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ELSEVIER Nuclear Physics B (Proc. Suppl.) 57 (1997) 334-337

Gravitational kinks in 2d dilaton gravity M. Vasilid and T. Vukaginac a *

~Department of Theoretical Physics, Institute "Vin~a" P.O. Box 522, 11001 Belgrade, Yugoslavia

The exsistence of gravitational kinks in 2d dilaton gravity, for topologically trivial spacetimes, is analysed. Hamiltonian analysis shows that such configurations are allowed, at least in a finite interval of time, provided some appropriate initial conditions are chosen. Investigation of the stationary case gives the conditions which the potential of the dilaton field has to satisfy in order that the theory admits gravitational kink solutions. As a special example, flat gravitational kinks are considered.

1. I N T R O D U C T I O N

Gravitat ional kinks are topologically nontrivial metric configurations which were studied for the first t ime by Finkelstein and Misner [1]. S- ince then their properties have been analysed in a number of papers [2-8], but a dynamical model possessing this type of solution in topologically trivial spacetimes is still missing. It has been shown that each homotopy class of four-dimensional metric configurations contains a spherically symmetr ic one [9], which makes it essentially a two-dimensional (2d) object. Some features of 2d gravitational kinks have already been investigated [10,11]. In this paper we shall search for a 2d dynamical model that would allow for gravitational kink solutions.

In section 2 the homotopy structure of the s- pace of metric components is analysed and a representative in each homotopy class is chosen. These are the examples of inequivalent gravitational kinks. Section 3 is devoted to a simple model of 2d dilaton gravity. A detailed analysis shows which conditions the dilaton field potential has to satisfy so that the theory admits s ta t ionary gravitat ional kink solutions. To generalize the previous analysis to n o n - s t a t i o n a r y solutions, in section 4 we perform the Hamiltonian analysis of the theory. It is shown that gravitational kink configurations can be compatible with constraints, but also that they can become divergent

*This work has been supported in part by the Serbian Research Foundation, Yugoslavia.

0920-5632/97/$17.00 © Elsevier Science B.V. All rights reserved. PII S0920-5632(97)00375-7

in a finite period of time. As an example of such a scenario, flat gravitational kinks are briefly analysed in section 5.

2. G R A V I T A T I O N A L K I N K S I N 2 D

The space of components of the 2d metric ten- sor gu~" is three-dimensional. There is, however, a forbidden region in this space defined by vanishing metric determinant. The cone-shaped surface g - g00gll - (g01) ~ = 0 divides the space of metric components into three separate regions, each characterized by a definite signature. Two of them, G2 and G-2 , are inside the cone and contain the metrics of Euclidian signature (+ ,+) and (- , - ) , respectively. The only topologically nontrivial region Go is outside the cone g = 0, with the metrics of Lorentzian signature ( - ,+) . It is obviously multiply connected with the first homotopy group being the group of integers.

We shall restrict our attention to topologically trivial spacetimes (R 1 x R 1) and asymptot ical- ly Minkowskian metrics. The configuration space C = {g,~,(a) : R 1 -+ G0} is represented by closed loops with the point g , , = ~ , , fixed and which complete!y lie in the connected region Go. A non- contractible loop m times wound around the forbidden surface g = 0 is called a kink and is at- tributed the kink number m. Let us choose the representative in each homotopy class of metric configurations to be

,~ ( c o s t a 0 s inm0 ) g ~ , - - s inm0 - c o s m 0 ' (1)

M. Vasilik, T. VukaSinac/Nuclear Physics B (Proc. Suppl.) 57 (1997) 334-337 335

where ~ - 7r + 2arctg c~, c goes from - ~ to +oo and m is an integer. This metric is represented by a loop which leaves the point g,~ = ~,~ in c = - o o , winds m times around the surface g = 0, and ends in ggv = r/~v when c = +oc. Continuous coordinate t ransformations cannot change the kink number, while the discrete transformation of t ime reversal maps kinks into antikinks (m -+ - m ) [1].

