We've been quite far in the analysis of the mathematical tools at our disposal in modern physics. We've investigated many aspects of the parasychological problem, in connection with neurobiology. It's now time to make a synthesis of all this and justify our choices.
 

We have a biological body, which is a (highly) complex autonomous system, able to self-regulate, interact with its surrounding environment and adapt to it. However, if we choose to consider it globally, we can "forget" about all this complexity and model it as a "macroscopic (touchable) matter", that is, as a mass distribution inside a finite volume of space with internal pressure p, temperature T and other thermodynamical parameters describing the different chemical concentrations. This is the "global" description of "substantial" matter.

Through its mass, this body produces a gravitational field. In addition, it has a nervous system, which produces an electromagnetic field through its electrical charges: the nervous system behaves like a "plasma", producing electrical currents, while keeping its global charge to zero [actually, all biological cells have an electromagnetic activity, but neurons are amongst those cells that are (re)activable].

These three complementary aspects of the animal body orientated us toward a macroscopic description of it with these three interacting components: substantial matter, G-field and EM-field. The complexity of the biological mechanisms inside the body demanded a generalization of the much-too-simple Maxwell model. We proposed such an extension, taking into account the feedbacks of the fields on their sources. As a result, any change in the behavior of, say, the field of consciousness, dynamically modifies the distribution of charges inside the nervous system, that is, active paths in the neuron graphs which, in turn, modify the field, and so on, until possible (but not systematic) equilibrium situations are found. Physically speaking, the memorization process can be described as a charge retention.

It goes the same for the G-field. Since its source is complex, so will it be. Any change in the distribution of substantial matter inside the body will then modify the G-field and back. This will be much less noticeable as the electromagnetic activity of the nervous system since the coupling constant of gravitation is far lower than that of electromagnetism. Still, it exists and we shouldn't forget about it, even if we can legitimately consider it's completely negligible in organic bodies.
 

Then, we introduced (or re-introduced) general relativity and we showed that elastic deformations of space-time were actually induced by material sources. The much larger freedom of choice in designating "matter" including, not only "substantial" one, but fields themselves, i.e. "non-substantial ones", showed that the induced deformations actually had nothing to do with gravity, which was now a source of space-time deformations like macroscopic matter and the EM-field. Our reviewed interpretation of GR was confirmed by the existence of non-trivial solutions of the Einstein equations keeping space-time plane (and not only flat) and that could therefore be produced by no kind of matter at all, may it be substantial or not.

When matter is present, it curves space-time. But we should be very careful which part of space-time we're dealing with. For the equations of GR with material sources actually describe the curvature of space-time inside matter.

Inside. Not outside.

Outside matter, the equations are those "in the vacuum".

And we can no longer superpose both types of solutions, for the equations of GR are no longer linear.
 

Why am i now strongly insisting on the qualitative difference between them?
 

Because i came to the conclusion that the only way to bring a physically consistent explanation of comas was to accept the idea that space-time was truly deformed inside the body.
 

I'd like to emphasize here the essential difference there is between psychotic behaviors and the NDE. Because there lays a possible explanation of the "Tunnel".
 

In psychotic behaviors, the patient has an altered perception of external surrounding space-time. These alterations are mostly due to dysfunctions inside the nervous system (severe disconnections of entire bundles of fibers in the brain, for instance). But it remains a mere perception of things. Actually, as we can check it, reality around us is altered nowhere. The patient incorrectly observes his environment, that's all. Inside and outside of him, there's no perceptible deformation of space nor time. The origin of this false, or pseudo-alteration, of both inner and outer realities is purely organic.

On the opposite, NDEs have nothing psychotic at all. The "Tunnel" can only form if, and only if, space-time inside the patient's body is significantly altered. In this case, the deformation of both space and time is not fictitious, but real. It's not perceived outside the patient, because it remains confined inside of him.

There can be no perceptive process anyway, since there's no more cerebral activity in the neocortex...

