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10.1 Bianchi I models and eleven-dimensional supergravity

On the supergravity side, we will restrict the metric and the electromagnetic field to depend on time only,
2 2 2 a b ds = − N (t)dt + gab(t)dx dx , (10.1 ) Fλρστ = Fλρστ(t).
Recall from Section 9.3 that with these ansätze the dynamical equations of motion of eleven-dimensional supergravity reduce to [61Jump To The Next Citation Point]
1- (√ -- − 1 ac ) -1- √ -- aρστ --1- √ -- a λρστ 2 ∂ gN g ∂gcb = 12 N gF Fb ρστ − 144 N gδ b F F λρστ, (10.2 ) ( √ -) 1 ∂ F tabcN g = ----ɛtabcd1d2d3e1e2e3e4Ftd1d2d3Fe1e2e3e4, (10.3 ) 144 ∂Fa1a2a3a4 = 0. (10.4 )
This corresponds to the truncation of the sigma model at level 2 which, as we have seen, completely matches the supergravity side. We also defined ∂ ≡ ∂t as in Section 9.3. Furthermore we have the following constraints,
1( ) 1 1 --gacgbd − gabgcd ∂gab∂gcd + ---F tabcFtabc + --N 2F abcdFabcd = 0, (10.5 ) 4 12 48 1N F tbcdF = 0, (10.6 ) 6 abcd ɛtabc1c2c3c4d1d2d3d4Fc c cc Fd dd d = 0, (10.7 ) 1 23 4 1 23 4
which are, respectively, the Hamiltonian constraint, momentum constraint and Gauss’ law. Note that Greek indices α,β,γ, ⋅⋅⋅ correspond to the full eleven-dimensional spacetime, while Latin indices a,b,c,⋅⋅⋅ correspond to the ten-dimensional spatial part.

We will further take the metric to be purely time-dependent and diagonal,

∑10 ds2 = − N 2(t)dt2 + a2i(t)(dxi)2. (10.8 ) i=1
This form of the metric has manifest invariance under the ten distinct spatial reflections
xj → − xj, i⁄=j i⁄=j (10.9 ) x → x ,
and in order to ensure compatibility with the Einstein equations, the energy-momentum tensor of the 4-form field strength must also be diagonal.

10.1.1 Diagonal metrics and geometric configurations

Assuming zero magnetic field (this restriction will be lifted below), one way to achieve diagonality of the energy-momentum tensor is to assume that the non-vanishing components of the electric field F ⊥abc = N −1Ftabc are determined by geometric configurations (nm, g3) with n ≤ 10 [61Jump To The Next Citation Point].

A geometric configuration (n ,g ) m 3 is a set of n points and g lines with the following incidence rules [117105Jump To The Next Citation Point145Jump To The Next Citation Point]:

  1. Each line contains three points.
  2. Each point is on m lines.
  3. Two points determine at most one line.

It follows that two lines have at most one point in common. It is an easy exercise to verify that mn = 3g. An interesting question is whether the lines can actually be realized as straight lines in the (real) plane, but, for our purposes, it is not necessary that it should be so; the lines can be bent.

Let (nm, g3) be a geometric configuration with n ≤ 10 points. We number the points of the configuration 1,⋅⋅⋅ ,n. We associate to this geometric configuration a pattern of electric field components ⊥abc F with the following property: ⊥abc F can be non-zero only if the triple (a,b,c) is a line of the geometric configuration. If it is not, we take F ⊥abc = 0. It is clear that this property is preserved in time by the equations of motion (in the absence of magnetic field). Furthermore, because of Rule 3 above, the products F⊥abcF ⊥a′b′c′gbb′gcc′ vanish when a ⁄= a′ so that the energy-momentum tensor is diagonal.

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