Spacetime, space, time. Spacelike, timelike and null coordinates. What is physical reality? (upated 03-11-21)

Spacetime: What it is not.

In relativity, spacetime is neither a concatenation of space and time and nor a mixture of space and time because, in relativity, time and space, unlike spacetime, are not physical entities. This is why suggesting that spacetime, in “Newtonian language”, is something in all its space extension and time extension is not correct, because space and times are “shadows” of spacetime and we cannot correctly describe something by its shadows!

Spacetime: A new fundamental entity of physical reality

The concept of spacetime is an indivisible entity, the only one able to represent “physical reality” reducing space and time to be only shadows of the spacetime, as Minkowski, in 1907, likely referring to the allegory of Plato’s cavern, claimed about special relativity, this remaining true in general relativity.

By physical reality, we mean that there will be agreement by all observers, whoever they are, on the nature and parameters of an observed phenomenon.

Spacetime synthesizes space, time and movement into an entity, this being illustrated by Einstein’s equation which defines the universe by a spacetime, whose geometry includes these three elements.

What best characterizes space-time is the nature of null geodesics (those followed by light), which is not surprising since they are the only ones that satisfy the limit (maximum ) which is the constraint which gives space-time its hyperbolic structure. Note that their proper “time” is zero (hence their name) and that they are always geodesic, whereas for the other geodesics and universe lines, not necessarily geodesics, (timelike or spacelike), this criterion is not met. It may seem surprising, then, that when we define a metric, we try to describe it with one coordinate of time type and three of space and not with null coordinates. This is a survival of Newtonian approach, where time and space are immediate concepts of our mind.

A critic about the representation by null coordinates is that the light geodesic is not a frame of reference where time and space can be defined. If this notion of frame of reference may have been useful in the early days of relativity with the Lorentz transformations, it is not essential. Coordinates are arbitrary and by coordinate transformations one can switch from one representation to another.

The Newmann-Penrose (tetradic) formalism, mentioned below, provides an excellent illustration and shows its interest. Articles describingin more detail the interest of null geodesics can be found in:

http://astromontgeron.fr/s%C3%A9minaire-maths-philo_2019.pdf http://astromontgeron.fr/SR-Penrose.pdf

The phenomenological explanation offered by the metric with one coordinate of the time type and three of spaces is biased, which may explain why it is difficult to understand it.

Bachelard, in his book, “the new scientific spirit”, stressed that relativity cannot be reduced to Newtonian mechanics and that their comparison in a weak field could only be done at the cost of mutilations of the theory of relativity!

The metric with null coordinates, which is more difficult to interpret, is more deeply and structurally linked to the phenomenon, which is obvious per the greater simplicity in the calculations as will be mentioned below.

What is called physical reality: Example.

For instance, let us suppose that we observe two explosions of stars (supernovas) in the universe. It is an event which has an intrinsic physical character.

Different observers will measure distances d1, d2,..,di,… dn, between them, all different, and, as well, will simultaneously measure time intervals t1, t2, .ti, .. tn, between their explosions, all different.

But if each observer calculates the spacetime interval si between the two explosions, from their own data di, ti, then all of them will find the same result (s1 = s2 = .. si = ..sn = s) .

Why are we not familiar with spacetime?

 It is this concept of spacetime, likely the most difficult to conceive, that has baffled the most the scientists, in the development of the theory of relativity. The concepts of time and space, generally considered as immediate knowledge of our mind, are so pregnant in our mind, that it is difficult to get free of them, when spacetime is involved.

Indeed, in common life, the amazing effects of spacetime are so tiny that it is difficult to distinguish it from space and time!

What is left of time and space: The coordinates.

When using analytic geometry for describing geometric entities, we need a basis of 4 vectors. Space and time are still used as the coordinates on this basis, allowing us to define the metric of spacetime for calculating the spacetime interval.

Obviously, these coordinates are arbitrary, and therefore we cannot attribute a physical character to them. This has been a trauma for many scientists at the beginning of the twentieth century. In fact, space and time only serve as temporary parameters in this calculation.

Even though, space and time are widely used as coordinates, they are not mandatory.  We can use null coordinates which have an intrinsic spacetime character, this allowing generally to simplify the calculations, at least in some interesting cases ( Kerr black Holes, for instance).

Note that at the beginning of special relativity, one tried to save these concepts of space and time, by tricks, such as the synchronization of Galilean frames. But, that applies only within one frame. Although, the Lorentz transformations allowed to extend the process to several frames, it was becoming quite complex.

The Newmann-Penrose formalism

Newmann and Penrose have developed a formalism using null coordinates which lead to a spinorial local 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. Therefore null coordinates more adapted for spacetime description.

This point of view is developped in : http://www.astromontgeron.fr/SR-Penrose.pdf A summary is given below.

Summary

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 an other 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 reflexion on the structural similarities with the group theory analysis that we have only briefly evoked in this document.

For more, see also : http://www.astromontgeron.fr/s%C3%A9minaire-maths-philo_2019.pdf