Well, the universal rule unfortunately is: each time it’s too easy, you look elsewhere…

I was far from thinking to such an easy solution to eq. (13):

 

(1)               W²(x) = -6c²/x²  i.e.  ¶ⁱ¶_i[W²(x)] = 0

 

As simple as that… It follows that, when x² > 0 (time-like), W² < 0 (Coriolis-like – magnetic type) and when x² < 0, W² > 0 (Newton-like, electric type). Moreover, W² < f_pl² implies that x² > -6R_pl².

(1) also gives the solution for the more general equation (3), ¶ⁱ¶_iF(x) = -(K/6)F³(x), F(x) = [1 – W²(x)/f_pl²]^1/2, when K = cte. One finds:

 

(2)               F²(x) = 6/Kx²  ,  W²(x) = f_pl²(1 – 6/Kx²)

 

Then the condition W² < f_pl² leads to x² > 0 only. For x² >> 6/K, which can be quickly realized for 1/K = R_pl², a first-order approximation gives W_ij(x) » ½ f_pl(1 – 3/Kx²)e_ij that is, G_i(x) » (3f_pl/2Kx)e_i, with x = (x²)^1/2 and e_i a vielbein. Such a solution still converges at space and even space-time infinity, but much slower than a Maxwellian one (in 1/x²). Now, for a large class of G-fields we observe in M_4-, the (1/x²) law is much more often verified than this law in 1/x. This seems to indicate that the condition K = cte does not hold very long. Maybe around scales of distances of order R_pl, near the birth of the Universe.

Indeed, I’ve made the calculations for a Maxwllian potential:

 

(3)               G_i(x) = -c(R_pl²/x²)u_i

 

assuming the u_is do not depend on the space parameters x (the rigid bodies model). The result is:

 

(4)               K(x) = (48R_pl⁶/x²⁰)[11x¹² – 8x¹⁰(x_iuⁱ)² - 16R_pl⁶x⁶ + 16R_pl⁶x⁴(x_iuⁱ)²]

 

or, under another form:

 

(5)               K(x) = (48R_pl⁶/x⁸)[11 – 8(x_iuⁱ)²/x² - 16R_pl⁶/x⁶ + 16R_pl⁶(x_iuⁱ)²/x⁸]

 

showing that the scalar curvature of the conformal 4D space-time strongly depends on the observation point xⁱ. For xⁱ = R_pluⁱ, one obtains:

 

(6)               K(R_plu) = (12/R_pl)²

 

However, as uⁱ(s) = dxⁱ(s)/ds, this relation on the xⁱs corresponds to a motion:

 

(7)               xⁱ(s) = xⁱ(0)exp(s/R_pl)

 

that is, an exponential growth of space and time distances in M_4- where s > 0, an exponential dump where s < 0 and oscillation modes when s² < 0.

None of these conclusions actually contradicts the Standard Model. Good news.

 

But a much better news now comes from what follows.

We want to connect the conformal metric ds₈² = [1 – W²(x)/f_pl²]ds_4-² to a true 8D metric with constant coefficients. This way, we will have a plane 8D space-time (in the classical vacuum) and, when it comes to a motion in M₄₊ only will we retrieve the conformal 4D space-time a biological observer perceives. We have to do this to make sure our model is compatible with a larger 8D frame.

Well, not only is it the case, but what I obtained was, once more, far from being expected.

We write the conformal deformation of flat Minkowski M_4- as:

 

(8)               ds₄₊²(x) = [W²(x)/f_pl²]ds_4-²(t) = [t_plW_ij(x)dx^k(t)][t_plW^ij(x)dx_k(t)]

 

The factorization introduces 3-tensor coordinates l_ij^k locally defined as:

 

(9)               dl_ij^k(x) = t_plW_ij(x)dx^k(t) = -dl_ji^k(x)

 

(we don’t need to precise the t-dependence of these coordinates, as it’s already contained in the xⁱ variables). A straightforward connection of the l_ij^ks to the xⁱ⁺⁴s, B104 eq (5) is given by:

 

(10)           l_ij^k = d_i^kx_j+4 - d_j^kx_i+4 = t_pl(d_i^kG_j - d_j^kG_i)

