Nuclear Physics 8273 (19X6) 73274X NorthHolland. Amsterdam
SYMMETRIC SOLUTIONS TO THE MAXIMALLY
GAUSSBONNET
EXTENDED EINSTEIN EQUATIONS* James T. WHEELER The Utliwrsitv
of Chiwgo Depurmlerlr of Plq~sits and G~rico Fermi Imtitute. Chiwgo,
IL 60637. USA
Received 9 January 19X6 (Revised 5 March 19X6)
The most general gravitational lagrangian which can be constructed from the curvature twoform. the vielbein oneform. and tensors invariant in the tangent space is a linear combination of dimensionally extended Euler characteristics. Several recent studies indicate that superstring lagrangians include such terms. In an arbitrary number of dimensions. with arbitrary torsion. we show that in the most general such extended theory the only static, spherically symmetric. massive solutions to the variational equations of motion contain gravitational singularities. The existence of an event horizon is proved for certain cases. and a bound on the location of singularities is found. A certain class of nonasymptotically flat solutions is found for which the metric and the torsion are not entirely determined by the field equations.
1. Introduction Consider
the class of all densities _yk =,g+$~~J
of the form* ..
If there are no factors of the vielbein,
.gyf,s,. . d’Eohul..rjg...h. these densities
become
(1) the total divergences
whose integrals are proportional to the Euler characteristic. When there are one or more vielbein forms present, they are known as dimensionally extended Euler characteristic densities. Such densities are no longer total derivatives, and may therefore serve as lagrangian densities in gravitational field theories. Several properties suggest them strongly for this role: (i) The firstorder density, _E”. is the usual Einstein lagrangian, while the lowest order density, PO, gives rise to a cosmological constant. Theories involving FL therefore make close contact with the usual theory of gravity. * Supported
in part
by the NSF, grant #PHY83.01221. Presented as a thesis to the Department of of Chicago. in partial fulfillment of the requirements for the PhD degree. tensor. yk *&“I” is the curvature Zform. p 4 is the vielbein lform and F, ,, is the LeviCivita contain5 k factors of &#‘lh and enough factors of the vielbein to saturate the remaining indices on the LeviCivita tensor. The wedge product is assumed between forms. Physics. The University
0550.3213/X6/$03.50 Elsevier Science Publishers (NorthHolland Physics Publishing Division)
H.V
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(ii) 5°~ is the only parity conserving lagrangian constructible from k factors of the curvature twoform, the vielbein oneform, and invariant tensors in the tangent space. (iii) All of the densities .£Pk are free of ghosts ([1], [2]), ie., there are no negative norm eigenstates in the quantized theory. All are divergences to lowest order in perturbation theory. (iv) .L~': arises as the order oe' correction in the lowenergy expansion of string models ([1]+[3], [4]). It is likely that higher order characteristics also occur in string theory *. We therefore consider the most general lagrangian composed of extended Euler densities /( m
~   E akL~'~ ,
(2)
k=O
where k,, is the integer part of ~(D  2) and the '~k are arbitrary constants. The series terminates at k,,, since ,.C,ek becomes a total divergence for 2k = D and vanishes for 2k > D. For generality the cosmological term c o is included. Solutions to the system with k,,, = 2 have been studied previously. Static, spherically symmetric solutions have been found by Wheeler [5] and by Boulware and Deser [6]. Boulware and Deser also studied the stability of the two classes of solution which arise due to the quadratic nature of this case. Cosmological solutions have been studied by M~llerHoissen [7] and by Wheeler [5]. In this work, solutions are found for arbitrary k,,, ~< D, in an arbitrary finite number of dimensions. The possibility of nonvanishing torsion is included. It is shown that the only static, spherically symmetric, asymptotically flat solutions to the variational equations of motion contain gravitational singularities. In sect. 2 below, the full lagrangian ,LP (eq. (2)) is varied and the resulting equations of motion are evaluated for the case of a static, spherically symmetric vacuum. The main results of the paper are proved in sect. 3. The full asymptotically flat solution, represented as a power series, is derived and shown to be singular in all cases. The existence of an event horizon is proved for certain cases, and a bound on the location of singularities is found. In sect. 4, special values of the coefficients g:k are chosen for which there exist further, nonasymptotically flat solutions. Some of these solutions are shown to be geometrically free in the sense that the metric and *
Based on the presence of .,LP~ and .L/'_, in string theories, Zwiebach [1] conjectured that tile gravitational part of the full string lagrangian is composed entirely of dimensionally extended Euler densities. However, as Z u m i n o [2] has pointed out, the threegraviton vertex includes a k (~ term which cannot arise from these densities. Nonetheless, R o m a n s and Warner show that partial supersymmetrization of the Lorentz ChernSimons term requires "~2 in the purely gravitational sector and suggest that full supersymmetrization may involve .La3,£,04, r,LPs and conceivably no further terms. If this is true, then these densities may form a substantial part of the string lagrangian, and an u n d e r s t a n d i n g of their contribution to the nature of solutions may be helpful in finding solutions to the full theory,
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the torsion are not entirely determined by the field equations. The generalization of this special case to any spacetime with a maximally symmetric subspace is also discussed. The final section includes a summary of these results.
