- Email: [email protected]
Effective gravity theories with dilations
~vblum.e 175, number 4
PHYSICS LETTERS B
14 Augue~ 1986
EFFECTIVE GRAVITY T H E O R I E S W I T H DILATONS David G. BOULWARE
:
Institute for Theoretical Physics, University of California, Santo Barbara, CA 93_t06,USA
and S. DESER Department of Physics, Brandeis Univers#y, Wa#ham, MA 02254, USA Received 20 May 1986 inclusion of the dilate: in the Eins:ein p:us Gauss-Bonnet model removes the de Sitter ground state pe:.~tted :,n its absence; ~he genetic case with higher order ir.variar.,'.s is discussed. Spherically symmetric static so:ufions are conNdered and ~heir asymptotic behavior established; they are nor.trivial only for a nor, vanishing dilaton field and are compafib.~e with the existence of an horizon at wbAch the diiaton is regular.
Recent progress in string theory has stimulated work on extensions of Einstein gravity inv'otving higher powers of curvature [1-6] wNch emerge in the slope expa~.siono Of particular interest is the leading q~adratic correction proporfionaI to the Gauss-Bonnet combination [7-9], whose terms quadratic in the de~atio,a from the flat metric vanAsh and therefore do not induce ghost graviton modes it: the propagator. :n a pre'~ious paper [1t, we studied this model and found its spherically symmetric solutions; the asymptofically fiat solutions have a structure quali~tively similar to the usual Schwarzschild metric: there is still an event horizon (at a smailer radius) and a (weaker) singuo larity a~ the origin. On the other hand, there are cosmological (de Sitter) solutions in addition to Minkows_V5 space~ with curvatures set by the string constant scale. De Sitter geometries are ~nstable (all sinai! excitations of the graviton have negative energy, h e n ~ o r d i n a u matter would be unstable against emission of these quanta) but higher order terms in the curvature could give rise to stabIe de Sitter soiutions. The string expansion, however, involves mass: Permanent address: Department of Physics FM-15, University of Washington, Seattle, WA 98195, USA.
less fields other than the grunion, the most releva~:t being the dilaton ~ which appears in the combination e ~ R ' + : as welt as in a kinetic (~e) 2 term. The purpose of tNs letter is to study the influence of the ditaton on the effective gravita° tional action both through its own field equation and in the gravitational equations. For spheficalIy symmetric solutions, a is found to vanish at spatial infimty i~ such a way as to !cave the asymptotic flatness unaffected. Although we cannot find the sotution explicitly, we can obtain its asymptotic form. We aiso show that it is (locaily) consistent with the existence of an event horizon, and that the ditaton is regular there. The existence of an horizon is a global question, which (in the absence of positivity conditions on the effective stress tensor) cannot be estab:ished without integrating in from spatiaI infinity, it is also impossine to d e t e r ~ n e completely the nature of the singuiafity at the origin by purely iocai considerations, so we have not attempted to list the possibilities there. The effect of the diiaton on the cosmo:oglca~ solutions is dramatic: they are excluded at the quadratic curvature level Since spacetime is ma×5really symmet:c, the diiaton must be a constant and both the gravitational and diIaton eqt~ations
0370-2693/86/$03°50 © Elsevier Science Publishers BoV, (North-He!land Physics Publishing Division)
409
14 August 1986
PHYSICS LETTERSB
Volume 175, number 4
constrain the cosmoiogicai constant A. Both equations may be expressed in terms of a singb function f ( A ) . I.o. the general case, a solution with nonvanisNng A wilI exist only if f has a double zero at A o The action including the diiaton fietd is
is the value of the :agraagian ~ / ¢ - g . Here, cpA p+: is the value of X '(p+~) at the Sitter space with curvature A. The ~ equation reads aD! (D-
o
A2+
Z
papCpAp+I
--i
f
,R
+
=
(:)
'" . . . .
