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Within the context of Newton’s theory of gravitation, restricted to point-like test particles and central bodies, stable circular orbits in ordinary space are related to stable circular paths on a massless, unmovable, undeformable vortex-like surface, under the action of a tidal gravitational field along the symmetry axis. An interpretation is made in the light of a holographic principle, in the sense that motions in ordinary space are connected with motions on a selected surface and vice versa. Then ordinary space is conceived as a 3-hypersurface bounding a
*n*-hypervolume where gravitation takes origin, within a
*n*-hyperspace. The extension of the holographic principle to extra dimensions implies the existence of a minimum distance where test particles may still be considered as distinct from the central body. Below that threshold, it is inferred test particles lose theirs individuality and “glue” to the central body via unification of the four known interactions and, in addition, 1) particles can no longer be conceived as point-like but e.g., strings or membranes, and 2) quantum effects are dominant and matter turns back to a pre-big bang state. A more detailed formulation including noncircular motions within the context of general relativity, together with further knowledge on neutron stars, quark stars and black holes, would provide further insight on the formulation of quantum gravity.

According to current QCDM cosmologies, the universe as a whole is increasingly expanding e.g., [

A simplified description can be made within the framework of Newton’s theory of gravitation, where spherical-symmetric matter distributions can be conceived as point-like outside their boundaries. The concept of space curvature may be introduced considering a vortex-like surface under the action of a tidal gravitational field along the symmetry axis, where the distance of a surface point from the symmetry axis relates to the distance of a test particle from a central body. The current paper aims to answer two specific questions restricted, for simplicity, to circular motions.

First, could stable circular orbits (radius, R) described by a test particle (mass, m) around a central body (mass, M) be related to stable circular paths described by a test particle (mass, m) on a massless, unmovable, undeformable vortex-like surface, subjected to a constant gravitational force,

Second, could the above mentioned holographic principle be extended to ordinary space, conceived as a 3-hypersurface, where test particles are subjected to a constant gravitational force from extra dimensions? If yes, could quantum effects and unification of the four known interactions be interpreted in this context?

The first and second questions raised are dealt with in Sections 2 and 3, respectively. The discussion is drawn in Section 4. The conclusion is shown in Section 5. A special vortex-like surface and an analogy for the invariance of light velocity are presented in the Appendix A1 and Appendix A2, respectively.

Let a test particle of mass, m, move along a stable circular orbit of radius, R, around a central body of mass, M. Let both the test particle and the central body be conceived as point-like. The balance of gravitational and centrifugal force at a generic point of the orbit, after little algebra reads:

where G is the constant of gravitation and V the circular velocity.

Let a vortex-like, massless, unmovable, undeformable surface (in short, surface) be subjected to a tidal gravitational field along the symmetry axis, implying constant attraction directed downwards, as depicted in

Let the surface be defined as

where the vertical axis is directed downwards. Let

The equation of the tangent is:

where

With regard to a test particle on a selected point, P, of the surface, the gravitational force,

the centrifugal force,

Motion along a stable circular path,

where v is the circular velocity.

The combination of Equations (1) and (5) yields:

and the substitution of Equation (4) into Equation (6) produces:

where the intercepts of the equation of the tangent,

for a formal demonstration, an interested reader is addressed to Appendix A1.

The above results may be summarized into a single statement.

Holographic principle. Given a central body of mass, M, and a vortex-like, massless, unmovable, undeformable surface subjected to a tidal gravitational field along the symmetry axis, implying constant attraction directed downwards, any test particle moving on a stable circular orbit of radius, R, and velocity, V, has a counterpart on the vortex-like surface, moving along a stable circular path of radius,

In general, the above statement should be conceived as a principle of corresponding states: the term “holographic” has to be intended in the sense that motions in ordinary space are connected with motions on a selected surface and vice versa.

The above considerations could, in principle, be extended to general relativity but, for simplicity, are restricted to classical mechanics, which implies velocities up to infinity. If light velocity is assumed as an empirical upper limit,

according to Newton’s theory of gravitation. Its counterpart in general relativity, using Schwarzschild’s metric, reads e.g., [

which is three times larger.

Similarly, stable circular paths on a vortex-like surface, centered on the symmetry axis, cannot occur below a critical radius,

where

Let ordinary space be conceived as a 3-hypersurface bounding a n-hypervolume, totally

With regard to a planet surrounded by a liquid shell, a global ocean say, the source of the gravitational field within the hypervolume is represented by the whole planet and matter within the hypersurface by the water surface, respectively. In absence of matter (empty ordinary space), the water surface is totally flat. In presence of negligible amount of matter, the water surface can be considered flat to a good extent. In presence of considerable amount of matter, the water surface is curved by gravitational interaction. A test particle on the curved surface is subjected to the tangential component of the gravitational force, as the normal component is balanced by the surface tension. In principle, the curvature angle with respect to the horizontal plane ranges from

Keeping in mind the above mentioned analogy, let the whole range of curvature be represented as an equilateral hyperbolic funnell i.e. a revolution figure related to the rotation of the bottom branch of an equilateral hyperbola around the axis, z, as depicted in

due to the symmetry with respect to the rotation axis, z.

