NEWTON'S GRAVITATION AND COSMIC EXPANSION

Part 2 of 3 Parts
James Constant
gravitation@coolissues.com

Introduction

COSMIC FORCES OF EXPANSION AND SLOWDOWN

        In the present theory, cosmic expansion appears due to a repulsive force, assumed to exist, whose source remains unidentified but defined by Newton's Second Law of Motion, that is constrained by Newtonian gravitation. As shown by equation (1.8), a cosmic force F_c acts upon the galaxies within the span of Hubble's Linear Law. This force is the difference between the force of Newton's second law of motion which provides expansion and Newton's gravitational force which provides slowdown. Beyond the span of Hubble's Linear Law, the cosmicforce provides an expansion and slowdown of galaxies. Before discussing the relativistic forces that act upon the galaxies, I find it convenient to develop a speed distance law from observations.

Speed Distance Law 

        Beyond 3.96 bly, the high end of Hubble's law, distance measurements become less accurate and reliable. Many cosmologists assume that Hubble's law is linear beyond 3.96 bly. Distances are inferred by measuring redshift and then finding the distance from Hubble's law v=Hr.¹ To do this, they give up the idea that doppler operates beyond 3.96 bly since speed is limited by special relativity and cannot become endless as distance increases indefinitely. These theories replace doppler by the idea that space itself expands. They claim that, because space expands, galaxies recede faster than light.² I reject this view in the present theory and hold that galaxies cannot recede faster than light and that doppler is the main phenomenon associated with the observed cosmic expansion and source for galactic redshifts.

        To find how speed v/c behaves with distance r, I note two facts, first, at nonrelativistic speeds speed follows Hubble's linear law v/c=Hr/c and, second, recent observations show that the farthest galaxies have redshifts of z=7 to z=12³ which imply relativistic speeds v/c as r. A suitable function which satisfies these observations is

                                                                                                    (2.1)

in which r₁~ 4.32 mly, the low end of Hubble's linear law. In equation (2.1) a and b are coefficients needed for best fit with actual data, but are set a=b=1 in this work. Hubble's constant H=1/t defines a cosmic time distance t which equals cosmic distance D_c=ct=10 bly. Moreover, the farthest known objects are galaxies about 7-12 bly from Earth. These distances are far beyond the high end of Hubble's linear law at 3.96 bly. Using equation (2.1) with a=b=1, v/c=0.377 when r=10 bly and v/c=0.46 when r=13 bly.

        By differentiating equation (2.1) I obtain the rate at which speed changes with distance

                                                                         (2.2)

        The relativistic velocity as a function of dimensionless quantity z is 

                                                                                            (2.3)

in which is doppler redshift m_o-m divided by the laboratory frequency m_o. For example v/c=0.988 when z=12 (93.9 bly). Separately published is Appendix 1 which provides some useful fundamental and conversion constants and Appendix 2 which is a table of distances with corresponding speeds, rates and redshifts based on equations (2.1) through (2.3). www.coolissues.com/gravitation/Appendix/app1.htm and app2.htm.

        Since dr=vdt, equation (2.2) can be stated in terms of acceleration

                                                                                           (2.4)

        Equation (2.4) is a kinematic acceleration based on the observations that galactic speed increases with distance but never exceeds the speed of light. In view of equation (2.1), galactic acceleration based on equation (2.4) increases to a peak and then decreases and approaches zero as . By differentiating equation (2.4) and setting the result H²(v/c)(1-2v/c)(1-v/c)=0, and since v/c never reaches 1, a maximum acceleration is found at v/c=0.5 which, in view of equation (2.1) occurs at about 14.7 bly.
        Equation (2.1) and its derivatives are subject to further confirmation by observations. Derivative equation (2.4) shows that galaxies accelerate at ever lower rates and then, when v/c=0.5 (14.7 bly), decelerate at ever higher rates. Recent observations of remote supernovae indicate that the expansion of the universe was decelerating (slowdown) before it began accelerating (speedup) about 5 by ago.⁴ Distance 14.7 bly, therefore, appears high by about a factor 3 but this may well be due either to a lack of setting coefficients a and b in equation (2.1) or to the inaccuracy of measurements both of which should be improved with time. Or it may be due to the effects of relativistic forces discussed next.