In the psychotic brain, there remains an activity, even if wrong.

In NDEs, we can no longer argue there's any. It just don't hold.
 

Why should space-time be altered inside the patient's body?

Because the functioning of the body is itself altered. As soon as it's placed in a coma state, even artificial, part if not all of the nervous functions are blocked. "Frozen". As they command all of the biological functions inside the body, this "freezing", even partial, has direct and quick consequences on the rest of the body. For instance, in "artificial coma" (anesthesia), both the sensor and the motor systems are blocked, so that the body is already no longer in «nominal» functioning.
 

The novelty holds in the fact that, opposite to what we could think, this restriction alone suffices to change the nature of space and/or time inside the body.

We indeed have to stick to the observational facts. And the facts tell us that, as soon as stage I, the patient looses the notion of time. Yet, outside his body, time keeps on being perceived the same. So, the change can only stand within him.
 

The physical explanation is to be found in the equations of GR inside his body, not outside.

We can restrict ourselves to a single thermodynamical parameter, the blood pressure p. Inside matter, the metrical coefficients describing the (local) deformations of space-time explicitly depend on p, g_ij = g_ij(x,p). The result is a family of metrics and not a single one, as would be in a vacuum: for each value of p, space-time inherits a geometry described with the metrical coefficients g_ij(x,p=cte). When p changes, these metrical coefficients can change and we go from a geometry to another one. For instance, g_ij(x,p=p₀) = g_0,ij(x) gives a geometry, g_ij(x,p=p₁) = g_1,ij(x) gives another.
 

If we agree with the classification of comas into 4 stages (which is not the only one possible), we need 4 critical values of p: p_c1, p_c2, p_c3 and p_c4. Each time the blood pressure falls down one of these critical values, the patient enters another stage of coma. At each critical threshold, something happens in what we call the «main directions» of the strain quadric, i.e. in at least one of the diagonal coefficients g_ii(x,p). Let us detail this, in the transition model based on pressure.
 

In the awaken state of consciousness we can classify as «stage 0», the sane patient perceives time and space as should be. This is described by g₀₀(x,p) > 0 and g_aa(x,p) < 0 (a = 1,2,3).

When p reaches the first critical value p_c1, g₀₀(x,p_c1) vanishes, while g_aa(x,p_c1) remains < 0. This vanishing is sufficient to deeply change the properties of space-time inside the body, because it means that, at p = p_c1, «the notion of time disappears».

Between p_c1 and p_c2, g₀₀(x,p) becomes < 0, meaning the patient looses the notion of time. He keeps that of space, since g_aa(x,p) remains < 0. He can remain perfectly conscious (at least in his subconscious) he’s having a surgery, he can even ask the medical staff questions sometimes (he won‘t remember), but he has no idea of how long it takes.
 

It should already be clear we changed geometry since, above p_c1, we had one time dimension and 3 space dimensions and, below p_c1, we get four space dimensions.
 

Let’s argue that, at p = p_c2, the patient also looses the notion of space itself. He enters stage II, «light coma», g₀₀(x,p_c2) remains < 0 and at least one of the three g_aa(x,p_c2) now vanishes.

Between p_c2 and p_c3, he’s in stage II, he has lost both the notion of time and that of space, because we now have g₀₀(x,p) < 0 and (at least one of the) g_aa(x,p) > 0. It wouldn’t mean much to interpret it saying «he now has the notion of only one space dimension and (between one and ) 3 time dimensions».
 

At p = p_c3, he enters stage III, «deep coma». There, g₀₀(x,p_c3) is still negative, but g_aa(x,p_c3) vanishes again. The patient «begins to recover the notion of space».

Between p_c3 and p_c4 we will set to zero, he still has no idea of time, but has that of space again, since g_aa(x,p) < 0 again.
 

At p = p_c4 = 0, he’s in stage IV. There, g₀₀(x,0) = 0, while g_aa(x,0) < 0.
 

