Good news for you, today, guys: i've discovered something extremely interesting only analyzing a bit more the equations of GR. A propagation mode even more fundamental than gravity itself, namely, propagation through deformations of space-time. There's nothing more universal in our known physics. Details.
 

If you take, for instance, Newton's equation for the gravitational potential, you find two possible solutions: a pure wave, independent of any material source, and a field emitted by a material source. This equation being linear, the complete solution is the superposition of these two contributions. The emitted part itself shows two behaviours: the confined one, which stays inside the source, and the unconfined one, which propagates outside the source. So is the Newton potential, inversely proportional to the distance from the emission centre. This unconfined field thus propagates outside the source, in the "vacuum", and carries with it the physical properties of the source: the static Newtonian potential produced by a "point-like" source with central symmetry is directly proportional to the mass of this source. So, what does the unconfined field is propagating through space the information "i've been produced by a (static) point-like body with mass (at rest) m". When a second body with mass m' receives this information, a "link" is created, we call "gravitational interaction between a body with mass m and a body with mass m'".
 

The same happens in the case of static charges: the unconfined static scalar potential carries with it the information on the electric charge of its point-like source.
 

Well, a similar process happens in GR. The "matter field" is the source, the "deformation field" is the field. When matter is present, the deformation produced, when unconfined, carries with it the physical properties of its emitting matter source. When this source is a macroscopic body or a system of macroscopic bodies, the deformation originates from a "touchable" source; when the source is a gravitational field or any other interaction, the deformation originates from this field.
 

The frame of GR is really "general". It also includes quantum fields: if we deal with boson or fermion fields, the "matter field" will simply be the energy-momentum tensor of the corresponding quantum field and the deformation produced in space-time will carry the bosonic or fermionic properties of the source.
 

It even holds for all quantum vacuum states, as the mean value of the matter field is non-zero (opposite to "classical" vacuums). The corresponding deformation will then be produced by vacuum states themselves. This mans that, even a vacuum can generate (local) deformations in space-time. However, these deformations will be pure fluctuations, as a quantum vacuum is a pure fluctuation.
 

As we can see, there's a single propagation mode, a deformation of space-time, but many possible origins of this deformation.
 

Deformation of space-time is the most fundamental propagation mode in our known physics. It holds for any kind of source, substantial or not, even in the total absence of particles.
 

The equations of GR even tell us something more than weird: if we place ourselves in the quantum context and, despite this, we still impose a zero matter field, there still remain a non-trivial solution to the wave equation. This is rather more than weird, for it means that, even in the absence of all quantum vacuum, we would logically expect the only solution to be plane Minkowski space-time. That's not the case.

Is this solution purely mathematical, "metaphysical", as it's produced by absolutely nothing at all, or is it just the property of deformations to self-produce, independent of anything else, thanks to the non-linearity of the equations of GR? Can be.

Whatever it is, this is a question for quantum cosmology we won't need, we can use this large freedom of origins to introduce the physical notion of PSI propagation without needing to introduce new dimensions of space nor time, simply saying that:
 

A PSI PROPAGATION THROUGH SPACE-TIME IS A DEFORMATION MODE ORIGINATING FROM A NON-SUBSTANTIAL SOURCE SUCH AS A GRAVITY FIELD, AN ELECTROMAGNETIC ONE,...

Substantial matter produces deformations, but it also produces a field. Seen the smallness of Einstein's gravitational constant, most of the time, the intensity of the field produced is many orders higher than that of the deformations, so that we can neglect them compared to the interaction. Mathematically, we will restrict ourselves to the equations for this interaction and won't matter about the "second-order" set of Einstein's equations.
 

We can no longer do that for non-substantial matter, as neither interactions nor fermion wavepackets produce any "ordinary" field. For those, we need another kind of information exchange and GR precisely offers us an "alternative".
 

At the same time, it proposes us an answer to the "Tunnel": a deformation of space-time outside matter (or even matter-free) with axial symmetry. As a propagator, it then carries a non-substantial information from one place to another in space-time, close to or far away from the starting location, an information that can mix a matter field originated from gravitation ("PSI masses") and a matter field originated from electromagnetism ("PSI charges").

This is quite an interesting possibility, only involving deformations in space-time, while "ordinary" fields play the role of the former "substantial body" in an "ordinary" motion.

For only elementary fields are not compact. As soon as complexity is introduced, especially in material sources, the produced field inherits this complexity, as well as the autonomy of the substantial source (carrying its features!) and non-linearities guarantee compactness, soliton solutions,...

