Today, two things: we’re first going to talk about observability. New insights about it, thanks to the enlarged frame. Then, a surprise. I’ll let you discover about it in the next paper.

 

We first rename our frames, for more convenience. We’ll call M_4- the usual 4D Minkowski flat space-time with coordinates xⁱ (in meters) M₄₊ what we called the “gravitational space-time”, that is, a second 4D Minkowskian space-time with coordinates Gⁱ (in meters per seconds) and M₈ = M₄₊xM_4- the Euclidian product of it, which makes the enlarged 8D physical frame. We can no longer say M₈ is Minkowskian: strictly speaking, a Minkowski space-time is a space-time with a single time-like dimension. We can only say M₈ is either “non-Euclidian”, “pseudo-Euclidian” or “hyperbolic”.

What we showed in the 3 last bidouilles (103, 104 and 105) suggests us to take a 8D flat metric on M₈ of the following form:

 

(1)               ds₈² = ds₄²[1 – W²(x)/f_pl²]

(2)               W²(x) = W_ij(x)W^ij(x) = W²(x)/c² - W²(x)

(3)               ds₄² = c²dt²[1 – v²(t)/c²]

 

The field W(x) is the “electric-like” part of the gravitational field on M₄₊, W(x) is its “magnetic-like” part.

Before going further, a little remark about physical dimensions. When we say M₄₊ is “over” M_4-, we have to precise what we mean. Mathematically speaking, none of the dimensions we use are “over” others: in any reference frame where we choose a given point as our origin, all the axes start from that point, so that all directions are actually “at the same level”. We can choose to use all of them or only a part of them. For instance, in a 3D reference frame, we draw rectangular axes x, y and z, all “at the same level”. They simply indicate that, in such a space, we can move along any or all of the dimensions available: along axes x, y or z; along planes (x,y), (y,z) or (z,x) or in the volume (x,y,z). The same holds for an arbitrary number of dimensions, with only more possible combinations.

Physically speaking, we’re allowed to say that M₄₊ is “over” M_4- provided we justify the physical conditions about this. In our present purpose, the central condition is: to reach the 4 next physical dimensions, we need our velocity v to be greater than c, at least, to an observer’s eyes who stayed in M_4-. Now, if we assume the world to be larger than only 4 dimensions, it becomes obvious that everything inside such a world is imbedded in the 8 dimensions. That’s what explains why we could discover the quantum behaviours: as we saw in B103, what we called the “PSI mass” is proportional to the mass density of a wavepacket in M_4-. Not all wavepackets are quantum. But the ones we considered precisely are, since they represent a probability amplitude of presence and a wavy extension of a point-like rigid body.

What we showed in B103 was nothing else than “any wavy object in M_4- corresponds to a PSI object in M₄₊”. If the wavepacket is material, the corresponding PSI object will be material (but obviously not substantial); if the wavepacket is radiative (boson-like), the corresponding PSI object will be radiative.

The metric (1) expresses what is observable and what is not in M_4-, M₄₊ and M₈.

To be observable in M_4-, we need to be causal: ds₄² > 0. This is done for 0 £ v £ c. The value v = c is an upper limit to observability in M_4-. It means that “ordinary” waves propagates at c in the vacuum and, beyond c, one finds no “ordinary” signals: the only components of a signal propagating at v > c are its nodes and since they have zero amplitude, they carry no information at all.

The same happens in M₄₊: we are causal for W² £ f_pl². The value f_pl is the upper limit. What this now means is that “PSI waves” propagate at f_pl in the vacuum of M₄₊, as any other “PSI signal” and, beyond f_pl, one finds no PSI signal anymore. No more PSI information of any kind. This actually corresponds to what is assumed to have occurred at the birth of the Universe, according to the Standard Model: frequencies f > f_pl correspond to temperatures T > T_pl and pressures p > p_pl. In this earliest age of the Universe, gravity was so gigantic that nothing could stand, no matter, no radiation, absolutely nothing, only a quantum vacuum. Now, we said that PSI signals had a quantum origin in the microscopic (in the macroscopic, we find more general wavepackets). We now see that, below the Planck time, that is, from t = 0 to t = t_pl, there was no ordinary signal, nor even PSI ones, i.e. nothing even quantum. Only fluctuations of the primordial vacuum. A t = t_pl, the PSI emerged from this vacuum and, with it, elementary quantum wavepackets.

