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B.3 Spherical symmetry

In the generic case, i.e., outside a symmetry center, of spherical symmetry we have S = SU (2), F = U (1) = exp⟨τ ⟩ 3 (⟨⋅⟩ denotes the linear span), and the connection form can be gauged to be
AS ∕F = (Λ (τ2) sin ϑ + Λ(τ3)cos ϑ)d ϕ + Λ(τ1)dϑ . (89 )
Here, (ϑ,ϕ ) are (local) coordinates on ∼ 2 S ∕F = S and, as usual, we use the basis elements τi of ℒS. Λ (τ3) is given by dλ, whereas Λ (τ1,2) are the scalar field components. Equation (89View Equation) contains, as special cases, the invariant connections found in [135]. These are gauge equivalent by gauge transformations depending on the angular coordinates (ϑ,ϕ), i.e., they correspond to homomorphisms λ, which are not constant on the orbits of the symmetry group.

In order to specify the general form (89View Equation) further, the first step is again to find all conjugacy classes of homomorphisms λ: F = U(1) → SU (2) = G. To do so, we can make use of Equation (83View Equation), to which end we need the following information about SU(2) (see, e.g., [115]). The standard maximal torus of SU(2) is given by

T(SU (2)) = {diag(z,z−1) : z ∈ U (1)} ∼= U (1)

and the Weyl group of SU(2) is the permutation group of two elements, W (SU (2)) ∼ S = 2, its generator acting on T (SU(2)) by −1 − 1 diag(z,z ) ↦→ diag(z ,z).

All homomorphisms in Hom (U (1),T(SU (2))) are given by

λ : z ↦→ diag(zk,z−k) k

for any k ∈ ℤ, and we have to divide out the action of the Weyl group leaving only the maps λk, k ∈ ℕ0, as representatives of all conjugacy classes of homomorphisms. We see that spherically-symmetric gravity has a topological charge taking values in ℕ0 (but only if degenerate configurations are allowed, as we will see below).

We will represent F as the subgroup exp⟨τ ⟩ < SU (2 ) 3 of the symmetry group S, and use the homomorphisms λk: exp tτ3 ↦→ exp ktτ3 out of each conjugacy class. This leads to a reduced-structure group ZG (λk (F )) = exp⟨τ3⟩ ∼= U (1) for k ⁄= 0 and ZG(λ0 (F )) = SU (2) (k = 0; this is the sector of manifestly invariant connections of [136]). The map Λ |ℒF is given by dλk : ⟨τ3⟩ → ℒG, τ3 ↦→ k τ3, and the remaining components of Λ, which give us the scalar field, are determined by Λ (τ ) ∈ ℒG 1,2 subject to Equation (84View Equation), which here can be written as

Λ ∘ adτ3 = addλ(τ3) ∘ Λ .

Using ad τ3τ1 = τ2 and ad τ3τ2 = − τ1 we obtain

Λ(a0τ2 − b0τ1) = k (a0[τ3,Λ (τ1)] + b0[τ3,Λ (τ2)]),

where a0τ1 + b0τ2, a0,b0 ∈ ℝ is an arbitrary element of ℒF ⊥. Since a0 and b0 are arbitrary, this is equivalent to the two equations

k[τ3,Λ (τ1)] = Λ (τ2) , k[τ3,Λ(τ2)] = − Λ(τ1).

A general ansatz

Λ(τ ) = a τ + bτ + c τ , Λ(τ ) = a τ + b τ + c τ 1 1 1 1 2 1 3 2 2 1 2 2 2 3

with arbitrary parameters ai,bi,ci ∈ ℝ yields

k(a1τ2 − b1τ1) = a2τ1 + b2τ2 + c2τ3, k (− a2τ2 + b2τ1) = a1τ1 + b1τ2 + c1τ3,
which have non-trivial solutions only if k = 1, namely
b2 = a1 , a2 = − b1 and c1 = c2 = 0 .

The configuration variables of the system are the above fields a,b,c : B → ℝ of the U(1)-connection form A = c(x) τ3dx on the one hand and the two scalar-field components

Λ |⟨τ1⟩: B → ℒSU (2), ( ) ( -- ) x ↦→ a(x )τ1 + b(x)τ2 = 1- 0 − b(x) − ia(x) =: 0 −w (x) 2 b(x) − ia (x ) 0 w (x) 0
on the other hand. Under a local U(1)-gauge transformation z(x) = exp (t(x)τ3) they transform as c ↦→ c + dt ∕dx and w(x) ↦→ exp(− it)w, which can be read off from
A ↦→ z− 1Az + z− 1dz = A + τ dt , ( 3 --) Λ(τ ) ↦→ z− 1Λ(τ )z = 0 − exp(it)w . 1 1 exp(− it)w 0

In order to obtain a standard symplectic structure (see Equation (92View Equation) below), we reconstruct the general invariant connection form

A (x,ϑ, ϕ) = A1(x )τ3dx + (A2(x)τ1 + A3(x)τ2)dϑ (90 ) + (A2(x)τ2 − A3(x )τ1) sin ϑd ϕ + cosϑ dϕ τ3.
An invariant densitized-triad field is analogously given by
(Ex, E ϑ,E ϕ) = (E1 sin ϑτ3, 1 sin ϑ(E2 τ1 + E3τ2), 1(E2 τ2 − E3 τ1)) (91 ) 2 2
with coefficients EI canonically conjugate to AI (E2 and E3 are non-vanishing only for k = 1). The symplectic structure
J J {AI (x ),E (y )} = 2γG δI δ(x,y) (92 )
can be derived by inserting the invariant expressions into ∫ (8πγG )− 1 d3xA˙iaEai Σ.

Information about the topological charge k can be found by expressing the volume in terms of the reduced triad coefficients I E. Using

ijk a b c a b c 3 2 1( 2 2 3 2) εabcε E i E jEk = − 2εabctr(E [E ,E ]) = 2 sin ϑE (E ) + (E ) (93 )
we have
∫ ∘ ------------------ ∫ 3 1 || ijk i j k|| ∘ --1----2-2-----3-2-- V = d x 6 εabcε E aEbE c = 2π dx |E |((E ) + (E ) ). (94 ) Σ B
We can now see that in all the sectors with k ⁄= 1 the volume vanishes because then 2 3 E = E = 0. All these degenerate sectors have to be rejected on physical grounds and we arrive at a unique sector of invariant connections given by the parameter k = 1.


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