A topological metric in 2+1-dimensions

Real-valued triplet of scalar fields as source gives rise to a metric which tilts the scalar, not the light cone, in 2+1-dimensions. The topological metric is static, regular and it is characterized by an integer $\kappa =\pm 1,\pm 2,...$. The problem is formulated as a harmonic map of Riemannian manifolds in which the integer $\kappa $ equals to the degree of the map.

The topic of metrical kinks has a long history in general relativity [1] which declined recently toward oblivion. On the other hand in a broader sense interest in topological aspects in non-linear field theory, for a number of reasons, remains ever alive. Although these emerge mostly in flat 3 + 1−dimensional spacetime with the advent of higher / lower dimensions the same topological concepts may find applications in these cases as well.
The aim of this note is to revisit this subject in 2 + 1−dimensions. Motivation for this lies in part by the discovery of a cosmological black hole [2] which became a center of attraction in this particular dimension. Does a topological metric also make a black hole? The answer to this question turns negative, at least in our present study. The derived metric is sourced by a triplet of scalar fields φ a (r, θ) , (a = 1, 2, 3, is the internal index) which satisfies the constraint (φ a ) 2 = 1, with topological properties. Let us add that besides the topological solution our system of triplet fields admits special solutions including black holes. Our interest, however, will be focused on the topological one. Unlike the geometrical kink metrics [1] which tilt the light cone, leading to closed timelike curves, our topological metric tilts the scalar field along its range. The metric admits an integer, κ = ±1, ±2, ... which can be interpreted (following Ref. [3]) as the topological charge / homotopy class. The total energy is a multiple of |κ| and relation with the harmonic maps (HM) of Riemannian manifolds [4] suggests that κ is at the same time the degree of the map.
Variational principle yields the field equation and the constraint condition (4).
In the sequel we shall make the choice with κ = ±(integer) for uniqueness condition. This reduces the action effectively, modulo the time sector, to in which a prime stands for d dr . With the energymomentum tensor variation with respect to α (r) and Einstein equations we obtain the following equations This system of differential equations admits a number of particular solutions. As an example we give the following.

A black hole solution
This is obtained by where C 0 is an integration constant that can be interpreted as mass. The scalar field triplet takes the form which is effectively a doublet of scalars. Ricci scalar of this solution reads which is singular at r = 0. Event horizon r h of the resulting black hole is so that it is characterized by the index κ. Similarly the Hawking temperature also is stamped by the integer κ 2 . Clearly this is a different situation from the BTZ black hole [2], where the parameter, i.e. cosmological constant (and electric charge) are not integers.

The topological solution
Our system of equations (10-13) admits a solution with the choice A (r) = 1. Accordingly, Eq. (10) reduces to the Sine-Gordon equation where in which r 0 is a constant that will be set r 0 = 1. In the new variable the solution for α (ρ) and B (ρ) become α (ρ) = 2 tan −1 1 sinh ρ so that the resulting 2+1−dimensional line element takes the form This represents a regular, non-black hole, static spacetime. The non-zero geometrical quantities are Ricci scalar: R = 4κ 2 cosh 2 ρ (25) and the energy-momentum tensor is As a result our triplet of scalar fields take the form This leads for ρ = 0, to It is observed that between 0 ≤ ρ < ∞ the angle α (ρ) shifts from −1 to +1, which amounts to the case of onekink. It should also be remarked that 'kink' herein is used in the sense of flip of the φ 3 component of the triplet, not in the sense of light cone tilt. The energy density of the kink is maximum at ρ = 0, which decays asymptotically whose energy E κ is We wish to add, for completeness that the α (r) equation can be described as a harmonic map, between two Riemannian manifolds M and M ′ [4] f A : M → M ′ which are defined by = g ab dx a dx b , (a, b = 1, 2).
The energy functional of the map is defined by which yields, upon variation the equation for α (ρ). Note that in this map we consider a priori that α = α (ρ) and β = β (θ). The degree of harmonic map (d) is defined in an orthonormal frame x i by which equals to the topological charge [3]. Although the maps in the original work of Eells and Sampson [4] were considered between unit spheres (in particular S 2 → S 2 ) in the present problem our map is from R 2 → S 2 . Unfortunately the non-compact and singular manifolds of general relativity create serious handicaps which prevented a wider application of the concept of degrees of the maps once they are formulated in harmonic forms.
In conclusion we comment that topological properties of field theory were well-defined in a flat space background. Due to the singular and non-compact manifolds of general relativity these concepts found no simple applications in a curved spacetime. In this note we have shown that at least in the 2 + 1−dimensional spacetime the problem can be overcome. The source of our metric is provided by a triplet of scalar fields which may find applications as multiplets of scalar fields in higherdimensions. It has been shown that the triplet source gives rise to other solutions, such as black holes, besides the topological metric. The technical problems such as the non-linear superposition of Sine-Gordon solutions in a curved space leading to the 'multi-kink' metric remains to be seen.