According to (1), we are observable in all M₈ for ds₈² > 0, that is for (v < c, |W| < f_pl) or for (v > c, |W| > f_pl). The first situation corresponds to current processes: we’re observable in both M_4- (v < c) and M₄₊ (|W| < f_pl). The second situation corresponds to “before the Planck age”: the Universe is then observable, but there’s nothing concrete to observe… (and there’s no observer yet, anyway! J).

We loose observability in M₈ for ds₈² < 0, i.e. (v > c, |W| < f_pl) and for (v < c, |W| > f_pl). The first situation corresponds to no ordinary signal in M_4- but PSI signals in M₄₊; the second situation, to ordinary signals in M_4- but no PSI signals in M₄₊.

We see our metric (1-3) is inclusive: if any over the two 4D space-times is out of observation, the whole 8D process is out of observation.

 

TO AN M_4- BIOLOGICAL OBSERVER’S EYES, (v > c, |W| < f_pl) CORRESPONDS TO THE END OF THE TUNNEL (B89): THE WAVY BODY REACHES THE GREAT WHITE LIGHT AT v = c, THEN GETS OUT OF OBSERVATION. BUT, AS A PSI BODY, ITS REAL SPEED IS NO LONGER v, BUT W (B105). AS A CONSEQUENCE, IT REMAINS OBSERVABLE IN M₄₊. WE LOOSE OBSERVABILITY IN M₈ BECAUSE WE LOOSE IT IN M_4-.

 

If there was no Tunnel, we would be able to observe (with the help of suitable instruments) a wavepacket with group velocity v_gr < c, accelerating up to v = c, then disappearing. Unfortunately, without the help of a convenient structure, relativity in M_4- prevents us from reaching c within a finite relative time. So, such a process would take “an infinity” (while the proper time attached to the wavepacket could remain finite).

Besides, we can compute the intensities and the source of a “limit” G-field verifying G_iGⁱ = c² (B89). If we made no mistake, we have W = 0, so that m = 0, that is, a constant G-wave. However, even waves in M_4- have a PSI mass. That of a G-wave is:

 

(4)               M_G(x) = (2p/k)G²(x)

 

and that of an electromagnetic wave,

 

(5)               M_EM(x) = (8p²e₀)A²

 

which can always be transformed back into a M_G(x) through the correspondence A² = G²/(4pke₀). Only when G² = 0, i.e. G_i is light-like, which is not the case of a Tunnel, is M_G(x) = 0. For a Tunnel:

 

(6)               M_G(x) = 2pc²/k = 2pm_pl/R_pl          (Tunnel)

 

is a constant. When G_i is time-like, M_G(x) > 0; it’s < 0 when G_i is space-like.

There’s a difference between the quantum mass and the PSI mass for massless fields: their quantum mass is zero, while their PSI mass is given by the general formula (10), B103, using the Lagrangian density of the massless free field in M_4-.

It means that massless fields in M_4- also become, in turn, PSI sources for PSI fields in M₄₊ and this is perfectly logical, since a massless field like G_i(x) or A_i(x) is its own quantum wavepacket. These are two examples of PSI bodies being non material: G_i(x) is made of gravitons, A_i(x) of photons.

 

ANY FORM OF LIGHT IN M_4- SENDS BACK TO A NON-MATERIAL PSI BODY IN M₄₊ AND THUS, TO A CORRESPONDING PSI SOURCE THERE.