The light-cone analysis shows that these topologically nontrivial metric configurations have a twisting light-cone structure with regions of anomalous causality. The light cones of the metric (1) twist along a -ax i s just like the light cones of Finkelstein and Misner kinks do. The m kink is characterized by m / 2 twists. Each twist is surrounded by a one-way surface through which causes can propagate in only one direction. For Iml > 1 these configurations represent nonsingular 2d black holes with horizons at the points where cos m0 = 0.

In 2d theory gravity two of the metric components can be gauged away by reparametrizat ion invariance. Any 2d metric of Lorentzian signature can be written in the form [10]

e~ ( c o s ¢ s i n e ) \sine -cos¢ ' (2) g ~

and ~ or ¢ could further be specified. Although every 2d metric can locally be put into the con- formally flat form, which corresponds to ¢ = 0, this cannot be done globally. We shall choose the gauge which includes k ink- type metrics:

~a r n g,~ = e g ,~ . (3)

This gauge is also incomplete. There are metric configurations, not covered by our gauge choice, which also have twisting light-cone structure even if their kink number is zero [12]. An example is a loop in the space of metric components, wound n times around the surface g = 0 and then unwound n times in opposite direction. This configuration is an example of m = 0 nonsingular black hole.

3. K I N K S I N D I L A T O N G R A V I T Y

In the search for a physical model accommo- dating gravitat ional kinks as analogues of four

dimensional nonsingular black holes we shall consider a simple dilaton gravity theory given by the action

I (4)

This kind of theory has already been considered in reference [13] where the form of the potential has been determined from the requirement that the theory admits nonsingular black hole solutions. Similar constructions of nonsingular black holes can also be found in references [14-18]. Not all of these, however, are kink type solutions. Fol- lowing [13], recently developed in [19], we search for a potential V(¢) which admits s tat ionary kink configurations. Variation of the action (4) yields the field equations

d V R + = 0, (5)

and

VuV~ ¢ + !g , , v ( ¢ ) = o (6)

only two of which are independent. Since we are interested in gravitaional kink configurations we shall choose the gauge (3). Then, the most general s tat ionary solution is obtained by substi tuting ¢ : ¢(0) and ~ : ~(a) into the field equation- s. The details of the analysis are given elsewhere [20] and here we shall only give the results.

The analysis shows that ¢ cannot be constant for ]m I > 1, and the equations of motion reduce t o :

e ~ = c~¢', a = const , (7)

and

¢' cos ,n0 = w ( ¢ ) , (8)

where we introduced the prepotential function W(¢) defined as

dW(¢) c~V(¢) = (9) d¢

According to (7), ¢ is a monotonic function of o, taking values in the interval D - (¢_, ¢+), with ¢+ - ¢(J=oo). From equation (8) it follows that W(¢) must have the same number of zeros

336 M Vasili~, T. Vuka~inac/Nuclear Physics B (Proc. Suppl.) 57 (1997)334-337

on D as cos ta0 has on the whole ¢ axis. Let us denote these zeros by q)i and ai, respectively (i = 1 , . . . ,2m).

It can be shown tha t the necessary and suffi- cient conditions for the field equations to satisfy, in order to admi t regular m-k ink solutions, can be wri t ten in the form:

d W d cos mO = ( d - -V- (10)

where ¢i < ¢i+1 and cri < c~i+1, together with

: 0 ,

W(¢) --+ 0 as ¢--+ ¢:F = finite, (11)

w ( ¢ ) -+ finite as ¢ --+ ¢~: = :700.

¢

The easiest way to construct a potential with given properties is to choose a monotonous function ¢(~r), subst i tute it into (8) to obtain W(¢(o ' ) ) as a funct ion of e, and then use (r = cr(¢) to find W(¢) itself. As an example, we can take the lin- ear di la ton vacuum ¢ = c~ as a s tar t ing point and find

W(¢) = Win(C) = cos [me(C)] . (12)

The potent ial V(¢) is determined f rom (9) up to a mult ipl icative constant .

In the end, let us comment on the possibility of having a potent ial which would allow for an arbi- t rary number of inequivalent kink configurations. If Dm (m = O, 1,..., k) are disjoint finite domain- s of the prepotential functions Wm (¢) then it is possible to construct the function W(¢) with the domain D D Um Dm such tha t its restriction to Dm is exactly Win. If Wm admits m-k ink solutions it follows tha t W will admi t k + 1 inequivalent kink configurations with kink numbers taking values f rom 0 to k. The theory of this kind will possess k - 1 types of nonsingular black holes.

