Not a lot to add today, rather miscellaneous.

I first searched for something interesting in a non-linear extension of the electroG model, such as those proposed by our PSI program, I didn’t find any possible way to generate both an electric current and an electromagnetic potential from possible couplings of weighting matter and its G-potential.

The “electromagnetic mass” cannot b confused with the weighting one, because it behaves as the electric charge: whereas two weighting masses with the same sign attract each other, two EM masses with the same sign repulse each other. There’s a mere equivalence between the electric charge and the EM mass, that does not change the physical behaviour of the sources for as much. For this reason, it would be hard to generate an electric charge from a weighting mass, even trying to extend the definition of this last one. The only case of the electron speaks enough: we’ve seen how different are the values of its weighting mass at rest and of its EM mass. Things don’t work like this.

When talking about unification, the sources involved should be able to transform into one another through a symmetry group. Here, an electric charge should transform into a weighting mass and conversely. Or should it? What actually gives a transformation group are linear combinations of already existing quantities. In our present purpose, a combination like am+bq should give a mass m’, while cm+dq should give an electric charge q’. However, this is not at all what quantum theory shows.

In the “isospace” theory, we start with a given particle that can be found under a certain number of “states” or “configurations”, may we assume some restrictions. In the first SU(2) model of the strong interactions, for instance, the particle was the “nucleon”, which could be found under two “states”, the neutron |n⁰>, electrically neutral, and the proton |p⁺>, positive. That these two states could refer to a single particle expressed the (electric) charge independence of the strong interactions. The restriction was then: “assuming we neglect electromagnetic interactions, much weaker, the strong nuclear interaction makes no difference between a neutron and a proton”. The number 2 in the symmetry group refers to the number of such states making a single particle, i.e. configurations for which the strong interaction acts just the same. The choice of the Special Unitary group SU is due to the particle wavefunction, which is obviously complex-valued (SU is a rotation group in a complex plane).

So, two states, a 2-state wavefunction y^1,2(x) with complex conjugate y*^1,2(x): as many components as a spin-1/2. Hence the name of “isospin ½”, meaning “two states of charge” in place of “two states of intrinsic angular momentum”.

When working on mathematical groups, people are interested in discovering their properties and, in particular, their real “dimension”, i.e. the number of rotations with real-valued angles. It so appears that special unitary groups SU(n) have n²-1 such angles: we say their dimension (or number of generators) is n²-1. As a consequence, SU(2) possesses 3 rotation angles. The transformation matrix applies on y^1,2(x) to transform it into another wavefunction y’^1,2(x), obviously of the same kind. It does not act upon “strong charges”. What happens is that, amongst the 3 available generators, which are 2x2 matrices (actually the same as the Pauli matrices), one of them (usually called I₃) is diagonal and gives the charge (or the charge operator). The two others transform one of the state into the other one, that is, the proton wavefunction into the neutron one and conversely.

The very same occurs for the larger symmetry group SU(3) of the new strong interaction theory. There, we have 3 fundamental states, u, d and s, representing three “quarks”. Again, we shouldn’t see these particles as different, but as different configurations of a single particle. SU(3) has 8 generators. It includes SU(2). The isospin is 1 (3 states). Each generator can be represented by a 3x3 matrix. Two of them appear diagonal, F₃ (containing I₃) and F₈. F₈ gives what is called the “hypercharge”. The charge Q, properly speaking, becomes a linear combination of F₃ and F₈ (with constant coefficients). It again applies on the wavefunction y^1,2,3(x) of the 3-state model. Two such symmetry groups have been built, one for quarks (“flavour dynamics”) and one for gluons (“color dynamics”), “gluing” quarks. However mathematically the same, these two groups shouldn’t be physically confused, as they refer to different properties inside the same frame: SU_q(3) relates to “quark flavour” (fermions), SU_c(3) to “gluon color” (bosons). Notice the isospin is the same for both. Another essential difference with the spin (½ for quarks, 1 for gluons).

