NEWTON'S GRAVITATION AND COSMIC EXPANSION

Part 1 of 3 Parts

James Constant

grav@coolissues.com

Introduction

The present theory in three parts is based on classical mechanics and special relativity. Each part of the work is presented as a separate web page. This first part of the work describes the cosmic force of gravitation, in terms of classical gravistatics, analogous to electrostatics, describes the action at a distance and retarded action types of universes, respectively, as likely and unlikely, Hubble's Linear Law, non cosmic and non relativistic forces of expansion and Newtonian gravitation within the span of Hubble's Linear Law. It also provides values for the cosmic mass density, cosmic rest mass and radius. The theory disproves Mach's Principle and considers the Big Bang Theory unlikely. The second part provides a nonlinear extension to Hubble's Linear Law, describes the relativistic cosmic forces of expansion and gravitation, and the effect of mass shielding by the negative field of galactic and cosmic mass, The theory is based on the idea that what we observe in the universe by redshift measurements exists today. gravitation.coolissues.com/Relativistic/ncosm2.htm The third part speculates about the nature of sources for the observed red shifts and microwave background and about the detectability of black holes. It also compares the present theory with Einstein's theory and gives a summary of the present theory. Included in the third part are Appendices for Useful Constants and Distance vs Speed Tables. gravitation.coolissues.com/Conclusions/ncosm3.htm

COSMIC FORCE OF GRAVITATION

Gravistatics

In gravistatics, the field E which is associated with a mass M is given in terms of a potential and satisfies the static gravi-Maxwell's equations

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (1.1)

in which G is Newton's constant and is the mass density of M seen at the field location of E. Using the last two of equations (1.1) gives Poisson's equation

.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. (1.2)

which must be satisfied by the gravitational potential at all times.

By way of example, the static potential of a sphere of uniform density and mass of radius R is found by solving equation (1.2) ¹

..................................................

.......................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(1.3)

from which we find

...............................................

.............................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(1.4)

which show that the field is an attractive one inversely proportional to r² when and proportional to r when . The potential energy and force are given by V=m and F=mE, respectively.

F = mE is a statement about the force F on the mass m of a body located in the field E of M. While equations (1.4) describe how the field E is related to the first mass M and distance r, F=mE describes how force F is related to a second mass m and to field E. In a simple case, m and M are constants and therefore E and F are functions of distance alone. In special relativity, m and M may also vary with speed and therefore E and F become functions of speed as well as of distance. Since F=mE, the forces are

............................................

.......................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1.5)

which say that force F_o applies outside mass M of radius R and force F_i applies inside mass M of radius R. F_o is Newton's force of gravitation known to apply in millions of applications locally, terrestrially, and on planetary and galactic scales. F_i is a less known unappreciated Newtonian type gravitational force which I claim applies on the cosmic scale. Equations (1.5) are different equations F_oF_i. Do F_o,F_i apply in the cosmos?

Action At A Distance Universe And Mach's Principle

Newton's law of gravitation slows the ascent of baseballs and speeds up their descent. It prevents our solar and galactic systems from flying apart and binds together galactic structures. We know, therefore, that it operates locally outside high density matter as it exists in earthbound applications, or as it exists to form solar, planetary and galactic systems, even the remotest ones. However, there is no proof that it works on a cosmological scale. A brief analysis is necessary. I note that I cannot apply equations (1.5) to cosmology unless I determine a cosmic mass M=Mc and a cosmic radius Rc. I also note that equations (1.5) were obtained on the assumption that where is the average density inside sphere R and is zero outside sphere R. As shown later, this assumption is roughly approximated from current observations of cosmic matter density.

In the present theory, cosmic distance Rc is divided into the Galactic Universe which extends to about dRc=14 bly and the Radiation Universe which begins about 1 bly and extends beyond 14 bly. In an action at a distance universe , in which dRc is the radial distance of the observable universe. The action at a distance universe requires that gravitational effects transmit instantaneously which is what we observe. From equations (1.5) I obtain

Fo = 0.....................................................

.............................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1.6)

Equations (1.5) and (1.6) both result from classical action at a distance gravistatic theory. While equations (1.5) work in our everyday terrestrial, planetary and galactic experience, equations (1.6) are strongly suggested by current observation to work at cosmic distances. Thus, subject to further observation, cosmic force Fi is an attractive force which increases linearly with distance r. As shown below, the physical evidence of a uniform density strongly supports an inference that action at a distance force is a cosmic force.

