IL NUOVO ( ' IMENTO VoL. 65B, N. 2 11 Ottobre 1981

Reintroducing the Concept of {{Force>> into Relativity Theory (*).

S. 3I. 3IAHAJ.~X

DQmrlme~lt o/ Physics, U~Hrer,~ily o/ Texas at Aust in - .lustiJ~, Y'ex. 78712

A. QADIll

_l[athcmalies Dclm~tmcJd. ( fuaid-i- . lzam U~,icer~ity - lsla~labad, l~akisla.n

I ' . )1. ~,:ALANJU

(.'e~tre /or Parlicle 7"hcorg, ]-)~3mrlmc~l o~ Physics Unicersil U o~ Texas al ~lu.sliJ~ -~lusli~l, Tex. 78712

'rieevu[o il 12 Agosto 198[)

S u m m a r y . - We suggest that reintroducing ~, forces ~ into re la t iv i ty theory may provide new insight~ mid results. A look at the Kerr-Newmaml geometry, and special ca~es of it, from thi~ viewpoint indicates tha t ~here can be a short-range repul.~ion in general. This repulsion suggests that (~ naked singularit ies ,> may be physical ly feasible. We also find tha~ there is a , gravito-electric repulsion ,~ which would he importan~ to consider in a grand-unification scheme of strong, weak and electromagnetic forces.

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

The usua l present t~ t ion of gene ra l r e l a t i v i t y avo ids t i l e c o n c e p t of <, fo rce >> in

f~vour of ~ c o m p l e t e l y g e o m e t r i c desc r ip t ion . This appro,~eh m i g h t h~ve been

fine if onr physiea.1 i n t u i t i o n was no t so f i rmly based on t h e c onc e p t of force.

W i t h o u t g iv ing up t h e p r o g r a m of g e o m e t r i z a t i o n we m a y ge t new ins igh t s

in to t h e work ings a n d p r e d i c t i o n s of gene ra l r e l a t i v i t y b y r e i n t r o d u c i n g t h e

c oncep t of force in to re la . t iv i ty t h e o r y .

(*) To speed up publication, the authors of this paper have agreed to not receive the proofs for correction.

4 0 ~

REINTRODUCING TI~IE CONCEPT OF( ( FORCE )) INTO RELATIVITY THEORY 4 0 ~

There is a general t endency to suppose t h a t an observer in a freely falling f r ame would not feel any force act ing on h im and would, therefore, be unaware

of the presence of the g rav i t a t ing source towards which he is falling. I t is

well known tha t this view is not really correct because the observer could de- tec t the presence of the g rav i t a t ing source b y measur ing t idal forces. The usual

view would be correct if the size of the measur ing device were negligible com- pared with its dis tance f rom the g rav i t a t ing source, thus m a k i n g the t idal force negligible as well. However , near the source, the t idal force can be quite

significant. As we show la ter , an observer equipped with a sufficiently ac-

curate device (to measure the t idal force) would see some interest ing changes is the readings on his device. I n the nex t section we shall consider h o w he

would in terpre t those changes and, in the two sections thereaf ter , wha t changes he would see in a Schwarzschild, Reissner-Nords t rom or K e r r - N e w m a n n (1) geometry .

We are thus led to consider the usual relat ivist ic generalization of the Newtonian concept of gravi ta t ional force as the inward, radial accelerations (times mass) due to a source. As we shall see in sect. 5, we get nongravi ta t iona t forces which can be repulsive, apa r t f rom the usual gravi ta t ional a t t rac t ion. This brings out the fac t t ha t general re la t iv i ty is a theory of acceleration and

not merely a theory of gravi ta t ion. I t also demonstra tes , as shown in sect. 6, t h a t even (( naked singularities ~) are not all t h a t naked.

l~inally, in sect. 7, there is a brief s u m m a r y and discussion of the results obtained. I t is hoped t h a t the above-ment ioned ideas m a y help to give a more intui t ive unders tanding of the demons t ra t ion by QADIR and W~EELER (~) of

the va l id i ty of Penrose ' s conjecture (3) t ha t in a closed cosmology the black-hole singulari ty and the big-crunch singulari ty will amalgamate . This result could

become more comprehensible in t e rms of the usual concept of ~orce, as an observer falling into the hole sees a (~ repulsive force ~) (in some sense), which keeps h im f rom falling into the s ingulari ty till the Universe collapses. We

m a y call this (( force ,) the (~ force of t ime ~). This approach m a y also help us to answer the question (~ wha t happens to the supposed ama lgamat ion of singularities if the b lack hole evapora tes ~). The (( repulsion ~) experienced by the incoming part icles m a y not allow the ac tual fo rmat ion of a singulari ty in a b lack hole which evapora tes as there is never enough t ime for its formation. Another quest ion t h a t the present view m a y help to answer in this connection

