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arXiv:gr-qc/0005055v1 15 May 2000

Topologically massive magnetic monopoles

A. N. Aliev, Y. Nutku and K. Saygılı

Feza Gursey Institute, P. O. Box 6 Cengelkoy 81220 Istanbul, Turkey

We show that in the Maxwell-Chern-Simons theory of topologically mas-

sive electrodynamics the Dirac string of a monopole becomes a cone in anti-

de Sitter space with the opening angle of the cone determined by the topo-

logical mass which in turn is related to the square root of the cosmological

constant. This proves to be an example of a physical system, a priory com-

pletely unrelated to gravity, which nevertheless requires curved spacetime

for its very existence. We extend this result to topologically massive gravity

coupled to topologically massive electrodynamics in the framework of the

theory of Deser, Jackiw and Templeton. The 2-component spinor formal-

ism, which is a Newman-Penrose type of approach for three dimensions, is

extended to include both the electrodynamical and gravitational topolog-

ically massive field equations. Using this formalism exact solutions of the

coupled Deser-Jackiw-Templeton and Maxwell-Chern-Simons field equations

for a topologically massive monopole are presented. These are homogeneous

spaces with conical deficit. Pure Einstein gravity coupled to Maxwell-Chern-

Simons field does not admit such a monopole solution.

1 Introduction

The principal result we shall present in this paper is a physical system which

at the outset is not related to gravity but which nevertheless requires curved

spacetime for its very existence. This situation is best illustrated with the

example of a magnetic monopole in the framework of both electrodynamical

and gravitational topologically massive theories in 3-dimensions. We find

that the essential new feature introduced by topological mass is to open

up the Dirac string of a monopole into a cone. The intuitive example of

this phenomenon takes place for Maxwell-Chern-Simons (MCS) theory in a

Riemannian 3-manifold with Euclidean signature which shows that solutions

of the MCS field equations naturally lead us into de Sitter (dS) space with

conical deficit. Three dimensional flat spacetimes with, or without conical

deficit do not allow such a solution.

In 4n+3 dimensions there exists the Chern-Simons action through which

we can introduce topological mass into Maxwell’s electrodynamics and Ein-

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stein’s gravity [1].

has no propogating degrees of freedom and no Newtonian limit [2]. On

the other hand, three dimensional gravity with a pure Chern-Simons ac-

tion is equivalent to the Yang-Mills gauge theory of the conformal group

and therefore is finite and exactly solvable [3]. There is, however, a very

interesting non-trivial theory of gravitation in 2 + 1 dimensions which has

been proposed by Deser, Jackiw and Templeton (DJT) [1] where the grav-

itational Chern-Simons action is added to the Hilbert action. This is the

theory of topologically massive gravity (TMG). It is a dynamical theory of

gravity unique to three dimensions and the geometry of its exact solutions

is non-trivial. Mathematically the DJT field equations pose an interest-

ing challenge in that they are qualitatively different from the Einstein field

equations while posessing their elegance and consistency. Clement [4] has

made the most thorough investigation of the solutions of DJT field equations

for TMG as well as TME which uncovered many interesting effects due to

topological mass. Self-dual solutions of TME coupled to Einsteinian gravity

were discussed by Fernando and Mansouri [5] and by Dereli and Obukhov

[6] who gave the general analysis.

This class of fields we shall consider falls outside the domain of solutions

considered earlier [4]-[6] and illustrate in its purest form some of the new

interesting effects that take place in the presence of topological mass. Earlier

we [7] presented the spinor formulation of TMG in terms of real 2-component

spinors which provides a very useful formalism analogous to the Newman-

Penrose formalism [8] of general relativity. We shall extend this formalism to

include topologically massive electrodynamics and gravity. This formalism is

helpful for constructing physically meaningful exact solutions of the coupled

DJT-MCS field equations. We shall use it to derive the exact solution for a

topologically massive magnetic monopole.

In the simplest case of n = 0 pure Einstein gravity

2 Dirac Monopole

It will be useful to start our considerations with a brief review of the Dirac

monopole and its extension to TME in order to explain the essential idea

we shall use throughout this paper. Maxwell’s electrodynamics is given by

the action principle

IM= −1

2

?

(F −1

2dA) ∧∗F (1)

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in Minkowski spacetime and leads to the Maxwell field equations

F= dA,⇒ dF = 0(2)

d∗F=0 (3)

the first one of which is the second Bianchi identity. Dirac [9] pointed out

that for a magnetic monopole eqs.(2) must fail at least in one point on every

Gaussian surface enclosing the monopole. The set of all such points forms

the Dirac string. The Maxwell potential that satisfies these requirements is

a generalization of the 1-form obtained for the polar angle dφ on the plane

which is closed but not exact. The U(1) potential 1-form and the field 2-form

for the Dirac monopole are given by

A=g [1 − cosθ]dφ,

g sinθdθ ∧ dφ,

(4)

F= (5)

the latter of which is the familiar element of area on S2. The semi-infinite

Dirac string is at θ = 0 and the surface integral

?

