{'fraction_non_alphanumeric': 0.04671532846715328, 'fraction_numerical': 0.015547445255474452, 'mean_word_length': 4.129539498315237, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Time-Dependent Open String Solutions in 2+1 Dimensional Gravity

We find general, time-dependent solutions produced by open string sources carrying no momentum flow in 2+1 dimensional gravity. The local Poincar\'e group elements associated with these solutions and the coordinate transformations that transform these solutions into Minkowski metric are obtained. We also find the relation between these solutions and the planar wall solutions in 3+1 dimensions.


Introduction
There has recently been much interest in 2 + 1 dimensional gravity [1]. Three dimensional Einstein gravity has the important feature that spacetime is locally flat away from sources. Nevertheless solutions can be globally non-trivial. It also admits a gauge formulation with the action given by the Chern-Simons action for the 2 + 1 dimensional Poincaré group ISO(2, 1) [2]. In this work we study general, time-dependent solutions produced by open string sources (which are translational invariant along the direction of the string) carrying no momentum flow. Static string solutions in 2 + 1 dimensions were previously studied by Deser and Jackiw [3]. Stationary string solutions were also studied [4].
The remainder of this paper is organized as follows. In Sec.II we obtain vacuum solutions away from the string, and study their gauge formulation. In Sec.III we match these vacuum solutions across the string using junction conditions. We then obtain the relation between these solutions and the 3 + 1 dimensional planar wall solutions studied by Vilenkin, Ipser and Sikivie [5] [6]. Brief conclusions are given in Sec.IV. We study the geodesic motion of test particles in a typical background metric in the appendix.

Vacuum Solutions and Their Gauge Formulation
In this section we shall find vacuum solutions for the following "line symmetric" metric ds 2 = −A 2 (t, z)dt 2 + C 2 (t, z)dz 2 + B 2 (t, z)dx 2 (2.1) This is the most general form one can have if one requires the metric be invariant under x translation x → x + a and inversion x → −x, the x-axis being where the string resides. One can further make A = C by appropriate coordinate transformations. (It is well known in elementary differential geometry that any twodimensional metric can be brought into this "isothermal" form.) The position of the string, z = 0, will in general be transformed into z = h(t). One can then transform it back to z = 0, while retaining the A = C form of the metric, by a transformation of the form where f and g are subject to the constraint and a prime denotes differentiation of a function with respect to its argument.
Therefore to find general solutions, we only need to find solutions with A = C.
However, we would like to leave A and C arbitrary as far as we can, so that results are directly applicable to solutions not in the A = C form. It is also easier to obtain some particular solutions when A and C are arbitrary. We will illustrate this later.
To proceed, we choose the dreibein e a ≡ e aµ dx µ to be Latin indices are raised and lowered by η ab = diag(−1, 1, 1) and Greek indices by g µν . It follows that the connection 1-forms ω ω ab ≡ ω µab dx µ and the curvature two-forms R ab ≡ R µνab dx µ ∧ dx ν are given by In 2 + 1 dimensions, the curvature tensor is proportional to the Einstein tensor Therefore vacuum Einstein equations can be solved by setting the curvature tensor R µναβ to zero. In the case of static solutions, one has To find time-dependent solutions, we define We then obtain the following five equations. Only four of them are independent, but all are needed to make the whole set of equations symmetric.
The last four equations tell us that P 2 − Q 2 is a (t, z) independent constant. Thus solutions fall into two classes:

Type-I solutions
The solutions are characterized by P 2 − Q 2 = ±β 2 = 0. With the + sign, the solutions are given by where β is an arbitrary constant and S = S(t, z) an arbitrary function. Interchanging cosh S and sinh S in the above also gives solutions. This corresponds to choosing the − sign. We will refer to these solutions as type-I(±) solutions. For simplicity, the distinction (±) will sometimes not be made.