(1) See, for example, C. W. MISNER, K. S. THORNE and J, A. WREELER: Gravitation (San Francisco, Cal., 1973). (2) A. QADIR and J. A. WHEeLeR: Amalgamation o] the black-hole singulariSy ~ith She big.crunch singularity (to appear in a volume in honour of Jules Gehenian, being edited by R. DE,EVeR and M. DE~EUR). (a) R. PE~ROSE: Confrontation o/ Cosmological Theories with Observationa~ Data, edited by M. S. LONGAIR (Dordrecht, 1974).

~06 S.M. MAHAJAN, A. QADIR and P. M. VALANJU

is ~ what happens to the supposed amalgamation in an open universe? ~. Since

there is no (~ big crunch ~, there can be no amalgamation of singularities. The

formation of the black-hole singularity, however, seems to be inescapable be-

cause it must happen in ,~/inite proper time. Though we are unable to suggest

any answer to the above question at present, it is hoped tha t it could be found

in the concept of force.

2. - The concept o f the force of a freely fall ing observer.

There is a problem in defining the ~ force ~ acting on a freely falling ob-

server, because, by definition, his frame is the one in which a point observer

cannot detect any force. However, if the spatial extent of the observer is not

negligible compared with his distance from the gravitat ing source, he can easily

tell, by measuring the tidal forces, whether his ~, free fall ~ is towards a grav-

itational source or not. Here we consider what the observer would measure

and how he would interpret his observations, particularly with reference to

at t ract ive or repulsive forces.

s /

I - . ( . . '

\/(+) " % (-)-,

Fig. 1. - A spring of length l connects two masses. The springs ends in a needle which rotates around on a dial which shows compression (--), st.retching (+) or no change (0).

Let us first consider tidal forces due to at t ract ive or repulsive forces in ~qewtonian physics. An ideal apparatus for measuring such forces would be

an arrangement of a spring of length 1 connecting two equal masses, one side

having a needle whose rotation on a dial measures the tension in the spring,

as shown in fig. 1. Thus the observer can tell whether the spring is being ~ stretched ~>, ~ left neutral ~, or ~ compressed ~ by observing the movement of the needle. In classical physics, the needle would be in a neutral position in

the absence of a central force, indicate stretching (+)/compression (--) for

an attractive/repulsive central force.

R E I N T R O D U C I N G T H E C O N C E P T OF (( F O R C E >) I N T O R E L A T I V I T Y T H E O R Y 44}7

Since our freely falling observer is now in a position to perceive (~ forces ,,

it is meaningful to ask what <~ fo r ce , he feels as he falls towards a naked sin-

gularity. Let us consider the Ker r -Newmann metric in Boyer-Lindquist co-ordinates,

(1)

where

ds s = A d t s + 2 B d t d ~ - - Gdr s - DdO s - E d g S ~

(2)

A = ( r S - - 2 m r + a ~ cos ~ 0 ~- Q') / ( r ~ + a ~ eos~0),

B = a sin~ O ( 2 m r - - Qs)/(r* + a 2 cos ~ 0) ,

C = ( r ~ + a* c o s ' O ) [ ( r * - - 2 m r + a ' -} - Q*) ,

D ---- r 2 - ~ a ~ c o s s 0 ,

sin s O[r" -4- aS(1 + cos s O) r s ~ 2 m a s sin s Or + a 4 cos s 0 - - aSQ" sin s 0] E ---- r s -~ a s cos s 0

m being the mass of the gravi ta t ing point particle, Q its charge and m a its

angular momen tum (in gravitat ional units). The acceleration vector is

(3) A ~ ~ . dSP~ - - ds* R~c~tbpct~ '

where t �9 is the uni t timelike vector tangent to a geodesic and p" the space-

like position vector f rom the geodesic to a neighbouring one. I f we consider

the front end of the measuring device (shown in fig. 1) to move along the first geodesic end the rear end along the other geodesic, the device being placed in

t h e , r ad ia l , (or r) direction, the acceleration becomes

(4) A ~ ---- -- 1R%lo.