p~2
]
where
The ~"P~ terms invoive appropriate powers of the curvature and its covariant derivatives. The factor Q, is the volume of the unit (D - 2)-sphere, G is Newton's constant in D dimensions, wi@_ units of L 17-2, while those of % are L -D+2". We use units in wNcb 2 ~ G = 1. It is straightforward to study the equations for cosmoiogicai solutions. The dilaton field can be shown to to be constant either by maximal symmetry arguments or by expiicit use of the field equations; by rescaling % it may then be set to zero. [Even the singular Iimk e - ~ - o o does not admit a de Sitter solution, since the theory effectively reduces to the pure Einstein theory.] The gravkafional equations effective!y reduce to their trace. Furthermore, since the curvature is covariantiy constant, oMy variations of the explicit metric dependence (but riot of the mtrvatures) in the action contribute. Herme, at these soiutior~s, the gravitational equation reduces to 0= (D-2)D(D-
:)A
+ a( D - 4 ) D ( D +
2p
-
1)(D- 2)(D-
3)A 2
2)%cM .+:
P
= ( D - 2'A d / d A ) / ( A )
g ~
upon use of R~.x~ =- (8~,g~ -82g.x)A. Here /(A)- =
D(D-1)A+
aD!
(D-4):
+ Z p=2
410
1
/
A2
(s)
(k d/dA
-
1)/(A)°
(4)
Eqso (2), (4) can be satisfied only if f.~Aj has doubte zero at some A 0 4= 0. This is manifestly impossible for the quadratic case where f ~ aA + bA 2, so oidy A o = 0 survives there. (An alternative statement of this result is that e = 0 implies vanisNng of the G a u s s - B o n n e t invarian< which in turn implies that of t~e scaiar curvature by the gravitational equation.) While there are indeed m o d d s involving higher curvature terms which permit AoV=0 solutions, they must further be stable for small excitations about the De Sitter vacuum. Here file discussion is sirmlar to that given in ref. [ii, except that there are now (coupled) equations for both the gravitational (h~,) and e modes, tn general, there wit~ be terms proportions1 to ~2h~, which imply ghost behavior and their coefficients from the various contributions to such terms must caned, tn addition, terms of the form R - . . ~ ..~ R will produce h ~ ~+'k contributions and must be absent for stabilky° Furthermore, the coefficients of the E.'h~, and Oe terms must (after diagonalization of the e - hi~ kinetic terms) retain their original, non-ghost, signsSWe have not systemaficaIty investigated the implications of these quite restrictive requirements on the Ngher order terms of the generaI action (t)o The simp!est and most illuminating explick solutions b. any gravitational model are the sphericaily symmetry "Schwarzsctdld" geometries. Our first remark concerns the absence of such solutions if ~ vanishes, in that case, the gravitational equations reduce to those of the e = 0 theory, but wifl~ the further constraint that the G a u s s - B o n n e t term, and hence the scMar curvature vanish. The known solutions of the gravita° tional equations with a = 0 violate the latter requirement and on!y flat space is permitted. We Mso remark that because there is now a scalar mode present, the Birkhoff theorem no b n g e r holds [1,6] a~d time dependence could enter
Votnme 175, number 4
PHYSICS LETTERS B
through o. For simplicity, we wili consider on!y static sphericaiIy symmetric spaces be!owo To order ~, the action t-~ evah~ated at a static spherically symmetric space in Schwarzschild coordinates, ds ~= -eZ('>-X) d t 2 + e Z X d r Z + r a d 2 f @ _ 2 ,
(5)
reads
14 August i986
so that s - s / r ~ - 3 o Using these leading terms in (e, ".~, ~) to eva!uate the remaining terms in the o equations~ we find that ~ behaves as o ~ s / r D-3 + m s / 2 r a(D-3~
Inserting the teading terms of (~5, o) in the ~> equation yieids + ~ _ as2/4rZ(D-3)
- a~(D
,
-
r ~ +. / ~o _ n / ., ,~i j .
(6)
Here primes denote d / d r , ,~ = r - 2 ( ! - e-2~), ~ a ( D - 3 ) ( D - 4). The di!aton equation obtained by varying a is therefore
, - 4" )~ +(a/~)e°(e°(~-~+~)'-[~/(~
× {e-%O-~ [e2.~(t- r2~b)]')).