2. Static, spherically symmetric equations of motion We wish to vary (see also Zumino, [2]) both the vielbein e" and the connection ~.~' in the action integral of the lagrangian density km
*~q~= ~ ?k ~"h ''' ~<'defe~''' e%,h .... df,~ ./,"
(3)
/, = 0
Since
~°#' =
d~0"h + ~o"'Gh, the connection variation of .L,°, 8,~£f, gives
,'~ a m ~ /;',~,'d 3~.£P= ~f~xk(d3~oob + zw ow,, b'~)~t ...
~ef~eg
" ..
e I1G#>
We integrate the first term by parts, while the identity (A.1) derived in appendix A is applied to the second term. Use of the Bianchi identity D ~ "a= 0 then reduces the connection variation to ~
(_) k q
k & ~ < , ~ j <~<~, . . "
~,i<~J,... "'e<,h<.<~. . . . . S.~... ~, '
where c ,  g k (   ) k ( D  2k), and 3 . . " = De" is the torsion 2form. This expression contains k  1 factors of ~,,b and D  2k  1 factors of e ". The vielbein variation, ~.£o, of ,.~ is straightforward since ~ h is independent of the vielbein.
Vanishing of the variation of the action therefore implies o = S__,c, (  ) *~l,°@"<...
0 = Z~(

~<%s...
)*kS~ " ~ 9 " ' ~
...
(4)
,,~.~<.... <,
~jg, h..
,
. ~e g ~ a h c d e . . . f g
h...~.
(5)
Eqs. (4) and (5) are the full field equations for the system. Next, eqs. (4) and (5) are evaluated for a static, spherically symmetric spacetime. Latin indices on forms are tangent space indices, while greek indices refer to coordinates on the manifold, with the exception that d x ° = dt and d x r = d r refer to the manifold. Indices from the beginning of either alphabet (a, b . . . . or a,/3 . . . . ) run over values 1,2 . . . . . D, while indices from the middle of either alphabet (i, j, k . . . . or #, p . . . . ) other than 0 and r take only the values 1, 2 . . . . . d, where d = D  2 is the dimension of the maximally symmetric subspace.
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For a static, spherically symmetric spacetime, the metric tensor may be written in the form
_g2 h 2
gaff =
(6)
r 2 ( ~ q X.x~,/(1X2)) where x ~ = x " x . and x ~'= x.. The components of the vielbein oneform are therefore . o =gdt,
(6a)
* r  h  1 d r ,
(6b)
.i=Sji{rdx"r(l
+(1x2)l/2x~'x~dx~/x2)}.
(6c)
In the same coordinate system the static, spherically symmetric torsion twoform will be (see appendix B): ,yO= TorO( r ) d t e ~ '
(7a)
~Yr= To r( r ) dt , ~ ,
(7b)
~"i= ( T o ( r ) dt + T~(r) d r ) 2 ,
(7c)
To = ro."(r),
(7d)
Tr=Tr,~(r).
(re)
where
Finally, the nonvanishing curvature twoforms are given by ,~ij = ~R~ iJ d x ~ d x ~ = _ F ( r ).i~j,
(8a)
~ o , = Ro~O,dt dx ~,= ( g  l g , h 2 / r ) e O g ,
(8b)
~ r i = R r "ri d r d x ~ = ( h h ' / r )e~'
(8c)
.~or = Ro,Or dt d r = h g  ~( hg')'e°e " ,
(8d)
where F ( r ) = (1  h : ) / r : .
