Is the Newmann-Penrose formalism in relativity, a new paradigm revealing the causal structure of the physics? 11/26/22


On the strong belief that the light [1], ruling the causality in relativity, is the key for understanding the physics, in 1961 Newmann and Penrose developed a new formalism using null coordinates, aimed to grasp better the structure of some solutions of the theory of general relativity (for instance, Kerr black holes), which lead to a spinorial local description of the structure of spacetime. As general relativity uses a tensorial formalism, this leads to a quite complex spinor-tensorial formalism. But the nice counterpart of the complexity of this formalism is an efficient description of the intimate structure of spacetime. This efficiency, likely comes from a morphism between the formalism and the physical phenomenology. This shows that null coordinates are more adapted for the relativistic spacetime description than the Minkowski ‘ones.

In quantum mechanics, a « light front quantization formalism » using the same formalism, but for another purpose, has been seriously considered , mainly at the turn of the century. This should have been forecasted because the Newmann-Penrose formalism can be described in terms of spinors formalism perfectly suited to quantum mechanics. As in general relativity, in quantum mechanics this approach looks to be of a great interest for grasping the structural properties of the solutions, for instance for the vacuum energy and the related cosmological constant problem.

Up to now, the synergies between the two applications of the null coordinates description have not be clearly identified, but this looks to open a way to investigations towards, at least, formal links between them.

The Newmann-Penrose formalism is developed in :

Tetrad formalism

The tetrad formalism, used in general relativity, is based on an orthonormal basis to define the local spacetime which is that of special relativity. Special relativity is generally defined in Minkowski coordinates (t, x, y, z), in a Galilean frame of reference, where the paths of light rays are also defined.

 Breaking with this point of view, we propose, by a new paradigm, to reverse the approach by using null coordinates, adapted to light paths, as a reference in special relativity instead of those of Minkowski which are a remnant of the heritage of Newtonian mechanics. This is motivated by several (correlated) reasons.

First, since it is the fact of the existence of a “maximum” velocity associated with the velocity of light that implies the hyperbolic structure of spacetime, the information related to this particular structure of spacetime should be included in the nature and structure of the null geodesics, making them more suitable for understanding such hyperbolic spacetime.

Second, as the number of different Galilean frames of reference is infinite and as they are all equivalent, the choice of one of them is arbitrary, while the fact that the celerity of light is the same in all, makes it unique, therefore not arbitrary.

One may object that a frame whose basis vectors is made up of null vectors is not a Galilean “frame of reference” like a Minkowskian frame of reference. But this “constraint”, inherited from Newtonian analysis, is not mandatory and may mislead us by its Newtonian approach. As the relations and the transformations between a Minkowskian base and a null base are straightforward, a result acquired in a null base can be transposed if necessary in a Minkowskian base, for a “Newtonian” interpretation and vice versa.

We claim that the interpretation of the results acquired in a null base will be more in accordance with the nature of relativity since this theory is constrained by the properties of light.

We claim also that, consequently, by this choice, as more information about the the phenomenology is included in the formalism, this will simplify the calculations and would enlighten the understanding of the structural properties of this spacetime .

As an example, we show that in null coordinates, as defined by the Newmann-Penrose (NP) formalism, the formalism  is simpler.

We will propose a phenomenological interpretation of this result.

The conclusion of the article is:

This non exhaustive survey was intended to demonstrate that another approach than that of Minkowski was possible and constructive in special relativity for deriving very simply the transformations between frames which allow to grasp more deeply the symmetries of this spacetime.

This analysis led us to assume an other point of view for explaining the phenomenology of spacetime in relativity where the observer is not the reference. It would also be interesting to develop a deeper thought on the structural similarities with the group theory analysis that we have only briefly evoked in this document.

For more, see also :

[1] It is the fact that there exists a upper limit in velocity in relativity which implies the hyperbolic (local) structure of the theory. The way it rules the causal structure of the theory is described by the « light cone ».

Some demonstrations show that, in Einstein’s two postulates for special relativity, the essential postulate is that there exist a upper limit in velocity. The value of this limit, illustrated by light, is just an experimental data.

Light and more generally electromagnetic waves, as well gravitational waves travel at this velocity. This property is bound to the infinite range of the interactions carried by these waves, associated to massless bosons in quantum mechanics.