Relativistic Forces 

        The cosmic force of the observed expansion is provided by relativistic force F_e=d(mv)/dt which, since m=m_o/(1-(v/c)²)^1/2 and v=dr/dt, can be differentiated

                                            (2.5)

in which m_A = apparent mass due to negative field mass, more fully discussed later, m_o = rest mass and a = the radius of m_o. The result of differentiating is

                                                          (2.6)

in which du_e/dt is the acceleration due to expansion force F_e obtained from relativistic mechanics.

        The inside cosmic force⁵ of gravitation is provided by equation (1.6) which, since m=m_o/(1-(v/c)²)^1/2 and can also be written as

                                                                                (2.7)

in which du_g/dt is the acceleration due to gravitation force F_i obtained from Newton's mechanics. In equation (2.7) G = gravitational constant, m_A = apparent mass, q = density. Equation (2.7) is "relativistic" since cosmic density q and apparent mass m_A change appreciably with relativistic speed at distance beyond r₂=3.96 bly, the high end of Hubble's linear law.

        Term in equation (2.7) is proportional to H² and this fact has been noted in several theories. For example, in Newtonian steady state theory and in relativistic Einstein-Sitter theory, the density Hubble constant relationship is given by

                                                                                                                          (2.8)

However, in the steady state theory H is a constant but in the Einstein-Sitter theory H is inversely related to Time. ⁶

        In the present theory, H is related to speed v. In view of equation (2.2), Hubble's linear law dv/dr=H is replaced by dv/dr=H(1-v/c) or, stated another way, Hubble's constant H is replaced by . Accordingly, I set

                                                                                       (2.9)

from which

                         r>r₂                                                     (2.10)

which can be used in equation (2.7). Equations (2.9) and (2.10) are dictated by the fact that vHr Hubble's law as v0 in equation (2.7). Note that equations (2.9) and (2.10) are valid only for galactic mass, since their v/c term originates from galactic speed distance law equation (2.1). Later, I show equation (2.8) applies only for photons.

        The total cosmic force is the sum of forces of expansion and gravitation

                     (2.11)

in which du/dt is the cosmic acceleration due to cosmic force F_c. Acceleration du/dt is a relativistic acceleration based on the theories of relativistic mechanics and Newton's theories of motion and gravitation.

        In view of equations (2.2) and (2.11),

                                                     (2.12)

which states that the cosmic acceleration du/dt obtained from theory does not exactly conform with cosmic acceleration dv/dt obtained from observation. At low speeds (nearest distances) du/dt~0 and at high speeds (farthest distances), since the m_A/m_o and d(v/c)/dt terms remain finite, du/dt~ + or - depending on the sign of the { } bracket term as . However, since the [ ] bracket term remains finite, d(u/c)dt . In contrast, d(v/c)/dt is always positive. At low speeds (nearest distances) d(v/c)/dt=Hv/c (v=Hr) and at high speeds (farthest distances) d(v/c)/dt as . Accordingly, d(v/c)/dt approximately conforms only when the { } bracket term in equation (2.12) is positive.

        Equation (2.12) permits computing actual values of du/dt, at distances short of du/dt~ and is particularly useful since all galaxies convert to radiation well before then. For example, in view of equation (2.4), equation (2.12) says that, the positive expansion force F_e initially zero increases with speed v/c against F_i the negative Newtonian type attraction inside force which increases with distance at the farthest distances. At some distance(s) the two forces cancel.

                                                                        (2.13)

        In the non relativistic region, below r₂<3.96 bly, v/c<0.203, the high end of Hubble's linear law, relativistic equations (2.12) and (2.13) reduce to non relativistic equations (1.8) and (1.9). In this region, the equality of the cosmic expansion and gravitation forces, and thus F_c=0, occurs at a distance of about r₁~4.32 mly, the low end of Hubble's linear law. In the relativistic region r₂>3.96 bly, assuming galactic mass still exists, the equality of forces and thus F_c=0 occurs at a distance r₄ determined by equation (2.13). Thus, while the negative Newton's attraction term predominates over the positive expansion term below distance r₁ and above distance r₄, the positive expansion term predominates between these distances. 