(8e)
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For later reference we also note here that the scalar curvature when g = h is given in terms of F by R =  [(d2+ 2)F+ ~(d+ 4)rr' + r2F'].
(8f)
The only components of ~50 which preserve the symmetry are ~e ° and Be". Consider first the 6e ° component. Note that Be° is proportional to dt. Therefore the 2forms .~oi and ~o,. cannot occur in 6..~°, since they each are also proportional to dt. Moreover, ~ r can occur only once since it is proportional to dr. The variation therefore has only two terms, for any value of k: 0 = Y~,ck()kS~ ij ... e k [  2 k ~ % " + ( D 
2k  1 ) ~ " % r] dteij..k,l,,~o.
(9)
The components of the volume form may be written as ele j .., e%'e ° =
g/.
k,odV '
(10)
where dV is the usual volume density. Substituting the expression in eq. (10) and the expressions for the curvature from (8) into eq. (9), then integrating over the maximally symmetric subspace, we find 0 = Y~C~rD[F(r)] k  l [ k r ( r Z F ) ' +
(D
2k  l)r2F].
(11)
Note that, for each value of k, the rdependent terms form a total derivative. Integrating term by term, with s an integration constant, gives sr~l)1) = y ' c / , F k  p ( F ) .
(az)
The variation with respect to ~, after integration over the subspace and subtraction of eq. (11), yields an expression proportional to g ' / g  h ' / h : 0 = ~ckrDP'(F)[g'/g
h'/h],
03a)
where the prime on P denotes differentiation with respect to F and h may be regarded as a function of F given by h 2= 1  r 2 F ( r ) .
(13b)
Next, we impose the static, spherical symmetry on eq. (5). We eliminate terms which vanish due to the particular structure of the curvature, torsion and vielbein forms given in eqs. (6)(8), and use the components of the volume element as before. Only four of the variations  &o,.°r, 6¢0o°', 6¢0~°" and 6¢0~ir  give nontrivial equations. The 8co °~ and 8¢0ffr equations may be further simplified by using the
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J.T. Wheeler / Einstein equations
3~, °' and &o0°~ equations. The final results, together with eqs. (12) and (13) from the vielbein variation, give the full set of equations to be satisfied by a static, spherically symmetric solution. We collect here the full set of variational equations: (14a)
sr 
0 = P ' ( F ) [ g ' / g  h ' / h 1,
(14b)
0 = P'(F)T o ,
(14c)
0 = P'(F)T~,
(14d)
0 = P'( F ) Tot ~  P " ( F ) T o r F ' ,
(14e)
0 = P'(F)Tor °  2P"(F)T,.[gXg'h2/r
 F].
(14f)
77~ese are the static, spherically symmetric field equations arising from the lagrangian in eq. (3). The parameter s, as defined in eq. (12), measures the mass of the gravitating system, since eqs. (8e) and (14a) imply that it is the coefficient of the gravitational potential at infinity. In the following section, asymptotically flat solutions to eqs. (14a)(14f) when s 4= 0 are found, and shown to be singular. When s = 0 the nature of the solutions depends crucially on the polynomial P ( F ) defined in eq. (12). Subject to certain special conditions, solutions other than flat space are possible. These exceptional solutions are discussed in detail in sect. 4.
3. Asymptotically flat solutions: a singularity theorem This section is devoted to the proof of the following results: (i) The only massive (s 4: 0) asymptotically flat solution to eqs. (14a)(14f) for any 5 ° given by eq. (3) has vanishing torsion and has a metric given by eq. (6) with
g2=h2=l
 F r 2,
(15)
where F is a solution of the polynomial equation sr ~Dx~= P ( F ) = ~ c k F ~.
(16)
(ii) A solution of eq. (16) for F always exists in a neighborhood of r = oo, for at least one value of the sign of s. (iii) The extension of the asymptotic solution for F increases monotonically with decreasing r until it terminates at small r in one of two ways. Either (a) there is a
738
J.T. Wheeler / Einstein equations
curvature singularity at the origin surrounded by exactly one event horizon, or (b) there is a curvature singularity at some finite value of r with at most one event horizon, and possibly none. (iv) If the extension of (iii) terminates at some nonzero value r = r 0 > 0, then r 0 is bounded by ro <
( O;M)M/(D1) ,
where a M is the largest coefficient in the normalized series expansion of F (see eq. (20') below).