There’s a symmetry group that have been interesting me for long, it’s SU(3,1). It’s larger than SU(3) and, as rotation groups in real space-times, it’s no longer Euclidian. It has (3+1)²-1 = 15 generators. But, 8 of them are space-like, 1 is time-like and the 6 remaining ones are space-time-like. The mathematical decomposition of SU(3,1) is as follows:

 

(1)               SU(3,1) » SU_S(3) x SU_ST(2) x SU_ST(2) x U_T(1)

 

(» = isomorphism – equivalence, if you prefer). 15 = 8 + 3 + 3 + 1. It’s extremely tempting to relate it to the symmetry groups of the four known interactions. How? SU_S(3) can be identified with the color group SU_c(3), no difficulty. U_T(1) could be associated with gravity. If we make these choices, then the symmetry will represent a 4-state particle with one time-like state and three space-like ones. The time-like state is to be associated with the mass operator M; the space-like states, with the charge operators. SU_S(3) has two diagonal matrices = 2 charge ops; SU_ST(2), 1 diagonal matrix = 1 charge op, doubled. The 4-state wavefunction is y^1,2,3,4(x). y⁴(x) is the mass state. So, SU_ST(2) x SU_ST(2) = [SU_ST(2)]^x2 should be devoted to the unified electroweak interaction field. The present symmetry group is that given by the GSW model, SU_W(2)xU_EM(1): 3+1 = 4 generators only. We have 2 generators more. So, whereas they should be devoted to the weak nuclear field, or they should be part of a quantum extension of electrodynamics (extended QED). It’s not easy to decide, as [SU_ST(2)]^x2 is both space and time-like in isospace-time and so, mixes charge and mass. This would imply the photon to become massive, as the weak bosons. Now, this goes reverse to the goal usually targeted: to obtain massless gauge bosons inside a wider symmetry…

Yeah, except that… we now have a mass operator… J and transformations of mass into charges and backward. This should allow massive gauge bosons, while massless ones could still be found in the single non-Euclidian SU(3,1).

Let me explain better.

Relation (1) is a mathematical equivalence: the group on the left has same number of generators (same dimension) as the Euclidian product of groups on the right.

Physically now, it rather describes a transition:

 

(2)               SU(3,1) -> SU_C(3) x SU_ST(2) x SU_ST(2) x U_G(1)

 

Each of the groups on the right are sub-groups of SU(3,1). As a result, the initial symmetry is reduced. Geometrically, we go from a space-time structure to a product of torus: SU_S(3) is an 8D-sphere, SU_ST(2) a 3D-sphere and U_T(1) a 1D-sphere (circle). The whole product gives a 4-frequency torus S⁸xS³xS³xS¹, where S is the topological Riemann sphere. The Standard Model foresees SU_C(3) x SU_W(2) x U_EM(1) x U_G(1), assuming spin-1 gravity. So, there should happen a second transition involving the electroweak field alone:

 

(3)               SU_ST(2) x SU_ST(2) -> SU_W(2) x U_EM(1) x G

 

where G is a two-generator group. Let’s reason in terms of isospins: SU(2) -> 2 states -> isospin ½ ; U(1) -> 1 state, isospin 0. On the left, we have a pair of two isospins ½. This should give an isospin 1. For the isospin to be conserved in (3), we need G to describe an isospin ½. On the other side, we cannot have more generators (more symmetry) than we had before. So, 3+3 = 6 should transform into 3 + 1 + 2 and dim_R(G) should be 2. The only possibility is G = Spin(1), the Clifford group of (real) dimension 2. It is true that, group-theoretically, there is a close relationship between SO(3), the rotation group of E₃, SU(2) and Spin(1). The point here is that we shouldn’t forget we’re not in ordinary space or space-time, but in isospace-time. Now, Spin(2s) are the spin groups of ordinary space-time… Here, we’re dealing with rotations in isospace(-time). We can recover such an equivalence, if we rename our Clifford groups Isospin(2s), keeping the same structure. Our second transition will then take the form:

 

(4)               SU_ST(2) x SU_ST(2) -> SU_W(2) x U_EM(1) x Isospin(1)

 

We aren’t safe for as much… Because introducing Clifford structures in isospace undermeans introducing Fermi-Dirac statistics… “isofermions”… Can we do this?

I think we can. Because we already have an example of such “isofermions” in the color wavefunction of QCD. Color dynamics is safe if and only if it includes Pauli’s exclusion principle in its isospace. Why couldn’t we find the same with the weak field? And would this have any influence on chirality violation? I can’t say. What I can tell is that SU_W(2) enlarged into SU_W(2)xIsospin(1) now has 5 generators instead of 3 + a skew-symmetry on the corresponding wavefunction (leptonic charges).