I t is of interest to consider the acceleration vector for which the rotat ional

effect is maximum, i .e. in the plane of rotation. As shown in appendix A, the

geodesic equation for 0 can be solved by 0 = ~]2, ~} = 0, which means tha t

the mot ion remains coplanar if it starts off in the plane of rotation. I n this

case the metric coefficients reduce to

(5)

A = (1 -- 2 m / r -}- QZ[r~) ,

B = a - - a A ,

= (A + a~/r,)-l,

D = r ~ ,

E = r ~ -}- 2a ~ - - a ~ A .

408 s . M . MAHAJAN, A. QADIR and P. M. VALANJU

i t in e~,sily ver i f ied t h a t now the on ly nonze ro c o m p o n e n t of t he a c c e l e r a t i o n

r e e l e r is A ~, which is e v a l u a t e d in a p p e n d i x B to y i e ld

(6) A~ = [2m 4m2 - - 3Q2 ' 'n(3a2 @ 2Q~) L_ m 2 a 2 - 4 Q 2 a 2 - 3 9 2

a~(2~(9 ~ - 10m~) 4ma~Q ~ 2a~q ~]

E v e n t h o u g h t h e expres s ion for A ~ is v e r y complic '~ted, i t is c lear t h a t th is

vec to r does n o t change m o n o t o n i c a l l y , in ~'eneral. To u n d e r s t a n d t h e impl i -

ca t ions of eq. (6) fu l ly , we need to cons ider some specia l cases, i.e. w h e n one

or two of t he q u a n t i t i e s a, m a n d Q arc zero.

3. - Tidal forces in the Reissner-Nordstrom geometry .

I f we cons ider t h e case a = 0 in eq. (6), we ge t t he t i d a l force in t h e Re i s sne r -

N o r d s t r o m g e o m e t r y . I~ is c lear f rom eqs. (1) a n d (2) t h a t t h e r e is no pre-

f e r r ed p l a n e a n d a:ny p l ane can be chosen as t h e 0 = ~/2 p l a n e w i t h o u t loss

of o 'eneral i ty . F i r s t , we cons ider t h e ease Q = 0, g iv ing a t i d a l a c c e l e r a t i o n

in t he Schwarz sch i l d g e o m e t r y . I n t he local L o r e n t z f r ame , we ge t t h e ef-

fec t ive acce l e r a t i on A ~,

(7) A -~ -= 2ml / r a ,

which is j u s t t h e N e w t o n i a n express ion (it m u s t be reca l led t h a t p a s t r == 2m

t h e local L o r e n t z f r a m e is s u p e r l u m i n a l r e l a t i v e to a n <~ obse rve r a t i n f in i t y ~)).

Thus the t i d a l force will a p p e a r to be due to a t i m e l i k e c o m p o n e n t and wil l

ba s i ca l ly cause a change of c lock r a t e s - - a , sor t of <~ force of time,>. I t is th is

(( force of t i m e ~> which shows co l lapse l o t he e x t e n t t h a t t h e s i n g u l a r i t y fo rms

on ly a t t h e end of t ime .

W e can cons ider now t h e resul t for t h e R e i s s n e r - N o r d s t r o m g e o m e t r y .

E q u a t i o n (6) w i th a - 0 o'ives US

I t is c lear f rom eq. (8) t h a t i h e inc lus ion of t he change i n t r o d u c e s a <~re-

lmls ion ,> for r < 3Q2/2m.

The a b o v e resu l t m a y seem odd in two ways . Tl le f irst is t h e ex i s t ence

of ~ <~ force ~) on a n e u t r a l p a r t i c l e due to e lec t r i c cha rge a n d the second is i ts

r epu l s ive n a t u r e . The ex i s t ence of t h e force is imp l i c i t in t he f ac t t h a t t h e

m e t r i c is d e p e n d e n t on t h e charge a n d m a y be u n d e r s t o o d as be ing due to t he

<~ e n e r g y ~> s to r ed in t h e field on ~reeount of t h e p re sence of t he charge . The

R I ~ I N T R O D U C I N G T H E C O N C E P T O F (~ FORC]~ ~) I N T O R ~ L A T I V I T Y T H E O R Y 4 0 9

repulsive na ture of the force can be unders tood b y noticing t ha t the energy

is s tored in the entire field around the point and pulls the part icle back more t h a n it pulls i t forward. This a rgumen t would not app ly if the energy dis-

t r ibut ion ac ted like a uni form sphere of m a t t e r in Y~ewtonian physics, bu t a slight deviat ion f rom this s i tuat ion could enable the energy to give the ~ re- pulsive ,> effect noticed above. Let us consider the effect in more detail.