        Solving equation (2.13) for null distance r₄ is not possible because term m_A/m_o is an unknown. All that can be said is that null point r₄ would likely exist if galaxies survived to that distance. The cosmic force F_c=F_e+F_i, equation (2.12), as a function of distance looks something like the graph in Fig 2.1.

         The graph shows that below r_1, F_c<0 is an attractive predominantly Newtonian force; beyond r₁=4.32 mly, the low end of the Hubble linear law, F_c>0 is predominantly an expansion force which accelerates and then decelerates until, assuming galaxies still exist, it drops to zero at r₄ and then to as . The farthest object detected presumably occurs at v/c=0.988 (z=12, r=93.9 bly).[6] However, measurements beyond r₂=3.96 bly, the high end of the Hubble linear law, are increasingly less reliable. The graph shows, better than does cosmic force equation (2.11), the roles galactic masses m and cosmic matter density q have in the acceleration and deceleration phases of the cosmic expansion of galaxies.

         In deriving equations (2.5) through (2.13), I have treated the cosmos as a sphere which contains the Galactic Universe with most galaxies, within a larger sphere which contains the Radiation Universe with few galaxies and begins beyond 3.96 bly, the high end of the Hubble linear law. All galaxies in Universe act instantaneously according to Newton's laws. There is no paradox between the action at a distance Universe and galactic redshifts suggesting galaxies at earlier times because, according to the present theory, what existed then exists now although in similar form. Relativistic equations (2.5) through (2.13) follow directly from the observed facts of the cosmic expansion and gravistatic equations (1.5) and (1.6) to which they reduce when v/c<<1. As shown in the next Fig 2.2, I distinguish between instant  and past or future times

        Fig 2.2 shows that we see galaxies from Past Time by redshift measurements and we also experience galaxies in Instant Time, as instantaneous gravitational action at the local, planetary and galactic distances which extend as far as observations permit. The Sun does not disappear after it emits photons 8 minutes ago and the Andromeda galaxy does not disappear after it emits photons 800,000 years ago. It is reasonable, therefore, to conclude that galaxies observed many bly ago do not disappear after they emit their photons although they have changed in form or have even been replaced. Equations (2.5) through (2.13), are based on the assumption that what we observe in the universe by redshift measurements also exists today in similar form, and are justified if they make correct predictions of observations.

Mass Shielding 

        In Newton's gravitation, mass density q is always finite inside a spherical mass M but is taken to be zero outside M. This is another way of saying that M is a constant. Brillouin, however, shows that the field E of mass M always has a negative mass so that when seen from a distance, apparent mass M_A is the sum of its actual mass M and negative field mass M_f. Brillouin provides the actual formula ⁷

                                                                             (2.14)

in which a is the radius of a sphere of actual mass M at the center of coordinates. Equation (2.14) reduces to M_A=M(1-µ/2a) when µ/2a<<1. When r=a, apparent mass equals actual mass M_A=M and field mass has no effect. When r>>a apparent mass M_A is inversely proportional to 1+µ/2a and becomes less than its actual mass M as µ/2a increases. In the limit as

        µ_A=µ+µf =2a         µ>>2a                                                                       (2.15)

which states that the self shielding of a mass µ by the negative mass µf of its field will prevent the apparent mass µ_A from ever exceeding 2a. Thus, while the actual mass µ of a cosmic object may be infinitely large, due to great mass, small radius and/or relativistic speed, its apparent mass µ_A can never exceed 2a. The effect is that the actual mass M of a cosmic object, galaxy or cosmic mass, may be much greater than its apparent mass M_A. Because of self shielding by negative field mass, the actual mass of galaxies or of the cosmos is gravitationally weakened. Distant cosmic objects believed gravitationally weak are associated with gravitational effects in their near fields which avoid the masking effects of their far fields. Brillouin's theory of self shielding of mass by the negative mass of its field may well account for what astronomers call dark energy, energy not seen but which is assumed from its gravitational effects, in the distant cosmos. 

