The speed of light is the low-dimensional limit

The singularity in a black hole is the higher-dimensional limit

With r_S\space =\space \frac{2\space *\space l_P^2}{\lambda}, the Schwarzschild radius is directly related to the Compton wavelength # Dimensional Physics

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Theory unifying general relativity with quantum field theories

Christian Kosmak, Germany Würzburg 2023 Version 4.1 – 05.30.2023 Binding energy as intersection of spacetime density.

## 2 Basic idea

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To put the GR and the QFT on a common basis it is reasonable to take one of both theories as given and to generate the other one, in a correspondence principle. At present it is often assumed that the ART must be adapted to the QFT in the area of the Planck scale. The interaction of gravity is supposed to be done by quantized exchange particles and/or the spacetime itself is supposed to be quantized. QFT is confirmed to many decimal places and can be well studied in a laboratory (albeit very large). In addition, the GR contains an ununderstood singularity in all solutions. Therefore, QFT is a good choice as a starting point.

In the DP, an opposite starting point is chosen to solve the problem. The GR is considered as correct. The problems within the GR (e.g. the singularity) are solved by the common basis. The QFT is generated, without an adjustment, from few more assumptions. Therefore, a continuous spacetime is basically assumed in the DP. An initial assumption is made so that a clean starting point is given for the following considerations.

Assumption A-01: The ART is correct.

### Approach in the DP

The field equation of Einstein must be assigned a clear definition of the terms. For the first approach the simplest form of the field equation is sufficient:

G_{\mu\nu}\space =\space k\space *\space T_{\mu\nu}

The proportionality constant k does not play a role in this chapter yet and is treated separately in the chapter „spacetime structure“. The Einstein tensor  G_{\mu\nu} and the energy-momentum tensor T_{\mu\nu} are the decisive quantities here. In order to better understand the idea of spacetime density, the equation is rearranged. The Einstein tensor is simply brought to the other side. This transformation may be done with every equation.

0\space =\space  k\space *\space T_{\mu\nu}\space -\space G_{\mu\nu}

As you can see from the equation, the terms must cancel each other out. The following picture is the first picture of a SL. That SL is in the core of the galaxy Messier 87, a „monster“ of about 6.5 billion solar masses and a Schwarzschild radius (SSR) of about 20 billion kilometers. Gravity is no longer hindered by the other fundamental forces. According to the rearranged field equation, this SL, from the point of view of spacetime, is the desired state. Explicitly a zero.

In the classical view for the field equation is:

• G_{\mu\nu}: A clearly localized and continuous geometrical expression of the spacetime as „space curvature“. It requires only spacetime itself. No separate objects.
• T_{\mu\nu}: A collection of quantized energy and mass objects, with no clear localization for each individual object. All objects exist separately in spacetime, but are not themselves spacetime. Spacetime is the dynamic stage, but not the actor.

With the given interpretation, the two tensors are so different that an identity cannot be logically reconstructed. If the equation is correct (assumption A-01), the tensors must get a „better“ meaning. Therefore, the DP makes the following approach:

• G_{\mu\nu}: Every gravity is a clearly localized, continuous and geometrical expression in the spacetime as spacetime extension. It needs only the spacetime itself.
• T_{\mu\nu}: Any form of energy and mass in the tensor (thus the whole tensor) is a clearly localized, continuous and geometrical expression in spacetime as spacetime density. It needs only the spacetime itself.
• A density in spacetime causes a stretching.

This interpretation is sufficient to generate all postulates and principles of GR. Therefore, this approach is formulated as an assumption in the DP. All further considerations are made on this basis.

Assumption A-02:

• Each mass-energy equivalent is a spacetime density.
• Gravity is a spacetime stretching
• A density creates an stretching.

The interpretation as stretching and density follows from the assumption A-01. This results from the mathematical solution of the GR. The path element towards a mass becomes larger, therefore a stretching. The geometrical counterpart for it is a density. Therefore, a density and an stretching have been chosen in the consideration.

One can also explain it as follows. If a density is to be produced at a place, then the „material“ must be drawn from the environment to the density, which corresponds to a stretching to the density. On the basis of these considerations it is correct that the gravity is directed to the mass. All masses attract each other, because the gravity is directed to the mass.