Let a test particle of mass, m, move on a stable circular orbit of radius, R, around a central body of mass, M, both conceived as point-like for simplicity. Let

where G is the constant of gravitation and V the orbital velocity. The combination of Equations (13)-(15) yields Equation (1) and, in particular, Equation (9) assuming light velocity as an empirical upper limit.

The gravitational force within the hypervolume via Equation (14) reads:

where

in terms of the orbit radius, R. On the other hand,

Let a finite matter distribution, spherical-symmetric for simplicity, be defined by a total mass, M, within a finite radius, R. Accordingly, curvature effects are directly proportional to the angle,

Curvature effects are larger on the surface, which can be related to the angle,

The velocity of a test particle in a stable circular orbit, on the surface of the matter distribution, is increasing for decreasing R until

In summary, stable circular orbits of test particles around a central body can occur for

According to the holographic principle inferred in Section 2, stable circular orbits of a test particle of mass, m, around a central body of mass, M, can be related to stable circular paths of a test particle of mass, m, on a vortex-like, massless, unmovable, undeformable surface, subjected to a tidal gravitational field along the symmetry axis, implying constant attraction directed downwards. In other words, the gravitational action on the ordinary space can be related to the gravitational action on an axisymmetric surface. By analogy, the gravitational action on a n-hyperspace can be related to the gravitational action on an axisymmetric 3-hypersurface, coinciding with the ordinary space. For reasons of simplicity, considerations have been restricted to circular motions within Newton’s theory of gravitation, but it can safely be expected an extension to generic motions within the theory of general relativity.

If gravitational attraction takes origin from a n-hypervolume, the origin of (ordinary) space curvature is no longer due to matter in itself, but to gravitation from extra dimensions. Then matter appears to be entirely passive, in the sense that it can no longer be thought of as source of gravitational field. In this view, extra dimensions extend as well as ordinary ones, contrary to assumptions of superstring theories, where extra dimensions are highly compressed e.g., [

The main consequence of the above mentioned analogy is the occurrence of quantum effects, in the sense that a test particle can approach a central body (both considered as point-like) up to a threshold,

Within the context of Newton’s theory of gravitation, for an assigned (point-like) test particle and (point-like) central body, stable circular orbits in ordinary space have been related to stable circular paths on a vortex-like, massless, unmovable, undeformable surface, subjected to a tidal gravitational field along the symmetry axis, implying constant attraction directed downwards, and a holographic principle has been proposed. If ordinary space is conceived as a 3-hypersurface bounding a n-hypervolume where gravitational interaction takes origin, the extension of the above mentioned holographic principle has implied a minimum distance behind which test particles lose theirs individuality (intended as collection of intrinsic properties) and “glue” to the central body via unification of the four known interactions, behind which 1) particles can no longer be conceived as point-like but perhaps as strings or membranes; and 2) quantum effects are dominant and matter returns to a pre-big bang state.

The generalization of the procedure to noncircular motions within the context of general relativity, together with further knowledge about neutron stars, quark stars, black holes, e.g., [

Thanks are due to the Editor and the referee for their comments.

RobertoCaimmi, (2016) Gravitation, Holographic Principle, and Extra Dimensions. Journal of Modern Physics,07,426-434. doi: 10.4236/jmp.2016.75044

As a guidance example, let the meridional section of the vortex-like surface coincide with branches of equilateral hyperbola, as:

where the right-hand side is positive or negative according if the hyperbola branch lies on the fourth or third quadrant, respectively, see

The intersection of the hyperbola with a generic straight line, expressed as:

can be inferred via substitution of Equation (19) into Equation (18), as solution of the related second-degree equation in w. The result is:

where the double signs are uncorrelated.

The straight line is tangent to an hyperbola branch provided the discriminant is null, as:

accordingly, the coordinates of the tangential point are:

from which the slope and the intercepts of the tangent straight line are inferred via Equations (18) and (19), as:

in terms of the coordinates of the tangential point, P. Finally, the substitution of Equations (25) and (26) into (4) yields:

accordingly,

The substitution of Equations (25) and (26) into (7) after little algebra produces:

which is equivalent to Equation (8) and, in particular,

Let bodies (particles, in particular) be conceived as vessels moving on a global ocean. Let light be conceived as a circular wave on the ocean which can be induced via e.g., vessel oscillations along the vertical direction. Then waves propagate on the ocean surface, regardless of vessel motion. The concept of “light velocity in vacuum” appears to be meaningless e.g., [

Conceptually, wave velocity can be determined in the following way. Let vessels move between equally spaced series of buoies, similarly to electric trains between high-voltage tralisses. Let

In addition, wave velocity cannot be exceeded by vessel velocity,

Finally, the reference frame comoving with the ocean i.e. at rest with respect to the ocean can be conceived as “privileged”.

As mentioned in Section 3, ordinary space can be related to ocean surface and extra dimensions to ocean volume.