Proof of(i). P'(F) can vanish for all r only if F is independent of r. But this would contradict eq. (14a), since s 4= 0. Therefore P ' =g 0 and it follows immediately from eqs. (14b)(14f) that g'/g  h'/h
= 0,
To= Tr= To/= To/=O.
(17) (18)
The first of these equations implies that, up to a constant rescaling of the time coordinate, g = h, while the second expresses the complete vanishing of the torsion. The only equation remaining to be satisfied after the conditions (17) and (18) are imposed is eq. (16) with s ~ 0.
Proof of (ii). Noting that a neighborhood about r = ~ corresponds to a neighborhood about F = 0, we expand F as a power series in inverse powers of r. Experimentation with a general series for F quickly leads to the conclusion that only certain powers of r will enter the final solution. In particular, let M be the smallest integer such that c M 4= 0, and let y be given by the positive M t h root of yM=r~Dt).
(19)
Then any series solution for F 0 is of the form
F(y) = ~ a,,y",
(20)
n~ 1
where the a , are constant coefficients to be determined in terms of the lagrangian coefficients c k. The restriction to n >/1 in eq. (20) is the necessary and sufficient condition for asymptotic flatness of the metric. Notice that the solution is asymptotically flat if and only if the cosmological constant c 0 vanishes. Rather than writing the explicit expressions for the a , in terms of the c k, we will demonstrate the existence of solutions quite generally. Substituting eq. (20) into eq.
J.T. Wheeler / Einstein equations
739
(16) and writing r in terms of y gives
y = + [ P ( r ( y ) ) / s ] 1/M
(21)
The form of eq. (21) makes it clear that the function F(y) defined in eq. (20) is the inverse of the function [P(x)/s] l/M, The series for F(y) will converge to a finite value everywhere that [P(x)/s] 1/M is welldefined and invertible. It follows from eqs. (15), (19) and (20) that h 2 is welldefined everywhere that the series for F(y) converges. Now consider the invertibility of [P(x)/s] a/M. The solution of physical interest is the region of small, positive x (x > 0 ensures attractive gravity). In this region P(x) is given approximately by P ( x ) = cMxM/s.
This gives a positive solution for F as long as cM/s > 0, for then =
This root always exists and is positive if sgn(s) = sgn(cM). With this choice [P(x)/s] 1/M is always invertible, since cM4:0 by definition. Therefore, an asymptotic solution always exists.
Proof of (iii). We first demonstrate that this solution always terminates in a curvature singularity. Then we study the existence of an event horizon. The solution (20) may be extended until [ P ( x ) / s ] x/M fails to exist or fails to be invertible. This can happen in one of two ways: (a) If P ( x ) / s changes sign, then [P/s] 1/M may fail to exist. This is signaled by P=0. (b) For any M, [P/s] 1/M fails to be invertible if P ' = 0. However, since P(0) = 0 and P'(x) > 0 for 0 < x << 1, condition (b) always occurs before condition (a), and we need only examine the zeros of P ' . In general, there can be zeros of P'(x) for real x > 0, and the metric will become singular at those points. By expanding P(x) about the smallest positive x 0 such that P'(xo) = 0, and computing the corresponding approximation to F ( y ) , it can be shown using eq. (Sf) that the curvature is also singular at x 0. However, if all of the zeros of P'(x) occur with x < 0 or x imaginary, then the solution m a y be extended all the way to x = ~ (r = 0). If this happens then we are guaranteed not to have a curvature singularity at any finite value of r. This follows from the horizon theorem proved in [5], together with the easily proved fact that F is monotonically increasing as long as P ~ 0.
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J. 7] W h e e l e r / Einstein equation~
We now show that even if there is no singularity at any finite value of r that there must be one at the origin. Suppose F is nonsingular for all positive r. Define N <~k,, <~½(D  2) as the largest integer for which cA..4 0. Then (16) is a polynomial equation for F of degree N. Multiplying both sides of eq. (16) by r D ~ and taking the limit as r ~ 0 gives S=cNFNr
D 1 +O(rDIFNI).