I finally remarked something, that the ratio q_e/m_pl of the electric charge q_e = 1,60219x10^-19 C by the Planck mass m_pl = 5,456019873x10^-8 kg gives:

 

(5)               q_e/m_pl very close to (4pe₀k/861)^1/2

 

and 861 is about the ratio between the strong and the electromagnetic interaction (£ 1000).

This gives a permittivity coefficient e = e₀/861 for the strong field, which would then be 861 times less conducting than EM. It’s consistent with the orders of, whether energy thresholds (Mev -> Gev) or, equivalently, the ranges (10^-15 m = 1F -> 10^-3 F).

To conclude all this, given mass and the three colors r,v,b, mixed in a pseudo-Euclidian hermitian isospace-time, we could get the charge symmetries for the four known fundamental interactions through at least two transitions.

 

What’s the connection with the PSI ?

 

Well, none if you “restrict” yourself to pure high-energy physics and, if you want to see a connection with neurophysics and the PSI, however you try to take the problem of fundamental forces in the Universe, as far as we understand things (or I myself), there seems to exist at least two different forms of charges from the very beginning: mass and a certain type of charge. They don’t react the same: mass with same sign attract, mass of opposite sign repulse each other, whereas all other known charges behaves like the electric ones, they attract when they have opposite signs and repulse when they have same signs. Space-time confinement is something different. It’s about the behaviour of the force field in space and time, not about the properties of its quanta. This is what expresses SU(3,1): we can transform the mass state y⁴ into a charge y^{1,2 or 3}, or any charge state into a y⁴, yet the mass operator remains distinct from the charge ones. They don’t lay on the same kind of axis. In general we will have combinations y’ⁱ(x) = T^ijy_j(x), mixing the 3 charges and the mass.

Just like for space and time in SO(3,1).

As a result, the gravitational potentials G_i(x), despite the most universal of all, appear not sufficient to describe a massive and electrically charge event. We would need introduce “new” gravitational-like coordinates (4pe₀k)^1/2A_i(x), that wouldn’t behave like a G-field anyway, but still as an EM field. It would give us 4 dimensions more… that would be 12… + the two other nuclear interactions…

Unreal.

But we can go back to dimension 4, while introducing restrictions on the tangent space-times. It will amount to exactly the same, less the question of the number of dimensions. Those restrictions are, up to now: v(t) £ c and W_ij(x)W^ij(x) £ f_pl², to what we can add F_ij(x)F^ij(x) £ 4pe₀kf_pl² » 3,8 – 4 x 10⁶⁵ T², that is, |F_ij(x)| £ 6 x 10³² T roughly: far enough for our needs… J

Nothing to complain about, then, but all these “physical constraints” actually open us doors to brand new physical phenomena, where the PSI can find its place. If not as a function over 8D space-time, at least as a functional (i.e. an operator) over M_4-.

So, we will simplify life rewriting M₄ and even M Minkowski space-time, for I think I won’t have to introduce new frames.

However, I would have never discovered such restrictions on the gauge fields if I hadn’t try the “8D experiment”. So, everything but a waste of time, it was.

 

Here’s a very good comparison that will help non-technicians understand how difficult parapsycholigical problems are to be modeled.

We are going to talk about electrogravity today. To build a satisfying model of this unification between gravitation and lectromagnetism, it took me 2 hours for the classical part and two hours more for the quantum one. 4 hours only. Technicians will judge the results.

In comparison, I’ve been working hard on parapsy pbs for now 3 years…

We’ll be back to them afterwards and we’ll be able to see how interesting it will have been to “diverge” a bit.

 

I’ve already pointed out several times the fundamental difference there is between what we’ve been doing since the 19^th century, as to know, grouping field theories altogether, and true unifications. JC Maxwell didn’t “unify” electricity and magnetism. He never pretended so, anyway. He added a fourth equation to the three existing ones to complete the model. The set of four field equations now known as the equations of electromagnetism is unfortunately not a true unified theory: we still have two different and only complementary aspects of the “electromagnetic” field, the electric one and the magnetic one. The equations give the laws under which the variations of one is proportional to another variation of the second, but nowhere is explained how electricity would transform into magnetism and conversely. For instance, Faraday’s law asserts that the circulation of the electric field around a closed curve is opposite to the time variation of the induced magnetic field. Induced: the laws of electromagnetism remain induction laws. Maxwell’s equation says that the circulation of the magnetic field is now proportional to the time variation of the electric field plus a contribution of the electric current.