 

 

 

Wow, wow, wow! Smells better and better, that stuff!... J

Well, let’s now get back to good old classical relativistic mechanics, but in G-space-time. How can we do it? Just as in M₄, we need a “speed limit”, a velocity over which events are no longer observable in higher dimensions. Why? For three reasons: first, to be able to introduce the notion of “PSI waves” and propagation in higher dimensions; second, to obtain finer results in a theory of “weak PSI fields” and third, to have a suitable frame for strong PSI fields.

In B104, we introduced such a universal constraint: f_pl, the Planck frequency. It plays the role of c in G-space-time. But now, we demand:

 

(1)               0 £ W_ijW^ij = W² £ f_pl²

 

to be the causal condition in that frame. The Lorentz-like correction factor is now:

 

(2)               (1 – W²/f_pl²)^1/2 » 1 – W²/2f_pl² + O(W⁴)

 

in the weak field approximation. Seen the enormous value of f_pl compared to c (= 2,99793 x 10⁸ m/s), the range for observable motions in G-space-time is rather large…

The rest consists in copying relativistic mechanics. We start from a Lagrangian density for free motion:

 

(3)               £ = -M(x)f_pl²(1 – W²/f_pl²)^1/2 » -M(x)f_pl² + ½ M(x)W_ijW^ij – O(W⁴)  in J/m³

 

showing the quadratic term ½ M(x)W_ijW^ij as a “Galilean motion” in G-space-time for weak gravitational fields in M₄ (W² << f_pl²). M(x)f_pl² is the “PSI energy at rest” of a PSI body in G-space-time. It behaves as an energy density in M₄.

 

(4)               P_ij = M(x)W_ij/(1 – W²/f_pl²)^1/2 = -P_ji

 

is its “PSI-momentum” tensor, in Js/m³. It behaves as a density of action in M₄. The corresponding PSI energy tensor is:

 

(5)               T_ij = ½ (P_ikW_j^k + P_jkW_i^k) – ¼ g_ij£ = T_ji

 

It has scalar invariant:

 

(6)               H = T_iⁱ = M(x)f_pl²/(1 – W²/f_pl²)^1/2

 

and behaves as an energy density in M₄, as requested. For weak fields:

 

(7)               H » M(x)f_pl² + ½ M(x)W_ijW^ij + O(W⁴)

 

We have the identity:

 

(8)               P_ijP^ijf_pl² + H₀² = H²

 

with:

 

(9)               H₀ = M(x)f_pl²

 

Action can be defined through a differential relation:

 

(10)           dS_i = T_ijdx^j - P_ijdG^j

 

or, for constant T_ij and P_ij, through the algebraic relation:

 

(11)           S_i = T_ijx^j - P_ijG^j

 

This is the case for motions with “definite PSI energy and momentum”. In G-space-time, “PSI action” is a 4-vector, due to the presence of four dynamical parameters (the 4 xⁱs). So, we have 4 components of a PSI action, one along each of the dimensions of M₄. Dimensional analysis shows that these S_is are in J/m². They behave like the strength coefficient of a spring in M₄. Finally:

 

(12)           I = ò S_idxⁱ

 

is PSI inertia. It’s measured in J/m = N:

 

PSI INERTIA IN G-SPACE-TIME BEHAVES LIKE A FORCE IN M₄.

CONVERSELY, A FORCE IN M₄ CAN SEND BACK TO AN INERTIA IN G-SPACE-TIME.

 

It’s legitimate to ask ourselves what kind of force in M₄ could correspond to a PSI inertia in G-space-time. To get an idea of it, let’s make the following hypothesis:

 

1)      Reference frame at rest: W_ij = 0 <=> G_i(x) = ¶_iD(x) in M₄, with D(x) in m²/s behaving like a scattering coefficient in M₄;

2)      Constant wavy density: r(x) = r₀. This seems to fit with the wavy body in stage IV.

 

Hypothesis 1 gives:

 

P_ij = 0,

T_ij = ¼ g_ijM(x)f_pl² = [(m_plc)²/12m]r₀g_ij = ctes

S_i = T_ijx^j

I = ½ T_ijxⁱx^j = [(m_plc)²/24m]r₀xⁱx_i

 

As we can see, I is proportional to the square of the space-time length in M₄. This is typical of a pullback force. Inertial confinement.