Clearly F must diverge at r = 0. Moreover, the strength of the singularity of the metric is given by h 2 = _ [S/CN]I/Nr(D2NI)/N
'
with the obvious consistency condition that sgn(c N)  s g n ( c M). It is easy to show using eq. (8f) that the scalar curvature is also singular. N o w consider the existence of an event horizon. If the only singularity occurs at r ~ 0, then since h 2,  o o as r ~ 0 and since h 2, + 1 as r, oc, there must be some finite value r 0 of the radial coordinate at which h, and hence g, vanish. This condition, together with the monotonicity of F, guarantees the existence of a single event horizon, located at r o. A singularity at a nonzero value r = r 0 m a y be naked, since it is possible that P'(ro) = 0 when r~F < 1. If ro2F> 1, there is exactly one horizon, by the m o n o t o n i c ity of F and the theorem of ref. [5].
Proof of(iv). It is possible to estimate the position of the onset of singularities in the solution (20). Given values for the a,, in terms of the c,, the series will converge for all y within the radius of convergence of the series. Since a solution exists near y = 0, the a,, must be bounded. Using a M to set the scale, eq. (20) m a y be rewritten in dimensionless variables as (am)2/'D3'F(~) ~ ~"~(a,,y)" <~~'~",
(20')
where a,, ~ [ a2M/( D 
and y is y times the maximum,
3)a,,] ~/"
aM, of the a,,. This means that the solution is
J.T. W h e e l e r /
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n o n  s i n g u l a r at least for ~ < 1, or
r> (aM) M/{D1). F o r typical string theories, M = 1 and D = 10 or 26. T h e scale a M will be on the o r d e r of the usual gravitational r a d i u s if the t h e o r y reduces to the Einstein theory, since q u a d r a t i c and higher order terms have coefficients on the o r d e r of a p o w e r of the P l a n c k length. This c o n c l u d e s the p r o o f of the claims s t a t e d at the b e g i n n i n g of the section. W e e n d the section with an example. F o r c I ~: 0, we list explicitly the values of the first few coefficients a,, in terms of the c k. E x p a n d i n g F in p o w e r s o f r ~D~) we find ~l I ~ S I C 1 ,
a 2 = (1/c,)[c2a2o],
a , =  ( 1 / c , ) [ 2 c 2 a o a I + c,aa] , a4 =  ( 1 / q ) [ c 2 a 2+ 2c2aoa 2+ 3c,a~a~ + c4a~], a, = (1/q)[2c2(a0a
, + a,a2) + 3 c , ( a o a ~ + aoa2) + 4c4a~a~ + csa~ ] ,
. . . etc. All o f the a,, with n > N are d e t e r m i n e d in terms of the a,, with n < N. N o t e t h a t the o n l y c o n d i t i o n s on the existence of this s o l u t i o n are that the c o s m o l o g i c a l c o n s t a n t vanish, a n d that c x 4= 0. W h e n c~ = 0, b u t c 2 ~ 0, the e x p a n s i o n m u s t be in p o w e r s of r (D~)/2, a n d so on, as d e s c r i b e d above.
4. Geometrically free solutions W h e n s = 0, the p o l y n o m i a l P ( F ) m u s t vanish for all F. This can h a p p e n o n l y if F(r) = a = const, so t h a t h 2 =
U n l e s s a = 0, choice h 2 = 1 than only D square of the
1

ocr 2 .
such solutions are not a s y m p t o t i c a l l y flat. N o t i c e that the p a r t i c u l a r  ar 2 gives the metric in eq. (6) m a x i m a l s y m m e t r y on D  1 rather 2 spatial d i m e n s i o n s . T h e c o n s t a n t a is seen to be the inverse of the r a d i u s of c u r v a t u r e of the m a x i m a l l y s y m m e t r i c subspace.
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J.T. Wheeler / Einstein equations
We now examine the field equations with s = 0 and F = a. Eqs. (14) now take the form:
O=P(a),
(14a')
0 = P'(a)[g'/g  h'/h],
(14b')
0 = P'(a)T o ,
(14c')
0 = P ' ( a ) T,.,
(14d')
0 = P'(a)Tj,
(14e')
0 = P'(a)To~ °  2 P " ( a ) T ~ [ g  l g ' h Z / r  a].