Let us proceed «　stage by stage　» :).
 

Each time one of the diagonal coefficient g_ii vanishes (i = 0,1,2,3), we stand on a «critical hypersurface». As we want GR to hold whatever the pressure, from above p_c1 down to zero included, the determinant of the metric, g(x,p) = det[g_ij(x,p)] is expected to remain < 0 for all p. This means we exclude «essential singularities», that would set GR into default. What we get instead, which is physically meaningful, is a «fictitious singularity» at each critical value of the blood pressure. Such a fictitious singularity, or critical hypersurface (of dimension 3), precisely indicates a change in the geometry of space or time.
 

However, we have to slightly modify the usual definition GR gave of a fictitious singularity.
 

In the original theory of GR founded on changes of coordinate systems, such a singularity was defined by:
 

THERE EXISTS A LOCAL COORDINATE SYSTEM IN WHICH THE ZEROS OF THE gii(x) ARE REPLACED WITH NON-ZERO VALUES.

In other words,
 

THE VANISHING OF AT LEAST ONE OF THE gii(x) IN A GIVEN LOCAL COORDINATE SYSTEM ONLY INDICATES THAT THIS SYSTEM IS NOT THE SUITABLE ONE.

In an adapted system, none of the g_ii(x) vanish. The (more than familiar) example of this is the static black hole: in the Schwarzschild system (spherical coordinates + time), a singularity appears at a distance-to-the-centre equal to the gravitational radius of the source body. This singularity is seen to be fictitious because g(x), the determinant of g_ij(x), does not vanish at r = r_g for as much. Going then from the Schwarzschild system to the more adapted Lemaitre system of local coordinates, the singularity in question disappears and light rays are found instead.
 

We have to modify this definition for we’re no longer playing with coordinate systems, but with true physical deformations (or strains) of space-time. In this context, we will rather say:
 

A SINGULARITY IN THE METRICAL PROPERTIES OF SPACE-TIME WILL BE CONSIDERED «FICTITIOUS» IF THE DETERMINANT OF THE METRICAL TENSOR REMAINS < 0, SO THAT ONE CAN ALWAYS FIND A LOCAL STRAIN FOR WHICH THIS SINGULARITY DISAPPEARS TO THE BENEFIT OF NON-ZERO, FINITE AND SIGN-DEFINITE VALUES.

Indeed, if you go back to the previous bidouille, you will realize that we have established the following result:

 

THERE EXISTS A NON-TRIVIAL LOCAL STRAIN FOR WHICH SPACE-TIME REMAINS PLANE.

Well, at each critical values of the blood pressure of our model of comas, there always exists a local strain y(x) for which the zeros disappear, regularizing geometry.
 

To obtain a Tunnel at p = 0, i have two options.

Whether i exclude that final critical value from the three other fictitious and consider we now face an essential singularity, for which g(x,0) = 0 and then i’ll have (unsolvable) problems with the equations of GR, as the inverse of my metrical tensor won’t be found, or

i consider it the fourth fictitious and its regularization should have the topological S²xR symmetry: a 3D spatial sphere as basis and time as axis. With g(x,0) < 0 and finite, there necessarily exists a local strain for which the tube structure appears.
 

And we get what we call in astrophysics a wormhole. A «tunnel» through space and time.
 

Why should this tubular structure appear only at stage IV? Good question. The answer might be found in the set of all coupled field equations in curved space-time: the 10 GRs, the 4 complex Gs, the 4 complex EMs + the equation of state and the 4 equations of motion for substantial matter. A total of 23 strongly non-linear coupled equations. Good luck...

All i can say is that no testimony so far as ever evoked a Tunnel in any other stage of coma. Not all NDEs have a Tunnel: it may not even be systematic. Some patients have nothing particular to report.

The answers surely lay in the complexity of the metabolism.

 

　

　

　

Mostly text today, as calculations are easy and i would prefer to discuss the point.

