PHYSICAL REVIEW D VOLUME 50, NUMBER 12 15 DECEMBER 1994

Mass inflation in (1 + 1)-dimensional dilaton gravity

J. S. F. Chan Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada NZL 3G1

R. B. Mann Department of Physics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

(Received 7 June 1994; revised manuscript received 16 August 1994)

We investigate the phenomenon of mass inflation in two-dimensional dilaton theories of grav- ity. We consider two distinct black hole spacetimes and construct the mass-idation solution for each. Our analysis is extended to include multihorizon spacetimes. We find that the mass function generically diverges in a manner quantitatively similar to its four-dimensional counterpart. However for multihorizon spacetimes with a certain geometry, we find that the mass function can remain bounded under identical physical conditions.

PACS number(s): 04 .70 .B~~ 04.20.Jbl 11.25.Pm

I. INTRODUCTION

Lower dimensional theories of gravity continue to at- tract the attention of theorists, in large part because many of the conceptually interesting features of their (3 + 1)-dimensional counterparts are retained while the corresponding calculations involved are much simpler. Investigations of ( I + 1)-dimensional gravity [I] have proven to be particularly rewarding, providing' further insight into black hole solutions [2,3], black hole radia- tion [4], cosmology [5], singularities [6,7], and quantum gravity [8].

It was first pointed out by Penrose [lo] that the Cauchy horizon of the Kerr-Newman solution of (3 + 1)- dimensional general relativity is unstable due to the in- finite blueshift of the ingoing radiation. This pathology of the Cauchy horizon raises a question of whether the Kerr-Newman solution can be analytically extended to some other asymptotically flat universes. Much more re- cently Poisson and Israel [9] made important progress on this problem by showing that the Cauchy horizon of the Reissner-Nordstrom solution forbids any evolution of spacetime beyond this horizon. They came to this con- clusion by noticing that the mass parameter inside the black hole becomes unbounded due to the presence of ingoing and backscattered outgoing radiation. As the causal structure of the Kerr-Newman spacetime is similar to that of the Reissner-Nordstrom case, it is believed that the Cauchy horizon in the more general Kerr-Newman case forms a similar obstruction to the evolution of the spacetime. Ori [ll] subsequently confirmed this mass- inflation phenomenon by constructing an exact solution of the Einstein-Maxwell equations in a simpler model. However, he argued that this mass-idation singularity is so weak that its tidal forces do not necessarily de- stroy any physical objects. This extensibility problem has initiated some controversy among physicists in the field [12,13].

Recently, Husain [14] showed that the mass-inflation phenomenon also occurs in (2 + 1)-dimensional space-

time. In this paper, we shall show that the same phe- nomenon has analogous partners in (1 + 1)-dimensional gravitational theories. Specifically, we consider a con- struction of the Ori model in dilaton gravity theories in 1 + 1 dimensions in which the background spacetimes possess more than two horizons. We shall show that in most cases these multihorizon spacetimes have inflation- ary mass parameters behind the outer horizon. However, for some multihorizon spacetimes with a particular ge- ometry we show that mass inflation may not occur; there is no singularity a t the Cauchy horizon, and the inner mass function remains finite.

We consider a noncritical string-inspired dilaton theory of gravity with the action of the form [15]

where R is the Ricci scalar, y , Q u, and an's are constants and Fp, is the Maxwell tensor. The last term CM denotes the matter Lagrangian such that ( b L M / b g P v ) / f i = Tpv. In the following we shall take TPY to be tihe stress- energy tensor of a null fluid. A choice of the values of the parameters y, Q, u, and the an is tantamount to a particular choice of theory. We shall consider two dis- tinct choices, each of which yields field equations whose solutions correspond to multihorizon spacetimes.

11. THE FIRST SOLUTION

When 7 = 4, u = 0, and the parameter Q is set to be a positive constant, the action (1) gives the field equations

7376 @ 1994 The American Physical Society

MASS INFLATION IN (1 + 1)-DIMENSIONAL DILATON GRAVITY

+ Q ~ - C a n ( n - l ) e Z n @

When these equations are expressed in Eddington- Finkelstein ingoing coordinates

they have a solution

where the stress-energy tensor with energy density p(v) is

where 1, is defined as 1, := - d,v. If the matter action were taken to be that of a massless scalar $(x), then p(v) = (d$/dv)'. The terms q and xo are integration constants corresponding to electric charge and the choice of origin of spatial coordinate. The function m(v) satis- fies the differential equation

As the parameters a, push M to be a higher order poly- nomial in exp(2 q5), the metric solution (8)' (9) allows a multihorizon structure for specific values of a,. (For example, the Born-Infeld black hole for an open string [16] has three horizons.) This solution is asymptotically flat as x + oo but is singular when x + -co. One can show that the function m(v) is the Amowitt-Deser- Misner (ADM) mass [17] of the spacetime and this so- lution represents a black hole which is irradiated by an influx of null radiation.