A stretching away from the mass, would distribute the density on the surrounding spacetime. By this stretching a „distributed density“ would arise in the spacetime and there is no more mass accumulation. This does not correspond to the observed universe.

Since a stretch is formed from the totality of spacetime, it must affect the entire spacetime and must not have a limited range. The density must represent a local behavior. If the stretch is generated spherically in a radial direction towards a mass, the length/extension of the radius is longer (as seen from the outside) than is possible in the given volume. This „extension“ of the radius must be accommodated in this volume. This is called the curvature of space.

In order to keep the text more understandable, only gravity is spoken of (if not necessary) and no more of a spacetime stretch or spacetime curvature.

### Space density is energy and mass

The identity between spacetime density and a mass-energy-equivalent is already fixed in the assumption A-02. Here, this equality, by the simple results from the SR, is developed further. The length contraction and the increase of the energy up to infinity, if one wants to reach the SL, is sufficient. This combination can be represented by a spacetime density and explains the difference between energy and matter.

If an object with rest mass is to be brought up to the SL, one must spend more and more energy for the further acceleration. Theoretically up to infinity. The SL can therefore never be reached. If an object corresponds to a spacetime density, one can increase the density by „compressing“ a space dimension more and more. Maximum up to zero. Then the object would have an infinite density. But an existing space dimension cannot simply disappear. Therefore this limit is not attainable. Thus the spacetime density can be equated with the energy. The length contraction from the SR is directly the increase of the density.

If an object possesses all 3 space dimensions in our spacetime, this object can never reach the SL. Therefore, an object with rest mass always occupies all space dimensions in our spacetime. The increase of the momentum is directly an increase of the spacetime density in a specific spacetime direction. As will be explained later in the chapter „Standard model“, the rest mass is for all elementary particles, the difference in the number of space dimensions in the low-dimensional image. The more space dimensions are involved, the more difficult it is to change the space density. Since only the number enters here, the rest mass is always identical for every sort (geometrical picture) of elementary particles.

In the reverse conclusion, an object, which moves with SL, must have one space dimension less, otherwise the SL is not attainable. This object must then, without external influence/interaction, always move with the SL. The SL is the existence condition for the object, since a space dimension is missing. The absence of a space dimension is like a switch for the SL. With only two space dimensions always SL, from three space dimensions never SL.

As an example an electron and a photon. The photon occupies only two space dimensions in its geometrical image in our spacetime and always moves with SL. There is no acceleration on SL. The SL is, without another interaction, the only possible form of existence. The electron occupies in its geometrical mapping in our spacetime three space dimensions and can therefore never reach the SL. The momentum is a „directed spacetime density“ and seems to increase the rest mass of an electron, because the density increases with the momentum. This difference, in the geometric mapping in our spacetime is, the distinction of energy and mass. Since both mappings are a form of spacetime density, they can be converted into each other via a interaction. Three important considerations follow from this:

• The length contraction from the SR is directly the increase of the spacetime density/energy of an object. This is absolutely real and not only an observation.
• If the length contraction from a motion represents a higher spacetime density and therefore a higher energy, then every object in our spacetime must have a state of motion. The spacetime density can be equated directly to a state of motion in our spacetime. The spacetime density itself represents motion. A completely resting object in a spacetime is not possible, because every object is an energy = spacetime density and therefore has a state of motion.

mass-energy equivalent = spacetime density = state of motion

• In the spacetime, the spacetime does not move. The spacetime density or energy itself is the movement. Thus, a rest mass must possess some kind of „inner“ motion. To the topic spin of an elementary particle at the chapter standard model more.

This is a different conception of energy, motion and spacetime than in all previous theories.

### Density of the spacetime definition DRD

It is not possible to recognize a density of spacetime locally in the density itself. All objects in the spacetime are as density, a geometrical mapping in the spacetime. Since spacetime is the definition of geometry, no change in its own spacetime can be detected directly. With the density the definition of the spacetime changes and with it the geometry constant with. Locally, in any density, a meter always remains a meter. In fact, the density in an object corresponds to the „density of the definition of spacetime or a density of the metric“. For this reason, in the DP, we do not simply choose the name spacetime density, but density of spacetime definition (DRD). The abbreviation DRD comes from the German as „Dichte der Raumzeitdefinition“.