 

We can get rid of the xⁱ-dependence in those relations. Or else, we easily retrieve (9). To conclude:

 

(11)           ds₄₊² = dl_ij^kdl^ij_k = 6dl_idlⁱ = 6t_pl²dG_idGⁱ

(12)           ds₈² = ds_4-² - ds₄₊² = dxⁱdx_i – dlⁱdl_i = dxⁱdx_i - 6t_pl²dG_idGⁱ

 

This is a first point, confirming (xⁱ,Gⁱ) is indeed the right coordinates frame. The second one is about velocity. We go back to the form (8) and separate the time-like terms from the space-like ones:

 

(13)           ds₈² = [c²dt² + (t_pl/c)²N²(x)dx² + R_pl²W²(x)dt²] – [dx² + t_pl²N²(x)dt² + t_pl²W²(x)dx²]

 

with N(x) the electric-like part of W_ij(x), in m/s², and W(x) its magnetic-like part, in Hz. We place this result together with (12), giving:

 

(14)           6dl² = [R_pl²W²(x) + (t_pl/c)²N²(x)v²(t)]dt²

(15)           6c²dt² = t_pl²[ N²(x) + W²(x)v²(t)]dt²

 

Notice that dl is now time-like and dt, space-like. This means that our 8D frame is endowed equal time-like and space-like dimensions: 4+4. I like that. J

From (14) and (15), we can calculate the velocity n(t) = dl(t)/dt:

 

(16)           n² = c⁴(W²/N²)(1 + N²v²/c⁴W²)/(1 + W²v²/N²)

 

We first check that n² £ c² does imply W²(1 – v²/c²) ³ 0 and conversely. Then, that the condition W² £ f_pl² implies:

 

(17)           n² ³ c²[t_pl²(N/c)²(1 + v²/c²) – 1]/[t_pl²W²(1 + v²/c²) + 1]

 

And now, the good surprises. To me, they definitely show that we’re in the right direction, working in the suitable frame.

 

(18)           W = 0  =>  n = v

 

IN THE ABSENCE OF THE CORIOLIS TERM, THE INSTANTANEOUS VELOCITY IN E₃₊(t) EQUALS THAT IN E_3-(t) AND SO, EQUALS THE GROUP VELOCITY OF WAVE MECHANICS.

 

This gives a direct connection between v, v_gr and n when W = 0 and one more link with quantum theory.

 

(19)           N = 0  =>  n² = c⁴/v² = v_ph²

 

THIS TIME, IN THE ABSENCE OF THE NEWTON TERM, THE INSTANTANEOUS VELOCITY IN E₃₊(t) EQUALS THE PHASE VELOCITY IN E_3-(t).

 

(20)           N/c = W  i.e.  W² = 0  =>  n² = c²

 

WHEN THE G-FIELD IS LIGHT-LIKE, THE INSTANTANEOUS VELOCITY IN E₃₊(t) EQUALS THAT OF LIGHT IN E_3-(t).

 

We can even show without difficulty that:

 

(21)           G_i = -cu_i  (Tunnel)  =>  W_ij = 0  (for rigid bodies)  => n² = c²

 

Well, I was far from expecting so much. We started from (1), B110, as a possible extension of M_4- giving back the Maxwell Lagrangian for weak G-fields (weak compared to f_pl) and we discover a lot of equalities with quantities usually used in signal theory in M_4-, in many different contexts. Specialized situations.

Excellent. I’m fully satisfied. Seldom enough to be noticed.

We can now seriously expect to move forward. That’s far the most satisfactory model I had.

 

A conscious bio observer would have the feeling that the Tunnel and “Beyond” all happen inside the Planck radius and then would send the experiencer back to the origin of our observable universe.

Esoteric vision of things. THIS is metaphysics.

The experiencer has no felling of all that, he just enter a Tunnel to a Great White Light. “Beyond” looks like a “Garden” to him. He never feels like he’s back to the birth of the universe. Or that he moves into a microscopic world. He only changed space-time. He entered a new frame, where distances and times are measured in different units, in a “new” body. That’s all.