Consider the matching of two patches of solution (6)- (9) along an outgoing null line S as shown in Fig. 1. As each patch of the solution describes a flux of ingoing null radiation, this combined solution models, in addition to

FIG. 1. A null line S divides the spacetime into regions I and 11.

an influx, a lightlike particle which propagates outward. As the null ingoing fluid is electrically neutral, the out- ward null particle carries no electric charge. Thus the static charge q is the same in both regions. However, the spacetime in region I is characterized by a mass function m = ml(vl) whereas region I1 is characterized by an- other mass function m = mz(v2). That there will be a mass-inflation solution is straightforward to see. Requir- ing continuity of the metric along S yields

whereas continuity of the influx across S implies

which is equivalent to the condition TYll ny ny = T,,,I2 n t ns, with ny,2 = (2/a1,2,1). Inserting (12) into (13) yields

showing that near the Cauchy horizon, where a1 + 0 as vl + co, the inner mass function m2 diverges, as all other quantities remain finite.

We now proceed to construct an explicit solution. From the coordinate system (5)' the outgoing null geodesic satisfies the equation

where X is an affine parameter common for both regions and the overdot denotes a derivative with respect to A. Without loss of generality, we choose the parameter X to be zero at the Cauchy horizon and positive beyond that.

We now define a function X to be the value of x along S such that one of the Euler-Lagrange equations implies

7378 J. S. F. CHAN AND R. B. MANN 50 -

(16)

along S. If we define

one can show that

and

Because X is a boundary function to regions I and 11, which are characterized by mi and vi, where i is either 1 or 2, we obtain the matching equations

which will determine the evolution of the spacetime equa- tion if the boundary function X is known. The terms Zi in Eq. (20) are integration constants and we shall see that they play an important role in the analysis.

The (A-dependent) "mass" of the null particle can be defined as

We also define a constant

as the final ADM mass of the black hole observed in re- gion I after the hole has absorbed all the ingoing radia- tion. Therefore, the term 6m is interpreted as the mass of the radiative tail of the ingoing radiation. Since the Cauchy horizon corresponds to the limit vl -+ my we expect that

lim i r l ( A ) = e - z + ( X c ) p l = , A+o-

where X , is the location of the Cauchy horizon. This implies that Z1 = 0, yielding

lim M (X) = M X + X ,

123 1

by applying the limit to Eq. (18). Equation (20) can be written as

where ko is defined as

It is always possible to make ko positive definite if M1(XC) # 0 as shown in Fig. 2 because the slope of M ( X ) at Xc is nonpositive. We shall restrict ourselves to this case, i.e., M1(Xc) # 0, in this and the next sec- tions and investigate the case where the Cauchy horizon locates at a point of inflection of M in Sec. IV.

Ifweapproximateml-M = Iml -MI = I m l - M / = 6m between the event and Cauchy horizons, Eq. (18) can be approximated as

FIG. 2. A sample graph of

M ( X ) .

50 - MASS INFLATION IN (1 + 1)-DIMENSIONAL DILATON GRAVITY 7379

when I A I << 1. Since Z1 = 0, Eq. (19) can be approxi- mated as

in some negative neighborhood of A = 0. However, when I A I << 1, zz x Zz, which implies

As Sm represents the radiative tail of the ingoing ra- diation, its relevant asymptotic behavior is

where h and p are positive constants (see the Appendix for a discussion on this point). When A is close to zero, we find that Am is approximately

Am(A) = - Q Z 2 x ( A ) x - - hZ2 e24(Xc) I v1 (A) . (30) ko A

As a result, we obtain

On the other hand, when I A I << 1, according to Eqs. (21), (22), (29), and (31), we have

where (27) and (28) were used. This shows that in the case M1(Xc) # 0 the mass in region 11 becomes un- bounded near the Cauchy horizon where v2 x 0.