Clearly we have a m a x i m u m of the effective t idal force when

(9) r = 2O*/m,

which can be seen f rom outside the black hole provided t h a t this value lies outside the event horizon. Thus, provided t ha t

(lo) Q > ~/5 m/2 ,

our t idal-force measur ing device would reach a m a x i m u m value outside the black hole and then s ta r t dropping. The ~ force ~ would reach zero outside the black hole if

(11) Q > 2V'2m]3 .

Thus the informat ion so observed could be re layed b y the freely falling ob- se rver to a d is tan t observer. I t becomes evident t ha t we can ta lk meaning- fully of ~, force ~ for observers falling freely toward a black hole. The m a x i m u m value of A -~ is

(12) A -1 ~ m4/16QS. msx

Thus for an ex t reme Re i s sne r -Nords t rom black hole

(13) -1 1/16m ~ ~4vaax

a t a r -value of 2m, instead of the value of 2/m 2 a~ the outer horizon, as would be expec ted classically.

Two points are now immedia te ly obvious: 1) t h a t the freely falling ob- server can see the ~ change-over ,~ effect a long way f rom the event horizon

and 2) t h a t the m a x i m u m acceleration (for the case Q ~ m) is a fac tor of 32 less t h a n the Newton ian t idal force a t the event horizon.

4 . - T h e t i d a l << f o r c e s ~> i n t h e K e r r - N e w m a n n g e o m e t r y .

We arc now in a be t t e r posit ion to discuss eq. (6) with a ~ 0. F o r a general orbit the ~ force ~ lies somewhere be tween the ro ta t ing and the nonro ta t ing cases

27 ~ I I N u o v o Oimento 13.

410 s. M. MAtIAJAN~ A. QADIlZ and F. M. VALA~'JU

depending on the local value of 0. I t is exactly the nonrotat ing one for an

approach along the axis of rotation. Therefore, it is entirely adequate to con-

sider the r foree ~ in the plane of rotat ion to get an idea of its general behaviour.

Let us consider the uncharged ease first (Q ~ 0). For simplicity of ana-

lysis we eonsider A rather th.~n A- l :

(14) A1 2ml [1 2m_~ a 2(3 m 2m2~t = ~-~- - - r W' +2r § ~'~/J"

I t is clear tha t the effect here is not to enhance the ~, repulsion ~>, but to

reduce it. However, there is nothing p~rticularly interesting about the change

in tidal ~ forces ~ due to rotat ion in this case.

Looking at the ease Q ve 0, we see tha t the spinning charge can give rise

to a greater enhancement of the repulsive effect than the nonspinning ease

for a sufficiently large value of Q. I t is not particularly instructive to do a

detailed analysis of the structure of the tidal ~ forces ~ in the general case,

but we see tha t there is a ~ repulsive effect *~ for sufficiently small r.

5. - ~ Forces ~, as s e e n in t erms o f geodes ic .

Having seen tha t a ~, force ~ can be operationally defined for a freely falling observer, we would now like to consider something more similar to the _New-

tonian central force which gives rise to the tidal forces we have been considering

and see whether it a.lso shows a structure similar to tha t s h o ~ by the tidal

force as we proceed towards the point particle. Since the tidal force is, in some

sense, ~ the gradient of a central force ~, one might expect a similar structure

for the central force also. Thus, roughly speaking, an ~ integration ~ of the

tidal force discussed earlier would be expected to give something similar to

the ~ central force ,~ we want to consider. A bet ter definition of the central force is obtained by considering the radial

geodesic equation for the part icular situation under discussion. The expres-

sion for ~ is the relativistic analogue of the Newtonian ~ force per unit mass ~

at the point. The geodesic equations in the plane of rotation are

0 5 )

"t" - - r -2 C[(EA,1 + BB1) t ~- (EB , I - - B E , l) ~] i" = O,

~' - - r -2 C [ ( B A I - - A B , ~ ) i ~- (BB,1 -[~ AE,1)@Iv = 0,

+ �89 AC-~,~)i ' + 2(e- 'B,~-- BC,-~, ) l r

-- ( C - ~ - - ~C-~)~, ~ + C-~1] = O,

where A , B , C, D , E are given in eqs. (5) (see appendix C). The first two equa-

