B.3 Spherical symmetry

In the generic case (i.e., outside a symmetry center) of spherical symmetry we have $S = SU(2)$ , $F = U(1) = exp<t > 3$ ( $<.>$ denotes the linear span), and the connection form can be gauged to be

$AS/F = (/\(t2) sin h + /\(t3) cosh) df + /\(t1)dh. (74)$

Here $(h, f)$ are (local) coordinates on $~ 2 S/F = S$ and as usually we use the basis elements $ti$ of $LS$ . $/\(t3)$ is given by $dc$ , whereas $/\(t1,2)$ are the scalar field components. Equation (74

) contains as special cases the invariant connections found in [91 ]. These are gauge equivalent by gauge transformations depending on the angular coordinates $(h,f)$ , i.e., they correspond to homomorphisms $c$ that are not constant on the orbits of the symmetry group.

In order to specify the general form (74 ) further, the first step is again to find all conjugacy classes of homomorphisms $c: F = U(1) --> SU(2) = G$ . To do so we can make use of Equation (68 ) to which end we need the following information about SU(2) (see, e.g., [81 ]): 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

c : z '--&gt; 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 $ck$ , $k (- N0$ , 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 $exp<t > < SU(2) 3$ of the symmetry group $S$ , and use the homomorphisms $ck: exp tt3 '--> exp ktt3$ out of each conjugacy class. This leads to a reduced structure group $ZG(ck(F )) = exp<t3> ~= U(1)$ for $k /= 0$ and $ZG(c0(F )) = SU(2)$ ( $k = 0$ ; this is the sector of manifestly invariant connections of [92 ]). The map $/\|LF$ is given by $dck : <t3> --> LG, t3 '--> kt3$ , and the remaining components of $/\$ , which give us the scalar field, are determined by $/\(t ) (- LG 1,2$ subject to Equation (69 ), which here can be written as

/\ o adt3 = addc(t3) o /\.

Using $adt3t1 = t2$ and $adt3t2 = - t1$ we obtain

/\(a0t2 - b0t1) = k(a0[t3,/\(t1)] + b0[t3,/\(t2)]),

where $a0t1 + b0t2$ , $a0,b0 (- R$ is an arbitrary element of $LF_ L$ . Since $a0$ and $b0$ are arbitrary, this is equivalent to the two equations

k[t3,/\(t1)] = /\(t2), k[t3,/\(t2)] = - /\(t1).

A general ansatz

/\(t ) = a t + bt + c t , /\(t ) = a t + b t + c t 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, c1 = c2 = 0.

The configuration variables of the system are the above fields $a,b,c : B --> R$ of the $U(1)$ -connection form $A = c(x) t3dx$ on the one hand and the two scalar field components

$/\ |&lt;t1&gt;: B --&gt; LSU(2), ( ) ( -- ) x '--&gt; a(x)t1 + b(x)t2 = 1- 0 - b(x)- ia(x) =: 0 -w(x) 2 b(x)- ia(x) 0 w(x) 0$

on the other hand. Under a local U(1)-gauge transformation $z(x) = exp(t(x)t3)$ they transform as $c '--> c + dt/dx$ and $w(x) '--> exp(- it)w$ , which can be read off from

$A '--&gt; z-1Az + z -1dz = A + tdt, ( 3 --) /\(t ) '--&gt; z-1/\(t )z = 0 - exp(it)w . 1 1 exp(- it)w 0$

In order to obtain a standard symplectic structure (see Equation (77 ) 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 $integral (8pgG) - 1 d3xAiaEai S$ .

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 $k = 1$ .