NEWTONIAN DYNAMICS

Cosmic Dark Matter and Energy

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NEWTONIAN DYNAMICS

EXTENDED NEWTONIAN DYNAMICS(END)

Space consists of an additional extremely large and bounded 4^th spatial dimension. The 4^th spatial dimension apparently forms a closed large cylindrical topology. This appears to be a crucial fact of the universe that cannot be easily dismissed, as it simply enabled me to explain and derive all the observed cosmological anomalies without resorting to fictitious matter and energy. I hope the following arguments would justify the need for a very large extra space dimension.

We live in our usual 3 dimensional brane if you will, and have no direct access with the 4^th spatial dimension. This must be accepted, but gravity plays a major role in the 4^th spatial dimension, as will be apparent in the following unusual analysis:

Figure 1 above depicts the new assumed topology of space. It has an additional extremely large 4^th spatial dimension which is cylindrically bounded. The 3 dimensional space(xyz) is shown extending to infinity on the cylindrical length. The cylindrical circumference is the 4^th dimension(w), which is therefore bounded and finite. Let ‘R’ denote half the circumferential distance of the cylindrical 4^th dimension of space and ‘r’ denote the distance in the 3 dimensional space(xyz). The radial extent points to the bulk 5^th dimension(u) where everything is embedded. Note that the bulk 5^th dimension, what ever it may be, plays a vital role in that it lets the 4 dimensional space and string particles, virtual room to flex into(Read my other publications, MECHANICAL ORIGINS of GRAVITATION, WEIGHT and INERTIA, and A RELATIVE MASS FORMULA for BARYONS).

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DECENDING into OUR WORLD

From this 5^th dimensional bulk vantage point, we see the whole picture. We see a 3 dimensional world with a fourth bounded cylindrical spatial dimension of four volume V(4) given by:

V(4) = k*R*r^3 ……………………………………………………1

Where V(4) is the hyper volume or call it 4 volume.

‘R’ = half the distance along the bounded cylindrical circumference dimension.

‘r’ = the distance in 3D space, example from a mass ‘M’.

‘k’ = a spherical factor, to be determined.

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i.e.

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It appears therefore that the gravitational field strength distribution is governed by two radial spherical surface areas surrounding a central mass ‘M’, one, due purely to a curvature term within our 3D world, and the other associated with a curvature from the 4^th dimension. These two together provides inverse square force laws as follows:

New Force law = > 1 / r^2 + 1 / rR …………………………4

For ‘r’ less than ‘R’, the two radii merge into a single distance ‘r’ and we have R = r therefore

Force law => 1 / r^2 + 1 / r^2 = 2 / r^2

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We normalize this to match the Newtonian Force Law, and equation 4 becomes

New Force law => 1 / 2r^2 + 1 / 2rR …………………………5

This derivation also supports my original analysis that I published in July 2005 at the Web site Cosmidarkmatter.com, where I resorted to Gaussian Curvature of Space to derive equation 5. At that time I failed to normalize equation 5 by using the mean curvature and dividing through 2. Having two ways to arrive at equation 5 naturally gives strong support to my hypothesis to Extend Newton’s gravitational law.

NEWTON to END

Let us consider what unfolds! if the 4^th dimension is bounded and fixed in size by the constant circumference ‘2R’. From equation 5, when ‘r’ is less than or equal to ‘R’ we have( where ‘R’ becomes ‘r’ ) the familiar Newtonian force law ‘1 / r^2 ‘.

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Effectively, when ‘r’ reaches ‘R’ in the cylindrical distance, ‘R’ becomes constant as far as gravity is concerned and consequently the force law is governed by equation 5, thereby extending the inverse square law with an additional effectively inverse distance law.

MASS and CURVATURE of SPACE

As Einstein postulated, matter curves space. Therefore, the curvature of space is governed by the central mass M. The curvature term ‘1/r’ enters the Newtonian force law and the corresponding acceleration as follows:

g(N) = (( Sqrt(GM))/r * ((Sqrt(GM))/r )

The second term in equation 5 therefore can be arranged as:

g(r) = (( Sqrt(GM))/r * ((Sqrt(GM))/R )…………………….