Type-II solutions
This case corresponds to P 2 − Q 2 = 0. The solutions are where P = P (t, z) is an arbitrary function. We choose either all upper signs or all lower signs. Eq.(2.11) with β set to zero also gives type-II solutions.
According to Eq.(2.7), the vacuum spacetime is maximally symmetric with zero curvature. Therefore despite its appearance, the spacetime is actually homogeneous and isotropic about all points. The metric can always be transformed into the Minkowski one We assume that this coordinate transformation is described by where x µ ≡ (t, z, x) T and q a ≡ ( t, z, x) T . This defines the (spacetime dependent) matrix Λ. That Λ is a Lorentz matrix follows from the fact that e a µ is the dreibein. To see this, consider a vector V µ and its transformation into Minkowski coordinates V a . We have Comparing this with one obtains in matrix notation There are six Killing vectors. In Minkowski coordinates q a , they are given by where c a are constant vectors and J the Lorentz generators. If we define the Lie bracket of two vectors l a and m a to be then the Lie bracketing of these Killing vectors reproduces the commutators of the Poincaré group [7]. Transforming these vectors into x µ coordinates we obtain the vectors which must satisfy the Killing equation Substituting Eq.(2.22) into this we obtain, with l a "stripped off", (The extra contributions obtained when the differential operators act on the l a given by Eq.(2.20) cancel due to Eq.(2.14) and the antisymmetry of the Lorentz Note that, as we will show later, (Λ −1 ∂ µ Λ) a b is anti-symmetric in a and b for any Lorentz matrix Λ. Eq.(2.26) determines Eqs.(2.14) and (2.27) reproduce in an instructive way the familiar equations in the Poincaré gauge formulation of the Einstein gravity. These can also be viewed as a general proof of Gerbert's claim that the vector q a satisfying Eq.(2.14) is the local coordinate transformation that transforms the metric g µν into η ab , which he made based on the study of a class of point source solutions [8].
In terms of the gauge formulation, since the gauge fields vanish (The connection 1-forms Eq.(2.5) already satisfy the torsion free condition, while Eq.(2.6) sets the Riemann curvature to zero.), the gauge potentials associated with these vacuum solutions must be pure gauges Here U is an element of the three dimensional Poincaré group ISO(2, 1), ω a µ = − 1 2 ǫ abc ω µbc , and J a , P a are group generators obeying with ǫ 012 = 1. We wish to present the Poincaré group elements U corresponding to the solutions Eqs.(2.11) and (2.12). In the meantime, we will find the coordinate transformations that transform these solutions into the Minkowski metric. We use a 4 × 4 representation for ISO(2, 1) where an arbitrary group element V has the where q a is a three-vector and Λ a b is a 3 × 3 Lorentz matrix. If we denote V by (Λ, q), then Eq.(2.28) reads with U −1 ≡ (Λ, q) are the group generators. More explicitly Since the right-hand side of Eq.(2.31) is known, we can solve the equation to obtain (Λ, q). To this end, we introduce We consider an arbitrary Lorentz group element Λ, parametrized by 1 Λ = e uJ+−vJ− e wJ2 (2.36) 1 The fact that any rotation group element can be brought into this form was used to construct representations for the three dimensional Euclidean group, see [10].
Using the commutators Eq.(2.35) and Comparing this with Eqs.(2.31), (2.5) and (2.11), we get for type-I(+) solutions or, more explicitly, To find out the desired coordinate transformations, we only need to integrate ∂ µ q = where S is given in terms of the metric via Eq.
For type-II(+) solutions, the Λ matrix is given by That is, The general transformations are One can easily obtain the transformation laws.
Next we specialize to solutions with A = C. We only consider "+" type solutions. The "−" type solutions can be similarly studied. For type-II solutions, the integrability condition for the first two equations in Eq.(2.12) tells us that ln P satisfies the wave equation Thus the general solution for P may be written as where a prime denotes differentiation of a function with respect to its argument.
The choice of this form for P is for future convenience. It follows that the metric is given by where f and g are arbitrary functions subject to the constraint Eq. (2.3). Under a coordinate transformation and where β is a constant and f, g are arbitrary functions satisfying Eq.(2.3). This can be transformed into Minkowski metric by the transformations in [3], solutions that are not simply related to those must be time-dependent ones.
This completes our study of the vacuum solutions.