A self-interacting field such as those already encountered in nuclear physics also lead to complex and autonomous configurations, without the need for any fermion source. So, we already have examples of "wavy patterns" in present-day physics.

The same will hold for gravity and electromagnetism even under a U(1) gauge group because, this time, of a curved metric: we've recalled that the matter field is not based on the plane Minkowski metric, but on the curved one; furthermore, the energy-momentum tensor is not linear, but quadratic, in the field variations, while the potential contribution may appear completely anharmonic.
 

Astronoms are activally searching for «　gravitational waves　» in the observable universe, basing themselves on Einstein’s interpretation.

Will they find the PSI instead?

That would be rather funny, wouldn’t it?

I think i found the answers to my questions, this time. It's not about brand new physical principles, frames or whatever. Present-days physics has all the ingredients to answer our questions. It's rather about finding the right interpretation. Once we've done that, we can stay within the four dimensions of space-time and find brand new types of matter. That's what we need before all, basically: matter of new natures.
 

It will surely surprise more than one, knowing my position on GR, but i'm forced to recognize that General Relativity brings us all those answers, if we're able to make the proper deductions from it.

The reasoning is as follows. Matter curves space-time. That's the principle of GR. So, we can use Einstein's field equations and i'm glad we can. With a slight but essential difference: the metric of curved space-time is not specific of gravity, as at Einstein's, but the result of the presence of matter in space-time. The curved metric is a local deformation of space-time. Therefore, the metrical tensor is a strain tensor and that's why it's dimensionless. The local expression of that strain is obtained solving for the Einstein's equations. Obviously, the solution will depend on the form of the energy-momentum tensor of matter, acting as a source of deformation. However, as that source itself has to be described within the curved frame and not the plane Minkowski space-time, the system of Einstein's equations "closes in": starting with a plane space-time, we introduce matter, that matter locally curves the frame, which in turn modifies the material source, modifying the curvature and so on, until a possible equilibrium is found. This is the geodynamical process.
 

The question is: what kind of matter are we to deal with?
 

For there are numerous types of physical matter. Einstein himself explained he derived his field equations following the same procedure that led Maxwell to the set of equations for electromagnetism, i.e. a set of second-order equations for the field potentials. He also accepted from the start that, in addition to macroscopic "touchable" matter, the electromagnetic field itself could serve as a source of curvature. So, physics does give us many different types of matter, according to our needs.
 

We have "substantial" or "touchable" matter, represented by the energy-momentum tensor for macroscopic bodies.

We have "electromagnetic matter", represented by the energy-momentum tensor for the electromagnetic field. The (44)-component divided by c² does give us a mass density, despite this matter has nothing "touchable" anymore. Stronger than that: even in a space-time entirely empty of electric charges, Maxwell's theory learns us there still remain electromagnetic waves, completely independent of charges, and these waves still give a mass density...

I already talked about this but i'd like to repeat it here, as the context is extremely favourable: we shouldn't confuse between waves and fields produced by matter, outside of it. Electromagnetic waves need no charges at all to exist, undermeaning they can perfectly pre-exist to any electrical charge in the Universe. On the contrary, electromagnetic fields produced by charges need material sources and, outside of them, they only behave as waves. Not only is the distinction important, it's essential.

In an empty Universe, GR tells us there can still be waves curving the frame and defining matter. Even a Universe empty of waves could still be curved.

The difference i make with Esintein's approach is that gravity is not a "pseudo-force", of "force of inertia", but a true force and as such, it does not enter the metric of space-time, but the energy-momentum tensor, just like the electromagnetic field.

Gravity is a source of local deformations of space-time and the metric contains the geometric informations on the deformations produced.

Under this condition, i totally agree with GR.
 

So, already at the classical level can we distinguish between three types of matter: "touchable" matter (macroscopic bodies), "electromagnetic" matter and "gravitational" matter. They all lead to mass densities, but only the first one is substantial. The two others have nothing substantial at all. If we have electric charges, they contribute to the energy-momentum tensor, through the charge-field coupling. If we have masses, they contribute to the energy-momentum tensor, through the mass-field coupling.

It's more appropriate to talk about "masses" in the case of gravity and about "charges" in the case of electromagnetism.
 

Let's now consider the simplest situation, that of pure waves.
 

We first take gravity, in its linear Maxwell model, as there's absolutely no need for looking at complex fields. On the contrary, we turn to basic elements. In the source-free model, there's no "ordinary" mass. The whole energy density is contained in the G-waves.
 