The very same has been done by Glashow, Salam and Weinberg on the electroweak field: we still have an electromagnetic field on one side, with its massless electrically neutral photon, and a weak-interacting nuclear field on the other side, with its three massive gauge bosons; W⁺ and W^- are electrically charged and have same mass, up to minor corrections, Z⁰ is electrically neutral and its mass is a bit heavier than that of the Ws. Besides, charge symmetry works under the “unification” group SU_w(2)xU(1) which clearly shows electromagnetism and the weak interaction remains, again, two different and only complementary processes: the charge symmetry of EM is U(1), that of W is SU_w(2); the Euclidian product of groups enlights the separation between the two.

To sum up this difference, we can enounce that:

 

GROUPING FIELDS TOGETHER INTO A SINGLE SET OF EQUATIONS REQUIRES TO ADD DIMENSIONS TO THE PHYSICAL FRAME.

ON THE CONTRARY, UNIFYING FIELDS NEEDS NO SUCH REQUIREMENT. BECAUSE FIELDS CAN THEN TRANSFORM INTO ONE ANOTHER AND REPRESENT DIFFERENT FORMS OF A SINGLE FIELD.

 

We can now turn to electrogravity. We already know the product of the electromagnetic potentials A_i(x) by the universal constant (4pe₀k)^1/2 behaves just like gravitational potentials. Reconsidered under a “unified optics”, we can say that (4pe₀k)^1/2A_i(x) is “another form of gravitation”, now induced by electrical charges, whereas “conformal gravitation” is induced by “weighting masses”.

This is absolutely equivalent to saying that the ration q/(4pe₀k)^1/2, where q is an electrical charge, behaves like a mass or is just another form of mass, or conversely, that the product m(4pe₀k)^1/2, where m is a mass, behaves like an electrical charge or is just another form of an electrical charge.

Now, these two forms of mass, despite equivalent, cannot be identified. An elementary calculation readily shows that q_e/(4pe₀k)^1/2, where q_e is the electron charge gives an “electromagnetic mass”:

 

(1)               m_em(e^-) = -1,60219x10^-19 x 1,160522537x10¹⁰ = 1,859377604x10^-9 kg

 

far from the weighting mass of the electron m_g(e^-) = 9,10956x10^-31 kg (and assumed to be positive).

To take these two forms of mass into account, we introduce the masses:

 

(2)               m_± = m ± q/(4pe₀k)^1/2

 

as a two-state extension of the traditional mass m. For electrically neutral bodies (q = 0), m_± = m. For gravitationally neutral bodies (massless bodies, m = 0), m_± = ±q/(4pe₀k)^1/2. We can now have m_± = 0 for:

 

(3)               -m = ±q/(4pe₀k)^1/2

 

as mass, like the electric charge, can take both signs.

The unified electrogravitational model is based on such an extension. The extended gravitational potentials are:

 

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

 

The derived field intensities are:

 

(5)               W_ij^±(x) = W_ij(x) ± (4pe₀k)^1/2F_ij(x)

 

and the energy-momentum densities (gravitational currents),

 

(6)               p_i^±(x) = p_i(x) ± j_i(x)/(4pe₀k)^1/2

 

All these extensions give the feeling that gravitational components and electromagnetic ones remain separate and merely add, up to a sign. This is not the case: in each of these extensions, what we add is two different forms of a single enlarged entity. Again, (4pe₀k)^1/2A_i(x) is a G-potential, G_i(x)/(4pe₀k)^1/2 is an EM-potential; (4pe₀k)^1/2F_ij(x) is a G-field, W_ij(x)/(4pe₀k)^1/2, an EM-field; j_i(x)/(4pe₀k)^1/2 is a G-current, (4pe₀k)^1/2p_i(x), an EM-current.

There’s no need to introduce additional dimensions of space-time.