(7)
Variation of 4~ gives the Einstein (constraint) equation
o = (~-,+)'
- ,~,'-~(~
+ a eo(r~-:¢)'
- ,%)o'V2,
(~)
while variation of ~ expresses the coordinate condition fixing ~: 0 = - re-~4S - 2 a r D - " + e ° ( ~ ' + +') + a r % ' 2 / 2
+ [ 4 f / ( D - 4)] e ' e'r~-~4~'
× [(~'~,' e°) ' - 4,' e°, " - : ( , * ' + ~.)].
- 4 ) , :,' ~ - ~,
(:.'2)
These terms do not aIter the behavior of e in (11)o Finally, the corrections to ~ due to (e, ~) iead to
× {eO~-~ ! ,'(~ - ,~., ) - ~ ' ~ ' ~ /}~" ;- ~ ') - a e%C'-~(i
- 3)/(D
(9)
in the absence of the dilaton [1], ~ b . - ( m / r D-~) and ~ = 0 (i.e, -g00 = g2~). This behavior is consistent asymptoticaiiy with the futl equations including a, but there are of course corrections at finite ro The leading term in the ~ equation is
- m / r e - ~ _ a s 2 ( D - 3)/2r2(v-2)
- a,~[~
+ 2~(D - 3)/(D
- 4)]/H ~-~;,
wbAch provides a self-consiste'nt set of leading terms in a!i three fie~d variables. There are two free parameters (m, s) in the asymptotic solution, the coefficients of the leading asymptotic terms of (~, a). Note that varAshing of o to the order displayed implies that m = 0, consistent with our result that a = 0 implies flat space. It is not possb b!e, from the asymptotics Aone, to prove that if m = 0 space is flat, since that is a global property requiring a positive s~ress tensor. Vanishing of the leading term in both ~ and o implies that the asymptotic expansions of both functions va~fish identically. Under the assumption that there is an event horizon, i.e, a simple zero at ; of e -~'~ = t - r2+ at some r~, the fieid equations can be analyzed loca'.,'iy. The event horizon is, by assumption, a coordinate (rather than curvature) si~gMarity and not a singular surface of the space, hence e ~* and e ° must be reguIar there and the zero of e -:x must be simple, in addition, the fie~d equation implies that o is also reguIar at r w In the neighborhood of r the o dependence of the effective action is w-- ~fdrI-A(r)
e°-
½(~- r~)o'2],
('_.'4)
and the equation for o is of Liouvit!e form; [(r-~)o']'=A
e °,
(15) 411
Voiume 175, :umber 4
PHYSICS LE~ERS B
where A is .the value (regular by assumption) of the G a u s s - B o n n e t term near r H. The indicial equation then implies that r - - YH
but e ~ can be reguiar oniy if p >t 0, hence
-=p I n t r - r~.) + a,
r.):']' =A(r-
(17)
et
(:8)
w N c h has a regular power-series expansion. However, ~;he field equations for ~ and ~ must also be reguiar, hence , ' e-aXe ' ~ p S i ( r -
rH) + regular terms,
r ~ rH
and p must vanish. Therefore e is regular at the event horizon and e ° is both regular and nonvanisNng. Since ~ is regular, the equations for @ and $ are Mso both regular and have regular solutions, at least :ocaity° A finn1 remark on exact solutions is thin gravitationai plane waves are unaffected by the diIaton; they remain solutions [1] of the coupled system with e = Oo Although i~dusion of the dilaton takes i m o a c c o u m one of the important new effects suggested by strings, the effective gravity model treated here is far from realistic for a n u m b e r of reasons. FirsL we have not considered compactification through which the space becomes M4X BD-4; since tNs symmetry breaking presumably occurs on a scale set by a, there is litde macroscopic significance go our solutions. A related point is that a e ' R 2 is but the first in an infinite series of corrections in the curvature and its derivatives. We have only discussed the impact of these high.or terms in connection with cosmological solutions. Keeping only the R a correction is reasonable in a weak-curvature approximation; however, ir~ considering exact solutions, including the cosmological one) their higher-order dependence on a can only be taken serioudy if these
412
14 August i986
higher-curvature corrections are included as well an impossibie task even it we knew their exact structure. The freedom of making field redefinitions [10! is really another aspect of this difficulty: If &~ is an Einstein solution, then the general redefinition ~,,~ ~ ' = g~ -~. R~f~,~~~(Rx,~,) wiil yield different metrics ofgshelt but .the same on-sheii soiutions. (New so!udons witI oniy appear if the redefinition becomes singuiar, in w N c h case it is not permitted!) T N s means that there is an intrinsic ambiguity in the string series corresponding to the off-shell choice of variabie; for our model this means that e *~'v R 2~ + R 2 ) terms are not significant. We conclude that, whi!e fl~e results of the truncated model may be indicative of the direction in which string corrections alter the expectations of the Einstein theory [1], they wo~ld not foreshadow solutions of the whole string expansion w N c h are not near Einstein space, e.g., by being devoid of horizons and singularities. Nevertheless, inclusion of di!atons seems to be a step in the right direction c o m p a r e d to the purely gravitational models° This work was sapported in part by the US Department of Energy under Contract Noo DEA C 0 6 - g l E R 40048, and by the National Science F o u n d a t i o n under G r a n t Noso PHY82-01094 and P H Y 8 2 d 7 8 5 3 , supplemented by N A S A .