         In view of equations (2.14) and (2.15), I distinguish three cosmic regions, first, the "no effect" region when µ/2a<1/2, second, the "some effect" region where 1/2<µ/2a<2 and, third, the "full effect" region where µ/2a>2. In the "no effect" region there is no appreciable negative field effect and apparent mass behaves like actual mass of our everyday experience (large mass, relativistic mass), µ_A~µ. In the "some effect" region negative field mass increasingly appears and apparent mass becomes a function of actual mass, equation (2.14). Finally, in the "full effect" region negative field mass maximizes its shielding effect and µ_A=2a, equation (2.15).

         If µ is the actual mass of a galaxy, then equation (2.14) represents the apparent mass, actual and negative field, of the galaxy. If µ/2a<<1, apparent mass µ_A approximately equals actual mass µ and it is only when µ/2a>>1 that apparent mass µ_A=2a in which case the gravitational effect of actual mass µ is weakened.

        The cosmological importance of negative field mass is that it puts an upper limit on actual mass, massive or relativistic in equations (2.5)-(2.13). This means that, as galactic actual mass m and cosmic actual mass M must be replaced by galactic apparent mass m_A and cosmic apparent mass M_A. As result, unlike actual masses M and m, apparent masses M_A and m_A in equations (2.5) through (2.13) do not become infinite when . Since the total cosmic, or galactic, apparent mass µ_A is the sum of the actual mass µ and negative field mass µ_f, µ_A=µ+µ_f =µ/(1+µ/2a), I conclude that in the limit, when µ/2a>>1, µ_A=µ+µ_f =2a which says that about 50% of the mass in the universe is positive mass µ and about 50% is negative mass µ_f. This distribution occurs at the farthest distances. A substantially small apparent mass µ_A means that mass terms in equations (2.5)-(2.13) and expansion force F_e and gravitation force F_i tend to zero. In Fig 2.1, I represent the possibility of this hypothesis occuring at the farthest null distance F_c=0 at r₄. In such case, cosmic force F_c0 as r. Whether it does so depends on whether cosmic mass converts to radiation before or after the farthest null distance F_c=0 at r₄.

        If µ is the cosmic actual mass, since µ=µ_o/(1-(v/c)²)^1/2 equation (2.14) can be restated as

                          r>>a                                                                  (2.16)

in which cosmic rest mass µ_o, a constant, is known from equation (1.9) and v/c is obtained from measurement. Equation (2.16) says that µ_A2a as vc. Unfortunately, with few exceptions for nearby galaxies, we have no direct ways of finding a and thus determining the value of galactic or cosmic apparent mass µ_A. Generally, since a is unknown, apparent mass µ_A is unknown. All we can say is that µ_A2a = assumed constant as vc. I say assumed constant because it certainly is constant for actual mass of our experience but can only be assumed constant for cosmic actual mass. Nevertheless, we do have some reasonable theories and indirect ways for thinking about apparent mass µ_A. Thus, if cosmic apparent mass µ_A is constant, as required by the law for conservation of mass, we have the result that

                                                                                                        (2.17)

which says that since µ_o=constant and since µ_A~2a is assumed constant in the "full effect" region of the universe, a maximum recession speed v~c is approximately constant in the "full effect" region. At lesser speeds v/c, the question is: beyond what speed does the square root, in equations (2.16) and (2.17), become smaller than 1 and actual masses M and m be replaced by apparent masses M_A and m_A in equations (2.5) through (2.13)?

        In other words, the effects of negative field mass are quite different in the "no effect", "some effect" and "full effect" regions of the universe. It is for this reason that some idea of the boundary between the three regions is necessary before the effects of negative field mass can be understood. At what speeds does the square root in equations (2.16) and (2.17) start having effect? In the "no effect" region, it is the actual mass ratio µ_o/2a alone that determines the value of apparent mass µ_A. In the "some effect" and "full effect" regions, it is the recession speed v/c as well as the actual mass ratio µ_o/2a that determine the value of apparent mass µ_A.