R E I N T R O D U C I N G TI~IE C O N C E P T OF (~ F O R C E ~) I N T O R E L A T I V I T Y T H E O R Y 411

tions can be integrated to give

(16) t = r + Be) r- ' ,

r = ~(B - - A~) r-~,

where ~0 is the angular momen tum per uni t mass (of the tes t part icle experiencing the force). For a (~ radially incoming ~) tes t part icle (~ = 0) we have t ---- CEl t 2,

= CB/r ~. As shown in appendix C, the (~ geodesic force* (per uni t mass) acting inwards takes the simple form

(17) a2 Q (1 2a2 . +r , ,

Let us consider expression (17) for part icular cases so as to unders tand the significance of the various terms appearing in it. Firs t we take Q = 0 ~ a. In this case the (~ force ~) reduces to m/r ~, the usual Newtonian force with no modifications.

The case of the charged black hole is al together different. In the absence of rota t ion (a-~ 0), expression (17) becomes m/r ~ - Q~/r 3, which has no Newtonian analogue. Thus, apar t f rom the gravi tat ional a t t rac t ion of the point mass, there is what we shaU call a Reissner-Nordstrom repulsion; as explained in sect. 3, the repulsion away from the black hole is due to an out- ward a t t rac t ion by the electric energy in the field around the tes t particle. Since it goes as Q~[r ~ and not Q2[r2, i t will not (( cancel out ,) with the a t t rac t ion from the other side and hence a net pull outwards remains. Clearly, the Reissner-Nordstrom (~ electrogravitic ~> repulsion, which is a new feature of the theory of relat ivity, overtakes the gravi tat ional a t t rac t ion at r = Q~/m, outside the inner horizon. Thus, for an extreme Reissner-Nordstrom black hole, the

change-over occurs a t the event horizon. For an uncharged, rota t ing black hole, expression (17)becomes (m/r).

�9 (l~-a~/r2), clearly enhancing the a t t rac t ive force due to the mass. This re- sult could have been expected by noticing tha t ro ta t ion affects geodesics b y a dragging of inertial frames.

The complete expression can be easily unders tood now. The ro ta t ion merely enhances the a t t rac t ion (or repulsion) due to the mass (or the charge). For an ext reme Kerr -Newmann black hole the (( force )> defined here, per uni t mass, a t the event horizon, is 2ad/m 5.

6. - The question of naked singularities in relativity.

So far we have been largely ignoring the case of Q # m or a ~ m so as to avoid naked singularities. Naked singularies are normally assumed to be nonphysical,

412 s.M. MAnAJAN, A. QADIR and r. M. VALANJU

as it is supposed tha t they would enable sigm~ls f rom the singularities (which are not describable by our prcscnt physics) to reach us. In part icular , one could

(~ see ~> a particle falling into the singularity and achieving an infinite energy density. I t is unfor tuna te tha t , f rom ~his point of view, any of the so-called

e lementary particles mus t be unphysieal if they are to be regarded as point particles. This fact is not given mos t impor tance because their Schwarzsehild radius is much less ( ~ 20 orders of magni tude less) t h a n the P lanek length (hG/c3) ~ - . 10 -33 era, ~md hence their gravi ta t ional effects are negligible com- pared with the vacuum fluctuations.

None of these argmnents is entirely convincing and the <~ cosmic-cen-

sorship hypothesis ,)(') tha t all singularities are <, clothed ~> by event horizons m a y be nothing more than a pious hope tha t we need not be bothered about the singularities themselves. We discard the cosmic-censorship hypothesis and show tha t the supposed problems associated with m~ked singularities are ab-

sent in the Ker r -Newmann geomet ry and m a y not arise in general. :Hence e lementary particles, like electrons, can safely be regarded as point particles. The impor t an t point to r emember is t ha t there arc electrogravit ie forces