Open String Solutions and the Relation to Planar Walls
We will look for reflection symmetric solutions of the form The energy-momentum tensor of an open string, positioned at z = 0, is given by We consider sources for which where σ and ξ correspond to the energy density and tension of the string, respectively. For the metric Eq.(3.1), the three velocity of the string is u µ = (1/A, 0, 0) and the normal to the z = 0 hypersurface is n µ = (0, 1/C, 0). Thus the induced two dimensional metric is Conservation of the energy-momentum tensor is given by ∇ µ T µν = 0. The x component of this equation is trivially satisfied. The z component is also identically satisfied for reflection symmetric solutions, due to the following prescription [12] Γ z αα (z = 0) = lim Finally, the t component gives us the conservation law In the regions z > 0 or z < 0, the solutions are given by the vacuum solutions found in the last section. We need to match these solutions using junction conditions. Junction conditions can be obtained by integrating the Einstein equations across the string. In an orthonormal basis, we have If we apply +ǫ −ǫ dz to these equations, only terms with a δ(z) singularity survive. On the left-hand side, only terms containing a double z derivative ∂ 2 z survive. Using the explicit expressions for the curvature tensor, Eq.(2.6), we obtain, for reflection symmetric solutions, For any solutions satisfying Einstein equations and the junction conditions, the energy-momentum conservation is automatically satisfied.
We present some special solutions as examples. These are obtained by choosing S and P in Eqs.(2.11) and (2.12) to be simple functions. We use Eq.(3.11) for the energy-momentum tensor, and thus Eq.(3.16) for the junction conditions. For type-I solutions we obtain where α is a constant. They correspond to P = αe αt , αe −αz , respectively.
One can transform these solutions into the A = C form. For example, from where f is still an arbitrary function. Similarly, type-II solutions are given by to Vilenkin's domain wall solution [5], which takes the form in terms of our convention. On the other hand, there is no four-dimensional solution corresponding to Eq.(3.18); and one can verify that is not a vacuum solution in 3 + 1 dimensions.
To establish the general relations, we use the reduction formulae where α, β, µ, ν = 0, 1, 2 or t, z, x and D α is the covariant derivative associated with the three dimensional metric g αβ We see that given a 2 + 1 dimensional vacuum solution Eq.(2.1), the metric will be a flat spacetime solution in 3 + 1 dimensions if and only if Performing a dimensional reduction once more to 1 + 1 dimensions, we see that only possibly non-vanishing terms are D α D x B. Using the explicit expressions for Γ λ αβ , we see that the only possible non-zero term is Next we consider junction conditions. If we normalize G as in Eq.(3.7), the junction conditions in 3 + 1 dimensions (with energy-momentum tensor given by Eq.(3.11)) are Ipser and Sikivie have noted the "gravitational repulsiveness" of a planar wall [6]. Consider a test particle, initially at rest, in the vicinity of a planar wall. From the geodesic equation we obtain its initial acceleration in the z direction: From the junction conditions we see that this is positive if −σ + 2ξ is positive.
The wall is then said to be repulsive. Essentially the same calculation applied to the 2 + 1 dimensional theory tells us that a string is gravitationally repulsive if the string tension ξ is positive.

Conclusions
In conclusion, we have found general, line symmetric vacuum solutions and we have matched them across the string using the junction conditions. For any metric, we have given the corresponding string energy density and tension. We have studied the gauge formulation of these solutions and found their local Poincaré group elements. The relation of these solutions to the four dimensional planar walls is also obtained.

APPENDIX
In this appendix, we study geodesics in the background metric ds 2 = −z 2 dt 2 + dz 2 + z 2 e 2t dx 2 , −1 < z < ∞ Since there is only one cyclic coordinate, it is difficult to solve the differential equations. Thus we will use the Hamilton-Jacobi equation, which turns out to be Letting W (t, z, x) = xp x + W 1 (t) + W 2 (z), we get where p x , E, η are arbitrary constants. The motion is described by where p x , η, E, x 0 , ζ, λ 0 are constants. The integrals can all be carried out. As a simple example, we consider the motion with η = 0. The geodesic is where α is a constant. The metric Eq.(A.1) can be flattened by the transformation In terms of these new coordinates, the above geodesic becomes manifestly a straight line t + z = 1 − αλ, t − z = 0, x = λ (A.9) On the other hand, an arbitrary straight line in ( t, z, x) coordinates is not necessarily a geodesic, since it may be out of the region that the spacetime of Eq.(A.1) covers.