        To gain some perspective, v/c<0.203 below distance r₂=3.96 bly, the high end of the Hubble linear law. This is the "no effect region" part of the Galactic Universe. Between this region and distance r₃=13 bly v/c<0.46 so that this is the "some effect" part of the Galactic Universe in which speed starts asserting itself on cosmic mass. Finally, the "full effect" part of the Galactic Universe extends beyond distance r₃=13 bly. Recent observations have detected redshifts of z=5 (v/c=0.946), z=7 (v/c=0.969), and z=12 (v/c=0.988) so that, at some distance beyond r₃=13 bly, the square root in equations (2.16) and (2.17) tends to zero and the ratio µ_o/2a dominates over the square root term. Thus, at speeds above about v/c=0.988 (r=93.9 bly), the distance of the farthest detected galaxy, µ_A~2a, an assumed constant. What this all means is that masses m and M are actual masses in the "no effect" region but must be replaced by apparent masses m_A and M_A in the "some effect" and "full effect" parts of the universe, with necessary adjustments of equations (2.5) through (2.13) within each region. It is for this reasoning that, in equations (2.11)-(2.13), I have replaced galactic actual mass m by apparent mass m_A, and cosmic actual mass µ by apparent mass µ_A.

        It has been theorized that black holes exist ⁸. Here, I speculate whether they can be observed and detected. By definition, no light escapes from a black hole, i.e., a photon's energy E=hv cannot overcome the potential energy V=-lhm/a of a black hole, E[-V , which results in the condition that the actual mass radius l of a black hole must be equal or be greater than its radius a, lma.

        Three important equations of cosmology are the cosmic force of expansion F_e=d(m_Av)/dt based on Newton's 2nd Law of Motion, Newton's force of gravitation F_g=-GM_Am_A/r², and Brillouin's apparent mass M_A=M/[1+l(1-a/r)/2a] =M+M_f which also applies to m_A,m replacing M_A,M.

Apparent Mass Effects

        Mathematically, Brillouin's equation allows a number of regions which depend from the radius a of actual mass M and the distance r from M.

        I. Near field                    

        II. Far field                                                                (2.18)

in which actual mass l>0 or l<0 and radius a is always positive a>0. In Region I, distance r is located in the near field of actual mass M, and apparent mass l_A equals actual mass l. In Region II, distance r is located in the far field of actual mass M. In both regions l_A always has the same sign as l. In Region II, if l>>2a, l_A=2a and if l=-2a l_A=-º.

        Physically, the actual mass M of our experience is always positive l>0. The case for negative actual mass l<0, therefore, is speculative. Region I is most familiar to us since no modification is required of Newton's laws. In Region II Newton's laws are modified by replacing actual masses M,m by apparent masses M_A,m_A,. In effect, while Region I represents no effect by field mass, Region II represents shielding by field mass. The value of l/a greatly affects Newton's force of gravitation F_g=-GM_Am_A/r² whose value and sign changes when the sign of M_A or m_A changes.

        Assume now that actual mass M is positive l>0. Since a black hole cannot be detected directly it may be possible to observe its gravitational effect upon nearby galaxies. However, if l>>2a then l_A=2a and a black hole will have a negligible effect in its far field because of shielding by its negative field mass. I conclude that if l>>2a black holes cannot be observed directly. This leaves open the possibility that a black hole can be detected indirectly by its effect on nearby galaxies in its near field. For example, in a two body problem involving a black hole M>>m of a nearby galaxy, the total mass is reduced mass Mm/(M+m)=m. In this case observations of light would originate in the nearby galaxy orbiting about the black hole. However, planetary duets with black holes have not been observed. At present all information about black holes is indirect coming from observations of gas swirling about unseen center.⁹ It is unlikely that a nearby galaxy orbiting about the black hole can survive long enough to send light that can be observed.

        Next, assume that actual mass M is negative l<0. If l=-2a l_A=-º. black holes would repulse matter and galaxies near and far and this could be observed directly. For example, jets have been observed from presumed black holes at the center of observed clusters.¹⁰ Whether black holes have negative mass is an open question.

Cosmic Effects

        The Newtonian term in equation (1.8) predominates below about 4.32 mly the low end of Hubble's linear law and, in theory if actual mass still exists, beyond about 93.9 bly, the farthest detected object distance. At distances between 4.32 mly and 93.9 bly the expansion term in equation (1.8) predominates.