There's already a bidouille about zero-energy motions. We're now going to talk about zero-curvature space-times. That is, plane space-times. I'll limit myself to 4D space-times, as generalization is straightforward.

Let then M be a space-time with a single time dimension and 3 space dimensions, x a point of M (anywhere) and y(x) a deformation sending that point x onto the point x'(x) = x + y(x). We compute the metrical tensor g_ij(x) from the surface element around x'(x) and derive the Christoffel symbols C_k,ij(x). The result is very simple:
 

(1)     C_k,ij(x) = [g⁽⁰⁾_kl + d_ky_l(x)]d_id_jy^l(x)

where g⁽⁰⁾_kl is, as usual, Minkowski metrical tensor.

The question i asked myself was: are there solutions of
 

(2)     C_k,ij(x) = 0     globally on M

other than the trivial translation y(x) = a = cte?

The answer is yes. There are even plenty of such solutions, with the form:
 

(3)     yⁱ(x) = aⁱ + S_n=1^N aⁱ_i1...inxⁱ¹...xⁱⁿ/n!

with N finite or not. And these are the only possible solutions. Calculating for the first and second derivatives and inserting them into (2) leads to a set of 2(n-1) conditions on the coefficients of (3). We can check that the first derivatives of the Christoffel symbols then globally vanish, so that the Riemann curvature tensor is indeed everywhere zero, making M plane. Yet, not only is there a deformation in M, but it can be highly non-linear. Physically, this means that not only is M elastic, but it's also plastic!

All this is essentially due to the pseudo-Euclidian geometry of M. In Euclidian geometry, we can easily verify that all coefficients of (3) but aⁱ and aⁱ_i1 vanish, reducing the deformation to a translation + a dilatation/contraction. On the contrary, in pseudo-Euclidian geometry, squares like V^lV_l are no longer positive-definite, leading to new possibilities (light-like ones).
 

It was quite natural for me to wonder what the hell could well be those deformations not affecting the geometry of M anywhere. Indeed, if M was to remain elastic and even plastic, we should expect it to curve...
 

The thing is, (3) describes a anharmonic (= multi-frequency) spring. The pullback is guaranteed by the non-negative powers of the xⁱ. It’s a typically confined solution. The only satisfying physical interpretation i found was that the higher the order N, the more rigid M.

In other words, the development of (3) in higher and higher powers reinforces the flatness of M, assuming, of course, the coefficients satisfy (2).

We can as easily check that all negative powers of the coordinates, leading to non-confined solutions, can never satisfy (2) everywhere, so that M is always curved.
 

This thoroughly changes the familiar picture of flatness we have: reasoning on the metrical coefficients as the «field potentials» of the geometry of M, we consider flatness as constant metrical coefficients.

But the deformation y(x) is more fundamental than the metric. And variable deformations of the form (3) all lead to flatness, despite the metric is variable. This means that these variations actually do not contribute to the curvature of M. In the opposite, they make it more and more rigid.
 

This mechanism might explain how a stochastic geometry of space-time induced by a genuine vacuum could lead to a flat geometry afterwards. The stress tensor field T_ij(x) can always be defined as the invariant of the elasticity tensor field T_ijkl(x) of M:
 

(4)     T_jl(x) = T_ijkl(x)g^ik(x)

If we want the Einstein equations to hold in M, we must connect this elasticity tensor field to the curvature tensor of M, with the same symmetry properties. This is done through a linear combination of R_ijkl(x) and its invariant, the Ricci curvature tensor R_ij(x) and the Gauss scalar curvature R(x). Now, if R_ijkl(x) is to vanish everywhere on flat space-time, the stress tensor field T_ij(x) must equally vanish. This confirms the fact that a plane space-time is necessarily empty of any kind of matter. But, more fundamentally, T_ijkl(x) vanishes everywhere, so that all elasticity coefficients of M are identically zero.