References [11 D.G. Ben!ware and S. Deser, Phys. Roy. Let*. 55 (i985) 2656. !2] G. (3ibbons, DAMTP preprint (1986). [3i J. Madore, Phys. Lett. Al!0 (1985) 289; Atli (1985) 283. [4] F. M~e!~eroHoissen, Phys. Lett B163 (1985) 106. {5] J.T. Wheeter, NucL Phys. B268 (i986) 737. {6~ D.L. WiltsNre Phys, Lett B169 (1986) 36. [7} B~ Zwiebach, Phys. Le~t. B156 (1985) 315. [8] A. Tseyflin, Lebedev preprim N6 (1986); M.J. Duff and B.E.W. NHsson, CERN preprint (1986). [9] S. Deser and N. Red¿ich, Brandeis preprint (1986). [10] G. 't Hoof* and Mo Vekman, Ann. tast. Henr~ Poincare, 20 (1974) 69; (3. 't Hooft Acts Universitafis Wradslavensis No. 368, Proc. XHth Winter School in Theoretical physics (Narpacz, Poiand). [ti] D.J. Gross and E. Witten, Princeton preprint (1986).
PHYSICS LETTERS B
14 Augue~ 1986
EFFECTIVE GRAVITY T H E O R I E S W I T H DILATONS David G. BOULWARE
:
Institute for Theoretical Physics, University of California, Santo Barbara, CA 93_t06,USA
and S. DESER Department of Physics, Brandeis Univers#y, Wa#ham, MA 02254, USA Received 20 May 1986 inclusion of the dilate: in the Eins:ein p:us Gauss-Bonnet model removes the de Sitter ground state pe:.~tted :,n its absence; ~he genetic case with higher order ir.variar.,'.s is discussed. Spherically symmetric static so:ufions are conNdered and ~heir asymptotic behavior established; they are nor.trivial only for a nor, vanishing dilaton field and are compafib.~e with the existence of an horizon at wbAch the diiaton is regular.
Recent progress in string theory has stimulated work on extensions of Einstein gravity inv'otving higher powers of curvature [1-6] wNch emerge in the slope expa~.siono Of particular interest is the leading q~adratic correction proporfionaI to the Gauss-Bonnet combination [7-9], whose terms quadratic in the de~atio,a from the flat metric vanAsh and therefore do not induce ghost graviton modes it: the propagator. :n a pre'~ious paper [1t, we studied this model and found its spherically symmetric solutions; the asymptofically fiat solutions have a structure quali~tively similar to the usual Schwarzschild metric: there is still an event horizon (at a smailer radius) and a (weaker) singuo larity a~ the origin. On the other hand, there are cosmological (de Sitter) solutions in addition to Minkows_V5 space~ with curvatures set by the string constant scale. De Sitter geometries are ~nstable (all sinai! excitations of the graviton have negative energy, h e n ~ o r d i n a u matter would be unstable against emission of these quanta) but higher order terms in the curvature could give rise to stabIe de Sitter soiutions. The string expansion, however, involves mass: Permanent address: Department of Physics FM-15, University of Washington, Seattle, WA 98195, USA.