111. THE SECOND SOLUTION

In the case when 7 = a = 2 and Q = 0, the action (1) yields a set of field equations which has a solution

f = F", = 9 (x - xo)2 '

in the Eddington-Finkelstein coordinate system (5) with the use of the same stress-energy tensor (10). The con-

stants q and xo are again the integration constants and the function m(v) which satisfies the same differential equation (11) is the ADM mass of the black hole. As the singularity is located at x = xo, the spacetime is divided into two disjoint sets: R X {(- oo, 20)) and R x {(xo, a ) ) . Thus the whole real line consists of two disjoint universes and each of them has a one-way-collapse solution (33)- (36). In the static case, as the ingoing coordinate v is defined as

where t is the usual Schwarzschild coordinate time, it tends to negative (positive) infinity when x in the left (right) universe approaches the Cauchy horizon.

Consider the matching of two patches of solution (33)- (36) along the outgoing thin null particle S as in the previous section. This time we define z;(A) as

such that one of the Euler-Lagrange equations yields

i;;(A) = - $ ( c ; ) ~ [ ( X - 20) M 1 ( X ) - M ( X ) + m;(v;)]

x ( X - (38)

along the null line S. Thus, the system analogous to (18)-(20) is

mi(v; (A)) = M (X(A)) - 2 z,(A) X(A) , (39)

Again we use the subscripts 1 and 2 to distinguish the quantities m, v, z, and Z in regions I and 11.

Making use of the same definition as in (21), we find

As Z1 is zero, it is clear that sgn(Z2) = sgn(X - xo) in order to have positive Am. Similarly to the previous case, Eq. (41) can be expressed as

where ko is defined as

A sample M - X graph is shown in Fig. 3. By examining the slope of M as before, we have ko > 0(< 0) in the right (left) universe if the Cauchy horizon is not at a point of inflection of M. The other case where ko = M1(Xc) = 0 will be studied in the next section. By realizing that m i - M = sgn(X - zo) 1 m l - M I between the event and Cauchy horizons according to (35), therefore in some small negative neighborhood of A = 0, Eq. (39) can be approximated as

J. S. F. CHAN AND R. B. MANN

because Z1 is zero. In region I, vl can be approximated as

in some negative neighborhood of X = 0. Nevertheless, when I X I << 1, z2 = Z2, which implies that

Notice that Eqs. (46) and (47) are similar to (27) and (28) in the last section and it is the logarithmic property of vl and linearity of 712 which trigger the mass-inflation mechanism in region 11.

By using Eq. (42), we find that Am can be approxi- mated as

which can also be written as

On the other hand, we can write

and so the mass in region I1 in this case is also unbounded

FIG. 3. A sample graph of

near the Cauchy horizon. Finally, when we calculate the dyad component Rlool

of the Riemann tensor using either the metric (8) or (35), we find that the leading term of this component has the form

which is exactly the same as the behavior for the corresponding component associated with the (3 + 1)- dimensional Reissner-Nordstrom metric. Hence the tidal distortion near the Cauchy horizon is also bounded in both solutions even though the Ricci scalar is infinite there.

IV. CAUCHY HORIZONS WITH VANISHING SURFACE GRAVITY

For spacetimes with multiple horizons, such as those with metrics (8), (9) and (35), (36), mass inflation pro- ceeds as described in the previous two sections for generic values of the parameters Q , a,, and q. However, since our metric solutions are basically polynomials of a degree higher than 2, for certain values of these parameters it is possible for ko [(25),(44)] to be zero; i.e., for M1(X) to vanish at X = X,. More generally, it is possible for the &st N derivatives of M ( X ) to be zero at the Cauchy horizon, where N is some positive integer. This will intro- duce qualitatively new behavior near the Cauchy horizon as we shall now demonstrate.

Since M ( X ) is a finite polynomial there will be some order of derivative of M ( X ) at X = X, that will be nonzero. As a result, we suppose there exists a posi- tive integer b 2 1 such that, for every integer i E [1, b], M(') (x,) = 0 but M(~+') (x,) # 0. Furthermore, as X, is the Cauchy horizon, we also have M(X,) = M.

In the first solution, the equation for the Cauchy hori- zon can be expressed as

50 - MASS INFLATION IN (1 + 1)-DIMENSIONAL DILATON GRAVITY 7381

where X, is the location of the event horizon and is some positive integer. The function P represents the zeros corresponding to the horizons within the second (Cauchy) horizon. As a result, p(e24(")) will not change sign for all x 2 X,. In order for & to switch sign when x - Xc switches sign it is necessary for 6 to be an odd integer. Thus p(e24(=)) must be positive for at least all x 2 Xc in order that & is positive beyond the event hori- zon.

