Can one define conserved Poincare charges in General Relativity?

There has been an interesting discussion in viXra blog about whether it is possible to define the notion of conserved energy, and more generally the notion of conserved Poincare charges in General Relativity. Also Lubos has participated. My conviction is that this is not possible without additional conditions on the metric (asymptotic Minkowski space property) and one must certainly give up the hopes of obtaining the conserved Poincare charges as Noether charges from standard action describing matter coupled to gravitation.

The following argument suggests that there are some hopes of getting non-conserved but well-defined Poincare charges in asymptotically Minkowskian space-time.

1. Entire Poincare algebra is needed in quantum theory and the Lie-algebraic realization in terms of space-time vector fields gives the only hope of achieving the goal. One could consider also the extension of Poincare algebra to an infinite-dimensional Lie algebra with generators approaching Poincare algebra generators asymptotically.

2. You would start with the identification of vector fields j^Ia defining infinitesimal translations, rotations, and boosts in asymptotic regions. In this region they define asymptotic Killing vector fields satisfying

   D_aj^I_b+D_bj^I_a=0

   and the currents

   (G-λ g)^abj_b

   are asymptotically divergenceless because Killing vector field property is true and G and g are divergenceless in covariant sense. If you can continue j^I to entire space-time uniquely ,you get well-defined Poincare charges, which are however not conserved.

3. You must replace Killing vector field property with something weaker and the condition that j^I define flows conserving only four-volume instead of distances is a natural generalization. This implies the condition

   ∇⋅ j^I=0

   and the infinite-dimensional Lie-algebra of volume preserving vector fields is obtained.

4. A further condition is needed and this is very natural. You must be able to define global coordinates along the flow lines of the vector fields in questions. This requires

   j^I = Ψ ∇ Φ.

   Φ defines the coordinate. This kind of vector fields are known as Beltrami fields.

5. In asymptotic region Φ would represent either a counterpart of linear M⁴ coordinate, rotation angle around some space-like axis , or hyperbolic angle around time-like axis. In the asymptotic region Ψ would be constant for translations in the asymptotic region. For the rotations around a given axis the orthogonal it would reduce to the orthogonal distance ρ from that axis. For the Lorentz boosts around given time-like axis to the orthogonal radial distance r from origin in the rest frame defined by that axis.

Let us look what volume preservation and Beltrami property give.

1. By simple calculation you obtain

   ∇² Φ +2 ∇ (log(&Psi);⋅ ∇ Φ=0.

   This is massless field equation with additional term which might relate to massivation. If one has two solutions with same Φ, one obtains the condition

   (∇ Ψ₁-∇ Ψ₂)⋅ ∇Φ=0 ,

   which suggests that you must have

   ∇ Ψ⋅ ∇ Φ=0

   quite generally.

2. The physical interpretation would be obvious. The solutions describe as special case the modes of massless gauge field. Φ defines the counterpart of a pulse propagating to local light-like direction and Ψ defines a local polarization vector orthogonal to it. There are also solutions which do not allow this interpretation and corresponds to the functions Φ and Ψ, which are relevant in the recent case.

3. The solution set is quite large for a given Φ. You can replace Φ with an arbitrary function of Φ if the additional condition

   ∇Φ⋅∇Φ=0

   having obvious interpretation holds true. Same applies to Ψ. Linear superposition holds true. You can also form the Lie-brackets for given Φ and one finds that they vanish. Therefore you have infinite-dimensional Abelian algebra. The natural interpretation is as commuting observables corresponding to polarization direction and propagation direction.

4. Can one obtain unique continuation of j^I from the asymptotic region to the interior so that unique conserved Poincare charges would exists for asymptotically Minkowskian space-time? The radiative solutions are the problem. If the condition that the radiative part vanishes in the asymptotic region implies that it vanishes everywhere, there are no problems.

   Minkowski space serves as a good test bench. In this case functions Φ(p⋅ m) are simplest propagating pulses: here p is light-like momentum. The condition that they vanish in all directions including the propagation direction in which p⋅ m is constant indeed implies that Φ vanishes. By choosing Ψ so that it vanishes far away does not allow to achieve the condition. Hence there are hopes that one can define non-conserved Poincare charges in asymptotically flat space-times. One can however imagine the presence of light-pulses which are emitted and absorbed and thus exists in a finite volume of space-time. These might course problems.

5. In the case of non-vanishing cosmological constant one would obtain infinite energy and the contribution to the charge would be the charge assignable to the vector field defining time translation. This does not favor cosmological constant.

As a matter fact, one ends up with the Beltrami fields from a general solution ansatz for a solution of field equations in TGD. The interpretation is that one has the analog of Bohr quantization for solutions of the extremely nonlinear counterpart of Maxwell's equations coupled to classical gravitation via induced metric. Only the superposition of solutions corresponding to same function Φ is allowed. They represent pulses of various shapes and different polarizations propagating in a particular local light-like direction.This conforms with what one knows about outcomes of state function reduction. These solutions have 3- or 4-D CP₂ projection. So called massless extremals with 2-D CP₂ projection have same physical characteristics. Cosmic strings and CP₂ vacuum extremals with Euclidian signature of metric describing massless particles are also basic solutions and the topological condensation of CP₂ type vacuum extremals to a space-time sheet with Minkowskian signature of the induced metric creates around itself a solution described by Ψ and Φ meaning that particle picture implies field picture. Note that the proposed identification of gravitational charges could make sense also in TGD framework.

These Abelian algebras and perhaps large algebras generated by them via commutators might be relevant also for the construction of the solutions of field equations in General Relativity. The construction of deformations of an existing metric by adding gravitons is what comes in mind first. The scalars Ψ would define polarizations in a given background metric used to build polarization tensor and the functions Φ could be used to build the analogs of plane waves. One would obtain gravitons and also gauge bosons localized in transversal directions. The algebra formed by the Beltrami flows could thus play a role analogous to Kac-Moody algebras. What is interesting that one could always interpret a many-graviton state as a background to which one can add new kind of gravitons! This all is of course speculation but because these algebras allow a concrete interpretation as classical representations of elementary bosons, I would not find it completely surprising if an algebra related directly to the metric would play a fundamental role in quantization of General Relativity.