Clearly, PSI inertia behaves in M₄ like a “spring”: the more distant you are from the centre xⁱx_i = 0 (the more stretched the spring), the stronger the force to bring you back.

 

REGARDING M₄, HOMOGENEOUS PSI MATTER AT REST (IN “HIGHER DIMENSIONS”) LOOKS “CONFINED” INSIDE A TYPICAL RADIUS OF [24m/(m_plc)²r₀]^1/2 » 0,2995078(m/r₀)^1/2, VERY CLOSE TO (1/3)(m/r₀)^1/2.

 

Numerical applications.

 

For m = 70 kgs, we find I = 0,1592522 r₀xⁱx_i Newtons;

For m = m_cell = m_pl and r₀ = 10¹¹ silencious neurons, I = 2,0431849 x 10¹⁹ xⁱx_i Newtons, typical M₄ radius = 7,3864876 A.

 

Crystal-clear: this time, the results are significant even in the macroscopic. This is mainly due to the value of the Planck mass (around 5 x 10^-8 kg). So, in the first situation, for a wave density of 1 at 1 m, we still find 0,16 N: far from being negligible, but rather weak; in the second situation, even at the Angström scale xⁱx_i = 10^-20 m², we still find 0,2 N: something like “asymptotic freedom in QCD” appears, but the strength is not negligible yet.

 

This is all very natural: inertia is a property of matter. Not to be confused with the force of inertia, which opposes motion in the D’Alembert Principle. What is particularly interesting here is that this property of PSI matter gives birth to a physical interaction that is interpreted as a pullback force by a biological observer of M₄.

The result is striking in the second situation: the base element is the neuron cell; we’ve assumed that the totality of these neurons were silencious at stage IV (that’s about 100 billions in the human nervous system – 60 billions in the brain + 40 billions in the rest). Not only is the “pullback force” still important at the atomic scale, but the biological observer would find a typical “range” of a bit more than 7A only. He will so deduce that “PSI matter is confined inside the atomic scale at best”.

 

Well, what we proved here is that all this good observation and deduction, that seem so natural and logical to us… is spoiled by our dimensional restriction: WE are “confined” into the first 4 dimensions of the universe and make wrong deductions on things happening “over or beyond our space-time” precisely because of this.

The result we got for m = m_cell and r₀ = 10¹¹ are correct and reasonable, they’re not many orders of magnitude greater or smaller than expected, like the ones I previously obtained, yet they have nothing to do with reality…

They only have to do with the reality we can perceive.

“PSI reality” in G-space-time reduces to say: “we have a PSI inertia I = 2,0431849 x 10¹⁹ xⁱx_i and it has to be measured in ‘kg-psi times (length-psi)²’, which is equivalent to the Newton in M₄”. But there’s no “force” of any kind there.

Force is G-space-time is measured in (kgm/s)/m³, it’s a momentum density in M₄.

 

We interpret as a “force” what is not a force. And there is our mistake! J

 

 

Thus, this “G-space-time” seems to be promising. Here’s now another and rather striking confirmation.

We have two different measures of length depending on which space we are working in: in M₄, we have meters; in G-space-time, we have meters per second. Just as in classical 4D relativity, it would be nice, as a synthesis, to work with a single unit. We can choose m/s or stay with meters. If we keep this last option, we can define additional coordinates through:

 

(1)               xⁱ⁺⁴ = Cte x Gⁱ

 

Obviously, for our new xⁱ⁺⁴ to be in meters, we need Cte to be in seconds. Now, as c, this constant must be universal. The only constant of this kind we know so far that is compatible with the Standard Model is the Planck time:

 

(2)               t_pl = R_pl/c = (hk/c⁵)^1/2 = 1,3512808 x 10^-43 s

 

