Gravitational kinks in 2d dilaton gravity

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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.


    Gravitational kinks are topologically nontriv- ial metric configurations which were studied for the first time 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 topo- logically trivial spacetimes is still missing. It has been shown that each homotopy class of four-dimensional metric configurations contains a spherically symmetric 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 gravita- tional 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 station- ary gravitational kink solutions. To generalize the previous analysis to non-s ta t ionary solutions, in section 4 we perform the Hamiltonian analy- sis of the theory. It is shown that gravitational kink configurations can be compatible with con- straints, 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 anal- ysed in section 5.


    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 van- ishing metric determinant. The cone-shaped sur- face 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 asymptotical- 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 for- bidden 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

    ,~ (costa0 sinm0 ) g~, - - sinm0 -cosm0 ' (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 = -oo , winds m times around the surface g = 0, and ends in ggv = r/~v when c = +oc. Continuous co- ordinate transformations cannot change the kink number, while the discrete transformation of time reversal maps kinks into antikinks (m -+ -m) [1].

    The light-cone analysis shows that these topo- logically nontrivial metric configurations have a twisting light-cone structure with regions of anomalous causality. The light cones of the met- ric (1) twist along a-axis 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 nonsin- gular 2d black holes with horizons at the points where cos m0 = 0.

    In 2d theory gravity two of the metric compo- nents can be gauged away by reparametrization invariance. Any 2d metric of Lorentzian signature can be written in the form [10]

    e~ (cos s ine ) \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 kink-type metrics:

    ~a rn 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.


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

    dimensional nonsingular black holes we shall con- sider 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 stationary kink configurations. Variation of the action (4) yields the field equations

    dV R + = 0, (5)


    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 gener- al stationary solution is obtained by substituting : (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 to :

    e ~ = c~', a = const, (7)


    ' 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 costa0 has on the whole axis. Let us denote these zeros by q)i and ai, respectively (i = 1,.. . ,2m).

    It can be shown that the necessary and suffi- cient conditions for the field equations to satisfy, in order to admit regular m-kink solutions, can be written in the form:

    dW 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 giv- en properties is to choose a monotonous function (~r), substitute it into (8) to obtain W((o')) as a function of e, and then use (r = cr() to find W() itself. As an example, we can take the lin- ear dilaton vacuum = c~ as a starting point and find

    W() = Win(C) = cos [me(C)]. (12)

    The potential V() is determined from (9) up to a multiplicative constant.

    In the end, let us comment on the possibility of having a potential which would allow for an arbi- trary 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 that its restriction to Dm is exactly Win. If Wm admits m-kink solu- tions it follows that W will admit k + 1 inequiva- lent kink configurations with kink numbers taking values from 0 to k. The theory of this kind will possess k - 1 types of nonsingular black holes.


    In order to investigate the more general situa- tion which also includes non-stat ionary solutions, we shall perform the Hamiltonian analysis. Since the time 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 particular kink config- uration satisfies the constraints in r = 0. If so it. should stay stable during the time evolution.

    The standard procedure leads to the following expression for the canonical Hamiltonian density:

    , , = - gg00 - v ( ) , (13)


    1 ~11 + ~[goo,gol], i t

    - ( -g ) -2g01, (14)

    where we have defined the functional [x, y] = ( -g ) -} (x'y - xy'). The primary constraints are:

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

    I x2 -- 01 + [gll,g00]- (15)

    The requirement that these constraints are con- served in time yields the secondary constraints:

    1 - al'+G2 +2" -gnV() ,

    2 - (16)

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

    Amongst all initial data that satisfy the con- straints (15) and (16) there are also gravitation- al 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 asymptotical ly Minkowskian kink will ensure that the boundary conditions stay fixed during the time evolution since gu~ = rh, ~ together with = const, represents the vacuum of the theory. This situation looks rather strange because it seems that every kink configuration satisfies the constraints under the above condi- tions. This also includes the case V() = 0, which has been analysed in [20] where it has been shown that it does not admit regular ]m] > 1 kinks. We shall briefly analyse this case in the following sec- tion.

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


    As an example of the previously mentioned sit- uation 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 nec- essarily 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 on- ly 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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