        Newton's law of gravitation which applies in the near field between a black hole and a galaxy. Assuming that M_A=M and m_A=m

                r~a_M+a_m,                                                                      (2.19)

which applies equally to all types of duet stars including black holes.

        Within the span of Hubble's Linear Law, between 4.32 mly and 3.96 bly, actual mass is nonrelativistic and galactic mass stays together. Beyond 3.96 bly, actual mass is relativistic, and galaxies begin disintegrating. Thus, galaxies become gravitationally unbound beyond 3.96 bly as vdc. The largest object observed being gravitationally bound is the cluster. The farthest cluster observed is the Hydra cluster at the farthest distance 3.96 bly (v/c=0.203), the high end of Hubble's linear law. Gravitational bounding disappears beyond this distance. The observation and detection of black holes in clusters, therefore, is increasingly less likely beyond about 3.96 bly.

        Since black holes cannot be seen and produce diminished effects in their far fields, they unlikely to produce effects upon nearby galaxies which can be distinguished from effects produced by ordinary galaxies. I conclude that black holes at the center of observed clusters cannot likely be proven to exist by indirect evidence on their nearby galaxies.¹¹ And, the explanation of jets observed from galaxies at centers of clusters, as being black holes, is an open question. Since gravitational forces unbind at the farthest distances beyond 3.96 bly, The likelihood of observing clusters and detecting black holes is at nearest distances where the unbinding effects of apparent mass and relativistic speeds are lowest. Indeed, the brightest cluster observed is the Perseus cluster at a distance of about 300 mly (v/c<<1). Nevertheless, some say that black holes can explode releasing radiation which can be detected to reveal their presence. Because it does not lose its gravitational pull, a black hole cannot explode.

² Charles Lineweaver and Tamara Davis, Misconceptions About the Big Bang, Scientific American March 2005 page page 40

³ NASA - Hubble Digs Deeply, Toward Big Bang (z=7-12) http://www.nasa.gov/vision/universe/starsgalaxies/hubble_UDF.html;
http://en.wikipedia.org/wiki/Redshift#Expansion_of_space See Extragalactic Observations. Ref. 44 (z=10); 
Wayne Hu and Martin White, Scientific American February 2004 page 47 (graph 13.7 bly); Adam Riess and 
Michael Turner above, Scientific American February 2004 page 67 (right graph 14 bly); http://en.wikipedia.org/wiki/Universe#Size_of_the_universe_and_observable_univ

⁴ Adam Riess and Michael Turner above, page 67 (right graph)

⁵ I call equation (1.6) an inside force F_i (because it operates inside a sphere of mass M) to distinguish it from equation (1.5) an outside force F_o (because it operates outside a sphere of mass M). Outside force F_o is Newton's force F_g.

⁶ D. Sciama, Modern Cosmology, Cambridge University Press 1971 pp 116-118

⁷ Leon Brillouin, Relativity Re-examined, Academic Press 1970 pages 93-95

⁸ Black Holes http://en.wikipedia.org/wiki/Black_holes

⁹ Prabhakar Gondhalekar, The Grip of Gravity, Cambridge University Press 2001, pages 246, 281.

¹⁰ Wallace Tucker, Harvey Tanenbaum, Andrew Fabian, Black Hole Blowback, p 43-49 Scientific American, March, 2007.

¹¹ Prabhakar Gondhalekar above at page 283, Gondhalekar reports researchers have determined that orbits of about 200 "Stars in the central cluster (of the Milky Way) follow a Keplerian velocity distribution around a black hole of 2.6 million solar masses". Since l=1.475 km for the Sun l=3.835x10⁶ km for the reported Milky Way black hole. According to the present theory, a black hole implies l/am1 which in turn implies that the radius of the reported Milky Way black hole is a[3.835x10⁶ about 5.5 times the radius of the Sun and almost the radius of planet Mercury's orbit. Is this proof of the existence of a black hole at the center of our Galaxy? Certainly it is not a direct proof. Nor can it be an indirect proof since the same result can be obtained by a star with l/a slightly less than 1, for example a neutron star with l/a=0.9.

Copyright © 2004, 2008 by James Constant

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