Introduction
The Newmann-Penrose formalism is described in Special relativity in Newman-Penrose Formalism 06/25/24 in this site. Have a look on it before reading the article.
Some issues about this formalism
A major issue is that, in this formalism, the special relativity needs only 3 degrees of freedom for its full description. This results from the fact that the basis is made of null vectors and that there are only 3 linearly independent null vectors in the 4- dimensional spacetime of the special relativity[1].
A trick, using a complex vector with its conjugate, allows the conversion of the components of any 4-vector (timelike, null or spacelike) described in a Minkowski 4-basis vectors into its components in the 3 null basis vector of the NP formalism and conversely [2].
Four coordinates for a three linearly independant basis!
These 4 coordinates allow the coordinate transformation between a Minkowski basis and a null basis of a 4-vector but this corresponds only to 3 degrees of freedom as the conjugate of a complex coordinate does not provide an additional degree of freedom, and as there are only 3 basis null vectors in the null basis!
But as this allows all that which is permitted in the Minkowski basis (4 degrees of freedom), this would be equivalent.
An other representations of the relativity?
Notwithstanding with the fact that a complex number has two parameters necessary for solving the number of degrees of freedom issue, the null basis has only 3 basis null vectors.
In the Minkowski representation, spacetime has a (-,+, +,+) signature. In the NP representation, would the signature of spacetime be (0, 0, 0,0) or (0,0,0,0*) , where the 2 last signatures of the metric (0, 0*) would be that of conjugates complex null vectors ?
Therefore does the NP formalism provide a representation of relativity in a different mathematical formalism?
I had the opportunity, some years ago, to ask this question to Penrose, at the end of his conference on « twistors », during a seminar about « science and philosophy « hold at the IHP in Paris.
In quite a long argumented answer , he says that the NP formalism was just an interesting and useful formalism but, per his opinion, was too limited for really being a new representation of general relativity unlike the twistors representation more consistent.
Anyway, in both of them, (in a more extensive way in the twistor representation, where the set of all null geodesics passing by a spacetime point, in the general relativity, defined in 4 dimensions, are represented by a single point ), the null geodesics are the key concept which drives the questionning about the laws of the physics.
Would the NP formalism simplify solutions in relativity?
Following Chandrasekhar, the NP formalism is more efficient than that of « Minkowski « for describing black hole for instance.
The interest is that a 4 degrees of freedom problem would be solved by a 3 degrees of freedom problem which should be simpler.
As there is no free lunch, either the problem is physically a 3 degrees of freedom problem and the use of a Minkowski basis just a poor solution or more likely some part of the complexity of the phenomenology of the relativity is included in the structure of the null geodesics.
More about null geodesics
As null geodesics looks to be at the heart of the issues for a solution for general relativity, one should investigate more deeply theirs properties.
In « Espace-temps. Coordonnées. Celles de type nul, conduisent à un nouveau paradigme. ». we defined some essential properties of null coordinates on a spacetime.
Unlike timelike or spacelike worldlines which may be not geodesic, null worldlines are always geodesics. One may wonder whether the set of all null geodesics fully define all the “spacetime points “ of the manifold resulting of the Einstein equation and therefore its geometry even though, they will not define all the possible curves on the manifold.
Null geodesics define the conformal structure of spacetime
It is a set of null geodesic, called principal null geodesics, that defines the different classes of spacetime as specified in the Petrov-Pirani classification. Let us point out that this property was found by E. Cartan as soon as 1922.
As pointed out by Penrose, as null geodesics, alone, in addition, rule the causal structure of the spacetime, one see that they include a huge information about the structure of the relativistic spacetime.
Null geodesics define the causal structure of spacetime
The causal structure is represented by the regions delimited by a local light cone. All of this suggests that the conformal structure is likely the most important stucture of spacetime as illustrated by the use of the Carter-Penrose diagrams.
Metric vs conformal properties
We know that the metric is the fundamental parameter in relativity, therefore one may wonder whether it exists a relation between it and the conformal description.
In the metric we use (arbitrary) coordinates where units are defined. But a measurement is a ratio between an arbitrary unit and the object to be measured, i.e a conformal operation.
We know that among the 16 parameters of the (symetric) metric tensor, per its symetry only ten shoud be independant but in fact it is only 6. We have 4 degrees of freedom totally free, this leading to select a gauge for fixing them.
We know that for type D spacetimes in the Petrov-Pirani classification (black holes, static or in rotation, with or without an electric charge), in the spacetime (in vacuum) the Riemann tensor which defines the curvature of spacetime becomes the Weyl tensor which is invariant by a conformal transformation. This means that there is no « absolute scale »factor in these spacetimes! [3]
The metric, used for defining the Riemann tensor, is usually a generic metric including some « a priori » symetries and parameters to be defined by the Einstein equation includes arbitrary « units ».
If « compact « spacetimes would exist, would they provide a « size » of the spacetime which may be relevant as a physical unit, as a cosmological constant provides (dimension [L]-2) for maximaly symetrical spacetimes?
Obviously, this would be different in a quanta theory where a quantum can be used as a physical unit, allowing to define ratios between the sizes of the objects.
About the Einstein equation
The Einstein (algebraic) equation (EE) includes the Ricci tensor and the Ricci scalar (its contraction) along with the metric, via the Einstein tensor but does not include the Riemann or Weyl tensor. Does this mean that the EE does not constrain these tensors which define the curvature of spacetime?
In fact, it does, but indirectly via Bianchi identity, the result being not an algebraic equation but a differential equation similar to that of gravitational waves.
Conclusion
One can see that, as the NP formalism description looks more powerful for describing the nature of the physical phenomenology of the relativity, at least for type D spacetimes, where it is the (conformal) Weyl tensor which defines them, it may improve our understanding of null geodesics in (conformal) spacetimes, this opening a way for improving our knowledge of the theory.
Notes
[1] Because there is a constraint on a null geodesic: ds² =0, this implying that a null vector is orthogonal to himself, it may be orthogonal to only 2 other null vectors.
[2] The (t, x, y, z) Cartesian coordinates on the 4 basis vectors of a Minkowski become U=(t+x)2-1/2, V= (t-x)2-1/2, W= (y+iz)2-1/2, W*= (y-iz.)2-1/2. The constant 2-1/2 is a normalization constant. Conversely, t =( U+V)2-1/2, x= (U-V)2-1/2, y = (W+W*)2-1/2 and z =(W-W*)2-i/2 .
In Cartesian coordinates, the Minkowski ds² = – dt²+dx²+dy²+dz². Its signature is (- +,+,+) and its metric tensor {{-1,0,0,0},{0,1,0,0},{0,0,1,0},{0,0,0,1}}.
In the U, V, W, W* coordinates,this becomes: ds²= -2dU.dV +2dW.dW* of signature (0, 0, 0, 0) and of metric tensor {{0,-1,0,0},{-1,0,0,0},{0,0,0,1},{0,0,1,0}}.
More information in Special relativity in Newman-Penrose Formalism 06/25/24.
[3] It is not the case for all spacetimes.