If we had treated the problem, whether with a 2-component 4-vector field [G_i(x) , (4pe₀k)^1/2A_i(x)] or with a single complex-valued field G_i(x) + i(4pe₀k)^1/2A_i(x), we would have been forced to double the number of real dimensions (4 -> 8 components). But this does not lead to a convenient model: the complex-valued extension gives the wrong sign in the Lagrangian density (if we want this functional to stay real-valued), the 8D-vector field leads to the same + pbs of convergence.

The Lagrangian density leading to correct field equations while avoiding all these difficulties is:

 

(7)               £_EG = (c²/8pk)W_ij⁺(x)W^ij_-(x) – ½ [p_i⁺(x)Gⁱ₊(x) + p_i^-(x)Gⁱ_-(x)]

= (c²/8pk)W_ij(x) – ½ e₀c²F_ij(x)F^ij(x) – p_i(x)Gⁱ(x) – j_i(x)Aⁱ(x)

 

The Lagrange equations:

 

(8)               ¶ⁱ(¶£_EG/¶W^ij_-) = ¶£_EG/¶G^j_-  ,  ¶ⁱ(¶£_EG/¶W^ij₊) = ¶£_EG/¶G^j₊

 

then gives the set of field equations:

 

(9)               ¶ⁱW_ij⁺(x) = -(4pk/c²)p_j^-(x) , ¶ⁱW_ij^-(x) = -(4pk/c²)p_j⁺(x)

 

Surprise: m^- is the source of W_ij⁺ and m⁺ that of W_ij^-…

Adding (9a) and (9b) gives the field equations for gravity back:

 

(10)           ¶ⁱW_ij(x) = -(4pk/c²)p_j(x)

 

while (9a) – (9b) gives the field equations for electromagnetism back:

 

(11)           ¶ⁱF_ij(x) = e₀c²j_j(x)

 

The conservation laws for extended masses are:

 

(12)           ¶ⁱp_i^±(x) = 0

 

giving back:

 

(13)           ¶ⁱp_i(x) = 0  <=>  m = Cte

(14)           ¶ⁱj_i(x) = 0  <=>  q = Cte

 

The other set of field equations is made of the Bianchi identities:

 

(15)           ¶_[iW_jk]^±(x) = 0  <=>  ¶_[iW_jk](x) = 0  ,  ¶_[iF_jk](x) = 0

 

Finally, the Lorentz gauges are:

 

(16)           ¶ⁱG_i^±(x) = 0  <=>  ¶ⁱG_i(x) = 0  ,  ¶ⁱA_i(x) = 0

 

Consequently, unification does not introduce magnetic charges nor Coriolis masses, as long as the field equations remain linear, of course. No “anomalies” of that kind.

We now turn to the equations of motion of an incident body perturbated by an electroG-field. The Lagrange function describing it is:

 

(17)           L = ½ mc²uⁱu_i + ½ c(m₊G_i⁺ + m_-G_i^-)uⁱ = ½ mc²uⁱu_i + ½ c(mG_i + qA_i)uⁱ

 

The Lagrange equations are as usual (d/ds)¶L/¶uⁱ = ¶L/¶xⁱ. They give:

 

(18)           du_i(s)/ds = (1/mc)[m⁺W_ij⁺(x) + m^-W_ij^-(x)]u^j(s) = (1/c)[W_ij(x) + (q/m)F_ij(x)]u^j(s)

 

There, we can no longer split the equations in a purely gravitational force and a purely electromagnetic one. Only for electrically neutral bodies is the perturbation solely due to gravity. As for massless bodies, we face the same difficulty as before, only indicating this approach is not the correct one for them.

What is particularly interesting here is that, when gravity and electromagnetism compensate each other, the right-hand side expression in (18) vanishes:

 

(19)           W_ij(x) + (q/m)F_ij(x) = 0

 

This is the equilibrium condition for electrogravity. The motion of the incident body then seems free. Actually, it’s not, since we still have both a gravitational and an electromagnetic force. But everything happens as if there was no force at all acting upon the body. It’s nothing else but a well-known phenomenon: levitation.

Here, it’s possible to levitate even when velocity is close to the speed of light.

More specifically, one can find only distinct points xⁱ where (19) is satisfied. The equilibrium condition is then local and these points are equilibrium or libration points.