After differentiating the first form of & in (52) several times, and evaluating the result at X,, the first nonzero derivative is

On the other hand, if we perform the same operations to the second form of & in (52) it will agree with (53) only if 6 = b + 1. That is to say,

is the first nonzero derivative at the Cauchy horizon. Furthermore, because exp[2 $(x)] is a decreasing func- tion, Eq. (54) says that db+l) X must be negative, which in turn implies that Mh+'l(Xc) is also nega- tive. By using similar arguments as above, one can show that M ( ~ + ' ) ( X ~ ) is also negative for the second solution (35),(36).

In the case when M1(Xc) = 0, the approximation in both (24) and (43) breaks down because ko is zero. We shall consider the first solution in this context; the second solution is straightforward. In the first solution, M1(X) of (20) can be Taylor expanded at X = Xc because M ( X ) is analytic at that point. The dominant term in the expansion yields an approximation

where e E (A, 0) comes &om the mean value theorem. As the term inside the square brackets goes to zero when A tends to zero, the slowest rate possible at which it could vanish is approximately equal to eX1(e) < 0 A, where 0 is some finite constant. As a result, the approximation of (20) is modified to

zi (A) = zi - IE e- 2 +(xe) ~ b + l (55)

where

is a positive constant because M(~+')(x,) was shown to be negative and b is an even integer. Notice that Eqs. (55) and (56) are the generalizations of (24) and (25), respectively.

Since %,(A) is no longer linear in this case, vl has an approximation

which is no longer logarithmic. However, as we are con- sidering a small A approximation and Z2 is still nonzero, the expression for z2(A) remains the same as (28). Fi- nally, because x is in this case given by

the mass of the outgoing particle becomes

where Z2 again must be positive in order to have a posi- tive mass.

From expression (59), if p < 1 + l / b , dm will be un- bounded as A + 0, and the inner mass parameter m2 will also inflate because ma = ml + 6m. However if p > 1 + l l b , there will not be any mass inflation at all be- cause the exponent of A in (59) is positive. When A + 0, the mass of the particle tends to zero when p is large enough; the boundary particle "deflates" as it approaches the Cauchy horizon. Although Eq. (59) is an approxi- mation, we believe that the nature of a at the Cauchy horizon does affect the mass-inflation behavior because (i) Eq. (59) is the dominant term in the process, and (ii) the inflation of the inner mass parameter, expression (32), strongly depends on the fact that vl is logarithmic. In turn, the logarithmic nature of vl requires the leading order of zl(A) to be proportional to precisely A. Thus a slight deviation hom this linear behavior is expected to alter the behavior of the mass parameter as shown pre- viously. Note that the matching condition (14) is still satisfied, since it implies [18]

where the last relation is valid near the Cauchy horizon, whose surface gravity is ko cx M1(Xc). Strictly speaking, the left side of Eq. (14) should read al(dm2/dvl)dvl. Even if dm2/dvl is bounded, it is not necessary for a 2

to vanish at the Cauchy horizon since the advanced time vl is infinite at that point. If ko # 0 then the i ~ e r mass function m2 diverges. However, if ko vanishes then rn2 x v , ~ . The constant N will be determined in the manner described above - if it is ~ositive then mass inflation will not occur.

For the second solution (35) and (36) with vanishing M1(XC), one can show that M (b+l) ( xc ) is negative; thus, -21 (A) becomes

7382 J . S. F. CHAN AND R. B. MANN 50 -

which is analogous to (43), where K is a constant with s g n ( ~ ) = - sgn(X, - xo). Thus vl (A) changes to (57) but v2 remains the same as (47). Consequently, the mass of the null particle reads

where Z2 is a constant with sgn(Z2) = sgn(X - .xO) for positive Am. Therefore, if the nature of the null fluid is such that the radiative tail has a power law decay with p > l+ l / b , the outgoing null particle will have a decreas- ing mass as it approaches the Cauchy horizon. Thus this horizon is no longer a barrier to the evolution of space- time.

Note that it is the shift of the Cauchy horizon [along with the matching conditions (14)] from the inner horizon that is responsible for mass inflation. This shift is a result of the presence of an outgoing null particle with positive mass 6m = m2 - ml. As long as m2 is greater than ml, this shift always occurs. Initially the null particle car- ries positive energy and so mz > ml even when ko = 0. However, in this case the mass of the particle "deflates" while approaching the Cauchy horizon, approaching zero mass at that point. Hence mz = m l at that instant. Al- though the Cauchy horizon is initially shifted away from the inner horizon, the two horizons move closer together as the particle deflates. When they overlap each other, the outgoing null particle loses all its energy and vanishes forever.