(14f')
If P'(a) ~ 0 then eqs. (15) and (16) follow as before and spacetime is torsion free with g2 = h 2 = 1  ar 2. In particular, for a = 0, we have flat space while for a > 0 ( a < 0) we have (anti)de Sitter solutions. However, suppose P ' ( a ) = O.
(22)
Then eqs. (14b')(14e') are satisfied for arbitrary values of g, T0, Tr and To~r. The remaining equation requires one of the three conditions T~=0, g = const, = o.
(23)
Regardless of which of these holds, there remain components of the metric or torsion which are not determined by the field equations. All of the solutions of this type require particular relations among the coefficients ck. As soon as any geometric quantity  metric component g or torsion component To, Tr, Tot° or T o /  is left arbitrary, it becomes necessary to specify at least one of the coefficients c k in terms of the rest. Solutions with one or more geometric variables undetermined by the variational equations will be called geometrically free. Let us examine the geometrically free solutions. This occurs when P', and possibly also P " , vanish at a. Let r be scaled so that a = 1. Then each of the conditions on P determines one of the coefficients ck in terms of the rest. Therefore, for D < 6, no solutions of this type exist, while for D = 6, a cosmological constant is required. In particular, the Einstein theory does not have free geometry solutions since eqs. (14a') and (22) together require Co= q = 0 . D   8 is the smallest dimension for which all three conditions (14a'), (22) and (23) can hold simultaneously.
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To better understand the nature of geometrically free solutions we return to the original class of lagrangians and consider the subclass which in even dimensions may be written in the form:
..,qg=(R"heqe%h)(R"da2e'ed)...(RefaD/2ee e f )e,,b, d .... /,
(24)
where the ~, are constants related implicitly to the ?k by k = 1I ( x 
Odd dimensions are treated by inserting an extra vielbein oneform in .L,°, contracted onto the extra index on the LeviCivita symbol. All lagrangians of the form of eq. (24) admit geometrically free solutions as long as at least three of the a~ are equal. For then every term in the variation of .£,o contains at least two factors of (R o h  aeOeb), where a is the common value of the a~. This allows the solution R ab = o t e a e b
to sa'tisfy the field equations, even if it holds on only a D  2 dimensional subspace. In general, let n be the number of ai which are equal. Then whenever the (even) dimension of the maximally symmetric subspace is greater than D  2n + 2 there will be a free part to the metric. The metric tensor can be block diagonal of the form
(25) g~h =
0
s,j(y)
'
where s~j is the metric of a maximally symmetric subspace with curvature 2form given by RiJ = oteie j .
The field equations will be identically satisfied, whatever the form of g~,,(x). Notice thatg~,, could equally well be a function of y2 =yiy, as in eq. (14). The important thing is that the curvature tensor should be block diagonal, vanishing unless all four of its indices take values in the maximally symmetric subspace or all four take values in the free space. This is sufficient, given the presence of the LeviCivita tensor, to guarantee that one of the factors (R oh ae"e h) always vanishes. It is easy to see that the conditions (14a'), (22) and (23) are equivalent to demanding equality of three or more of the a k in eq. (24), so that the solutions with those conditions are special cases of this more general approach. We now consider briefly the stability of the free geometry solutions. Boulware and Deser [6] have already noted that the effect of higher order GaussBonnet terms
744
J.T. Wheeler / Einstein equations
varies, depending on the slope of the polynomial
at its real roots. They observe that the quadratic GaussBonnet term cannot produce a stable antide Sitter branch, while higher GaussBonnet terms produce both stable and unstable solutions, depending on the values of the parameters ~kUnfortunately, it is difficult to say more than this for the free solutions. Because the metric is partly undetermined, no complete statement can be made about the energy of the solutions. However, we can at least write an expression for the stressenergy tensor. If we separate out from the variational equations the usual Einstein tensor, G,,h, and call the remaining part the stressenergy, then since the metric in eq. (25) is a solution to G,h = Tub, we can find the stressenergy by substituting eq. (25) into G,~: T ~ = G~,, = R ~ , , ( x )  ~_&,,(x)( R~,~'(x)  D ( D  1 ) a ) , T,./ = G,2 = ½[(D  1)(D  2)a  R~"(x)] g , . / ( y ) . Since g , ~ ( x ) is arbitrary, the classical energy associated with the generic solution will not be positive definite, and will therefore be unstable. Only if some physical principle beyond the variation of the action restricts the class of solutions further could these solutions be admissible. For example, it is easy to show from the fact that the lagrangian in eq. (24) factors into two parts ,L°=.~r,_eeA°~ymm~t,.~ , that a sufficient criterion for stability is definiteness of the sign of ~free. Finally, we note that whenever the Einstein equation is satisfied on the free space, Two is timelike and the overall sign of T,a, is the same as the sign of a.