(enhanced by spin) which must be considered. This above point is of par t icular relevance if Einstein 's world view (s) (of

regarding particles as space-t ime singularities which interact through the eur-

vat.ure of space-time) is to be t aken seriously. One of us (AQ) has considered the quantum-theore t ic implications of this world view (6). A more explicit connection between the fundamenta l s of rel~tivity and the quan tum theory is being considered. For this world view to bc iutcrnally consistent, it is nec-

essary t ha t the e lementary particles be describable as singularities. Let us now consider an electron as ~ spinning, charged, gravi ta t ing point

particle. I t s Sehw~rzsehild radius is ~ l . 3 5 - 1 0 - ~ a c m , its charge (QG~/c 2) is ~--1.38.10 -a~ era, while its angular m o m e n t m n per unit m~ss ale is half of its Compton wave- length ~ 1.93.10 - n era. I t is clear t ha t a >>Q >>m in this ease. Thus, whereas the mass of the electron cannot be considered gravi ta- t ionally, it is not valid to ignore the curvature of space-t ime due to the electron

charge and spin. The force due to a Ke r r -Ncwmann singularity with negligible mass is

- - (Q2/r2) (I 45 2a~/r~-), which is entirely repulsive. For r>>a, the force ~ - Q2/ra, which is indeed very small for an electron. However , for r << a, the force is approxim~tely -- 2a~-Q~/r s ~ 1.4~9-10-aP/'r s, which rapidly increases as r de-

ere:~ses. I f we assume tha t the eleetrogravit ie repulsive force between an

(4) See, for example, R. P~\-ROSE'S article in Physics (~(1 (~o~temporary Needs, Vol. ], edited by RL~ZUDm~ (New York, N.Y., 1977). (~) A. EInSTEIn-: Albert Einstein: Philosopher Scientists, edited by P. A. SC,LIPP (The Library of Living Philosophers, Tudor Publishing Company, 1951). (8) A. QAmn: A particle i~terpretation o] the qua~tum ]ormalism (unpublished).

R E I N T R O D U C I N G THE CONCEPT OF (( :FORCE )) INTO R E L A T I V I T Y T H E O R Y 413

electron and a positon is simply twice the above value (which m a y not be a very good approximation), it exceeds the Coulomb a t t rac t ion at r,-~ 4.65. �9 10 -26 cm. For a p ro ton being probed by an electron this effect occurs at r ~ 3.12.10 -28 em, as the more massive pro ton has a shorter Compton wave- length. Thus, for shorter distances, there is a ne t repulsion between the op- positely charged particles. I t should be noted tha t this effect is of relevance in the grand-unification scheme of strong, weak and electromagnetic forces, as the interact ion distances talked of there (7) are ~ (1028--10 -26) em. General- relativistic effects cannot be ignored there.

We see now tha t , in terms of the orbit (or geodesic) of the tes t particle, there would be a repulsion from a naked singularity. Therefore, we could still not see a part icle falling into a singularity. Moreover, nothing could emerge out of the singulari ty due to the infinite force at the singularity. (Note tha t the force could not be repulsive as tha t would break up the singularity.) Thus, even wi thout censorship, a Kerr -Newmann singularity is effectively c lo thed- - a sort of censorship wi thout censorship. We conjecture tha t there is no need for censorship of any ~ naked singularities ~) as they are repulsive instead of being a t t r a c t i v e - - a t least when one gets sufficiently close to them.

I t appears tha t the e lementary consti tuents of m a t t e r (whatever they m ay be) can be consistently regarded as space-time singularities. The more massive they are (up to a point), the closer can they be approached by a probe particle (as a----h/2mc), the distance of approach vary ing as m - t in our model. I t would be interest ing to see if this expectation~ fits with observation.

7. - Summary and discussion.

We saw tha t for a freely falling observer, (( forces ~ acting on him can be operationally defined by using the device depicted in fig. 1. This force is usually not considered in re la t ivi ty theory, as the size of the device is generally much less t han its distance f rom the gravitat ional source. However , the t idal force displays a surprising amount of s tructure, giving a deeper insight into the working of relat ivi ty theory and leading us to consider forces in terms of radial acceleration of a tes t particle. The relativistic analogue of the Newtonian force (which we have called the geodesic force) thus obtained also displays struc- ture, which makes (( naked singularities ~) physically acceptable.

The geodesic forces bring out an impor tan t feature of general relat ivivity, viz. the eleetrogravitie forces and the enhancement of forces by spin. These forces do not have a counterpar t in classical physics. I t is clear t ha t a fully relativistic quan tum field theory must contain a coupling between charged

(7) See, for example, S. W~INBERG: Phys. Today, 30, 42 (1977).