less fields other than the grunion, the most releva~:t being the dilaton ~ which appears in the combination e ~ R ' + : as welt as in a kinetic (~e) 2 term. The purpose of tNs letter is to study the influence of the ditaton on the effective gravita° tional action both through its own field equation and in the gravitational equations. For spheficalIy symmetric solutions, a is found to vanish at spatial infimty i~ such a way as to !cave the asymptotic flatness unaffected. Although we cannot find the sotution explicitly, we can obtain its asymptotic form. We aiso show that it is (locaily) consistent with the existence of an event horizon, and that the ditaton is regular there. The existence of an horizon is a global question, which (in the absence of positivity conditions on the effective stress tensor) cannot be estab:ished without integrating in from spatiaI infinity, it is also impossine to d e t e r ~ n e completely the nature of the singuiafity at the origin by purely iocai considerations, so we have not attempted to list the possibilities there. The effect of the diiaton on the cosmo:oglca~ solutions is dramatic: they are excluded at the quadratic curvature level Since spacetime is ma×5really symmet:c, the diiaton must be a constant and both the gravitational and diIaton eqt~ations
0370-2693/86/$03°50 © Elsevier Science Publishers BoV, (North-He!land Physics Publishing Division)
409
14 August 1986
PHYSICS LETTERSB
Volume 175, number 4
constrain the cosmoiogicai constant A. Both equations may be expressed in terms of a singb function f ( A ) . I.o. the general case, a solution with nonvanisNng A wilI exist only if f has a double zero at A o The action including the diiaton fietd is
is the value of the :agraagian ~ / ¢ - g . Here, cpA p+: is the value of X '(p+~) at the Sitter space with curvature A. The ~ equation reads aD! (D-
o
A2+
Z
papCpAp+I
--i
f
,R
+
=
(:)
'" . . . .
p~2
]
where
The ~"P~ terms invoive appropriate powers of the curvature and its covariant derivatives. The factor Q, is the volume of the unit (D - 2)-sphere, G is Newton's constant in D dimensions, wi@_ units of L 17-2, while those of % are L -D+2". We use units in wNcb 2 ~ G = 1. It is straightforward to study the equations for cosmoiogicai solutions. The dilaton field can be shown to to be constant either by maximal symmetry arguments or by expiicit use of the field equations; by rescaling % it may then be set to zero. [Even the singular Iimk e - ~ - o o does not admit a de Sitter solution, since the theory effectively reduces to the pure Einstein theory.] The gravkafional equations effective!y reduce to their trace. Furthermore, since the curvature is covariantiy constant, oMy variations of the explicit metric dependence (but riot of the mtrvatures) in the action contribute. Herme, at these soiutior~s, the gravitational equation reduces to 0= (D-2)D(D-
:)A
+ a( D - 4 ) D ( D +
2p
-
1)(D- 2)(D-
3)A 2
2)%cM .+:
P
= ( D - 2'A d / d A ) / ( A )
g ~
upon use of R~.x~ =- (8~,g~ -82g.x)A. Here /(A)- =
D(D-1)A+
aD!