Equation (1.6) brings to mind Mach's principle which holds that local matter has inertia because of its interaction with the rest of matter in the universe. However, equation (1.6) disproves Mach's principle because F_i=0 when r=0.

Retarded Action Universe And Big Bang

As stated, equations (1.6) are equations from which a reasonable inference can be made that F_i is an action at a distance force which applies at all distances in the cosmos as well as locally. This is not the case if cosmic forces are retarded. To see this, follow the procedure leading to equations (1.5) but now using non uniform density. This, in turn, leads to equations for the gravitational potentials, fields and forces

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . . . F=mE .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(1.7)

in which case spherical mass M of radius R has a non uniform density . Note that if density is uniform equations (1.7) reduce to equations (1.3). The point I make is that potentials, fields and forces which result from a non uniform density are not likely to describe an action at a distance universe. This brings to mind the big bang theory which holds that the universe started as an explosion in which matter density decreases repidly in time. The retarded action universe requires that gravitational effects transmit at finite speed which has never been observed.

Much but not all of what was said for equations (1.6) applies to equations (1.7). If equations (1.7) are applied to the cosmic scale, mass M=Mc is a sphere of non uniform density and radius R_c and m is at distance r from the center of coordinates. Force F in equations (1.7) is not likely to exist in the retarded action universe because, the physical evidence of a uniform density strongly supports an inference that action at a distance, not a retarded action, best describes the correct cosmic force.

Hubble's Linear Law

In 1929, Hubble formulated the expansion of the universe by observing that other galaxies are moving away from ours in a predictable manner. The more distant ones were receding faster than the nearby ones. He established his Hubble's Law v=Hr (velocity = Hubble's constant x distance), which is a linear relationship between the velocity of recession and distance. It says that the universe expands at a constant acceleration dv/dr=H. However, recent observations of remote supernovae indicate that the recession velocity is not exactly linear as Hubble's Linear Law requires. In 1998, astronomers discovered that the rate at which the cosmic expansion occurs is dv/dr<H. This discovery was revolutionary because it suggested that the big bang theory should be modified. To explain this slowdown of the cosmic expansion, some cosmologists invoked Einstein's cosmological constant that he invented in 1917, a fudge factor to balance the attractive gravity of matter, others attributed the slowdown to a mysterious entity called dark energy, while others said the law of gravity works differently at gigantic distances than it does in everyday life.² A non metaphysical theory is needed, based on proven physics, which predicts and explains the observations.

Hubble's Linear law v=Hr spans between coordinates r₁,v₁ at the low end and r₂,v₂ at the high end. Observations tell us that at the high end galaxies reach to a distance of about 3.96 bly at speeds of 6.1x10⁴ km/s (nebula Hydra).³ At this distance, v/c=0.203, z=0.22 and (1-(v/c)²)^1/2=0.98 so that the forces which act upon the galaxies throughout the span of Hubble's linear law v=Hr, valid through a distance of about 3.96 bly, are essentially non relativistic. Observations also tell us that at the low end of Hubble's linear law galaxies begin receding at a distance of about r₁'= 6.5 mly (2 Mpc) at speeds of v₁'=80 km/s (typical galaxy M81). At shorter distances most galaxies approach the Earth at distances less than about r₁''=2.12 mly (0.65 Mpc) at speeds of v₁''=-270 km/s (typical galaxy M31).⁴

Cosmic Matter Density

Current ideas are that the cosmos has evolved from an early uniform density to later non uniform density concentrations of matter⁵. It turns out that the matter density at different distances in the cosmic sphere appears as a power spectrum with the highest densities at the latest times and the lowest densities at the earliest times of the cosmic expansion. The density distribution looks something like that shown in the graph

		Relative
		10
		1	o	o	o
		0.1				o
		0.01					o
		0.001					o
		0.0001						o
			10-3	10-2	10-1	1	10	102	billion light years6

The graph above shows that beyond distance of about 1 bly, the density drops off precipitously. The decline coincides approximately with the high end of Hubble's linear law. Recall that equations (1.5) and (1.6) were obtained on the assumption that where is the average density inside sphere R and is zero outside sphere R. This assumption is rougly approximated in the graph above. Unquestionably, equations (1.6) can be modified for the case where the density follows the graph.

In the present work, I assume galactic mass m is in the static field of cosmic mass M in a sphere of radius dR_c. The apparent mass m in the far field of field of of any mass m⁷ is m=m_o/(1+µ_o/2a)(1-(v/c)²)^1/2 in which µ_o=Gm_o/c², m_o is the rest mass of m Here, in a first approximation, the term µ_o/2a is neglected and apparent mass becomes the familiar relativistic mass m=m_o/(1-(v/c)²)^1/2.