It’s the only quantity solely made of universal constants. Reversing (1) would give:

 

(3)               Gⁱ = f_plxⁱ⁺⁴ , f_pl = 1/t_pl = 7,4003864 x 10⁴² Hz

 

We’ll find a much wider use of that Planck frequency later on. For the time being, we investigated the limit case GⁱG_i = c² in B89. If we stick to this physical limitation in G-space-time, something that seems quite legitimate after all, being still about velocities, then the condition:

 

(4)               GⁱG_i £ c²

 

in G-space-time, this time, that is, on 4D lengths in that frame, together with:

 

(5)               xⁱ⁺⁴ = t_plGⁱ

 

implies that:

 

(6)               xⁱ⁺⁴x_i+4 £ c²t_pl² = R_pl²

 

Makes you remind of something?...

 

THE OSCAR KLEIN HYPOTHESIS.

WE RETRIEVE THE OSCAR KLEIN HYPOTHESIS ON ADDITIONAL DIMENSIONS OF SPACE-TIME.

 

What inequality (6) means is that, is we want to stay causal, i.e. observable, in higher dimensions of space-time, we need 0 £ xⁱ⁺⁴x_i+4. But (6) now requests that xⁱ⁺⁴x_i+4 £ R_pl². So,

 

FROM WHAT AN OBSERVER OF M₄ CAN PERCEIVE,

THE 4 HIGHER DIMENSIONS OF SPACE-TIME, EVALUATED IN METERS, CAN ONLY STAY CONFINED INSIDE THE PLANCK RADIUS IF TO REMAIN CAUSAL.

 

Actually, what is likely to happen?

Simply, (6) shows meter is no longer the appropriate unit to measure distances in the higher dimensions. If we use (3) instead, the only constraint we have is (4), which is interpreted as a kinematical constraint in M₄. No confinement anymore.

 

DIMENSIONAL CONFINEMENT IS AN ARTEFACT APPEARING WHEN WE, BIOLOGICAL OBSERVERS OF M₄, TRY TO KEEP ON MEASURING PHYSICAL OBJECTS AND EVENTS IN THE HIGHER DIMENSIONS WITH THE SAME SYSTEM OF UNITS WE USE IN THE FIRST 4 DIMENSIONS.

 

I’m particularly satisfied with this result, totally unexpected, since I’ve criticized the Klein hypothesis from the very beginning, arguing, as many others, that there was no justification at all for such a choice. Indeed, it was nothing else than getting rid of these additional dimensions that jeopardized the behaviour of fields in the macroscopic.

We now have a justification and even an explanation.

The hypothesis is justified to an M₄-observer’s eyes.

But G-space-time enables us to explain why it’s actually a mere artefact, that disappears in higher dimensions.

 

 

YES-WE-CAN. We can retrieve quantum field theory and quite easily. Here’s how.

We take advantage of the fact that M(x) is a density of inertia to search it under the form:

 

(1)               M(x) = Iy*(x)y(x) = Ir(x)

 

where I is an inertia (Js² = kgm²). In the bosonic model, we get this:

 

(2)               ¶ⁱ¶_iM(x) = I(y*¶ⁱ¶_iy + y¶ⁱ¶_iy* + 2¶ⁱy¶_iy*)(x) = (4mI/ħ²)£_y

 

where £_y is the Lagrangian density for the free field y:

 

(3)               £_y = (ħ²/2m)¶ⁱy¶_iy* - ½ mc²yy*

 

To establish (2), I made use of the field equation:

 

(4)               ¶ⁱ¶_iy = -(mc/ħ)²y

 

derived from (3). We get a better idea of the thing with y = r^1/2e^iq. The field equation (4) and the Lagrangian (3) become the system:

 

(5)               ¶ⁱr¶_iq + r¶ⁱ¶_iq = 0

(6)               (1/2r)²¶ⁱr¶_ir - (1/2r)¶ⁱ¶_ir + ¶ⁱq¶_iq = (mc/ħ)²

(7)               £_y = (ħ²/8mr)¶ⁱr¶_ir + (ħ²/2m)[¶ⁱq¶_iq - (mc/ħ)²]r

 

so that, if we use (2):