(D-4):
+ Z p=2
410
1
/
A2
(s)
(k d/dA
-
1)/(A)°
(4)
Eqso (2), (4) can be satisfied only if f.~Aj has doubte zero at some A 0 4= 0. This is manifestly impossible for the quadratic case where f ~ aA + bA 2, so oidy A o = 0 survives there. (An alternative statement of this result is that e = 0 implies vanisNng of the G a u s s - B o n n e t invarian< which in turn implies that of t~e scaiar curvature by the gravitational equation.) While there are indeed m o d d s involving higher curvature terms which permit AoV=0 solutions, they must further be stable for small excitations about the De Sitter vacuum. Here file discussion is sirmlar to that given in ref. [ii, except that there are now (coupled) equations for both the gravitational (h~,) and e modes, tn general, there wit~ be terms proportions1 to ~2h~, which imply ghost behavior and their coefficients from the various contributions to such terms must caned, tn addition, terms of the form R - . . ~ ..~ R will produce h ~ ~+'k contributions and must be absent for stabilky° Furthermore, the coefficients of the E.'h~, and Oe terms must (after diagonalization of the e - hi~ kinetic terms) retain their original, non-ghost, signsSWe have not systemaficaIty investigated the implications of these quite restrictive requirements on the Ngher order terms of the generaI action (t)o The simp!est and most illuminating explick solutions b. any gravitational model are the sphericaily symmetry "Schwarzsctdld" geometries. Our first remark concerns the absence of such solutions if ~ vanishes, in that case, the gravitational equations reduce to those of the e = 0 theory, but wifl~ the further constraint that the G a u s s - B o n n e t term, and hence the scMar curvature vanish. The known solutions of the gravita° tional equations with a = 0 violate the latter requirement and on!y flat space is permitted. We Mso remark that because there is now a scalar mode present, the Birkhoff theorem no b n g e r holds [1,6] a~d time dependence could enter
Votnme 175, number 4
PHYSICS LETTERS B
through o. For simplicity, we wili consider on!y static sphericaiIy symmetric spaces be!owo To order ~, the action t-~ evah~ated at a static spherically symmetric space in Schwarzschild coordinates, ds ~= -eZ('>-X) d t 2 + e Z X d r Z + r a d 2 f @ _ 2 ,
(5)
reads
14 August i986
so that s - s / r ~ - 3 o Using these leading terms in (e, ".~, ~) to eva!uate the remaining terms in the o equations~ we find that ~ behaves as o ~ s / r D-3 + m s / 2 r a(D-3~
Inserting the teading terms of (~5, o) in the ~> equation yieids + ~ _ as2/4rZ(D-3)
- a~(D
,
-
r ~ +. / ~o _ n / ., ,~i j .
(6)
Here primes denote d / d r , ,~ = r - 2 ( ! - e-2~), ~ a ( D - 3 ) ( D - 4). The di!aton equation obtained by varying a is therefore
, - 4" )~ +(a/~)e°(e°(~-~+~)'-[~/(~
× {e-%O-~ [e2.~(t- r2~b)]')).
(7)
Variation of 4~ gives the Einstein (constraint) equation
o = (~-,+)'
- ,~,'-~(~
+ a eo(r~-:¢)'
- ,%)o'V2,
(~)
while variation of ~ expresses the coordinate condition fixing ~: 0 = - re-~4S - 2 a r D - " + e ° ( ~ ' + +') + a r % ' 2 / 2
+ [ 4 f / ( D - 4)] e ' e'r~-~4~'
× [(~'~,' e°) ' - 4,' e°, " - : ( , * ' + ~.)].
- 4 ) , :,' ~ - ~,
(:.'2)
These terms do not aIter the behavior of e in (11)o Finally, the corrections to ~ due to (e, ~) iead to
× {eO~-~ ! ,'(~ - ,~., ) - ~ ' ~ ' ~ /}~" ;- ~ ') - a e%C'-~(i
- 3)/(D
(9)
in the absence of the dilaton [1], ~ b . - ( m / r D-~) and ~ = 0 (i.e, -g00 = g2~). This behavior is consistent asymptoticaiiy with the futl equations including a, but there are of course corrections at finite ro The leading term in the ~ equation is
- m / r e - ~ _ a s 2 ( D - 3)/2r2(v-2)
- a,~[~
+ 2~(D - 3)/(D
- 4)]/H ~-~;,
wbAch provides a self-consiste'nt set of leading terms in a!i three fie~d variables. There are two free parameters (m, s) in the asymptotic solution, the coefficients of the leading asymptotic terms of (~, a). Note that varAshing of o to the order displayed implies that m = 0, consistent with our result that a = 0 implies flat space. It is not possb b!e, from the asymptotics Aone, to prove that if m = 0 space is flat, since that is a global property requiring a positive s~ress tensor. Vanishing of the leading term in both ~ and o implies that the asymptotic expansions of both functions va~fish identically. Under the assumption that there is an event horizon, i.e, a simple zero at ; of e -~'~ = t - r2+ at some r~, the fieid equations can be analyzed loca'.,'iy. The event horizon is, by assumption, a coordinate (rather than curvature) si~gMarity and not a singular surface of the space, hence e ~* and e ° must be reguIar there and the zero of e -:x must be simple, in addition, the fie~d equation implies that o is also reguIar at r w In the neighborhood of r the o dependence of the effective action is w-- ~fdrI-A(r)