414 s. 5i. MAHAJAN, A. QADIR and r. M. VALANJU

and neutral m a t t e r to deal with interact ion distances --~ 10 -~-5 era. I t is in-

t r iguing to ponder on a possible connection between the a,pproaeh t aken here and ~ grand-unificat ion scheme of all interactions.

To sum up, there arc four ma jo r points made in this paper : 1) the useful- hess of the concept of <, force )) in relat ivi ty, 2) the absolute necessity of con- sidering general re la t iv i ty in a unification scheme, 3) the possibility of cen-

sorship wi thout censorship as a means of main ta in ing the feasibility of naked singularities, and 4) tha t re la t iv i ty theory is a theory of a c c e l e r a t i o n and not merely a theory of gravitat ion.

A. QADIR expresses hir gra t i tude to the Council for In te rna t iona l Exchange of Scholars under grant :No. 78-169-A. The work of P. M. VA~A~aV was

suppor ted by the U.S. D e p a r t m e n t of Energy under Contract No. EY-76- 8-05-3992.

A P P E N D I X A

To derive the geodesic equat ion for 0, we evaluate all the nonzero Chri-

stoffel swnbols Now the components of the inverse metr ic tensor, a b "

g% for the Ker r -Newmaam metric, given in eqs. (1) and (2), are found to be

(A.1) g o o = C E r - ~ , gOa=BCr-2 ' g 1 1 = _ _ C - , , g22____D-1, g a a = _ _ A C r - 2 ,

where C-~,r a is the value of the de te rminant . Hence the required Christoffel symbols are

(A.'2 )

2 1 D _ t A o , 0 0 '2 '" 0 3 7, ,~,

= - - - D - 1 C o ~ = - D - 1 D , 1 , 1 1 ') '~ ' 1 2 2

2 2 = '2 ' - ' 3 3 = 2 D - 1 E ' ~ "

Thus the 0 geodesic equat ion

becomes

1D 65 1 E ~ = 0 (A.4) "O + D - I ( - - � 8 9 1 8 9 D, , i 'O @ 2 ,2 - - ~ , v r J �9

R ~ I N T R O D U C I N C ~ TIL~ C O N C E P T OF (~ F O R C E ~) I N T O R E L A T I V I T Y T H E O R Y 415

l~ow, taking the initial value of 0 as ~/2, we note that the derivatives of A, B, C, D and E with respect to 0 vanish initially, as they all depend on sin20 or cos ~ 0. Thus eq. (A,4) becomes

(A.5) ~i+ D-1D,,~0 ---- 0 .

Now, if the initial value of 0 is taken to be zero, i.e. the motion is started in the plane of rotation, 0" is initially zero. Thus it must always remain zero, i.e. the motion, once started in the plane of rotation, always remains in the plane of rotation.

APPENDIX ]~

To evaluate A 1, the only nonzero component of our acceleration vector, we need to evaluate the component of the Riemann-Christoffel tensor

~oll ~1 ~ {010}1_/01 }.~{o 11}{0 o0}_{ol 0/{0 ~ The Christoffel symbols appearing in eq. (B.1) are

(B.2)

{1}1 1}=o 0 0 ~ -2 C-1A'I ' 0 1 '

{~}~ {~} ~1~ 0 3 ~ 2 C-1B'I' 1 1 = 2 '

{1 1 2 } ~ 0 ' {1 1 3 } = 0 '

1 1 C(BA,I__ABn)r_ 2 {0 0 1 } ~ C ( E A a + B B , , ) r - ' , {0 3 1}----~

Thus eq. (B.1) reduces to

(B.3) R*olo= ~C A,n--

- - J~ [(EA., + B B a ) A ,, + ( B A a - - A B , OB, t ] r - ' ,

bow, inserting the values of B, C and E, given in eq. (5), into eq. (]3.3), we obtain

(B.4)

Since

(B.5)

R'olo = �89 ~- a2/r~)A,n - (a'l r3) A,1] .

I A,, = 2(m/r~--Q*/r~) , A,,1 ~ 2 ( - - 2 m / r ~ + 3Q~/r4),

416 s . M . MAHAJAN, A. QADIR and P. M. VALANJU

eq. (B.4) f u r t h e r r educes to

(B.6) R~o~0 ---- - - (1 - - 2 m/r @ Q2/r 2 + a2/r 2) (2rely 3 - 3Q2/r ~) -

- (a~/r~)(m/r~- q2/r~).