V. CONCLUSIONS

We have shown that ( I + 1)-dimensional charged dila- tonic black holes with simple double horizons have yet another feature in common with their (3+ 1)-dimensional counterparts, namely, they mass inflate as a consequence of the interaction between incoming and backscattered radiation. This property holds in most cases even if the black holes have multiple horizons [as in Eq. (9)]. The singularity a t the Cauchy horizon associated with it is also as mild as the one in 3 + 1 dimensions. However, for solutions with vanishing surface gravity a t the in- ner horizon mass inflation will not necessarily take place. These solutions can be understood as arising from multi- horizon solutions as their parameters approach limiting values, much in the same way that the extremal Reissner- Nordstrom solution can be understood as the Q + M limit of the nonextremal case. However, it should be re- membered that the vanishing of ko is a local property of the metric function a; it is the stability (or lack thereof) of the first inner horizon that is relevant. In 1 + 1 dimen- sions mass inflation will be absent whenever the radiative tail of the outgoing null particle has a power law decay exponent p > 1.5. These effects should carry over to any (3 + 1)-dimensional black hole with vanishing sur- face gravity a t the inner horizon. Since the quantization of lower dimensional gravity is considerably easier than the (3 + 1)-dimensional case, it should also be possible

to extend the results of this paper to include quantum mechanical effects. Work on these issues is in progress.

Note added. After completion of this paper, we became aware of related work by Droz [18] and Balbinot and Brady [19] on mass inflation in 1 + 1 dimensions.

ACKNOWLEDGMENTS

We wish to thank Dr. V. Husain for drawing Refs. [18] and [19] to our attention. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada.

APPENDIX: SCALAR PERTURBATION IN (1 + 1) DIMENSIONS

In 1 + 1 dimensions, the usual scalar wave equatlor~ V2$ = 0 is not appropriate to use in the perturbatio~r problem because there is no potential barrier and hence there is no backscattering [20]. As a result, we use a11

alternative wave equation

where ( is a constant, to describe a radiating scalar field. If we put the wave equation into a conformally flat coor- dinate system

where a ( x ) is taken to be

for simplicity, the wave equation (Al ) will become

where the potential barrier B ( X ) is given by

and the function 0 is defined implicitly as n (x ) exp[O(x)] = x. Figure 4 is the graph of this barrier and one can see that it has a peak at about X = M + M ln(M/2).

Let us concentrate on a static background in an out- going coordinate system

In this coordinate system, our wave equation (Al) has a form

MASS INFLATION IN (1 + 1)-DIMENSIONAL DILATON GRAVITY

FIG. 4. The graph of the po- tential barrier B(X).

a (x) a,, lC) - 2 a,,$ + a' (2) a,$ + < a" (x) lC) = 0 . (A7) a difference equation

As the potential barrier of (Al) is of order 1/x3, one will expect that the resultant backscattering in this spacetime is similar to the one in 3 + 1 dimensions in which the leading order of the potential barrier is of the order l/r3. This can be verified mathematically as follows.

In Kruskal coordinates, the scalar field is expected to be well behaved a t the event horizon because the coordi- nates are well behaved there. As U := - exp(- u/2 M ) and V := exp(vl2 M ) , we have

As a result, we expect that

Thus the primary outgoing waves in 1 + 1 dimensions decay to zero exponentially like those in 3+ 1 dimensions.

Since the primary outgoing waves have the form

can be obtained by using Eqs. (A7) and (AS). This difference equation is the same as the one below part l (b ) in box 3, Ref. [21], when I = 0 and Bn = 0 except B1 = <. One can therefore follow the rest of the calculation in [21] to show that the backscattered waves die out as t - P.

When a ( x ) has more than one horizon, the dominant part of the metric is still given by (A3) because we are considering the backscattering which takes place just out- side of the event horizon [20] where x is still large. Thus the 1/x term dominates the terms 1/x2, l /x4, and so on. As a result, we still expect that the backscattered waves decay as t - P . Finally, in the case when the spatial func- tion in the metric is exponential rather than the inverse of x, the potential barrier is thinner than in the case considered above because the barrier in the exponential falloff is not as widespread as the barrier in the case of inverse decay. Thus the decay of the backscattered waves is even faster than t - P because more waves can tunnel through the barrier and propagate to infinity.

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