ~ ~0~ c t g h ~ eosh

6' ~ s i n h ~ ~ q- sin s ~ ,

D = Z ( e tgh~ l (slnh,)lv 'C '-~' �9

�9 RIs (*)

$i presenta una rassegna ed una discussione di alcuni articoli relativi alla singolarit~ di Sehwarzsehild. Si citano le indctcrminazioni di scala, con particolare rfforimento alia metrica di Bach e Yqcyl, come prova che la singolarit~ di Sehwarzsehild ~ reale. Si deduce un'analogia con il lavoro di Janis, Newman e Winicour.

(*) Trad~zione a eu,ra della Redazione.

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