Spherical symmetry

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

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 (89

) 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 (89 ) 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 (83 ), 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 (84 ), 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

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 (92 ) below), we reconstruct the general invariant connection form

An invariant densitized-triad field is analogously given by

with coefficients $EI$ canonically conjugate to $AI$ ( $E2$ and $E3$ are non-vanishing only for $k = 1$ ). The symplectic structure

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

we have

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 $k = 1$ .