The Casimir identity P_iPⁱ = m²c² is to be replaced with:

 

(20)           P_i⁺Pⁱ_- = m₊m_-c² = m²c² - q²c²/4pe₀k

 

It seems to be a general feature that the kinetic parts are products of both states, while the potential parts are sums (or traces, in the matrix language) of products of each state. Remember (20) is a purely kinetic identity. We can write P_i^± under the form:

 

(21)           P_i^± = m^±cuⁱ = mcuⁱ ± qc/(4pe₀k)^1/2 = P_i ± I_i/(4pe₀k)^1/2

(22)           I_i = qcu_i  in Am

 

This is almost everything we can say about the basics of the classical theory. Quantum electrogravity is built on the covariant derivative:

 

(23)           D_i = ¶_i – (i/2ħ)[m₊G_i⁺(x) + m_-G_i^-(x)] = ¶_i – (i/ħ)[mG_i(x) + qA_i(x)]

 

When m = 0, m is to be replaced with ħw₀/c², where w₀ is the pulse at rest of an oscillator.

The fermionic model is described by the Lagrangian density:

 

(24)           £_FEG = ½ iħc[(D_iy_F)*gⁱy_F - y_F*gⁱD_iy_F] – V(y*_F,y_F) + (c²/8pk)W_ij⁺(x)W^ij_-(x)

 

The field equation for the wavepacket is:

 

(25)           gⁱD_iy_F(x) = (i/ħc)¶V/¶y*_F

 

Those for the unified electroG-field, (15) and:

 

(26)           ¶ⁱW_ij⁺(x) = -(4pk/c)m_-y_F*g_jy_F  ,  ¶ⁱW_ij^-(x) = -(4pk/c)m₊y_F*g_jy_F 

 

The bosonic model has Lagrangian density:

 

(27)           £_BEG = -(ħ²/2m)(D_iy_B)*(Dⁱy_B) – V(y*_B,y_B) – (c²/8pk)W_ij⁺(x)W^ij_-(x)

 

Field equations:

 

(28)           DⁱD_iy_B(x) = (2m/ħ²)¶V/¶y*_B

(29)           ¶ⁱW_ij⁺(x) – (2pR_g^-/m)[m₊G_j⁺(x) + m_-G_j^-(x)]r_B(x) = -4pR_g^-r_B(x)¶_jq(x)

(30)           ¶ⁱW_ij^-(x) – (2pR_g⁺/m)[m₊G_j⁺(x) + m_-G_j^-(x)]r_B(x) = -4pR_g⁺r_B(x)¶_jq(x)

(31)           R_g^± = km^±/c²  ,  r_B(x) = y*_B(x)y_B(x)

 

As (29-30) are still linear, we can again split them into:

 

(32)           ¶ⁱW_ij(x) – 4pR_g[G_j(x) + (q/m)A_j(x)]r_B(x) = -4pR_gr_B(x)¶_jq(x)

(33)           ¶ⁱF_ij(x) + 4pR_em[(m/q)G_j(x) + A_j(x)]r_B(x) = (4pħR_em/q)r_B(x)¶_jq(x)

(34)           R_em = q²/4pe₀c²m

 

When q = 0, we get an EM wave; when m = 0, a G-wave. The rest is rather deceiving: nothing really new…

So, that will be all for today.

 

I may be a bit tired, but i have a structural problem with field theory… L

The Lagrange function supposed to describe the motion of an incident ordinary body submitted to the action of a gravitational field (or any other vector field) is:

 

(1)               L[x(t),v(t),t] = -m(t)c²[1 – v²(t)/c²]^1/2 + m(t)G[x(t),t].v(t) – m(t)f[x(t),t]

 

In this functional, m(t) is the mass of the incident body in its reference frame at rest and v(t) = dx(t)/dt, its velocity at time t along its trajectory x(t) in mobile Euclidian 3-space E₃(t). I want to be as precise as possible. At any time t of the motion of this incident body, x(t), its position at time t, coincides with the observation point x = x(t) where an observer external to the interacting system evaluates the influence of the G-field.

This G-field is whether a free wave propagating into E₃ or, more likely, a field emitted by a source body. This source body is ordinary too, but different from the incident body. It has mass at rest m’(t) and trajectory x’(t) in E₃(t) and there’s no reason why x’(t) should coincide with x(t) at any time (or there would be an instant at which the two bodies would collide).