Will we find a single energy density for all G-waves?
 

Of course, not. The general solution of D'alembert's wave equation depends on integration constants. These constants play the role of parameters, since they can a priori take arbitrary values. So, for a given set of values of this "integration constants", we will get a corresponding G-field. All we can assert is that all G-waves obey D'Alembert's equation. That diversity of G-waves according to the values of their parameters implies a diversity of "G-masses", since the energy-momentum tensor for G-waves will necessary depend on these parameters and so will mass densities.
 

We now consider electromagnetism, in the same charge-free Maxwellian model. There, there's no "ordinary" charge at all, the whole energy density is contained in the EM-waves. More than well-known. Again, we won't find a single EM-wave, but a variety of them, according to the values of their parameters. The energy-momentum tensor for these waves will give us a "mass density", but it's not really appropriate anymore, for the reason that electromagnetism behaves opposite to gravity. So, we should rather talk of an "equivalent mass density", we usually call an "electromagnetic mass" and convert it into a charge density.

Now, such a charge distribution, despite still in 4D space-time, has nothing to do with an ordinary charge density anymore, since we assumed that this last one was absent everywhere!

As you can see, our answers can actually come from that astonishing property of energy to take numerous forms that can transform into one another and from the relativistic property that matter can directly derive from energy. As a result of both, a variety of energies leads to an equivalent variety of masses and thus, forms of matter...
 

If we now go a step further and make the hypothesis according to which:
 

MASSES PRODUCED BY G-WAVES = "PSI-MASSES"

CHARGES PRODUCED BY EM-WAVES = "PSI-CHARGES"

 

not only can we stay in usual space-time, but we can define Andrade's "PSI-matter".

We can even get close to a suggestion from Changeux!
 

What do we need to make PSI-matter? We need PSI-particles. Can we get them? Yes: we have a variety of masses produced by G-waves and a variety of charges produced by EM-waves. Classically, we actually need only 3 charge state: (+), (-) and (0). According to the values we give to a PSI-mass and the value of a PSI-charge, we can at least classically define a «PSI-particle» as a «basic constituent that is assumed not to contain more fundamental constituents».

What do you find in chemistry, after all? Protons, neutrons and electrons making atoms. For chemistry, what are they? Protons are particles of matter with mass m(p) and charge +e, neutrons are particle of matter with mass m(n) very close to m(p) and charge 0 and electrons are particles of matter with mass m(e) << m(p) and charge -e. We need no additional informations to define those particles in chemistry. All that interests us is that protons and neutrons can combine to make nuclei and nuclei can combine with electrons to make atoms.

Well, we don’t need a whole encyclopedia of informations to define PSI-particles, it’s the very same procedure: we give ourselves PSI-masses and PSI-charges. We can even have the same values as for their ordinary counterparts: it all depends on the values of the parameters in D’Alembert’s equation!
 

Why should we look for a more complex equation if we want to define basic elements?...
 

So, let’s assume that, similar to supersymmetry, where we add «super-partners» to known particles, we add «PSI-counterparts» to our catalog of particles. Thre can perfectly exist other «PSI-particles» than those, but we don’t need them for our purpose. We just need «PSI-protons», «PSI-neutrons» and «PSI-electrons», with same mass and charge values as their «ordinary» counterparts.
 

But of a radically different nature!
 

PSI-masses take their origin in gravitational waves: it means there’re made of gravitational light.

PSI-charges take their origin in electromagnetic waves: it means there’re made of electromagnetic light.
 

If i assemble PSI-protons with PSI-neutrons and surround them with PSI-electrons, i obtain PSI atoms. Obviously, they will look very different from ordinary atoms, since we have nothing substantial anymore, but everything wavy.

Assembling PSI atoms makes PSI molecules and so on.

Now, look at what links PSI matter: it’s PSI charges. And PSI charges are EM light.

Adding complexity, we find that mental objects are EM productions, with an ordinary source this time. No problem: it enters the energy-momentum tensor and gives matter the same. But we now have a second category of PSI charges, those produced by a source of ordinary matter. These charges will behave as EM light if and only if they are radiations, i.e. only when their sources will be accelerated. They can’t behave as EM light, for they are confined within the biological body.

There can we see the neat distinction there is between a true wave and a field propagating outside its source. A mental object is none of them, but it still gives birth to a mass density and thus, to electromagnetic matter.

So, Changeux was not so «eccentric» when he suggested that, just like in atomic structures, «we could imagine» that mental objects could serve as «chemical links» between neuron cells in the nervous system, building a «bridge» between neurobiology and psychology.