But that’s the beginning of everything for us. J

 

Not a lot to say today, but i did prepare a little something. Complements on the 8D metric ds₈². We built it as a conformal (local) deformation of the Minkowski 4D metric ds_4-² on M_4-:

 

(1)               ds₈²(x) = [1 – W²(x)/f_pl²]ds_4-²

 

The context is the following one: a PSI body with PSI mass M(x) is moving into space-time M₄₊ with “velocity” W_ij(x). For a biological observer in M_4-, such a displacement is perceived, not as a motion, but as a gravity field with intensities W_ij(x). I’d like to insist on the difference between space-time M₄₊ in itself, with its coordinates Gⁱ and motions of PSI objects inside M₄₊: our bio observer has no direct perception of such objects, he only perceives effects of gravity. I’m not saying for as much all gravity fields originate from PSI motions, only that PSI motions behave like gravity fields in M_4-, and still, in the model we are investigating. Assume he made himself to the possibility of the existence of additional dimensions. What he will observe will be something like (1). That is, a G-field acting on flat space-time M_4- to curve it locally and to do it a conformal way.

This is not the Einstein theory of gravity. This is only differential geometry. And differential geometry restricted to the first four dimensions of the Universe we built. For the only independent dynamical variables in (1) are the xⁱs.

So, let’s turn to geometry and tensor calculus. When a PSI body is fixed in M₄₊, the W_ijs are zero over all M_4- and our bio observer observes… no gravity field (of that kind). The metric ds₈² reduces to ds_4-². A PSI motion should translate into local deformations of both space and time in M_4-. Now, how to distinguish them from other possible G-fields, I don’t know. This is not today’s subject anyway.

Since the metrical coefficients of (1) stay symmetric:

 

(2)               c_ij(x) = [1 – W²(x)/f_pl²]g_ij  ,  c^ij(x) = [1 – W²(x)/f_pl²]^-1g^ij

 

we can keep on applying the Riemannian geometry. I’ll preserve you from the details of the calculations, easy but painstaking and I’ll just give the main result, as to know, the scalar curvature of the metric (1):

 

(3)               K(x) = -6F^-3(x)¶ⁱ¶_iF(x)  ,  F(x) = [1 – W²(x)/f_pl²]^1/2

 

It has the merit of being a very simple formula. The inconvenient is that it gives no indication about the G-field. The behaviour of the G-field in M_4- is given by its equations of motion [B105 for the kinetic part of the PSI Lagrangian].

An interesting case is conformal flatness:

 

(4)               K(x) = 0

 

over all M_4- or only a domain of it. Since it would be physically meaningless to set F(x) to infinity, (4) implies:

 

(5)               ¶ⁱ¶_iF(x) = 0

 

To the reader not familiar with local geometry, I’d like to precise that flatness in curved spaces don’t mean at all these spaces are plane: the condition (4) is far from being sufficient. It only means the “total” or “Gaussian” curvature of the space is zero, whether globally or only in a domain, or that the points involved are parabolic. However, the space is curved. In our present context, it’s plane only when W_ij(x) = 0. Still, it does not suppress the G-field, for its potentials G_i(x) remain. W_ij(x) = 0 only means the G-potentials derivate from an even more fundamental potential D(x), in m²/s. When this is so, there is a G-field (as a real force, it cannot be suppressed everywhere in M_4-), but it has no effect on the motion of ordinary bodies in M_4-.

There’s something rather interesting in the most general solution of (5) that we will take over all M_4-, to avoid the delicate question of bounded values. The Newtonian form is not the only possible one. We actually have:

 

(6)               F(x) = F_-1R_pl²/x² + F₀ + (F₁/R_pl)x_iuⁱ + (F₂/2R_pl²)[x² - 4(u_ixⁱ)²]

 

as the most general solution satisfying (5). We suppose the uⁱs do not depend on the xⁱs, maning we’re in the rigid bodies context. F_-1R_pl²/x² is the Newtonian part. The rest is linked with space-time confinement. Coefficients F_-1 to F₂ are dimensionless.