4. H A M I L T O N I A N A N A L Y S I S

In order to investigate the more general situation which also includes non- s t a t iona ry solutions, we shall perform the Hamil tonian analysis. Since the t ime evolution is continuous it can not unwind the kink provided the boundary conditions stay fixed. This means that , in principle, it would be

enough to check whether a part icular kink configuration satisfies the constraints in r = 0. If so it. should stay stable during the t ime evolution.

The s tandard procedure leads to the following expression for the canonical Hamil tonian density:

, , = - gg00¢ - v ( ¢ ) , (13)

with

1 ~11 + ~[goo,gol]¢,

i t - ( - g ) - 2 g 0 1 , (14)

where we have defined the functional [x, y] =

( - g ) - } (x'y - xy'). The pr imary constraints are:

1 )CI ~--- 7"00 q- ~[gOl, g11]~ ,

I x2 -- 0 1 + [gll,g00]¢- (15)

The requirement tha t these constraints are conserved in t ime yields the secondary constraints:

¢1 - al¢'+G2 + 2 ¢ " - g n V ( ¢ ) ,

¢2 - (16)

where Gi = Gi(gu,,rr¢). Here we shall not need their explicit expressions. It can be shown tha t these constraints are conserved in time. Al- l of the constraints are first class, reflecting the reparametr iza t ion invariance of the theory.

Amongs t all initial da t a tha t satisfy the constraints (15) and (16) there are also gravi ta t ional kink configurations. As an example, we shal- l mention the initial configuration ¢ ( r = 0) = ¢o = const., which we choose to be the zero of the dilaton potential , V(¢0) = 0. Together with ~(r = O) = O, the constraints are satisfied for an arbitrary metric configuration guv(r = 0). The choice of an asymptot ica l ly Minkowskian kink will ensure tha t the boundary condit ions stay fixed during the t ime evolution since gu~ = rh, ~ together with ¢ = const, represents the vacuum of the theory. This s i tuat ion looks rather s trange because it seems tha t every kink configurat ion satisfies the constraints under the above conditions. This also includes the case V(¢) = 0, which has been analysed in [20] where it has been shown that it does not admi t regular ]m] > 1 kinks. We shall briefly analyse this case in the following section.

M. Vasili~, T. Vuka~inac/Nuclear Physics B (Proc. Suppl.) 57 (1997) 334-337 337

5. F L A T G R A V I T A T I O N A L K I N K S

As an example of the previously mentioned situation we shall consider the special case V(¢) = 0. Then, the equations of motions give ¢ = const. and

R= 0. (17)

We shall search for kink-type solutions of this equation in the gauge (3). Equation (17) is most easily solved in light-cone coordinates y+ and y - , which are well defined only between the zeros of cos mO [20]. The general solution is

~a = A(y +) + B(y- ) - In I costa01, (18)

where A and B are arbitrary functions of their ar- guments which can be chosen separately in each region of nonvanishing costa0. The question is whether this can be done so as to ensure the reg- ularity of T. It turns out that this is possible only for m = 1. In that case one can perfor- m the coordinate transformation that maps the m = 1 kink into the Minkowski metric, but this cannot be done without moving the points of c~- infinity. For [rn I > 1 the situation is different and a careful analysis of equation (18) shows that this solution can not be continuous and finite in the whole (r,~r) plane. The crucial demand is the existence of the regular solution for every r [20]. But, if one relaxes this demand, it turns out to be possible to find a ~a which is a regular function in some finite interval of 7", though necessarily nonanalytic. This is in agreement with the Hamiltonian analysis, since if the constraints are initially satisfied one can only deduce that the regular solution exists locally, for some finite time. It follows from the Hamiltonian equations of motion that the evolution can be irregular only in the points of vanishing metric determinant, g = 0. Further, analysis of these equations could tell us whether a specific initial configuration will remain regular during the global time evolution. In conclusion, there are flat Iml > 1 gravitational kinks in topologically trivial spacetimes but they are necessarily incomplete.

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