Here, we are before all concerned with the Spiritual. And we discover, out of known physics, that we already have to distinguish between:
 

- «wave-made» PSI matter;

- PSI matter originated from fields «freely» propagating outside their ordinary source;

- and PSI-matter originated from fields inside their ordinary source;

The «purest» are the first ones: they depend on no ordinary matter, they can pre-exist to it. I now believe that what we call the «evolution of the Universe» is the material phase. I believe that, after the Planck time began the production of «ordinary» matter out of fermionic vacuums (i don’t say «classical», but «ordinary», including quantum matter) and, before the Planck time, there was fermionic vacuums and boson waves. Matter under a non-substantial form.

I believe the PSI pre-exist to the Universe after the Planck time.

And i believe in «Light Beings», because physics tells us how to build them...
 

Finally, i strongly suspect that, during a NDE, we (temporarily) go from the third category of PSI-matter to the second-one (OBE) and back.

That we do meet with «Light Beings» of the second category («souls of deads»).

But that our fate could be decided by a Light Being of the first category.

Could be.
 

 

 

Just a little bit of overwork, these last days... Not bad, as I could reorder my way of working a bit. I may not go deep enough in the analysis of the objects I introduce. Nothing striking today, but a closer review of the components of gravity. I’d like to talk about matter, as it is at the centre of everything.

Without entering sophisticated models, we can already give an important extension of the linear model of gravity introducing 10 field functionals on M₄, W_ij[G(x),x] and V_i[G(x),x], with W_ji = -W_ij:

 

(1)               £_G[G(x),W(x),x] = (c²/8pk)W_ij(x)W^ij(x) + W_ij[G(x),x]W^ij(x) – V_i[G(x),x]uⁱ

 

As £_G is in J/m³, the 6 W_ijs are densities of action, while the 4 V_is are densities of energy. Th unitary vector uⁱ = dxⁱ/ds can be taken here on fixed points of M₄, as a ratio of infinitesimal quantities of same order, 1. Indeed, the dynamics isn’t about point-like material bodies, like in systems of masses interacting via gravity, but about the gravity field itself. As was already pointed about in former bidouilles, the working frame in (1) is no longer the physical space-time M₄, but a “gravitational space-time over M₄”, where “distances” are measured in m/s and “velocities” in s^-1 = Hz. So, to a M₄-observer, such “distances” and “lengths” are felt as velocities, while “velocities” are felt as frequencies.

I chose to talk once again about gravity because of its universal nature: if I strongly insist on that, it’s precisely because, from a given model of gravity, one can build similar models for other types of interactions and, in particular, electromagnetism.

The coefficient c²/4pk is a universal constant in the vacuum of M₄, but usually varies inside material media. If we compare (1) with the Lagrange function describing the motion of a material body of mass m and velocity v(t) inside a G-field,

 

(2)               L[x(t),v(t),t] = ½ mv²(t) + mG[x(t),t].v(t) – mcG₀[x(t),t]

 

assuming that |v(t)| << c, then c²/4pk plays the role of the “mass” of an “incident body” in (1). However, the similitude is not purely mathematical, since c²/4pk is a density of inertia and mass, whatever its physical nature, can always be defined as the scalar curvature of an inertia. So, in the present context, the locally-defined quantity:

 

(3)               m(x) = ¶ⁱ¶_i[c²(x)/4pk(x)]

 

is indeed a mass density. Outside matter, it’s identically zero so that, if we wanted to keep an analogy with (2), we would have to redefine our mass density as 1/4pkt_pl² = c²/4pkR_pl², from another universal constant, the Planck time (or radius).

 

This mass density (3) has obviously nothing to do with an “ordinary mass density” anymore, despite it’s still a function of x, and that’s a point I didn’t spend enough time on.

 

1)      the quantity m in (2) is global on Euclidian space E₃; should it varies, it would only be with respect to time t;

2)      an “ordinary” mass density m_ord spreads in 3-“ordinary” space; it models a (finite) collection of “point-like particles” (typically, atoms, cells, for our purpose – basic elements).

 

This is absolutely not what (1) represents.

 

1)      the G_i(x,t) rather looks like the velocity 4-vector of a “fluid” in M₄; it’s a continuous medium, it’s no longer a discrete collection of point-like elements;

2)      the mass density (3) must then be a global quantity on this “gravitational space-time”; whether it’s constant and, when it is, it’s a universal constant in M₄, which is not the case of m, or it varies from point to point, as m(t) does along time.