I’m still searching for suitable behaviours of (6). For instance, when xⁱ = R_pluⁱ, involving x² = R_pl², it would be quite nice to have W² = f_pl² and thus, F = 0. It would be as nice to have ¶_iF = 0 at the same time, for regularity. The first condition leads to:

 

(7)               F_-1 + F₀ + F₁ – 3F₂/2 = 0

 

the second one, to:

 

(8)               F₁ = 3F₂ + 2F_-1

 

This gives:

 

(9)               F₂ = -2(F_-1 + F₀/3)  ,  F₁ = -4(F_-1 + ½ F₀)

 

In the case of a finite domain of M_4-, we can fix two additional boundary conditions that can set the values for F_-1 and F₀. In the case of M_4-, these values are left free of choice so that we can take F₀ = 0 (replacement of the origin) and F_-1 = -1, giving:

 

(10)           F(x) = -R_pl²/x² + (4/R_pl)x_iuⁱ + [x² - 4(u_ixⁱ)²]/R_pl²

 

Nonetheless, real-valued solutions of (5) are justified for W² £ f_pl². If this should hold for x² ³ R_pl², should we expect a complete solution to show a complex-valued extension for x² < R_pl², i.e. W² > f_pl² or should we rather stop at R_pl? I’d bend for the second option, for we are in the “classical” context (smooth geometry) and this context holds down to R_pl, no smaller.

At small distances, the Newtonian behaviour is enhanced. At large distances, confinement is enhanced. This, for a conformally flat space-time, of course.

 

If our bio observer now believes he sees the four additional dimensions of M₄₊ to stay confined within the Planck radius, as in the O. Klein Hypothesis, then the metric involved is slightly different from (1), it’s:

 

(11)           ds₄₊²(x) = ds₈²(x) - ds₄² = -[W²(x)/f_pl²]ds_4-²

 

that is, only the deformed part. We don’t need to redo all the calculations, since the metric derivatives are the same and we juts have to replace 1 – W²/f_pl² with –W²/f_pl² in the scalar curvature (3):

 

(12)           K’(x) = -6F’^-3(x)¶ⁱ¶_iF’(x)  ,  F’(x) = [-W²(x)/f_pl²]^1/2

 

In the region W² £ f_pl², F’(x) is now imaginary-valued:

 

(13)           F’(x) = +i|W(x)|/f_pl  ,  |W(x)| = [W²(x)]^1/2  ,  F’*(x) = -i|W(x)|/f_pl

 

so that:

 

(12)           K’(x) = +6|F’(x)|^-3¶ⁱ¶_i|F’(x)| = 6f_pl²|W(x)|^-3¶ⁱ¶_i|W(x)|

 

Well, this total curvature is assumed to be everywhere constant, equal to 1/R_pl². As a consequence, the equation for the (absolute) amplitude |W(x)| of the G-field as seen by our bio observer will be:

 

(13)           ¶ⁱ¶_i|W(x)| = |W(x)|³/6c²

 

It comes in addition with the field equations for W_ij(x). I got hints of possible analytical solutions involving trigonometric functions, I’ll see next time.

 

A PSI body B1 sends a PSI signal to a PSI body B2. The process happens in M₄₊, but the dynamics also involves points of M_4-, so that the whole process is better described in M₈. Classically, propagation is second-order. First-order is rather for matter waves, in a microscopic description. The most general propagation equation one finds in M_4- writes:

 

(1)               L²f(x) = S[f(x),x]

 

where L² is a second-order linear operator with constant coefficients,

 

(2)               L² = ½ a^ij¶_i¶_j + bⁱ¶_i + cId

 

f(x) is any tensor field and S[f(x),x] a functional over M_4-. The process (1) is non-linear in general. For a^ij = 2g^ij, bⁱ = c = 0, L² is the d’Alembertian operator. We have ¶ⁱ¶_i = d²/ds₄², with ds₄² = c²dt²[1 – v²(t)/c²]. Solutions of ¶ⁱ¶_if(x) = S₀(x) can be found either in M_4- or in E₃. In M_4-, the “propagator” is in 1/s₄² = 1/xⁱx_i; in E₃, it’s in 1/|x| and involves propagation delays. I want to talk about them today. The value f(x,t) of the field f at point x in E₃ we observe at time t has been emitted by its source S₀(x,t) at time t’ = t - |x|/c = t(1 – v_moy/c), where v_moy = |x|/t is the mean velocity of the moving material source.

