Einstein's equations and second variation of volume element
Lubos had an interesting posting about how Jacobsen has derived Einstein's equations from thermodynamical considerations as kind of equations of state. This has been actually one the basic ideas of quantum TGD, where Einstein's equations do not make sense as microscopic field equations. The argument involves approximate Poincare invariance, Equivalence principle, and proportionality of entropy to area (dS = kdA) so that the result is perhaps not a complete surprise. One starts from an expression for the variation of the area element dA for certain kind of variations in direction of lightlike Killing vector field and ends up with Einstein's equations. Ricci tensor creeps in via the variation of dA expressible in terms of the analog of geodesic deviation involving curvature tensor in its expression. Since geodesic equation involves first variation of metric, the equation of geodesic deviation involves its second variation expressible in terms of curvature tensor. The result raises the question whether it makes sense to quantize Einstein Hilbert action and in light of quantum TGD the worry is justified. In TGD (and also in string models) Einstein's equations result in long length scale approximation whereas in short length scales stringy description provides the spacetime correlate for Equivalence Principle. In fact in TGD framework Equivalence Principle at fundamental level reduces to a coset construction for two superconformal algebras: supersymplectic and super KacMoody. The fourmomenta associated with these algebras correspond to inertial and gravitational fourmomenta. In the following I will consider different more than 10 year old  argument implying that empty space vacuum equations state the vanishing of first and second variation of the volume element in freely falling coordinate system and will show how the argument implies empty space vacuum equations in the "world of classical worlds". I also show that empty space Einstein equations at spacetime level allow interpretation in terms of criticality of volume element  perhaps serving as a correlate for vacuum criticality of TGD Universe. I also demonstrate how one can derive nonempty space Einstein equations in TGD Universe and consider the interpretation. 1. Vacuum Einstein's equations from the vanishing of the second variation of volume element in freely falling frame The argument of Jacobsen leads to interesting considerations related to the second variation of the metric given in terms of Ricci tensor. In TGD framework the challenge is to deduce a good argument for why Einstein's equations hold true in long length scales and reading the posting of Lubos led to an idea how one might understand the content of these equations geometrically.
2. The world of classical worlds satisfies vacuum Einstein equations In quantum TGD this observation about second variation of metric led for two decades ago to Einstein's vacuum equations for the Kähler metric for the space of lightlike 3surfaces ("world of classical worlds"), which is deduced to be a union of constant curvature spaces labeled by zero modes of the metric. The argument is very simple. The functional integration over configuration space degrees of freedom (union of constant curvature spaces a priori: R_{ij}=kg_{ij}) involves second variation of the metric determinant. The functional integral over small deformations of 3surface involves also second variation of the volume element �g. The propagator for small deformations around 3surface is contravariant metric for Kähler metric and is contracted with R_{ij} = lg_{ij} to give the infinitedimensional trace g^{ij}R_{ij} = lD=l×∞. The result is infinite unless R_{ij}=0 holds. Vacuum Einstein's equations must therefore hold true in the world of classical worlds. 4. Nonvacuum Einstein's equations: lightlike projection of fourmomentum projection is proportional to second variation of fourvolume in that direction An interesting question is whether Einstein's equations in nonempty spacetime could be obtained by generalizing this argument. The question is what interpretation one should give to the quantity g_{4}^{1/2}T_{μν}dx^{μ}dx^{ν} at a given point of spacetime.
That lightlike vectors play a key role in these arguments is interesting from TGD point of view since lightlike 3surfaces are fundamental objects of TGD Universe. 5. The interpretation of nonvacuum Einstein's equations as breaking of maximal quantum criticality in TGD framework What could be the interpretation of the result in TGD framework.