 

For these two reasons, that mass distribution (3) in M₄ has nothing “ordinary” at all. It’s in no way the mass density of an “ordinary” fluid, since an “ordinary” fluid as v(x,t) for velocity field in E₃, with time as a parameter of motion and even v(x,t) has no physical common point with G(x,t), which refers to nothing substantial at all: G(x,t) is a radiation produced by a substantial source of velocity v(x,t). Precisely…

The m(x) in (3) is the global quantity of “something” that is dispatched inside a “gravitational volume” G¹G²G³. So, it’s out of the question it could be made of atoms or even particles encountered in M₄. On the contrary, in M₄, it always appears diffuse… There’s no way to globally define it without destroying the field model itself.

 

The question is: what is that kind of “matter”? Can we even talk about “matter”?

 

I think we can talk about some kind of “matter”, since (2) describes the gravitational interaction between a source of mass m’, not represented, and an incident body of mass m. So, if we wanted to keep the analogy with (2), we would be forced in (1) to interpret (3) as the “mass” of an “incident body” moving in “G-space-time”, and W_ij[G(x),x] and V_i[G(x),x] as the “potentials”, in this “G-space-time”, of a new “gravitational interaction”, produced by another source “mass” m’(x).

 

The worse is that, if we extended quantum field theory to that “G-space-time”, we could find a fermionic statistic for these m(x) for, independent of the xⁱs, the frame has the very same structure as M₄. So, as we exhibited a spin “sub-structure” out of M₄, we would exhibit a similar sub-structure out of G-space-time and that would enable us to define a “matter state” in that frame.

 

Could it be the kind of matter we’re (desperately) searching for?

 

I’m now extremely careful in asserting anything. What draws my attention is that universal feature of gravity: if we replace G_i with (-4pke)^1/2A_i, (1) applies to electromagnetism and thus, to consciousness.

And what really gets interesting is that all these coefficients that define the new matter vary in ordinary matter: it means that this new matter takes all its significance inside ordinary matter and, in particular, biological matter…

 

The human brain working on analogies, the temptation is rather great to try a link between this “G-matter” and a “PSI-matter” made of “Andrade’s PSI atoms”: we have quantum theory to define that “G-matter”; so we can build “G-atoms” from “G-particles”; and so on.

 

I’m not saying we got the right answer this time, only that it’s not without interest.

We couldn’t build a new matter from the electromagnetic field because of its Bose statistics in M₄ and, anyway, that “matter” would be made of a single particle specie, namely photons, which would be far from enough to make autonomous complex systems.

However, we can find “consciousness experiences” using a model like (1) for the electromagnetic field. But we have to keep in… mind that, whatever the interaction involved, “G-matter” is different from it: the interaction makes the frame, G-matter expands inside that frame.

 

I’ll come back on this idea of Andrade in more details in a following bidouille. I need to think about it first.

 

 

 

I’m gonna stop there, for the time being, with non commutative geometry, and turn to something else, much easier to grasp and also much clearer. Field theory in a NC geometry can yield to extended properties of matter and interaction fields, but we quickly loose ourselves in technical details and anyway, I doubt it can have possible applications to large scales and the present universe. Should it be useful to something, it would probably concern the (very) early universe. Obviously, through cascades of symmetry breaking, it could then help understanding organization of our observable universe at different stages. But this has to do with cosmology, not parapsychology.

I shall only give a general method for constructing the non commutative version of the Riemann integral. The easiest way is to start from the definition of the first NC derivative of a NC-scalar function F_i(X) of a NC-vector variable X, formula (1), last bidouille. Integration (i.e. continuous summation) over dF_i(X) leads to:

 

(1)               F_i(X) = F_i(X₀) + ò_X0^X dF_i(Y) = ò_X0^X [Tr(dY.¶)]F_i(Y)

 

where the limits X₀ and X have to be taken matrix-valued [elements of M₄(R)]. In components:

 

(2)               F_i(x^lm) = F_i(x₀^lm) + ò_x0^x dy^jk¶_kjF_i(y^lm) = F_i(x₀^lm) + ò_x0^x dy^jkF’_kji(y^lm)

 

with x₀ and x to be understood as (x₀)^lm and x_lm and ¶_kj = ¶/¶y_kj is the derivative with respect to the integration variable y_kj.

Following this procedure, multiple integrals as well as the Lebesgue integral should not be difficult to extend.

 

 

I was right not to insist any longer yesterday, as i faced a little but unimportant technical pb.