As the theory of PSI signals can be built exactly the same way as for signals in M_4-, we would like to find the equivalent of these propagation delays in M₄₊.

In B106, we gave a metric in M₈, formula (1). We would now prefer to express it in m/s. So, we introduce the new metric:

 

(3)               dG²(x) = f_pl²ds₈² = [1 – W²(x)/f_pl²]f_pl²ds₄²

 

In place of ¶ⁱ¶_i = d²/ds₄² we now have:

 

(4)               d²/dG²(x) = [1 – W²(x)/f_pl²]^-1f_pl^-2¶ⁱ¶_i

 

We are interested in solutions of:

 

(5)               d²Y(G,x)/dG²(x) = S(G,x)

 

in M₄₊, where S(G,x) is now a PSI body. To achieve this, we factorize (3). Factorizing ds₄² already gave c²dt²[1 – v(t)/c][1 + v(t)/c] = (cdt_-)(cdt₊). With mean values, one found t_- = t(1 – v_moy/c) and t₊ = t(1 + v_moy/c). In both cases, there’s a conformal transformation on relative time: t_- is a contraction of t, t₊ is a dilatation of t. Both are conformal because their product remain invariant under the Lorentz rotation group SO(3,1). We can already enounce:

 

PROPAGATION DELAYS IN E₃(t) ARE CONFORMAL TRANSFORMATIONS OF RELATIVE TIME.

 

Turning to (3), we get :

 

(6)               dG²(x) = [½ e_ij – W_ij(x)/f_pl][½ e^ij + W^ij(x)/f_pl]f_pl²ds₄²

 

with e_ij the skew-symmetric constant tensor such that e_ije^ij = 4. This gives:

 

(7)               dG_ij-(x) = f_pl[½ e_ij – W_ij(x)/f_pl]ds₄

(8)               dG_ij+(x) = f_pl[½ e_ij + W_ij(x)/f_pl]ds₄

 

Again, with mean values, we find:

 

(9)               G_ij-(x) = f_pl[½ e_ij – W_ij^moy(x)/f_pl]s₄

(10)           G_ij+(x) = f_pl[½ e_ij + W_ij^moy(x)/f_pl]s₄

(11)           W_ij^moy(x) = G_j(x)/xⁱ – G_i(x)/x^j

 

This time, we have skew-symmetric 2-tensor transformations (4 dynamical parameters) of the space-time distance s₄. Again, these transformations are conformal, since their product G_ij-(x)G^ij₊(x) is invariant.

 

G_ij-(x) IS A CONTRACTION OF f_pls₄ A FACTOR [½ e_ij – W_ij^moy(x)/f_pl].

G_ij+(x) IS A DILATATION OF f_pls₄ A FACTOR [½ e_ij + W_ij^moy(x)/f_pl].

 

That’s what replaces the “transmission delays” for PSI signals: a PSI signal emitted by a PSI source at [½ e_ij – W_ij^moy(x)/f_pl]s₄ will be received by another PSI body at s₄, corresponding to a point xⁱ in M_4-, since s₄² = xⁱx_i.

 

THE PROPAGATION OF PSI SIGNALS IN M₄₊ CONFORMALLY TRANSFORMS M_4-.

 

I can perform a gauge transformation living the mean field W_ij^moy(x) invariant. According to B108, I have xⁱ¶_iD(x) = -2D(x) = -2cV_pl/x². So,

 

(12)           G_i(x) -> G_i(x) – 2cV_plx_i/x⁴

 

will satisfy. As a result, I can fix the “mean gauge”:

 

(13)           G_i(x)/x_i = 0            (summation included)

 

Now, (13) is equal to ½ e^ijW_ij^moy(x). So, if I factorize (9) and (10) by e_ij, I’ll always be able to gauge the conformal transformations of M_4- away. Once more, this is a freedom we don’t find in E₃(t). The result is [in the gauge (13) only!]:

 

(14)           G_ij-(x) = G_ij+(x) = ½ f_ple_ijs₄

(15)           ½ e^ijW_ij^moy(x) = G_i(x)/x_i = 0

 

IN THE GAUGE (13), PSI SIGNALS PROPAGATES “INSTANTANEOUSLY”, THAT IS, POINT-LIKE IN M_4- OR, AT THE SAME POINT OF M_4-.