The source body has matter distribution:

 

(2)               m’(x’,t) = m’(t)r’(x’)

 

where x’ is a point of E₃ inside a finite volume V’₃ of E₃. One usually assumes that r’ vanishes at the boundary of V’₃.

The same can be done for the incident body:

 

(3)               m(x,t) = m(t)r(x)

 

with x a point of E₃ inside another finite volume V₃ of E₃, still with r vanishing at the boundary of V₃.

Let us assume the source is not fixed in space, but moving. It thus produces a current density p’_i equal to what? Logically, we should expect:

 

(4)               p’_i[x’(t),t] = m’[x’(t),t]v’_i(t)  ,  v’_i(t) = dx’_i(t)/dt = [c , -dx’(t)/dt]

 

or even a v’[x’(t),t], since the source is represented as a compact set of point-like particles.

Now, the wave equation for the G-field is already given on fixed points x (“observation points”) and the solution of ¶ⁱ¶_iG_j(x,t) = -(4pk/c²)p’_j(x,t) in E₃xR is, for the emitted part only (i.e. up to a free wave):

 

(5)               G_i(x,t) = (-k/c²)ò_V’3 p’_i(x’,t - |x – x’|/c)d³x’/|x – x’|

 

Thanks to the boundary condition on the matter distributions, one can always analytically extend the volume integral to all E₃, it only makes its calculation easier.

According to what has been recalled above, all these points x’ in (5) can be identified with the points x’, since the integral is non zero only inside V’₃.

The point is clear: any current can only be defined from a motion; (4) and (5) are different: how do you want to define a velocity in (5)? Only extending the formula to:

 

(6)               G_i(x,t) = (-k/c²)ò_V’3(t) p’_i[x’(t),t - |x – x’(t)|/c]d³x’(t)/|x – x’(t)|

 

Then? What would become our velocity? According to (4), it should be, at the simplest:

 

(7)               v’(t) = dx’(t)/dt

 

or we can use x’(t) only if we extend it into V’₃, to include all the x’(t). For, if we start it from the boundary of V’₃, we’ll find no current at all…

No, finally, I don’t have any specific problem with field theory. So good… J

I even got a better formula than (6):

 

(8)               G_i[x(t),t] = (-k/c²)ò_E3(t) p’_i[x’(t),t - |x(t) – x’(t)|/c]d³x’(t)/|x(t) – x’(t)|

 

that can be directly used in (1), which describes the gravitational interaction between two ordinary bodies.

We can build a similar functional for the same kind of interaction between now two PSI bodies: one of PSI mass M(x) and one of PSI mass M’(x). The effect will be called “PSI gravitation”. It occurs in M₄₊ and the parameter space-time is M_4-:

 

(9)               £[Gⁱ(s_4-),dGⁱ(s_4-)/ds_4-,s_4-] = -M(x)f_pl²{1 – 6t_pl²[dGⁱ(s_4-)/ds_4-][dG_i(s_4-)/ds_4-]}^1/2 + M(x)G_I[G(s_4-),s_4-]dG^I(s_4-)/ds_4-

 

with G^I = (Gⁱ,Gⁱ⁺⁴ = xⁱ/6^1/2t_pl), G_I = (-G_i,G_i+4 = x_i/6^1/2t_pl). The G_Is are the new potentials, in Hz. We first notice that M(x) = (h²/3mc²)r(x) -> 0 for h -> 0 (classical limit) and/or 1/c -> 0 (Galilean limit). Conclusion:

 

THE PSI MASS CAN ONLY BE OBSERVED INSIDE THE QUANTUM RELATIVISTIC FRAME. IT’S NEGLIGIBLE IN BOTH CLASSICAL PHYSICS AND GALILEAN MECHANICS.

