By Martin C. Gutzwiller

Describes the chaos obvious in easy mechanical platforms with the target of elucidating the connections among classical and quantum mechanics. It develops the appropriate rules of the final 20 years through geometric instinct instead of algebraic manipulation. The old and cultural heritage opposed to which those medical advancements have happened is depicted, and lifelike examples are mentioned intimately. This booklet allows entry-level graduate scholars to take on clean difficulties during this wealthy box.

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5 Geometrical interpretation Let us first compute the scalar curvature R(3) of the slices with constant time η with the induced metric g (3) = a2 (η) (1 + 2D)γij + 2E|ij dxi dxj . 24) If we drop the factor a2 , then the Ricci tensor does not change, but R(3) has to be multiplied afterwards with a−2 . For the metric γij + hij the Palatini identity (eq. 25) gives δRi i = hij |ij − △h (h := hii ), hij = 2Dγij + 2E|ij . We also use h = 6D + 2△E, E |ij |ij = ∇j (△∇j E) = ∇j (∇j △E − 2K∇j E) = △2 E − 2K△E (0) (we used (∇i △ − △∇i )f = −Rij ∇j f , for a function f ).

48) (w := p/ρ). Similarly (ξ i = γ ij ξ|j ): (Lξ T (0) )0 i = 0 − T (0)j i ξ 0 |j + T (0)0 0 ξ 0 ,i = −ρξ 0 |i − pξ 0 |i ; so v − B → (v − B) − ξ 0 . 49) Finally, (Lξ T (0) )i j = p′ δ i j ξ 0 , hence δp → δp + p′ ξ 0 , Π → Π. 50) we also obtain Γ → Γ. 52) We see that Γ, Π are gauge invariant. Note that the transformation of δ and v − B involve only ξ 0 , while v transforms as v → v − ξ ′. For Q we get Q → Q − a(ρ + p)ξ 0 . 53) We can introduce various gauge invariant quantities. It is useful to adopt the following notation: For example, we use the symbol δQ for that gauge invariant quantity which is equal to δ in the gauge where Q = 0, thus δQ = δ − 3 HQ = δ − 3(1 + w)H(v − B).

One can show that these decompositions are respected by the covariant derivatives. 8) (prove this as an exercise). Here, the first term on the right has a vanishing divergence (show this), and the second (the gradient) involves only f . For other cases, see Appendix B of [29]. Is there a conceptual proof based on the isometry group of (Σ, γ)? 2 Decomposition into spherical harmonics In a second step we perform a harmonic decomposition. For K = 0 this is just Fourier analysis. The spherical harmonics {Y } of (Σ, γ) are in this case the functions Y (x; k) = exp(ik · x) (for γ = δij dxi dxj ).