 

There’s indeed propagation but, because of the non-Euclidian nature of M_4- and M₄₊, we can get rid of the “transmission delays”.

 

K… powerful… K

 

Leaves me voiceless…

 

Today, we’re going to talk a little bit about PSI signals but, before that, i’d like to make some additional notes on what has been said earlier.

 

About “unified field theory”

 

In B106, I recalled a relation between the potentials of an electromagnetic field (EM field, in short) and a gravitational one (G field), involving only universal constants. Strictly speaking, the substitution of 1/4pe₀ is –k and not k. So, the relation should rather be:

 

(1)               G_i(x) = (-4pe₀k)^1/2A_i(x) = ±i(4pe₀k)^1/2A_i(x)

 

which corresponds to a rotation of ±p/2 (i = e^ip/2) in a complex plane having axes A_i and G_i. We can keep G_i(x) = (4pe₀k)^1/2A_i(x) if we restrict ourselves to the amplitude alone. But we can do much better, saying any EM field is formally equivalent to a purely imaginary G-field (or conversely) and extend the real-valued G-field to a complex-valued one:

 

(2)               G_i(x) = G_1i(x) + i(4pe₀k)^1/2A_i(x) = G_1i(x) + iG_2i(x)

 

If doing so, we then get a true unification of gravity and electromagnetism, since EM can transform into G and back thanks to G_2i(x) = (4pe₀k)^1/2A_i(x), without the need to enlarge the number of dimensional parameters. We use to talk of Maxwell’s theory as a “unifying theory”, this is not really exact: it’s rather a “grouping” of electricity and magnetism. Sure, the set of 4 equations show that electricity and magnetism are two complementary aspects of a single entity called the “electromagnetic field”, but these two aspects only couple to each other, they don’t transform into one another: we have “electricity” as one feature of the EM field, produced by fixed charges and “magnetism” as the other feature, produced by currents, that is, charges in motion. We could say the same of the electroweak GSW model: the weak field only couples to the EM field, they don’t transform into one another: their “internal” symmetry groups remain different.

Opposite to this, (2) enables us to build a real “electrogravitational” theory. Mass is equivalent to a purely imaginary electrical charge and conversely.

 

About gravity in M_4-

 

The frame we built throughout B102 to 105 included brings new insights in what we call “gravitation” in M_4-, as to know:

 

MOTION G_i(x) IN M₄₊(x) = GRAVITY FIELD IN M_4-

 

Meaning what a biological observer restricted to M_4- will perceive as a “G-field” will send back to the motion of a PSI-object in M₄₊. I checked how pertinent was this choice of G_i as new coordinates, despite their universal nature, trying other possibilities for the xⁱ⁺⁴ in B104, I found the G_is the most suitable ones. The gauge transformation Gⁱ(x) -> Gⁱ(x) + ¶ⁱD(x) not changing the intensities W^ij(x) of the G-field in M_4-, we then deduce that

 

MOTIONS Gⁱ(x) AND Gⁱ(x) + ¶ⁱD(x) IN M₄₊(x) HAVE SAME “VELOCITIES”.

 

This was impossible in Euclidian 3-space E₃: a motion x(t) only had same velocity as x(t) + x₀, this motion translated from a fixed point. This was due to the “poverty” of the dynamical parameters: a single one in E₃(t) against 4 in M₄₊(x).

Consequence: since we can fix a gauge where ¶ⁱG_i(x) = 0, the only prescription on D(x) is:

 

(3)               ¶ⁱ¶_iD(x) = 0

 

meaning the G-field is defined “up to a wave” in M_4-. Hence the behaviour of D(x), which has units m²/s (a scattering coefficient):

 

(4)               D(x) = D(R_pl)R_pl²/x² , x² = xⁱx_i

 

and if we choose D(R_pl) = (4p/3)R_plc (choice is free), we get:

 

(5)               D(x) = cV_pl/x²

 

giving:

 

(6)               ¶ⁱD(x) = -2cV_plxⁱ/x⁴

 

We conclude that:

 

MOTIONS Gⁱ(x) AND Gⁱ(x) – 2cV_plxⁱ/x⁴  IN M₄₊(x) HAVE SAME “VELOCITIES”.