 

It’s not even enough to be quantum. In the weak field approximation W² << f_pl², we should find back the Maxwellian model:

 

(10)           £_{W² << fpl²} » ½ M(x)W^ij(x)W_ij(x) - p’_i(x)Gⁱ(x)

 

This is achieved in the first-order Fermi-Dirac model, where p’_i(x) = m’(t)y*(x)g_iy(x), and in the second-order Bose-Einstein model, where p’_i(x) = ħy*(x)y(x)¶_iq(x). FD gives:

 

(11)           £_int,F = ½ iħc[(D_iy)*gⁱy - y*gⁱD_iy] – V(y*y)

      = [½ iħc(¶_iy*gⁱy - y*gⁱ¶_iy) – V(y*y)] - m’(t)y*g_iyGⁱ

 

it corresponds to the classical relation gⁱP_i = mc. BE gives:

 

(12)           £_int,B = [ħ²/2m’(t)](D_iy)*(Dⁱy) – V(y*y)

= {[ħ²/2m’(t)]¶_iy*¶ⁱy – V(y*y)} - (ħy*y¶_iq)Gⁱ + ½ m’(t)y*yG_iGⁱ

 

and corresponds to the classical relation PⁱP_i = m²c². It’s therefore interesting to investigate higher orders. At the third order, the only way to make a scalar, Lorentz-invariant quantity is:

 

(13)           gⁱP_iP_jP^j = gⁱg^jg^kP_iP_jP_k = m³c³

 

The related wave equations then turn out to be:

 

(14)           gⁱg^jg^kD_iD_jD_ky = m³c³y

 

and send back up to a Lagrangian density:

 

(15)           £_int,3 = [1/2m²(t)c]{iħ³[(D_iy)*gⁱD_jD^jy - y*gⁱD_iD_jD^jy] + m³(t)c³y*y}

 

You can easily check it’s cubic in the G_is. Fourth-order is:

 

(16)           P_iPⁱP_jP^j = gⁱg^jg^kg^lP_iP_jP_kP_l = m⁴c⁴

(17)           gⁱg^jg^kg^lD_iD_jD_kD_ly = m⁴c⁴y

(18)           £_int,4 = [1/2m³(t)c²]{ħ⁴(D_iy)*DⁱD_jD^jy + m⁴(t)c⁴y*y}

 

Again, it’s quartic in G_i. One sees that the “base brick” is gⁱP_i and the n-th order is (gⁱP_i)ⁿ. One indeed can build a spin 1 from a pair of Fermi spinors, assuming these spinors are different. With 3 of them, one builds a spin 3/2 and so on. The spin 0 is obtained from a pair (up-down).

It means that all these models can be included in M(x)G_I[G(s_4-),s_4-]dG^I(s_4-)/ds_4- and even in M(x)G_i+4[G(s_4-),s_4-]uⁱ(s_4-)/6^1/2t_pl only.

 

The rest follows the same procedure as for establishing the field equations in M_4-. We have (9) in J/m³, G_i+4f_pl = F in s^-2. Just as f(r) = -km’/r for static G-scalar potentials with spherical symmetry and conserved sources, we let F(G) = -KM’/GⁱG_i for global G-scalar PSI potentials with Lorentz-invariant symmetry and conserved PSI sources (4D => Newtonian potentials are in 1/GⁱG_i). This gives K in m³/kgs⁴ and ¶²F/¶Gⁱ¶G_i in 1/m². Then we turn to the field equation for this F, ¶²F/¶Gⁱ¶G_i = KR(G) and we get R(G) in kgs⁴/m⁵ = (kg/m)(s/m)⁴, or:

 

(19)           R(G,x) ~ M(x)/|G|⁴

 

as expected for a PSI mass density. The corresponding current densities are:

 

(20)           P_i[G(s_4-),s_4-] = R[G(s_4-),s_4-]dG_i(s_4-)/ds_4-

(21)           P_i+4[G(s_4-),s_4-] = R[G(s_4-),s_4-]dG_i+4(s_4-)/ds_4- = R[G(s_4-),s_4-]u_i(s_4-)/6^1/2t_pl

 

They are in (kgm²/sm³)(s/m)⁴ = (density of action)/(4-volume of M₄₊). Consequently, the equations for the gravitational PSI field are:

 

(22)           ¶^I¶_IG_J(G,x) = -4pKt_pl²P_J(G,x) = -4pkP_J(G,x)   ,  ¶_I = ¶/¶G^I

 

Surprisingly, k (and not k/c²) is here the coupling constant.

 

It remains to describe the interaction between an ordinary body and a PSI body. That will be the next step.

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.