5. Summary In an arbitrary number of dimensions, with arbitrary torsion, we have found static, spherically symmetric solutions to the most general gravitational theory in which the lagrangian is composed entirely of dimensionally extended Euler forms (eq. (3)). All of the asymptotically flat solutions with nonvanishing mass parameter (s ¢ 0) have curvature singularities. All of the remaining solutions ( s = 0 ) have metrics with additional symmetry. These remaining solutions include flat space, de Sitter and antide Sitter spaces, and a special class we have called geometrically free solutions.
J. 7". Wheeler / Einstein equations
745
For the singular (s ¢ 0, asymptotically flat) solutions the following results hold: (i) The solution is unique and has vanishing torsion and a metric given by eq. (6) with g2 = h 2 = 1  Fr 2,
where F is a solution of the polynomial equation sr  ( 0  1 ~ = P ( F ) = ~_,ckF k.
(ii) A solution of the equation for F always exists in a neighborhood of r = o~, for at least one value of the sign of s. (iii) The extension of the asymptotic solution for F increases monotonically with decreasing r until it terminates at small r in one of two ways. Either (a) there is a curvature singularity at the origin surrounded by exactly one event horizon, or (b) there is a curvature singularity at some finite value of r with at most one event horizon, and possibly none. (iv) If the extension of (iii) terminates at some nonzero value r = r 0 > 0, then r0 is bounded by ro <~ ( a M ) M/(O1),
where a M is the largest coefficient in the normalized series expansion of F. Notice that the possibility of naked singularities is expected, since the quadratic and higher terms in the lagrangian can always be thought of as source terms for the usual Einstein theory and suitable choices of the coefficients can always make the resulting effective energy density negative. However, these solutions are qualitatively different from negative mass Schwarzschild solutions, since they all have negative gravitational potential (  r 2 F < 0) at infinity. For s = 0, possible solutions include fiat, de Sitter and antide Sitter spacetimes. In addition, there are special values of the coefficients c k for which there exist geometrically free solutions, characterized by the existence of a maximally symmetric subspace and a curvature tensor which is block diagonal. Thanks are due to Professor P.G.O. Freund for helpful discussions and his careful reading of early drafts of this paper.
Appendix A MATRIX P R O D U C T MANIPULATIONS
We prove here a matrix identity used in sect. 2 to reduce the variational equations. The proof is a generalization of one given by Zumino [8] for even
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dimensions. Let Mi~, i = 1, 2 . . . . . n be a collection of vectors and N °b an antisymmetric matrix, and form the product
A = M1~M2h... M,,'e~b .... . Now consider the effect of the rotation produced by exp()~Nah). Define Mi~()~) = [exp()~N)] a,.M[(0). Since A is a scalar, there will be no net effect of the rotation, so A(0) = A ( ; ~ ) = m ~ o ( ~ , )
• • •
M,,b(~)[exp()~N)] 'a
" " •
[ e x p (  ; ~ N ) ] ~b E c . . ,
d "
But e ~ / g h is rotationally invariant, so the group elements acting on it have no net effect, leaving
A(0) = A ( x ) =
....
The derivative of A with respect to )~ therefore vanishes. Explicitly taking the derivative, then setting ~, = 0 gives the identity
[ NadMlJM2b... M,,c+ MlaNbdM2d... M,C+ . . . + MlaM2b... N'dM,, d] eab .... .= O. (A.1)
The identity still holds if some or all of the M ' s are differential forms, as long as appropriate signs are inserted. Also, any pair of vectors may be replaced by an antisymmetric matrix with the inclusion of a factor of two. For example, if M1 ~ and M2 b are replaced by M ~h then eq. (A.1) is easily shown to reduce to [2 N"eMebM3"... M," + M~bN'eM3e... hi,, d + . . . + M"bM3C... N d M , ~] e,,hc ..d = O. The relationship holds in any number of dimensions.