Limiting ourselves to N = 2, which gives a harmonic oscillator, we can reduce (3) to the canonical form and see no coordinate transformation can globally eliminate the deformation: there is a deformation of M, and a variable strain tensor (the metrical one), but it’s induced by no matter, leads to no curvature and gives no information at all on the elastic properties of M.
 

I’d like to end this short paper making a remark about Einstein’s equations. We use to say that they only give informations on the local geometry of space-time. I disagree. Einstein’s formalism is based on the Riemann axiom. Locally, i.e. in the immediate neighbourhood of any point of M, we can always find a coordinate system in which the Christoffel symbols will vanish. However, their first derivatives won’t vanish simultaneously. First of all, it’s therefore a question of reasoning on fixed points of M (despite it has also been shown that this holds on geodesic paths) and, in any of such neighbourhoods, the Riemann curvature tensor is linear and so are its invariants. Consequently, Einstein’s equations are locally linear. So, if we deal with the non-linear set of equations, it automatically refers to the global geometry of space-time, since only globally can’t the Christoffel symbols be eliminated.
 

Another feature surprises me too.
 

Einstein’s reasoning is based on Newton’s Equivalence Principle. From the equality between the inert mass and the «heavy» mass, he deduced that gravity behaved like a «pseudo-force», a «force of inertia», according to D’Alembert’s terminology, and this led him to bring gravity back to a mere effect of the geometry of space-time. The argument was that we could always find a local reference frame in which «we no longer feel the force of gravity», orientating toward the Riemann axiom of geometry. Kaluza and al went even much further, turning all physical forces into geometrical effects.

This really surprises me since, when you feel gravity («feet on the ground»), whatever the coordinate system you choose, you keep on feeling gravity, or you would «loose yourself in space»...

Now, if you have a force opposing gravity, as soon as this force has equal magnitude than that of gravity, you obviously «levitate». Whatever the coordinate system you choose...

It’s not a question of coordinate systems, but of equilibrium: if the system of forces is in equilibrium, if there’s a force exactly compensating for gravity, you no longer feel it.

Because the point is, if you all bring back to mere geometrical effects, you end in doing «pseudo-physics», with «pseudo-fields» and «pseudo-forces»... everything resulting from inertia...

But, if gravity was a «pseudo-force» then, according to Einstein’s equations, the «matter field» would necessarily be «pseudo» as well.
Well, that’s precisely the idea underlying supersymmetry, with its Kähler geometry!!!
Turning all real physics into «pseudo»... fundamental interactions as curvatures of space-time, fundamental matter as torsion of space-time...
 

So, what are we to do? (3) above shows without ambiguity that deformations of space-time have nothing to do with coordinate transformations, nor with any matter... The presence of multiple-order deformations in flat space-time is only due to its non-Euclidian nature. Nothing else.

If we are to review the context of GR, we have to come back on the concept of motion in a curved space-time.

We started from the principle that a matter field, whatever its nature, causes local deformations in the space-time structure. A point x of space-time is therefore sent onto a point x'(x) = x + y(x), according to deformation theory. That point x is now fixed, it's not mobile anymore. If dx is a small variation around this fixed point x, the corresponding variation around x'(x) will be dx'(x) = dx + dy(x) and the metrical tensor of curved space-time will be expressed in terms of the derivatives of y(x) along all four directions of space-time.

But this has nothing to do with the motion of an incident point-like body in a gravitational field. This now refers to the elastic property of space-time under an "external" constraint, namely, the presence of matter within the frame, or even "intrinsically", in the absence of any «perturbation».

This is a radically different approach of the problem of GR. The equations of motion for material bodies are contained in the conservation law for the energy-momentum tensor of the matter field:
 

(1)     DⁱT_ij(x) = 0

where D is the Levi-Civita covariant derivative associated with the curved metric. They have nothing to do anymore with the geodesic equations:
 

(2)     Duⁱ/ds = 0
 

in curved space-time.