Since ‘R’ is a bounded space with it’s own extrinsic and fixed curvature, the curvature potential will be constant having the value Sqrt(GM)/R.

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g(r) = GM/2r^2 + GM/2rR ……………. 6

As seen from equation 6, Newton’s inverse square law is gradually turned in to an inverse distance law after a very large distance ‘r’ >>‘R’ is reached. Therefore the acceleration

transitions from inverse square to an effective inverse distance law after r = R.

g(r) => (GM / 2rR )………………………………………..7

It appears that after all, gravity is diluted right under our nose from the very beginning by the 4^th spatial dimension, all the while, at everyday scales, and we would never have guessed(Let alone Newton) why gravity is so week. Now imagine, if ‘R’ was only even as tiny as atomic size, Gravity would be as strong as all other forces in nature.

Distance scale ‘R’ in the 4 dimensional space therefore sets up a smooth cutoff point for the action of Gravity at long range. This therefore gives a natural reason for MOND, but,

as I will describe below, I am going to have to modify MOND for more reasons to follow. The major draw back for me with MOND is to have a sharp fixed cutoff acceleration point, leading to momentum/energy conservation problems. My equation 6 is what operates in nature with a smooth transition to the Extended Newtonian regime. MOND is a limiting case of Extended Newtonian Dynamics(END) of equation 6.

END to MOND

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This therefore gives us the MOND equation, providing, the MOND conjecture is true.

In reality, MOND should be modified to END equation 6 giving us:

g(r) = (1/2)( g(N) + Sqrt (g(N)*g(0))…….………9

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An END to DARK MATTER

I will now show that there is no need to invoke a fictitious dark matter to account for the flat Rotation curves of Galaxies. If equation 6 is applied to orbital motion surrounding

a large mass ‘M’ we obtain:

V(r) = Sqrt ( (GM/2)*( 1/r + 1/R))………………….10

Where ‘V(r)’ is the rotational velocity at distance ‘r’ from mass M.

It is clear that V(r) approaches Sqrt( (GM/2)*(1/R)) as ‘r’ become very large.

Therefore the rotation curve becomes flat at large ‘r’.

In order to establish the extent of ‘R’ at present time, I tried to fit equation 10 to

numbers available for the Milky Way Galaxy.

For example we know the Sun’s orbital velocity should be 160 Km/s if only Newtonian law applies. And the Sun is at a distance of 8.5 Kpc from the center of the Milky Way.

This allowed me to figure out using Newton’s equation, the value of GM as approximately about 218,000 (Km/s)^2*Kpc.

Using this value in equation 6, I worked out that ‘R’ is approximately 3.0 Kpc,

so as to match the actual rotational velocity of the Sun, i.e. 220 Km/s.

Therefore, at present, the cylindrical circumference of the 4^th dimension is 6.0 Kpc.

This gives the long range flat rotation velocity of the Milky Way galaxy at 190 Km/s.

This also gives us the cut off acceleration g(0) at the distance R = 3.0Kpc at

8.0*10^(-13) Km/s^2.

I would now like to modify equation 10 to take into account the central rotation velocity part of the galaxy like:

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Where ‘x’ is a distance squared value peculiar to galaxy type. Low values of x<12 would emulate HSB galaxies and high values x>12 would emulate LSB galaxies. If x=0 then equation 10 is recovered.

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Modified END

R = 3.0 Kpc was obtained using the Milky way parameters and therefore is a distance in local or present space in the 4^th dimension. But if you read my other publication "MODIFIED HUBBLE DYNAMICS(MOHD)", I have suggested that the 4^th bounded spatial dimension is contracting in time. Therefore R is not constant back in time. So ‘R’

was larger if we go back in time. Therefore the END formula predictions should vary with look back distance.

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It appears that the MOND cutoff acceleration(1.0 Km/s^2) was selected as a compromise to match the observed rotation data, which is close to the 0.9*10^(-13)Km/s^2 obtained above at a time when R was 9Kpc.

This is not all, from equations 6 and 9, I stated

g(0) = GM / R^2

where GM is the current local mass value corresponding to R = 3Kpc

and g(0) = 8.0*10^(-13) Km/s^2.

At large distances equation (9) gives us that the acceleration tends to

g(r) = Sqrt( (GM/r^2)*g(0))

The rotational velocity is therefore related as:

(V(r))^4 = GMg(0)

This therefore predicts the Tully-Fisher Relation.

