Ashtekar’s representation

4.3 Ashtekar’s representation

The example of the one dimensional wave equation 2.2 can be slightly improved by the following construction: We define a two dimensional vector $u := (u1,u2) := (ϕ,ϕt − αϕx )$ . We then have $u1 = αu1 + u2 t x$ , while, $u2 = − αu2 + (1 − α2 )ϕxx t x$ . Thus we have a diagonal – and so symmetric hyperbolic – system if $α = ±1$ , namely,

[ ] [ ] ut = α 0 ux + 0 1 u, 0 − α 0 0

The two possible values $α$ can take correspond to the two characteristic directions the wave equation defines. This trick can not be extended two more dimensions, basically because the space derivative of $ϕ$ is a vector, and so can not be properly mixed with its time derivative. But in other dimensions one can implement similar schema if the fields are not scalars but appropriate tensors.

Einstein’s equations in Ashtekar’s variables [7 , 8 ] is one beautiful example of this, since they have the remarkable property of naturally constituting a first order evolution system. Because of this reason it is also a compact system with twenty-seven unknowns, even before imposing the reality condition on the connection variable. Recently, it has been proven [40 ] that such a system is symmetric hyperbolic if suitable combinations of the constraint equations are added to their evolution equations, thus effectively changing the flow outside the constraint sub-manifold of phase-space.

In Ashtekar’s representation the basic variables are a densitized $SU (2)$ soldering form, $a B &tidle;σ A$ and a $SU (2 )$ connection $AaAB$ which are tangent to a space-like foliation of space time determined by given “lapse”-shift pair $&tidle;N = N ∕detσ$ , $N a$ .

The symmetric hyperbolic evolution equation system is:

ℒ &tidle;σb = −√-i𝒟 (N&tidle;[&tidle;σa,&tidle;σb]) + √-i-&tidle;N [C &tidle;, &tidle;σb] + 2𝒟 (N [a&tidle;σb) + N bC &tidle; + [A N a,&tidle;σb] t 2 a 2 a a

i i i ℒtAb = 𝒟b (AaN a) + N aFab + √--N&tidle;[&tidle;σa,Fba] + -2√--N&tidle;&tidle;σbC + -4N&tidle; 𝜖bdc&tidle;σcCd, 2 σ 2 σ

where $(𝒟)$ is the $SU (2 )$ derivative whose difference with respect to a flat connection is $AaAB$ . $C$ , $Ca$ , and $&tidle;C$ are the constraint equations,

C (&tidle;σ,A ) := tr(&tidle;σa&tidle;σbFab) (5 ) b Ca (&tidle;σ,A ) := tr(&tidle;σ Fab) (6 ) &tidle;C (&tidle;σ,A ) := 𝒟a &tidle;σa, (7 )

Note that here there is an extra vectorial constraint, a $SU (2)$ valued scalar, which corresponds to the fact that the system has extra degrees of freedom, the $SU (2)$ rotations. The extra constraint is just a strange way of asserting the symmetry of the second fundamental form, and is of the type of substitutions we made above to improve on the wave equation system. The constraints on themselves satisfy a symmetric hyperbolic system of equations.

Note that the principal part of the system is block diagonal and the eigenvectors-eigenvalues are very simple combinations of $&tidle;σ$ with the elements of an orthogonal basis ${ωa, ma,m¯a }$ , where $ωa$ is the wave vector.

In this new system, the “lapse”-shift pair can be chosen arbitrarily. But in fact the “lapse” that appears here is a scalar density which has already incorporated the square root of $h$ on it. So the freedom is actually the same as in the ADM representation. As in the frame representation, the lapse-shift pair appears only with derivatives up to first order. In this case it is relatively easy to see the freedom in making up evolution equations for the lapse-shift pair. As said above, the system is symmetric for Ashtekar’s variables, since the lapse-shift pair enters as terms with up to first derivatives, one can take those terms from the non-principal part of the system and promote them into the principal part of a bigger system which incorporates the lapse-shift as extra variables. Thus, these terms constitute an off-diagonal block of the bigger principal part matrix. Imposing symmetry to the bigger matrix fixes the opposite off-diagonal part of the matrix. The only freedom left is on the lapse-shift block-diagonal part, which can be chosen to be any symmetric matrix we like. The non-principal part of the equation system on the lapse-shift sector can also be chosen arbitrarily. Of course, in contrast with the ADM representation results, one can also choose a gauge condition via elliptic equations on the lapse-shift pair. In this case, the elliptic system can be of first order in the lapse-shift or related variables. For instance, one could use Witten’s equation to evolve them.

As we have seen, the generalized harmonic time gauge seems to appear naturally in most attempts to get well posed evolution systems. Thus it seems to be really a key ingredient, perhaps with some physical content. One could argue in that direction from the circumstances in which it appears in [30 ], namely effectively improving the estimates of the behavior of solutions admitting a Newtonian limit that is in a way related to the longitudinal modes of the theory. This longitudinal modes are part of the evolution, although they are not expected to behave in a hyperbolic manner. See also the comments around Equation (9) in [11 ].