e°-
½(~- r~)o'2],
('_.'4)
and the equation for o is of Liouvit!e form; [(r-~)o']'=A
e °,
(15) 411
Voiume 175, :umber 4
PHYSICS LE~ERS B
where A is .the value (regular by assumption) of the G a u s s - B o n n e t term near r H. The indicial equation then implies that r - - YH
but e ~ can be reguiar oniy if p >t 0, hence
-=p I n t r - r~.) + a,
r.):']' =A(r-
(17)
et
(:8)
w N c h has a regular power-series expansion. However, ~;he field equations for ~ and ~ must also be reguiar, hence , ' e-aXe ' ~ p S i ( r -
rH) + regular terms,
r ~ rH
and p must vanish. Therefore e is regular at the event horizon and e ° is both regular and nonvanisNng. Since ~ is regular, the equations for @ and $ are Mso both regular and have regular solutions, at least :ocaity° A finn1 remark on exact solutions is thin gravitationai plane waves are unaffected by the diIaton; they remain solutions [1] of the coupled system with e = Oo Although i~dusion of the dilaton takes i m o a c c o u m one of the important new effects suggested by strings, the effective gravity model treated here is far from realistic for a n u m b e r of reasons. FirsL we have not considered compactification through which the space becomes M4X BD-4; since tNs symmetry breaking presumably occurs on a scale set by a, there is litde macroscopic significance go our solutions. A related point is that a e ' R 2 is but the first in an infinite series of corrections in the curvature and its derivatives. We have only discussed the impact of these high.or terms in connection with cosmological solutions. Keeping only the R a correction is reasonable in a weak-curvature approximation; however, ir~ considering exact solutions, including the cosmological one) their higher-order dependence on a can only be taken serioudy if these
412
14 August i986
higher-curvature corrections are included as well an impossibie task even it we knew their exact structure. The freedom of making field redefinitions [10! is really another aspect of this difficulty: If &~ is an Einstein solution, then the general redefinition ~,,~ ~ ' = g~ -~. R~f~,~~~(Rx,~,) wiil yield different metrics ofgshelt but .the same on-sheii soiutions. (New so!udons witI oniy appear if the redefinition becomes singuiar, in w N c h case it is not permitted!) T N s means that there is an intrinsic ambiguity in the string series corresponding to the off-shell choice of variabie; for our model this means that e *~'v R 2~ + R 2 ) terms are not significant. We conclude that, whi!e fl~e results of the truncated model may be indicative of the direction in which string corrections alter the expectations of the Einstein theory [1], they wo~ld not foreshadow solutions of the whole string expansion w N c h are not near Einstein space, e.g., by being devoid of horizons and singularities. Nevertheless, inclusion of di!atons seems to be a step in the right direction c o m p a r e d to the purely gravitational models° This work was sapported in part by the US Department of Energy under Contract Noo DEA C 0 6 - g l E R 40048, and by the National Science F o u n d a t i o n under G r a n t Noso PHY82-01094 and P H Y 8 2 d 7 8 5 3 , supplemented by N A S A .
References [11 D.G. Ben!ware and S. Deser, Phys. Roy. Let*. 55 (i985) 2656. !2] G. (3ibbons, DAMTP preprint (1986). [3i J. Madore, Phys. Lett. Al!0 (1985) 289; Atli (1985) 283. [4] F. M~e!~eroHoissen, Phys. Lett B163 (1985) 106. {5] J.T. Wheeter, NucL Phys. B268 (i986) 737. {6~ D.L. WiltsNre Phys, Lett B169 (1986) 36. [7} B~ Zwiebach, Phys. Le~t. B156 (1985) 315. [8] A. Tseyflin, Lebedev preprim N6 (1986); M.J. Duff and B.E.W. NHsson, CERN preprint (1986). [9] S. Deser and N. Red¿ich, Brandeis preprint (1986). [10] G. 't Hoof* and Mo Vekman, Ann. tast. Henr~ Poincare, 20 (1974) 69; (3. 't Hooft Acts Universitafis Wradslavensis No. 368, Proc. XHth Winter School in Theoretical physics (Narpacz, Poiand). [ti] D.J. Gross and E. Witten, Princeton preprint (1986).