 

Up to know, nobody saw any reason to impose a restriction on the xⁱs. But we could well have:

 

(7)               xⁱx_i = x² ³ R_pl²

 

in opposition to formula (6) B104. Such a condition would simply mean what we call “classical” fields in M_4- are defined from a space-time distance R_pl, which corresponds to what the Standard Model supposes. Below R_pl, assuming we can extend the notion of “distance” (expressed in meters) down to zero, we should rather turn to the xⁱ⁺⁴. In other words, the “classical” coordinates xⁱ become meaningless under R_pl, while the “Klein” coordinates xⁱ⁺⁴ become meaningless over R_pl.

However, as I already underlined it, such a “dichotomic” vision of the Universe is only due to our will to maintain the meter as the unit of length and restrict our observations to M_4-. When we turn to M₈, we see (B104-105-106) that, for velocities v > c, one finds no more “ordinary” waves and, for frequencies W > f_pl, one even finds nothing quantum anymore. In this optics, it will mean that, “inside” the Planck volume, one would only find vacuum and fluctuations: no more matter, no radiation, not even PSI matter or PSI radiations. Absolutely nothing realized.

Well, one can agree or not with this scenario, but it is consistent with the Standard Model.

 

About the motion in the “Tunnel”

 

Third and final note I wanted to make, the motion in the Tunnel. If we assume Tunnels have the physical property GⁱG_i = c² or Gⁱ(x)G_i(x) = c² expressed in terms of their G-field in M_4-, then Gⁱ(x) = -cuⁱ as we showed in B89. Examined a bit deeper, this property is really nice and interesting, for it expresses the fact that, in such structures, the “potential velocity” G_i(x) exactly equilibrates the “kinematical velocity” cu_i of an incident object:

 

(8)               v_kinⁱ + v_potⁱ = cuⁱ + Gⁱ(x) = v_totⁱ = 0

 

ALL VELOCITIES BEING OBSERVED BY A BIOLOGICAL OBSERVER IN M_4-, THE TOTAL VELOCITY IS THE RESULTANT OF THE VELOCITIES OF AN INCIDENT BODY MOVING IN M_4-: ITS KINEMATICAL VELOCITY cuⁱ AND THE ADDITIONAL VELOCITY THE G-FIELD COMMUNICATES IT. THIS RESULTANT BEING ZERO, A BIOLOGICAL OBSERVER WILL SEE NO MOTION AT ALL: EVERYTHING WILL HAPPEN AS IF THE BODY WAS AT REST.

 

There’s no paradox at all, only two velocities compensating for each other.

 

Now, add all this:

 

1)      no apparent motion of any “body” in any “Tunnel”;

2)      “PSI” phenomena supposed to hold only at distances lower than the Planck radius;

3)      Nothing concrete subsisting under the Planck horizon, according to the Standard Model;

 

And you’ll be naturally led to conclude that, not only can’t you observe anything “PSI” from M_4-, but that there can be no “PSI” at all!!!

 

If we don’t change frame and add a new system of units, we can even use the Klein hypothesis to extend the dimensions of our world, we won’t find anything “PSI” for as much, and will never be able to link it to the quantum, because this would simply contradict the Standard Model…

 

So, biologists are not the only one: even quantum physicists may not believe in the PSI, if they have no suitable frame to work into.

 

We, at last, come to our first practical application of our new model. And not the faintest one.

 

We take two animals with a central nervous system. Animal A1 will be the emitter, A2 will be the receptor.

The central nervous system (CNS) of A1 produces a mental object. Such an object being electromagnetic, it’s a A_i(x) distributed in a volume V₁ of his CNS. According to what we saw in the previous bidouille, A_i(x) can be transformed into a G-field G_i(x) in M_4- and therefore corresponds to a non material PSI object in M₄₊. So:

 

ANY MENTAL OBJECT PRODUCED BY A CNS IN M_4- IS A NON MATERIAL PSI OBJECT IN M₄₊. ITS PSI MASS IS M_MO(x) = 8p²e₀A²(x) = (2p/k)G²(x).