If then, we focus on flat spiral galaxies:

GM is proportional to := G*Pi* R^2 *rho(1)

Where rho(1) is the mass per unit surface area of flat spiral.

Therefore g(0) is proportional to G*Pi*rho(1), and this is constant.

Therefore for flat spirals, g(0), the cut off acceleration is constant irrespective of the extent of the 4^th space dimension. This would explain the very good fit of spirals to MOND.

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As for globular galaxies and galactic clusters, GM is proportional to :

G* (4/3)Pi*R^3*rho(2), as the system structure can be considered to be populated more or less evenly throughout the volume.

Where rho(2) is the mass per unit volume of the system. In this case, g(0) is proportional ‘R’. Thus globular galaxies and galactic clusters, rotation curves flat part will appear to increase with look back time.

Long Range Gravitational Space & Dark Energy

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This equation shows that at short range the potential energy is negative and after a certain

distance greater than ‘R’ the potential energy becomes positive for ever. Note, the sign convention does not matter as long as we realize that the short range potential energy is gravitationally binding and therefore the opposite sign of the long range potential is expansive. At very large distances the potential energy increases logarithmically according to:

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Dark Matter Revealed, through //////////////////////////////////

I will now show, using a very different argument, how and where the illusive dark matter comes into reality.
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Dividing equation 18 by 19 we get the fractional dark matter to total mass content.
The actual dark mass profile would be somewhere in between the two extreme systems I derived above. It is easily seen that the percentage of dark matter compared to total mass increases from 50% when ‘r=R’ to nearly 100% for very large ‘r’. For practical system sizes, the value is around 95%, as currently believed.
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Fig 5 shows the rotation velocity
profile obtained from
equation 21(dotted) and
equation 20(solid).
It is seen that at long range, equation
21 becomes flat, whereas equation 20
increases at first, then decreases to
a minimum and rises again more slowly,
after which it settles to a flat straight line.
Interestingly the solid curve which can represent a distributed cluster velocity profile is seen to have higher initial rotation velocities and then drop before rising again and flatten out at long range. Of course, equations 14 and 19 represent the two extremes in radial mass profile. In reality, the actual mass of the system becomes constant at a large distance, where equation 14 applies. So the actual enhanced rotation curves depend on the visible mass distribution profile of a system. Like someone noted, it is as if there is a conspiracy between real matter and dark matter, and now you know why.
Gravitational Lensing? No Problem
Unlike MOND, my hypothesis actually enhances gravity by effectively multiplying the actual mass of a galactic system at large distances from the center. This would naturally produce conditions necessary for gravitational lensing effects.

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A Corollary
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This looks a wild conjecture, but maybe that is why we observe the 8.0 Km/s^2 Pioneer acceleration anomaly at long range from the Sun, which my equation for g(0) precisely gives.
In Conclusion
My hypothesis therefore predicts no mass discrepancies on any system below a distance ‘R’ where Newton’s law prevails. R = 3Kpc in local space, therefore there will not be any mass discrepancy anywhere below a distance of minimum 3Kpc. Since ‘R’ was larger back in time per my article "MODIFIED HUBBLE DYNAMICS(MOHD)", the flat rotation velocities of systems should be lower with look back time. This means also that there is no constant acceleration parameter as suggested by MOND.
I have put in a lot of years of thinking into this pure research. I have no facilities and time to investigate my hypothesis further. I would leave this work to other academics to verify against observational data. An interesting line of work would be to simulate an N body problem using equation 20 in a computer, to see how the system settles down to a stable dynamic structure after a long period of time, with the following derived DM density radial profile rho(r) equation 22, where, ‘d’= radius of visible bulk of real matter & rho(0)= average density of real visible matter.
rho(r)/rho(0) = (1/4)/(R/d)/(r/d)^2 – (1/2)/(r/R)/(r/d)^3 + (1/4)/(r/R)^3/(r/d)^3 …22
Another line of work would be to, actually re-investigate on Kaluza’s five dimensional relativistic formulation of gravity, without compactification of the 4^th space dimension, i.e. with the cosmic size 4^th dimension as I suggest.
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