 

(8)               ¶ⁱ¶_iM(x) = 2Ir[(1/2r)²¶ⁱr¶_ir + ¶ⁱq¶_iq - (mc/ħ)²] = I¶ⁱ¶_ir = (I/m)¶ⁱ¶_im

 

with m(x) = mr(x) the mass density. Integration gives:

 

(9)               M(x) = (I/m)m(x)

 

hence the result:

 

IN THE BOSONIC MODEL, THE “PSI” MASS M(x) IS INDEED LINKED AND PROPORTIONAL TO THE LOCAL MASS m(x) = m|y(x)|² OF A WAVY BODY GIVEN BY QUANTUM FIELD THEORY.

 

We can be led to the very same conclusion in the fermionic case. However, as it is a first-order model, the simplest way to proceed is to start from the general relation:

 

(10)           ¶ⁱ¶_iM(x) = £_y/c²

 

holding in both situations.

Our constant of inertia I is left free. If we choose:

 

(11)           I = h²/3mc²

 

we obtain:

 

(12)           M(x) = 1/3 (h/mc)²m(x)

 

This choice is motivated by what follows. We first have the identities:

 

(13)           m_pl² = hc/k , R_pl² = hk/c³ , c²/k = m_pl/R_pl , hc = m_plR_pl

 

These identities show very useful in the (important) case of:

 

(14)           M(x) = cte = c²/4pk

 

the coupling constant of gravity (in M₄). We then get:

 

(15)           m(x) = cte = m²/m_plV_pl = (m/m_pl)²m_pl, V_pl = (4p/3)R_pl³

 

and when m = m_pl,

 

(16)           m(x) = m_pl = m_pl/V_pl

 

that is, precisely, the Planck density for a spherical Planck particle.

 

What does all this mean?

It shows that the body we search for at stage IV is confirmed to be purely wavy, with mass density in M₄ given by RQFT (Relativistic Quantum Field Theory). This mass distribution m(x) = mr(x) then indicates the way wavy matter distributes in space and time. Notice that, opposite to classical mechanics, the time dependence is no longer contained into m, but reported into the number r(x) of elements involved, per unit volume.

Following that, what we showed here is that m(x) is actually proportional to what we defined in B102 as the “PSI mass” (to stick to our context), that is, the mass of a physical body, no longer in M₄, but in the higher dimensions of a gravitational space-time. The calculation from (11) to (16) above is interesting in itself, but its aim is to prove that we retrieve all the properties of the original Planck particle, the existence of which is intimately associated with the birth of our Universe. It was necessary to show this in order to consolidate our reasoning.

 

Just a tiny correction before we go on: in the bosonic vacuum state, the source does not necessarily vanish, since its phase can still vary in space-time.

Okay. We’ve already established something: that “PSI” bodies, if existing, can only be made of wavepackets. No substance of any kind anywhere.

Next question: under such conditions, what could be the physical frame inside which wavy bodies could form and evolve?

Forget about space-time. We have to find something else.

We identified PSI matter: organized systems of material wavepackets. It’s already something.

Ordinary matter being distributed in 3-space, PSI matter can only distribute in a space of wavepackets. If it distributed in 3-space as well, it would be observable.

But a “space of wavepackets” is a functional space, a mathematical space. If we think in terms of wavepackets, our physical frame remains ordinary space-time M₄. We can do mechanics, quantum field theory and thermodynamics inside it, but nothing “PSI”: in M₄, PSI is desperately trivial and leads to nothing interesting at all.

Now, we know that an expression like x(t) means “a motion of a substantial body in ordinary 3-space E₃ (x-space) in time”.

So, we deduce from this that y(x,t) means “a motion of a wavy body in a space of wavepackets (y-space) in space and time, that is, ‘over’ M₄”.