When doing geometry, the most frequent difficulty lays in mental representation of things: as long as your mental image is not clear, you face difficulties.

I should have noticed this:

 

NC-SCALARS ON M₄(R) ARE DIAGONAL MATRICES.

 

As simple as that, and it greatly simplifies things, as we can extend all constructions in commutative tensor theory (symmetric and skew-symmetric products, traces and invariants) without difficulty.

Whatever we do in M₄(R), we should keep in mind that space-times M_4j are actually states of the space-time M₄. As long as we don’t “lift degenerescence” by any means, there seems to be a “single” space-time M₄, more precisely, a 4D space-time in a single state. This is the so-called “degenerated state” of space-time. “Lifting degenerescence” means we separate energy levels so that different states appear. These physical states can correspond to configurations. In the SU(3,1) unified gauge model [or even U(3,1), if we lift the restriction on unimodularity, which does not restrict generality for as much], 4 such states or configurations are considered. Each of them is assigned a 4D space-time M_4j (j = 1,2,3,4): it’s nothing else than M₄ “in the j-th state or configuration”. “Reality levels”, would say some.

So, fixing ourselves a scalar quantity, say x, on M₄ undermeans this scalar is somehow “degenerated”: it’s a mixing of purer states. If we separate the 4 states of M₄, we find a 4-vector quantity x^j: for each state j, x^j is a C-scalar on M_4j.

On another hand, we can always extend any scalar quantity into a 4-vector. Let x¹ be such a C-scalar on M₄₁. It’s equivalent to the 4-vector of components (x¹,0,0,0). Take x² another C-scalar on M₄₂, it corresponds to it the 4-vector with components (0,x²,0,0). Similarly, we have x³ = (0,0,x³,0) on M₄₃ and x⁴ = (0,0,0,x⁴) on M₄₄. It amounts exactly to the same to saying the x^js are “states” of x on “states” M_4j” of M₄.

“Gluing together” those four 4-vectors, we obtain a diagonal matrix (or tensor) on M₄(R): this is what we call a “NC-scalar” on M₄(R). Writing X = (x^ij)_i,j=1,2,3,4 a matrix on M₄(R), it’s always possible to identify a C-4-vector (xⁱ)_i=1,2,3,4 of M₄ with the diagonal matrix D(X) = (xⁱⁱ)_i=1,2,3,4, so that we have, in components, xⁱ = xⁱⁱ (i = 1,2,3,4).

 

And this considerably simplifies our task.

 

To begin with, consider a C-scalar function f of the C-scalar variable x on M₄. The image y = f(x) is again a C-scalar on M₄. All these scalars must now be seen as “degenerated” quantities of NC-scalars xⁱ, yⁱ and fⁱ on M₄(R), so that y^j = f^j(xⁱ): commutatively, we now find a 4-plet of scalar functions f^j, each depending on the 4-plet of scalar variables xⁱ. Or else, a 4-vector field over M₄. If we now identify all these 4-vectors with NC-scalars on M₄(R), i.e. diagonal matrices, we equivalently find y^jj = f^jj(xⁱⁱ): this is a particular case of the much more general functional relation y^ij = f^ij(x^kl) on M₄(R), between NC-quantities.

 

ANY VECTOR FIELD OVER M₄ IS A NC-SCALAR FIELD OF A NC-SCALAR VARIABLE OVER M₄(R).

 

Consider now a C-tensor y^ijk of order 3 on M₄. We can rewrite it as (yⁱ)^jk. On M₄(R), yⁱ is a NC-scalar, while (.)^jk is a NC-vector. As a result, y^ijk transforms as a NC-vector on M₄(R), just like y^ij. A single 4-component index plays no role in the transformation on M₄(R).

Let’s take an example, to be clear.

Take a C-scalar a and a C-vector xⁱ on M₄. The tensor product of a and xⁱ identifies with algebraic multiplication: aÄxⁱ = xⁱÄa = axⁱ = xⁱa. A tensor of order 0 (i.e. a scalar) on M₄ adds no order to a given tensor through the tensor product. What we have is a dilatation of xⁱ a factor a: if |a| > 1, it’s a dilatation; if |a| < 1, a contraction. In no way will we make a matrix out of such a product.

Well, the same holds in M₄(R). The tensor (or the matrix, whatever) product of a NC-scalar aⁱ and a NC-vector x^ij on M₄(R), not only is commutative, since aⁱ identifies with the diagonal matrix (aⁱⁱ)_i=1,2,3,4, but gives a tensor (or matrix) of same order, as to know a^ijx_jk = å_i=1⁴ aⁱⁱx_ik = yⁱ_k.