In the absence of deformations, x’(x) = x, space-time is plane and the metrical tensor is Minkowski’s. The dual of xⁱ = (ct,x) is x_i = g⁽⁰⁾_ijx^j = (ct,-x) and it’s everywhere the same.

In the presence of deformations, i can build my curved surface element two equivalent ways: whether keeping Minkowski’s metrical tensor g⁽⁰⁾_ij and using coordinates x’ⁱ(x) or keeping my genuine coordinates xⁱ and introducing a variable metrical tensor g_ij(x),
 

(3)     ds²(x) = g⁽⁰⁾_ijdx’ⁱ(x)dx’^j(x) = g_ij(x)dxⁱdx^j

Developing the first expression gives me:
 

g⁽⁰⁾_ij[dxⁱ + dyⁱ(x)][dx^j + dy^j(x)] = g⁽⁰⁾_ij[dxⁱdx^j + dxⁱdy^j(x) + dx^jdyⁱ(x) + dyⁱ(x)dy^j(x)]

= g⁽⁰⁾_ij[dxⁱdx^j + dxⁱ(dy^j/dx^k)(x)dx^k + dx^j(dy^i/dx^k)(x)dx^k + (dyⁱ/dx^k)(x)(dy^j/dx^l)(x)dx^kdx^l]

= g⁽⁰⁾_ij[dⁱ_kd^j_l + dⁱ_l(dy^j/dx^k)(x) + d^j_l(dyⁱ/dx^k)(x) + (dyⁱ/dx^k)(x)(dy^j/dx^l)(x)]dx^kdx^l

so that
 

(4)     g_kl(x) = g⁽⁰⁾_kl + 2g⁽⁰⁾_j^l(dy^j/dx^k)(x) + g⁽⁰⁾_ij(dyⁱ/dx^k)(x)(dy^j/dx^l)(x)

Careful: this is not a usual coordinate transformation, as in the Einstein-Grossmann context, but a true local disturbance of the Minkowski metric!
 

The unit tangent vector is defined as the ratio of the variation dx’(x) around x’(x) on the variation ds(x):
 

(5)     uⁱ(x) = dx’ⁱ(x)/ds(x) = dxⁱ/ds(x) + dyⁱ(x)/ds(x)

when y(x) is everywhere zero or even constant, which corresponds to a mere displacement in space and time, g_kl(x) = g⁽⁰⁾_kl, ds²(x) = ds(0)² = g⁽⁰⁾_ijdxⁱdx^j and uⁱ(x) = dxⁱ/ds(x) is the unit tangent vector defined in the neighbourhood of the fixed point x.

You’ll have noticed that this unit vector is no longer a function of the curvilinear distance s, as in the problem of motion, but a field over points of space-time. There’s no motion of any material objects there, substantial or not. That kind of motion is in (1). The «motion» (5) stands for deformations of the frame. It can happen with or without matter. However, as before, equations (1) tell us that a given matter distribution will locally affect the flatness of space-time, which in turn, will affect the matter distribution, and so on. Consequently, the motion of any point-like body will be affected by these local deformations.
But the dynamics of fields as well.
 

It’s still possible to link uⁱ(x) to a velocity. But this will now be a velocity field, first, and it will relate to the speed at which space-time deforms. The velocity of point-like bodies stands in the energy-momentum tensor. It’s like making the difference between the velocity of electrical charges and the phase velocity of the electromagnetic field produced. The geodesic equations:
 

(6)     Duⁱ(x)/ds(x) = duⁱ(x)/ds(x) + Cⁱ_jk(x)u^j(x)dx’^k(x)/ds(x) = duⁱ(x)/ds(x) + Cⁱ_jk(x)u^j(x)u^k(x) = 0

now serve determining the smallest deformations and can give indications on the «accelerations» of deformations.
 

This autonomous dynamics is obviously independent from any mass. It has nothing to do anymore with the Equivalence Principle asserting that the «　weighting mass　» equals the «　inert mass　», a principle that concerns gravitation, not space-time elasticity...

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.