written and published by Renzo Diomedi // UNDER CONSTRUCTION

3 space-dimensions cannot explane the dark matter and the dark energy.

Conventionally, we use two series of cartesian complex conjugate planes intersecting each other and on each of them.

Then we fix a point in every plane.

We name J-planes the first set of planes and we name K-planes the set of intersecting planes.

the set of J-planes is the physical three-dimensional space made by particles of energy imprisoned in the mass.

The set of K-planes is the electromagnetic space made by photons, particles of massless energy.

So, the K-planes intersect J-planes and vice-versa.

The distance beetween the planes is quasi null, so, the hypercube consists of two Strings intersected among themselves

Every String is composed by points (one for each plane) , the string join all the points of all the J and K-planes vibrating in a two-dimensional surface called

Every set of planes creates a worldsheet. So the hyper cube created is composed by 4 space dimension created by 2 worldsheets intersected.

But the fourth spatial dimension can not be perceived by our senses created to perceive 3 dimensions only.

Considering 2 points on a sheet of paper, their distance along the surface will remain unchanged also bending the sheet, as it will always remain the same also curving the whole book, id est the geometric variety, of which it is part. We note that the whole universe is dotted with curved surfaces, and even when we draw a straight line on a sheet we have to remind us that the Earth's surface is bent and that the euclidean plane considered, is valid only by approximation locally.

If the measuring system is a Cartesian metrics, in a two-dimensional space, the

Length of the Strings:

The massless electromagnetic particles and particles with mass constitute the universe. If the distance among every point on J-plane and every point on K-plane is zero, the massless particle intersects and coincides with particle with mass .

using the metric tensor we can calculate the sum of the vectors of coincident points of J and K :

we can calcualte the distance among every point of every plane J and every point of every K:

we could also consider the two sets of crossed planes as cross-cut and intersected lattices considering the Strings belonging a worldsheet

we need to use tensors that in a geometrically bendable structure keep equal distance beetween the lattice points

then, also using the Einstein notation

then

then

then

then

The Metric Tensor

then contravariant metric tensor

then covariant metric tensor

to be continued...........................

to be continued............

the series of the points of the j-planes creates a curve function that we can use as a Line Integral =

the series of the points of the k-planes creates a curve function that we can use as a Line Integral =

TO BE CONTINUED....................

the series of the planes J is a cube, also the series K

each cube composes a string that is the result of the action of so many vector fields. Each cube, therefore, can be seen as a tensor. 2 intersecting cubes (hypercube) are a super-tensor

considering , then we have an Identity tensor:

then

to be continued....

NS, composed by a conservative continuityequation and a non-conservative Momentum equation not exactly measurable as a scalar field, but divisible by 3 scalar equations laid and projected along the directions

; = Density , = Viscosity

But the fourth space dimension could be considered as the synthesis of other 3 so in substitution of

Demonstration:

Considering the 4 independent variables

then, if the components of the viscous stress state are linearly linked to the components of the deformation velocity through Stokes' relations,

whereas

then

so, using we have

then

so,

momentum scalar equations:

using we have

TO BE CONTINUE...

Let the Times of the series of the planes , not coinciding

TO BE CONTINUE...

the axes joining the coordinates to the point, remain perpendicular at the main axes, (but not parallel)

conventionally, indicated with lower index

the axes joining the coordinates to the point, remain parallel at the main axes (but not perpendicular)

conventionally, indicated with upper index