28/08/2022
Quantum gravitation has two levels of problems. The first is with Petrov types D (black holes) II and III (Robinson-Trautman spacetimes) and type N (gravitational radiation). These have Killing vectors, which are 4-vectors K_a = K∂_a, that when projected on 4-momentum (momentum-energy) vector P^a give K_aP^a = constant. This is an isometry that can be used to define conservation laws. There is then a Hamiltonian, or similar constraint, that can define an equation of the form
HΨ[g] = iN^aK_aΨ[g],
which is a Schrödinger-like equation. Here N^a is a lapse or shift function. These types of spacetime solutions then have an asymptotic condition of some type which allow for the localization of mass-energy on some scale. These equations may also define Jacobi-like θ-functions, which will define division algebraic structure used to model quantum spacetime, such as quaternionic-octonions.
In general, with spacetime there is no localizability condition. A spatial surface with a global distribution of mass-energy does not permit a Gaussian surface from which one can evaluate the mass-energy source of spacetime. The general constraint equations are NH = 0 or N^iH_i = 0, the Hamilton constraint or momentum constraint. The quantum correspondence is then HΨ[g] = 0, which is the Wheeler-DeWitt equation. This occurs with cosmologies, which have no Killing vectors. These are sometimes called the Petrov type O solutions. The expanding observable universe can lead to non-conservation of mass-energy, as a tether between two distant galaxies could in principle generate energy from nothing. There are no conservation laws.
Cosmologies are then just plain weird. Quantum cosmology is another domain that is far more difficult. String theory can only solve spin-2 fields that deviate from a classical background that is flat or with negative curvature. Loop Quantum Gravitation, LQG, attempts to solve everything, but is unable to get far. The objection to string theory is this classical background, but LQG is not able to even work a perturbative and renormalizable scheme for black holes or gravitons, which string theory can, and is then stuck.
Another epistemological issue with cosmologies is that we only have one observable cosmos. We can observe many black hole collisions, and I think find signatures of quantum gravitation there, so we can have many systems and data points. Cosmology does not afford us this luxury required in science, which makes quantum cosmology difficult. Quantum mechanics requires a statistical set of measurements, but in the end we have only one cosmos to observe. It is unclear whether the multiverse leaves signatures on the CMB so maybe some statistics can be found. This means quantum cosmology is simply weird and we may never come to a mastery of the subject.
