Is the interpretation of Einstein’s equation rigorous, and correct? A way to quantization 9/09/22

On dynamics, in gravitation

What interests the scientist are the laws of motion of bodies in mutual gravitational interaction: the dynamics of a system. Indeed it is always the action of a phenomenon that we are interested in.

Newtonian mechanics

Newton, who located these bodies in a three-dimensional « absolute » Euclidean space (which therefore serves as a reference), assigned to each body an active mass ma, generating a « gravitational field » extending into this space according to laws he specified, a passive mass mp, characterizing the coupling of the mass with the field generated by the other masses (but not by his own:  no self-coupling). He also assigned an inertial mass mi, according to the famous law f = m i.a. where a is the acceleration that the body undergoes, mi, its inertial mass and f the « force » applied to the body.

This « invisible » force” f, exerted at a distance in a vacuum, derived from a scalar potential (therefore additive), which made it possible to easily calculate the potential generated by distant masses at any point. Knowing this potential, the position of a body in space and its momentum (a vector) we could calculate the trajectory of this body called « geodesics » (trajectory of the body when it is subject only to the gravitational force).

The dynamic parameter of this trajectory is Newtonian « absolute » time, independent of everything.

The principle of equivalence stipulated that mp = mi, (Galileo experiment at the Tower of Pisa) and the principle of action reaction that ma = mp. This made the 3 types of masses equal, with a suitable parameterization (in fact they are « proportional »).

This gravitational force at a distance seemed a little mysterious, but as the theory gave good results, (there was just a small anomaly for the advance of the perihelion of Mercury that we thought to explain) only the sorrowful spirits were upset.

General relativity

After special relativity in 1905, as early as 1907, Einstein became interested in gravitation, as he was convinced that the principles he had used should also apply to gravitation.

The problem was difficult and after unsuccessful attempts using the principle of equivalence, he will be interested in another type of approach: a geometric description to define the dynamics, namely the « geodesics » followed by the bodies under the mutual gravitational interaction.

These « geodesics » will no longer be trajectories, resulting from forces that apply to interacting bodies in Euclidean space, which are not geodesics, in the geometric sense, of Euclidean space, because geodesics of Euclidean space are straight lines);

They will be  geometric geodesics of non-Euclidean geometry.

It took him 10 years to get there, but at the end of 1915 he published his famous equation.

Gmn = k.Tmn.

Gmm is Einstein’s tensor, a geometric object in a 4-dimensional spacetime (t,x,y,z), provided with a metric, defining the curvature of geometry, k is a dimensioned constant (related to planck’s force, see pages of this site) ensuring the homogeneity of the equation and Tmn is a tensor, the energy-impulse tensor which represents « physics » (matter-energy).

It is this equation that will define the structure of spacetime resulting from the properties of a metric (which can have symmetries) and the matter-energy that will constrain it.

In this approach, all the masses and energy contribute to defining the spacetime to which, in return, all these masses and all this energy will couple and describe the geodesics. This includes implicit self-coupling.

Therefore, it is the geodesics of this spacetime (mathematically represented by a « manifold ») that will define the dynamics of the system.

Rigorously, should we consider all spacetime or only a class of geodesics?

There is a problem. The mathematical model defines a geometric object « a manifold » that can be considered as a set of points on which one can define any curve. But, the Einstein equation which is related to gravitation only, will select some physical geodesics among all the curves (a specific class).

Therefore, it seems natural to consider only the minimum class of geodesics, which will be called structural geodesics, which define a subset of the points of the manifold, in a structured way, because generated by these geodesics.

We know the difficulties presented, for example, in quantification. But in general, the methods consider the global manifold, not a much more constrained subset where one can hope that it is simpler.

Therefore, it would be interesting to study these possibilities with the convenient smallest subset because, rigorously, submitted to the gravitation alone, which is the object of Einstein’s equation, only these geodesics and the subset of points they define will be used.

Example of the « Schwarzschild » solution

This case is very simple because the spacetime, thus defined, is empty: the only mass at the center is a singularity. As E. Cartan had already described, in 1922 [1], there are two (infinite) classes of null geodesics, one radially incoming, the other radially outgoing.

Let us add that to reduce the subset as much as possible we can consider only the geodesics of a given frequency at infinity, because for different frequencies, the affine parameter (the momentum in this case of null geodesics) induces different geodesics.

These classes define the causal structure of space-time and generate a subset of the points of the 4-dimensional manifold (t, x, y, z) in a structured way.

This is what the solution of Einstein’s equation says.

The idea is that, even though, this minimum subset covers only a small part of the manifold, nevertheless, it would capture (provide and allow to define) all the physical properties of the solution to the  Einstein’s equation.

Should we also add the incoming and outgoing radial timelike geodesics (double infinity), without boost, which generate other points in a structured way of space-time, well described in the solution of Painlevé (1921)?

Strictly speaking, if there is only vacuum, one may wonder whether it is necessary.

We may also wonder if there are structural spacelike geodesics, as this class of geodesics are not considered in our physical world.

The selected classes define the restricted spherically symmetrical spacetime. We can use this subset, which although multiplely infinite, is much smaller than that which would be generated by the set of all points obtained with the set of all possible curves.

The hope is that this restrictive subset would help to quantization and mathematical improvements.

Note that quantization operates a restriction by a constraint, an operation similar to that which we propose, by a constraint also.

All other solutions (circular, non-circular, any geodesic and non-geodesic worldlines) are not « native » solutions of Einstein’s equation, because they require non gravitational phenomena.

Quantization of the geodesic equation

As, per our assumption, that the geodesic equation alone will define the useful physical spacetime provided by the Einstein equation in this solution, the problem is quantization of the geodesic equation. A smart solution was given by Painlevé in 1922.

see : https://astromontgeron.fr/Painleve-article-english.pdf

for more details.

Example of the standard model of cosmology.

In this case, the universe is not empty, since the momentum-energy tensor is not null. In addition to null geodesics, it will be necessary to include the geodesics of the matter « co-mobile » of the expansion.

All other worldlines of the universe, geodesic or not, do not natively match the solution given by Einstein’s equation.

Again, a reduced subset generates all the natively physically possible points of the manifold. The same type of remarks as before applies.

What about non-structural geodesics and non-geodesic worldlines?

We may, of course, also deal with non-structural geodesics and non-geodesic worldlines, but we must understand that this will be done, in general [2], by a « perturbative » coupling (without influence on spacetime) between phenomena, of a non-gravitational nature, for example boosts, in a local space at a point of the manifold with space-time (represented by the manifold) defined by Einstein’s equation, hence the fiber structure, of which space-time is the base and the local spaces, the fibers.

This will make possible physical curves passing through other points (t, x, y, z) of the four-dimensional manifold (an extension).

In general, all of this is related as general relativity, but this is not rigorous, and as such can be the source of confusion or even errors.

[1] Petrov and Pironi, will find and complete the contribution of E. Cartan, far much later.

[2] In Kerr-Newmann space-time, global space-time is defined by the coupling of 2 equations, Einstein’s and Maxwell’s, this is an exception of the general case since this constraint is global.