 

THE NC-PRODUCT OF A NC-SCALAR WITH A NC-TENSOR OF ANY ORDER IS COMMUTATIVE AND GIVES A NC-TENSOR OF SAME ORDER.

 

The difference with the commutative situation is striking: a C-tensor T^i(1)…i(2p) of order 2p and a C-tensor U^{i(1)…i(2p+1)} of order (2p+1) both transform as a NC-tensor W^i(1)…i(p) of order p. If you had to transform back, giving you a NC-tensor of order p, what would you choose, a C-tensor of order 2p or 2p+1? There’s a freedom there, equivalent to saying in the commutative, that C-vectors xⁱ and axⁱ, with a ¹ 0, “belong to the same class” or that xⁱ “is defined up to a multiplicative factor a (or a scale a)”. In M₄(R), a “scaling coefficient” is a 4-vector on M₄.

 

THERE’S AN EQUIVALENCE CLASS IN M₄(R) BETWEEN 2p-TENSORS AND (2p+1)-TENSORS ON M₄ IN THE SENSE THAT THEY BOTH LEAD TO A p-TENSOR ON M₄(R).

 

Remain careful with the sets you work in: a p-tensor on M₄ is a C-tensor; on M₄(R), it’s a NC-tensor.

 

We can now talk about non-commutative differentials and differential forms on M₄(R). This helps describing and understanding local properties of bodies, motions or even frames. Global properties are described by integration theory.

 

Let F_i(x^j) be a NC-scalar function of a NC-scalar variable x^j on M₄(R). The differential dx^j of x^j is a small variation around x^j. So, it’s one more NC-scalar (dx)^j, equivalent to the diagonal matrix (dx)^jj the non-zero components of which are all small variations around each of the x^j = x^jj. In clear, we have a 4x4 real-valued matrix made of zeros off-diagonal and of dx¹, dx², dx³ and dx⁴ along the diagonal. However, d being a C-scalar operator, it becomes meaningless in M₄(R), so dx^j is actually not the differential of x^j: what’s meaningful is the NC-scalar or C-vector (dx)^j. This being said, we have, with a slight abuse of notation that should have no consequences, precising the context: F_i[x^j + (dx)^j] = F_i(x^j + dx^j) = F_i(x^j) + (dF)_i(x^j). The quantity (dF)_i = dF_i is assumed to be a small variation of the function, of same order than that on the variable. We’d like to express it in terms of the derivative of F_i. For that, we would write contributing terms in the same order as for matrix product:

 

(1)               dF_i = dx^kj¶_jkF_i = [Tr(dX.¶)]F_i

 

This is the general expression for the first derivative of a NC-scalar function of a NC-vector variable on M₄(R). In particular, when X is diagonal, we find:

 

(2)               dF_i = (å_j=1⁴ dx^jj¶_jj)F_i = (å_j=1⁴ dx^j¶_j)F_i

 

since only the diagonal terms ¶_jj of the matrix ¶_jk = ¶/¶x^jk contribute. It follows that, for a NC-scalar function of a NC-scalar variable:

 

(3)               ¶_jF_i(x^k) = ¶F_i(x^k)/¶x^j = F’_ji(x^k)

 

is no longer a NC-scalar, but a NC-vector function. This is quite easy to understand: once again, C-scalar variables and functions can be seen as “degenerated” and so will be their derivatives at any order (i.e. as long as the function is derivable); on the opposite, NC-scalar functions and variables are 4-vector fields on M₄. So, to each state M_4j of M₄ is now associated a derivative of F_i and this is what (3) expresses: ¶_jF_i(x^k) = ¶_jF_i(x¹,…,x⁴) is the derivative in M_4j of the scalar function F_i in M_4i. As we now have 4 states of M₄, we find 4 states of any scalar function over M₄ and 4 states for the derivative of each state of F, giving 4x4 = 16 derivated numbers at each point of M₄ where F is derivable.

Still more generally, we know a C-1-form on M₄ is a C-scalar infinitesimal quantity:

 

(4)               a = a_i(x^j)dxⁱ

 

that remains invariant under coordinate transformations (a appears the same in all coordinate systems of M₄). The coefficients a_i(x^j) of this 1-form are not necessarily derivatives ¶_if(x^j) of a C-scalar function f(x^j). When this is so, a is said to be “exact”: it’s simply the differential of f; otherwise, it’s “inexact”.

