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 light-like 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 space-time correlate for Equivalence Principle. In fact in TGD framework Equivalence Principle at fundamental level reduces to a coset construction for two super-conformal algebras: super-symplectic and super Kac-Moody. The four-momenta associated with these algebras correspond to inertial and gravitational four-momenta.

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 space-time 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 non-empty 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.

1. The first variation of the metric determinant gives rise to

   δ g^1/2 = ∂_μg^1/2dx^μ propto g^1/2 C^ρ_ρμdx^μ.

   Here C^ρ_μν denotes Christoffel symbol.

   The possibility to find coordinates for which this variation vanishes at given point of space-time realizes Equivalence Principle locally.

2. Second variation of the metric determinant gives rise to the quantity

   δ² g^1/2= ∂_μ∂_νg^1/2dx^μdx^ν = g^1/2R_μνdx^μdx^ν.

   The vanishing of the second variation gives Einstein's equations in empty space. Einstein's empty space equations state that the second variation of the metric determinant vanishes in freely moving frame. The 4-volume element is critical in this frame.

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 light-like 3-surfaces ("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 3-surface involves also second variation of the volume element Ög. The propagator for small deformations around 3-surface is contravariant metric for Kähler metric and is contracted with R_ij = lg_ij to give the infinite-dimensional trace g^ijR_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. Non-vacuum Einstein's equations: light-like projection of four-momentum projection is proportional to second variation of four-volume in that direction

An interesting question is whether Einstein's equations in non-empty space-time could be obtained by generalizing this argument. The question is what interpretation one should give to the quantity

g₄^1/2T_μνdx^μdx^ν

at a given point of space-time.

1. If one restricts the consideration to variations for which dx^m is of form k^me, where k is light-like vector, one obtains a situation similar to used by Jacobsen in his argument. In this case one can consider the component dP_k of four-momentum in direction of k associated with 3-dimensional coordinate volume element dV₃=d³x. It is given by dP_k= g₄^1/2T_μνk^μk^νdV₃ .

2. Assume that dP_k is proportional to the second variation of the volume element in the deformation dx^m =εk^m, which means pushing of the volume element in the direction of k in second order approximation:

   (d²g₄^1/2/dε²)dV₃= (∂²g₄^1/2/∂ x^μ∂ x^ν) k^μk^νg₄^1/2dV₃= R_μνk^μk^νg₄^1/2 dV₃ .

   By light-likeness of k^μ one can replace R_μν by G_μν and add also g_μν for light-like vector k^μ to obtain covariant conservation of four-momentum. Einstein's equations with cosmological term are obtained.

That light-like vectors play a key role in these arguments is interesting from TGD point of view since light-like 3-surfaces are fundamental objects of TGD Universe.

5. The interpretation of non-vacuum Einstein's equations as breaking of maximal quantum criticality in TGD framework

What could be the interpretation of the result in TGD framework.

1. In TGD one assigns to the small deformations of vacuum extremals average four-momentum densities (over ensemble of small deformations), which satisfy Einstein's equations. It looks rather natural to assume that statistical quantities are expressible in terms of the purely geometric gravitational energy momentum tensor of vacuum extremal (which as such is not physical). The question why the projections of four-momentum to light-like directions should be proportional to the second variation of 4-D metric determinant.

2. A possible explanation is the quantum criticality of quantum TGD. For induced spinor fields the modified Dirac equation gives rise to conserved Noether currents only if the second variation of Kähler action vanishes. The reason is that the modified gamma matrices are contractions of the first variation of Kähler action with ordinary gamma matrices.

3. A weaker condition is that the vanishing occurs only for a subset of deformations representing dynamical symmetries. This would give rise to an infinite hierarchy of increasingly critical systems and generalization of Thom's catastrophe theory would result. The simplest system would live at the V shaped graph of cusp catastrophe: just at the verge of phase transition between the two phases.

4. Vacuum extremals are maximally quantum critical since both the first and second variation of Kähler action vanishes identically. For the small deformations second variation could be non-vanishing and probably is. Could it be that vacuum Einstein equations would give gravitational correlate of the quantum criticality as the criticality of the four-volume element in the local freely falling frame. Non-vacuum Einstein equations would characterize the reduction of the criticality due to the presence of matter implying also the breaking of dynamical symmetries (symplectic transformations of CP₂ and diffeomorphisms of M⁴ for vacuum extremals).

For the recent updated view about the relationship between General Relativity and TGD see the chapter TGD and GRT of "Physics in Many-Sheeted Space-time".