Appendix B
STATIC,
SPHERICALLY
SYMMETRIC
TORSION
A symmetry in a spacetime is a consequence of the existence of a Killing vector field which at each point gives the direction of the symmetry. For a tensor field to possess the given symmetry then means that its Lie derivative in the direction of the
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Killing field vanishes. For the torsion this gives (B.1) where a comma denotes a partial derivative and all indices (a . . . . ) are world indices taking values 1,2 . . . . . D. Indices from the center of the alphabet (i .... ) will take only values 1, 2 . . . . . d = D  2, with the remaining indices denoted by 0 and r. A semicolon will be used to denote covariant differentiation. First, consider the effect of the Killing fields in the time direction and in the coordinate directions of the maximally symmetric subspace. Given a value for T,h , at any location in spacetime, these fields may be used to find the value of Tub,. at any other point on the orbit of one of these Killing fields by Lie transport. This includes all points of spacetime with the same radial coordinate r, so T~h,. can be a function of r only. Now consider the remaining (rotational) Killing fields of the maximally symmetric subspace. Following Weinberg [9], we consider a point P, at which ~i, but not ~: s., vanishes. Then at P ~,(P) = O, gives 0 = ~i: j(P) + ~j: i(P) = ~ i , j ( P ) + ~j.i(P), ,~i: j(P) = g i k ( P ) ~ k , j ( P ) . These relations imply that ~i.j(P) is an arbitrary antisymmetric matrix. We may therefore write eq. (B.1) at P as 0 = ~t~oTJab,. + ~ t ' b T J l , +
~I'cT~f.
(B.2)
When a, b and c all take values on the maximally symmetric subspace, it is easy to show from eq. (B.2) that T,.jk= 0 except when D = 5. The exception arises because when D = 5 the maximally symmetric subspace admits the rankthree totally antisymmetric invariant tensor field e~jk. This allows the possibility T~jk = T(r)e~j k. Most of the remaining components of the torsion can also be determined from (B.2). For example, we compute ToiJ as follows. Let a = k, b = 0 and c = l in (B.2). Then 0 = 8t'kTJlol + 8t'tTkd ] . Contraction on i and k yields, after lowering indices
0 = ( d  1) 0, + T,o j  gj, 0'
(B.3)
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The antisymmetric part of To, j must therefore vanish unless d = 2, when it m a y be proportional to e,1. The part symmetric on i and j is given by: To~, , , = ( 1 / d ) gi,To~ k = ( 1 / d ) gi /To ,
(B.4)
where T0 is an arbitrary function of r. With To, ~= T0¢,j ~ as given by (B.4), eq. (B.3) is identically satisfied. The full static, spherically symmetric torsion tensor can be shown in this way to have c o m p o n e n t s T/j j,. = 0,
(B.5a)
(B.5b)
= T,/0 = 0 , Tlo 0 = Tir r = Tio r =
Tro i =
Yir 0 = O,
(B.Sc)
To,
(B.6a)
T~,v. = _ TirJ = ( 1 / d ) g i j T r ,
(B.6b)
Vo,j =  T, oj = ( 1 / d ) g ,
Tr= T~(r), Trot = Tror( r ) ,
To=To(r),
Voro = Toro( r ) .
(B.7)
(B.8)
Eqs. (B.5) through (B.8) give the full consequences of static, spherical s y m m e t r y for the torsion.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
B. Zwiebach, Berkeley preprint UCBPTH85/10 B. Zumino, Berkeley preprint UCBPTH85/13, LBL19302 L.J. Romans and N.P. Warner, DOE R&D Report and Caltech preprint CALT881291 J. Scherk and J.H. Sehwar'z, Nucl. Phys. B81 (1974) 118 J.T. Wheeler, Nucl. Phys. B268 (1986) 737 D. Boulware and S. Deser, US DOE preprint 4004827 P5 F. MfillerHoissen, Max Planck Institute preprint MPIPAE/PTh 45/85 B. Zumino, lecture notes, Berkeley (Spring, 1982) S. Weinberg, Gravitation and cosmology (Wiley, New York, 1962)