On e x p a n d i n g t h i s e x p r e s s i o n for R~o a n d i n s e r t i n g i t i n to eq. (4), we o b t a i n eq. (6).

A P P E N D I X C

To o b t a i n t h e t, ~, r geodes ic e q u a t i o n s for t h e K e r r - N e w m a n n g e o m e t r y , we n e e d t h e fo l lowing (nonzero) Chr is tof fe l s y m b o l s , no t g i v e n in eqs. (A.2) a n d (B.2) :

(C.1)

cD1 1 3 = 2 C ( E B ' ~ - - B E ' ~ ) r - 2 ' 2 2 2 '

3 3 - - - - 2 C - 1 E ' * ' 1 3 ~- '~ ' "

I n s e r t i n g eqs. (A.2), (B.2) a n d (C.1) in to t h e geodes ic e q u a t i o n s

}2b~P = (a =: O, 3) , 6t 1

(C.2) ~ ~- b c 0 )

we o b t a i n t h e f i rs t two of eqs. (15). Now, f r o m s y m m e t r y c o n s i d e r a t i o n s t h e r e a re two i n v a r i a n t s of t h e m o t i o n :

(C.3) Po = ~: , P~ ---- 55,

wh ich a r e t h e ene rgy a n d a n g u l a r m o m e n t u m (of t h e t e s t p a r t i c l e ) (( as seen f rom in f in i t y ~). Thus t h e c o v a r i a n t v e l o c i t y c o m p o n e n t s a r e

(c.4) 2 o - - - - p 0 / e = ] ,

23 ---- ~v = ct~ (say) .

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

(c.5) i - - - 2 0 - C - , ( E + B r ) r - : ,

c~ ~ 2 2 C - ~ ( B - - Aq~)r-~.

W i t h a l i t t l e b i t of a l g e b r a t h e s e e q u a t i o n s can be seen to s a t i s fy t h e f i rs t t w o of eqs. (15).

R E I N T R O D U C I ~ C T T H E C O N C E P T OF (~ F O R C E )) I N T O R ] ~ L A T I V I T Y T I L E O R Y 4 1 7

The radia l geodesic equat ion is

(C.6) ~-~ �89 � 8 9 1 8 9

The t e rms [ and ~b have a l ready been eva lua ted in t e rms of the met r ic com- ponents and ~ is ob ta ined b y rewri t ing the met r ic equat ion, eq. (1), as

(C.7) ~2 = (At2 _~ 2 B l ( o - E~o 2 - 1 ) C -1 .

(Recall t h a t 0 = 0 in the case under consideration.) Inse r t ing eqs. (C.5) and (C.7) into eq. (C.6), we ob ta in the last of eqs. (15). We replace now B, C and E in eq. (15) b y the i r values in t e r m s of A, as given in eq. (5), to ob ta in

(C.8) ~*-~ �89 2A/r)(r2-~- 2a2--a~A)2-~ -

-~ 2aa{(a2[r2)(1 - - A ) ~ (1 -~ a~/r2)A,1}(1 - - A ) ( r ~ + 2a 2 - a~A) +

+ {a~( 2 -t- a ~ / r ~ ) A , ~ - - 2 r ( 1 - a4/r~)A-- (4a~/r ) (1 + a~/r~)}(1- A)~].

�9 (a s ~- r~A) -~ ~- (A,~--2a~/ra)} = O.

After some tedious algebra, this ex t remely compl ica ted looking equat ion reduces to the beaut i fu l ly s imple one

(C.9) ~ - - (m/r2)(1 + a~/r ~) - - (Q2/r3)(1 ~- 2a/r ~) = o ,

giving us expression (17) for the force defined in t e rms of the geodesic.

�9 R I A S S U N T O (*)

Si suggerisce chela reintroduzione delle ~( forze )) nella teoria della relativitg pub fornire nuove possibilitg e risultati. Un esame della geometria di Kerr-Newmann, e di aleuni casi speeiali di questa, da questo punto di vista indica che, in generale, ei pus essere una repulsione a eorto raggio. Questa repulsione suggerisce che le (( singolariti~ nude ~ pos- sono essere fattibili dal punto di vista fisico. Si trova anche che e'~ una (~ repulsione gravito-elettriea )> ehe sarebbe importante considerare in uno schema di grande unitlea- zione delle forze forti, deboli ed elettromagnetiehe.

(*) Traduzione a cura della l~edazione.

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