Since the forces acting upon the galaxies are non relativistic within the span of Hubble's linear law, equation (1.6) is a negligible contribution, v/c~0 and the expansion force F_e=m_odv/dt is obtained from Newton's second law of motion and, the attraction force F_i=-GM_om_o/r², in which M_o is the cosmic rest mass within distance r, is obtained from Newton's force of gravitation. Since dt=dr/v and dv/dr=H, the cosmic force which acts upon the galaxies within the span of Hubble's linear law is

. . . . . . .. . . . . (1.8)

which says that cosmic force F_c increases linearly with distance while Newton's attraction force decreases approximately as the inverse square of distance. At some distance r₁ the two forces cancel

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .(1.9)

Observations show that the cancellation distance occurs between r₁'= 6.5 mly (2 mpc) (galaxy M81) and r₁''= 2.12 mly (0.65 mpc) (galaxy M31) or at about r₁= 4.32 mly (1.325 mpc). Accordingly, I solve equation (1.9) to obtain values for the cosmic rest mass ratio µ_o/r₁=4.1825x10^-8, the cosmic rest mass radius µ_o=GM_o/c²=1.71x10¹⁷ cm and the cosmic rest mass M_o=2.3x10⁴⁵ g. I further compute the cosmic mass density in the sphere of radius r₁, the lower end of Hubble's linear law, as =8x10^-30 g/cm³. This density compares favorably with the cosmic mass density of ~4x10^-30 g/cm³determined by other methods⁸.

The Hubble Horizon

According to the graph above, the cosmic mass density is an approximate constant to distance of about 1 bly, just a few billion light years short of the high end of Hubble's linear law. Accordingly, the cosmic mass in a sphere with radius r₂=3.96 bly is M_o(r₂)= (r₂/r₁)³M_o(r₁)= 1.77x10⁵⁴g. This is somewhat less than the size and mass of the observable universe determined by other methods reported at 14 bly and M_o(14 bly)= 3x10⁵⁵ g, respectively⁹.

In formulating equations (1.8) and (1.9), I use Newton's attraction force F_i=-GM_om_o/r² instead of equation (1.6) which can also be written as attraction force F_i=-GM_om_o(r₃/dR_c)³/r².The two forces are equal approximately when r₃=dR_c=14 bly in which dR_c is the radius of the observable Galactic Universe. The Galactic Universe is Newtonian and nonrelativistic up to r₂=3.96 bly, the high end of Hubble's linear law, and extends beyond that distance to r₃=dR_c=14 bly. In the Radiation Universe, which extends beyond about 1 bly, part or all of galactic mass is replaced by radiation.

As shown in the graph, beyond about 1 bly, the cosmic mass density drops off indicating that. as relativistic speed increases, mass density drops off beyond that point and, presumably, mass being replaced by radiation energy. One can speculate that the sum of mass density and radiation energy density remains constant at about the same level, i.e., extending the level part of the curve to the right in the graph.

The Earth was born about 4.5 by ago¹⁰, at the high end of Hubble's linear law. By this time the violent past had settled and the galactic universe emerged from its radiation ball of fire.

¹ A. P. Wills, Vector Analysis, Prentice=Hall, Inc. 1950 pages 116-119

² George Dvali, "Out of the Darkness", Scientific American, Four Keys to Cosmology, February 2004 page 68

³ Fred Hoyle, Astronomy and Cosmology, W.H.Freman and Company, 1975 page 250 Fig II.56 Hydra nebula

⁴ Steven Weinberg, Gravitation and Cosmology, John Wiley & Sons, 1972 page 638 Table shows typical galaxies M81,M82,M83 receeding and M31,32,33 approaching the Earth

⁵ Michael Strauss, Reading the Blueprints of Creation" Scientific American, Four Keys to Cosmology, February 2004 page 60 (computer simulations)

⁶ Michael Strauss, Reading the Blueprints of Creation" Scientific American, Four Keys to Cosmology, February 2004 page 60 (computer simulations) page 59 (graph)

⁷ Leon Brillouin, Relativity Re-examined, Academic Press 1970 page 93 equation (7.22)

⁸ Density of the Universe http://hypertextbook.com/facts/2000/ChristinaCheng.shtml

^{9 http://en.wikipedia.org/wiki/Mass_of_the_Observable_Universe}

¹⁰ Weinberg, above page 487

-------------------------------

Copyright © 2004 by James Constant

By the same author: http://www.coolissues.com/gravitation/sameauthor.htm