A NC-1-form on M₄(R) will be a NC-scalar infinitesimal quantity:

 

(5)               A_i = A_ijk(x^lm)dx^kj = Tr[A_i(X).dX] = Tr[dX.A_i(X)]

 

Similarly, if A_ijk(x^lm) = ¶_jkF_i(x^lm), A_i = dF_i and A_i will be said “exact”. If not, “inexact”. Assume now that X is diagonal: X = (xⁱⁱ)_i=1,2,3,4. Then (5) will give:

 

A_i = å_j=1⁴ A_ijj(x^ll)dx^jj = å_j=1⁴ a_ij(x^l)dx^j

 

Is it (4)? Yes, if we take into account that we now must have 4 C-1-forms, one on each M_4i. As soon as we “degenerate”, A_i reduces to a single component and so do its coefficients a_ij(x^l): that’s precisely (4) on M₄.

Let’s move on. A C-2-form on M₄ is a C-scalar invariant quantity:

 

(6)               f = ½ f_ij(x^k)dxⁱÙdx^j  ,  f_ij = -f_ji

 

When f = da is the outer derivative of a C-1-form a like (4), that is, when f_ij(x^k) = ¶_ia_j(x^k) - ¶_ja_i(x^k), then f is closed: we have the Bianchi identities ¶_if_jk + ¶_jf_ki + ¶_kf_ij = 0, that can also write df = 0 independently of any basis or d(da) = d²a = 0.

A NC-2-form on M₄(R) will write:

 

(7)               F_i = ½ F_ijklm(x^np)dx^mlÙdx^kj  ,  F_ijklm = -F_ilmjk

 

When F_ijklm = ¶_jkA_ilm - ¶_lmA_ijk, F_i will be closed: Bianchi identities are obvious. Let’s take X diagonal, (7) reduces to:

 

F_i = ½ å_j=1⁴å_k=1⁴ F_ijjkk(x^ll)dx^kkÙdx^jj = ½ f_ijk(x^l)dx^kÙdx^j = -½ f_ijk(x^l)dx^jÙdx^k

 

Again, this is (6) under 4 states, with a change of sign due to our choice of ordering components in (7). This sign being global, it only changes orientation on all M₄. As orientation is a choice of ours, it changes nothing on the physics of M₄, but the convention we gave ourselves (if the change of sign was local, it should be completely different).

The quantity dxⁱÙdx^j is a surface element on M₄, making a skew-symmetric coordinate 2-tensor ds^ij = -ds^ji: 6 components only over 16, all in m² (2-forms are infinitesimal quantities of order 2). That’s a NC-vector on M₄(R). This NC-vector corresponds to the 6 plans of M₄: each component of ds^ij is on a M₄-plane. I’ve established yesterday evening a correspondence between this surface element on M₄ and the dx^ijs on M₄(R):

 

(8)               dxⁱÙdx^j = ds^ij = ½ g_kl(dx^ildx^kj – dx^jldx^ki) = ½ Tr[(dX)² - (^tdX)²]^ij

 

where dX = (dx^ij)_i,j=1,2,3,4 and ^tdX = (dx^ji)_i,j=1,2,3,4 is the transpose matrix (obtained from dX inverting lines and columns). It seems to work. On the right, we have a skew-symmetric tensor product of all the dx^ijs, making a NC 2-tensor dS^iklj = ½ (dx^ildx^kj – dx^jldx^ki) we take the (kl)-trace of. On the left, we have a skew-symmetric tensor product of all the dxⁱs = dxⁱⁱs, making a C-2-tensor ds^ij = dxⁱÙdx^j.

Notice in passing that the metrical tensor g_ij on M₄ is a constant symmetric C-2-tensor on M₄ and therefore a NC-vector on M₄(R), with 6 zeros.

 

We have all the base ingredients for a generalization to n-forms. Higher orders follow the same procedure. I have also, for instance, established the equivalence (8) for a volume element on M₄, it’s:

 

(9)               dxⁱÙdx^jÙdx^k = (1/3!)g_lmn[(dxⁱⁿdx^mj – dx^jndx^mi)dx^kl + (dx^jndx^mk – dx^kndx^mj)dx^il + (dx^kndx^mi – dxⁱⁿdx^mk)dx^jl]

 

that is, ordinary cyclic permutation. One last formula for the 4-volume element dxⁱÙdx^jÙdx^kÙdx^l and that’s all. We can’t go over in M₄.

We can’t go over in M₄(R) either, for what concerns NC-forms: NC 4-forms are the highest we can build. All higher order NC-forms should be identically zero.