A New Cosmic Global Gravitational Law ….by Tissa Perera 7/20/05 P4
1. A total solution of major cosmological problems, it’s all in the Curvature of Space
2. I will now assume the topology of the Universe to be cylindrical in nature, where the 4th spatial
dimension (w) Occupies the bounded radial circular extent and the regular 3 dimensional(xyz)
space occupies the very large length extent of the cylindrical space as depicted below, where
R is the radius of curvature of the 4 dimensional space.
1. In the light of Einstien’s geometrical spatial curvature interpretation of Newton’s gravitational forces,
I further assume that the correct interpretation of the inverse square distance law is obtained simply
by using the Gaussian Curvature of space.
2. In the absence of matter this space has zero curvature as it should be.
3. Let us see what happens to the Gaussian curvature of this space in the presence of matter. If a mass M
is placed in this space it creates a local curvature of all surrounding 4D space. In the radial plane of
the cylinder there are two superimposed curvatures at a distance r from the mass M, one from the
local curvature 1/r and the other from the global curvature 1/R due to the cylindrical boundary.
Hence the total curvature in the radial plane is (1/r + 1/R). The other curvature in a plane along the
cylindrical axis is only 1/r.
These two curvatures on orthogonal planes are the two principal curvatures at any point distant r
from a mass M. Therefore the Gaussian curvature at that point is 1/r ( 1/r + 1/R ).
This should replace and modify the inverse square distance law of the Newtonian
Gravitational formula. We therefore obtain the new Global gravitational force law as:
F = GM ( 1/ r^2 + 1 / R*r ) …………….(1)
This contains the additional global curvature force term GM/R*r which varies as an inverse distance term with a much reduced force due to the large global radius of curvature R.
Let us apply this formula to orbital motions around the central mass M. We therefore obtain the orbital velocity V as:
V = Sqrt GM( 1/r + 1/R) ……………….…….(2)
This therefore gives us a flat rotation curve at large r distances from the central mass M. Within my limited resources and information available I fitted this equation to a crude match of the rotation velocity curve of the Milky Way Galaxy and obtained the following parameters:
V = Sqrt 195000(r/(22+r^2) + 1/6.5) ……………..(3)
Where distances are in Kpc and velocity in Km/s.
Note the modification of the inverse square law term just so to simulate for the distributed galactic central mass, which gives a more realistic inner rotation curve part for the velocity curve.
Approximately, GM = 195,000, and R = 6.5 Kpc or 22,000 LY
The observed rotation velocity curves of Galaxies are a natural consequence of the new space topology.
Note, that the Radius of Curvature of the 4th (w) dimension is established a value approximately 22,000 LY’s. Which is fairly small compared to the galactic dimensions, but enough to make a difference to gravity at large scales. At smaller distance scales the inverse square law completely swamps the cosmic curvature part 1/R but may still be detectable. The circumference of the curved dimension is 2Phi R approximately 133,200 LY. It appears that a galaxy can almost occupy the curved 4th dimension circumferential distance.
3. Yes to Dark Energy(Problem of acclerated cosmic expansion)
I will now show that Dark energy is a natural consequence of the new gravitational force law. Curvature Potential Energy in space surrounding the central mass is given by, E = Integral F*dr This gives:
Curvature Energy = Integral -GM( 1/r^2 + 1/rR )
Therefore Curvature Energy = GM( 1/r - (1/R)ln r )………….……(4)
This equation is telling us that at large distances r the spatial energy is negative.
It appears that this kind of spatial topology naturally has negative energy every where, except near mass. Therefore it is a natural answer to the so called Dark Energy problem. That is, Dark Energy is nothing but a natural consequence of the cylindrical 4D topology of space curvature.
Now, at a certain distance ‘r1’ from a central mass M, where R = r1 ln (r1), a bubble of zero energy exists.
Below this distance r1 the spatial energy is positive. So surrounding all objects there is a halo of positive Energy trying to shrink space and outside this zero energy bubble space wants to expand. This zero energy bubble radius works out to be approximately 14,675 LY in the case of the Milky Way Galaxy.
There appears to be a Positive Dark Energy sphere surrounding the center of Galactic objects.
4. Yes to Red Shift Periodicity (Problem of quantized red shifts)
This last problem of red shift periodicity happens to be an illusion created by the new spatial topology.
It just so happens that light from the very distance stars are seen directly as well as after circumnavigating
the Cylindrical space once, twice and a multiple times. Each time the light wraps around the cylindrical Circumference and travels the Length, it additionally travels a multiple amount of distance exactly equalto the Circumference of the 4th dimension. We therefore observe an apparent equidistant Spacing of each star image and hence conceive a pattern of equally spaced red shifts according to the general expansion red shift rule of space. I think I have clarified the idea. But lets look at some details.
The figure above shows a star S and an observer O at a large distance receiving light that circumnavigated
the additional cylindrical extent of the 4th Dimension once and twice around and so on a multiple number of this circumference.
Since the extra distance around the cylindrical space is relatively small, the apparent spacing is of the order of 2 phi * R Which is approximately 133,200 LY’s, This gives us a basic apparent distance quantization.
According to Hubble’s Law the basic apparent velocity quantization therefore is about 2.665 Km/s. This I believe is the basic source of periodic red shifts. This is exactly what is observed, and encourages further analysis of the constant parameters in the equation using a wide set of galactic data. Additionally this may
also explain why super clusters appear to form into long thin filaments and leave large voids in between.
I have not yet worked out exact numbers to clearly justify this view of the Universe. I have made a crude effort to find a match of the formula to a sketch of the rotation curve of the Milky way I came across. I am sure, further refinements are needed and it would be interesting to even see a relativistic extension of this formula, bringing in the missing time dimension to formulate a 5 dimensional extension of GR. Finally, I am convinced that such a simple concept that naturally explains all major cosmic mysteries,needs to be scrutinized further.
Contact: Tissa Perera,
eMail: firstname.lastname@example.org Web Site: CosmicDarkMatter.com
Phone: 214-263 9166
Cosmic Darkness Revealed