The definition of the recognizable geometry (metric) is always adapted to the density. The change of the geometrical basis is not detectable under any circumstances, locally in the density itself. With it all possibilities disappear to recognize this directly itself. This is an important aspect of the DRD. Thus the DRD can be determined only in the comparison with another DRD. Everybody can choose his DRD as zero point. From this the principle of relativity is derived. It does not have to be postulated. As a comparison value, for example, the energy is chosen. With an electron the rest mass is the zero point and the further changes of the DRD (other energies) come by the movement (momentum). With a photon the motion (SL) is the zero point and the frequency gives the change of the DRD. However, all indications always correspond to a DRD.

To keep the text more understandable, any mass-energy equivalent or object will be referred to only as DRD. A mass-energy equivalent could be called a positive DRD and gravity a negative DRD. To avoid confusion, gravity will continue to be referred to as gravitation rather than negative DRD. DRD is always only a higher density of the spacetime definition than the vacuum.

With gravitation almost the same statements are valid. Therefore, a free falling DRD in a gravitational field does not feel any force (change of the DRD). But there are two decisive differences:

• The gravity is directed to the mass possesses and from space time to space time a different value. The „pulling of the material“ dilutes in the volume.
• The stretch must take place in a curvature of the spacetime. The „extension“ of the way element must be taken up in the spacetime itself. This happens by the curvature of the spacetime in itself.

The energy represents the measure for the DRD. The curvature is the indication for the gravity. There is no need for additional new units.

### Force as change of DRD and gravity

The old term „force“ is dug out again and used in a new view.  Everything recognizable in our spacetime is a DRD or a gravity. A change of the DRD or the gravity can be understood, completely in the sense of the classical mechanics, as a force. However, the change happens on two very different ways.

The first three basic forces: strong nuclear force, weak nuclear force and electromagnetic force, have in common that between the DRD directly an exchange of DRD takes place. This process is described by the QFT. The description of the exchange in quantized form is part of the chapter „Quanta and waves“ and is not presented here further.

With the gravity the change is completely different. In a gravitational field, every DRD is from the own view, completely force-free. Gravity is not directly detectable locally for a DRD itself. One needs, as with the change of a DRD, an external feedback, in order to be able to recognize this. There are three possibilities for this:

• Something stands in the way of the DRD. On earth, this is taken over by the earth’s surface.
• The gravity changes from space time to space time in the direction of mass. At a very strong gravity, like at a BH, this difference can become large. A structure of DRD (e.g. a human being) feels then a force, within its expansion, without a feedback from the outside.
• The recognition of the change of a movement by the spacetime curvature on a geodesic. For example, the gravitational lensing effect.

A DRD with a density into only one space dimension is a vectorial DRD and corresponds to an momentum. A rest mass with all space directions corresponds to a scalar DRD

### Lagrangian formalism as compensation

In theoretical physics, many (almost all) equations and principles are derived via the Lagrangian formalism. Therefore, this formalism must be based on the principle of „density generates stretch“ as well.

The simplest form is L = T – V. With L as effect, T as kinetic energy and V as potential energy. If we choose as an example for the kinetic energy the momentum of a DRD and for the potential energy the gravitational potential, we can see the balance directly. The effect L, is the balance between the kinetic energy „DRD“ and a potential energy „gravitation“ in our spacetime.

Better one sees the comparison with the Lagrange equation of the second kind. With  \cfrac{d}{dt}\space \cfrac{\partial L}{\partial \dot{q_i}}\space -\space \cfrac{\partial L}{\partial q_i}\space =\space 0  one can see that a variation in the system must always balance. The dynamics of a DRD always runs with the smallest possible variation of the DRD. A variation of it must run toward zero.

A DRD behaves in the dynamics in such a way that this is balanced as few as possible by a potential. Since everything consists of DRD, every object follows this logic. For the geodesic in gravitation the exact opposite results. Gravity wants to balance as much as only possible DRD by the geodesic. Therefore, the geodesic is always in the direction of the center of gravity.

In classical mechanics, the Lagrangian formalism produces the least action. In relativistic physics, this corresponds to the largest possible proper time for an object. In DP, for a DRD, this becomes the least change. For gravitation the greatest possible change.