F = 4π g(6)

determines the monopole magnetic charge.

We shall now consider the Euclidean Maxwell-Chern-Simons topologi-

cally massive electrodynamics in order to illustrate the essential new idea

brought in by making the Dirac monopole topologically massive. With the

inclusion of the electromagnetic Chern-Simons term the action is given by

IMCS= −1

2

? ?

(F −1

2dA) ∧∗F − ν dA ∧ A

?

(7)

which yields the MCS field equation

d∗F = νF (8)

and the Bianchi identity (2) where ν is a coupling constant, the electromag-

netic topological mass. In order to satisfy these field equations with a U(1)

potential 1-form satisfying the properties of the Dirac monopole (4) we must

introduce a deficit in the polar angle θ that led to the Dirac string. That

is, topological mass has the effect of turning the string into a cone. Thus we

should consider a field 2-form of the type

F = g sin(bθ)dθ ∧ dφ (9)

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where b is a constant deficit parameter which will be related to topological

mass. Now it is clear that the potential 1-form (4) must be modified but

still lead to eq.(9) as the field. This suggests that we consider a Riemannian

manifold with the co-frame consisting of a modified form of the left-invariant

1-forms of Bianchi Type IX parametrized in terms of Euler angles

σ0

σ1

σ2

= dψ + cos(bθ)dφ

=−sin(bψ)dθ + cos(bψ) sin(bθ)dφ

cos(bψ)dθ + sin(bψ) sin(bθ)dφ

(10)

=

satisfying the Maurer-Cartan equations

dσi=1

2Ci

jkσj∧ σk

(11)

with non-vanishing structure constants

C0

12= −C0

Then a U(1) potential 1-form of the type

21= b,C1

20= −C1

02= b,C2

01= −C2

10= b. (12)

A = −gσ0

(13)

will have all the desired properties and lead to the field 2-form (9). The

clue to the satisfaction of the field equation (8) for TME lies in the fact

that with the co-frame (10) the Cartan-Killing metric ds2= ηikσi⊗σkwith

ηik= diag.(1,1,1) becomes

ds2= dθ2+ dφ2+ dψ2+ 2cos(bθ)dψ dφ(14)

which is simply de Sitter space with the polar angle suffering a defect. The

duality relations for the basis (10) immediately leads to the result that for

the potential 1-form (13) eqs.(8) of TME will be satisfied identically in dS

provided

b = ν,(15)

the deficit in the Eulerian polar angle is identified with topological mass.

From the curvature of (14) we find

λ =ν2

4

(16)

relating the cosmological constant to electromagnetic topological mass. For

the case of Lorentzian signature, c.f. section 6, this would be anti-de Sitter

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spacetime. The field 2-form (9) is shown in figure 1 where because we are

in the Euclidean sector the deficit in the polar angle caused by topological

mass can be given an explicit illustration. The maximal analytical extension

of dS space is given in the chart

cos

?ν

2θ

?

cos

?ν

2(φ + ψ)

?ν

1

2(φ − ψ)

?

?

= tanh(να),

sin

2θ=

sin(νβ)

cosh(να), (17)

=γ

whereby the metric (14) becomes

d˜ s2= Ω−2?

dα2+ dβ2+ sin2(νβ)dγ2?

2cosh(να). The incomplete Einstein static

(18)

with the conformal factor Ω =1

cylinder is manifest in eq.(18).

In the discussion of TME monopole we started out with MCS field equa-

tions (2) and (8) which are written in a general background. The expectation

was that these field equations will admit a solution in flat background space-

time for a physical system which is electrodynamic in nature and a priory

completely unrelated to gravity. This proved to be impossible. With the

missing cone in the field 2-form (9) eqs.(2) and (8) could only be satisfied

in curved space (14) with a corresponding conical deficit.

Thus we arrive at a remarkable conclusion that a physical system of

electrodynamic type requires curved spacetime for its existence.

3 Spinor formalism in (2 + 1)-dimensions

We shall now turn our attention to Lorentz signature and introduce the

Newman-Penrose version of TME equations. This type of study for topo-

logically massive gravity was given by Hall, Morgan and Perjes [10] and its

2-component spinor description with differential forms was constructed in

[7] which will henceforth be referred to as I. Here we shall extend this work

by first presenting the spinor formulation of TME and in section 5 couple it

to TMG.

We begin by recalling some basic relations from I. At each point of a

three dimensional space-time with the metric of Lorentz signature we can

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