 

This PSI object is located at point Gⁱ in M₄₊. The aim is to connect it to a PSI object G’ⁱ of the same nature. This can only be done through a trajectory Gⁱ(x) in M₄₊(x), with G₁ⁱ = Gⁱ(x₁) and G’ⁱ = G₂ⁱ = G(x₂). So, A1 is located at point 1 in M_4- and A₂ at point x₂. The trajectory Gⁱ(x) is a G-field in M_4-. However, it should be obvious it’s different from the G-fields produced by both animals. Actually, we don’t need any source for this Gⁱ(x). So, why introducing any? The simplest scenario can use a G-wave.

So, we have a first animal, A1, emitting a mental object MO1, which is a PSI object and we want this PSI object to be sent to a second animal A2, the receptor. In our scenario, this can be done with the help of a G-wave in M_4-: this wave carries the information of MO1 from A1 to A2. For MO1 to be properly received by A2, we then need:

 

(1)               G₁ⁱ = Gⁱ(x₁) = G₂ⁱ = G(x₂)

 

First conclusion:

 

THE G-WAVE SUPPOSED TO CARRY THE MESSAGE FROM A1 TO A2 CANNOT BE NEWTONIAN.

 

For if it was, it would continuously decrease as 1/xⁱx_i from A1 and we could never obtain (1). So, whether Gⁱ(x) is constant, at least from x₁ to x₂, but A1 and A2 can a priori be arbitrarily distant from one another, or this Gⁱ(x) is oscillating. I prefer the second possibility, more general. So, I consider a monochromatic G-wave (for it’s enough):

 

(2)               Gⁱ(x) = gⁱcos kx

 

with constant amplitude gⁱ and constant k_is. (1) is satisfied for:

 

(3)               cos(kx₁) = cos(kx₂)

 

that is:

 

(4)               kx₁ = kx₂ – 2np  ,  n Î Z*

 

or:

 

(5)               x₂ⁱ = x₁ⁱ + 2np/k_i = x₁ⁱ + nlⁱ  ,  lⁱ = 2p/k_i

 

FOR THE MENTAL OBJECT A1 PRODUCED TO BE FAITHFULLY SENT TO A2, A1 AND A2 HAVE TO BE DISTANT OF AN INTEGER MULTIPLE OF A 4-WAVELENGTH.

 

If n > 0, A2 will stand “on the right of A1”; if n < 0, he will stand “on the left of A1”.

What if this condition is not satisfied? We then have:

 

(6)               x₂ⁱ = x₁ⁱ + alⁱ  ,  a Î R*

 

This corresponds to a phase in (2). Consequence:

 

IF THE ABOVE CONDITION IS NOT SATISFIED, THE MENTAL OBJECT PRODUCED BY A1 CAN STILL BE RECEIVED BY A2, BUT ROTATED FROM AN ANGLE 2pa.

 

This explains why Mrs Cant-Remember-Her-Name-Gotta-Review-My-Literature received a horseshoe with a screw in the middle where a marine anchor was sent to her: rotation 90° with respect to the vertical. This case corresponds to a = ±¼.

 

IN OUR NEW 8D FRAME, TELEPATHY CAN FIND A POSSIBLE EXPLANATION AS A COMMUNICATION USING GRAVITATIONAL WAVES.

 

Tricky! You can isolate A1 in a closed room protected against electromagnetic field, the process will still be possible: gravitational fields and waves have nothing to do with those kinds of “obstacles”. Fortunately enough! For we would be at zero gravity as soon as we would be in such a room…

Furthermore, G-waves are far less energetic than electromagnetic ones: if truly this way, the process is extremely weak.

Finally, you’ll never be sure the message is correctly transmitted, for there are interferences between the G-field carrying the message, the G-fields produced by the animals and the G-field of the Earth. All these interferences, and many others, “second-order”, regarding the weakness of the process, can seriously damage the content of the message or jeopardize its transmission. Dissipation is easy.

No surprise there’s a lot of failures with mediums.

And no surprise you don’t do it every day.