We’re sent back to a new physical frame.

And, as this space can only stand “over” M₄, we have no other choice than to introduce new physical dimensions.

Let it be. What kind of dimensions wouldn’t change well-established behaviours in both classical and quantum dynamics?

We have 3 space dimensions measuring lengths and distances, unit meter; one time dimension measuring… time, unit second. So, we already have no problem with using different physical dimensions expressed in different physical units.

Let’s introduce new physical dimensions with still other physical units. No distances, no time, something else. y-units, m^-3/2? No: first, we need something the most general possible; second, we need something “in a vacuum”, i.e. outside of any matter. y-coordinates are made for PSI bodies (material waves), they can help localize PSI matter inside this new frame. But, as in M₄, space-time is defined independent of any ordinary matter. That’s why we can’t use y.

The best candidate I found is the gravitational field in M₄. It’s the most general, as any other interaction can express with it [example: A_i(x) = (q/m)G_i(x)]. It’s spin 1 and therefore a wavefunction for quantum theory. This G-field G_i(x) sends us back to a wave space with coordinates Gⁱ, units m/s. A kind of “potential velocity”.

If we do this, then we get 4 new physical dimensions Gⁱ, to add to the four xⁱ. Or to the 3 x and the single t.

A “motion” in this “gravitational space-time” is a gravitational field Gⁱ(x): that’s what a biological observer in M₄ can observe. If you go from a point G₁ⁱ to another G₂ⁱ, you’re on a trajectory Gⁱ(x) such that Gⁱ(x₁) = G₁ⁱ and Gⁱ(x₂) = G₂ⁱ: your M₄-observer will see a given G-field taking two different values at two different points of space-time.

What we now say is that such an observation sends us back to a motion in G-space.

Do I face difficulties with physical units in this G-space? Not really:

 

-         “distances” and “lengths” are measured in m/s, they are observed as velocities in M₄;

-         “time” is now 4D-space-time, in m;

-         “velocities” ¶_iG_j(x) to replace dx(t)/dt: Hz, frequencies in M₄;

-         “accelerations” ¶_i¶_jG_k(x), in Hz/m;

-         “energies”? Given by the field Lagrangian in M₄, £_G = (c²/8pk)W_ij(x)W^ij(x), with W_ij(x) “velocities”: £_G in J/m³. So “energies” are energy densities in M₄;

-         “masses”? Consequence of “energies”: the coupling constant c²/8pk is measured in Js²/m³ in M₄, hence a density of inertia; “masses” in G-space are only generalizations of this factor, so that £_G reproduces Galilean motion in G-space, £_G = ½ M(x)W_ij(x)W^ij(x); we deduce M(x) in Js²/m³;

-         “forces”? Deduced from the Maxwell-like potential part of £_G this time: p_i(x)Gⁱ(x) in J/m³ => p_i(x) in (kgm/s)/m³: densities of momentum and energy.

 

Do I have to fear modifications in the behaviour of fields in M₄? Obviously, no, since I’m now gonna work with “PSI fields” Y(G,x) in the total 8D frame. First terms near Gⁱ = 0, corresponding to no gravity or a weak one in M₄ gives me:

 

(1)               Y(G,x) = Y₀(x) + Y_1i(x)Gⁱ + ½ Y_2ij(x)GⁱG^j +…

 

Zeroth-order term Y₀(x) = field in M₄. For other M₄-observable contributions, just consider the functional Y[G(x),x]. Y can, for instance, express a PSI body, than Y₀(x) is its material wavepacket in M₄. It can be statistical or not. If we completely neglect gravity, Y will reduce to Y₀ in this (rough) approximation.

I’m now working on functionals like Y[G(x),x] when Y is in m^-3/2, to see if we properly retrieve quantum field theory in M₄. Answer next time.

 

NB: a G-space over M₄ will mean that a biological observer “imbedded in these higher dimensions” would see everything around him “plunged into a gravitational light”. That’s the way he would perceive the vacuum there, with his 4D senses.