Gravity, Caused by Stretching

          of the Universe?

 

                                                Abstract

The observed stretching of the universe, long associated with the Hubble rate of expansion, is used to formulate a new theory for gravitational attraction.  The basic law of gravity, as originally derived by Newton, is used and modified to reveal a revised concept of the universal gravitational constant—one that uses the Hubble constant and the average mass density of space.  All spatial distances and objects are assumed to be expanding at a rate that depends upon their current dimensions. There are also consequences for the apparent, measured speed of light.  The cause of the stretching is the hand of God.

 

                      Milton  R. Seiler

                                                                 Venice, Florida

                                                  May, 2008

 

 

                                                Introduction

Isaac Newton’s law of universal gravitational action between two masses, m₁ and m_2, separated by distance R, provides the following equation for the force, F, of attraction:

 

F = G m₁m₂ / R² newton,                                                         (1)

 

where the masses are expressed in kilograms (kg) and the distance is in meters (m).  The term G is the universal gravitation constant, generally accepted as 6.67 x 10^--11 newton m² per kg².  Henry Cavendish⁽¹⁾ first measured the value of G in a torsion balance experiment in 1798.  P. R. Heyl⁽²⁾, of the National Bureau of Standards, improved upon these measurements. 

 

There remains the problem of the source of the gravitational attraction.  Attempts over the years have not revealed this source, although much discussion has been produced about the delicate balance between gravitational attraction and the expansion of the universe. E. Hubble⁽³⁾ reported this velocity of expansion, related to the distance of the observed source of radiation, in 1929.  His simple expression was

 

v = H D,                                                                               (2)

 

where v is the indicated velocity of recession away from the observer, H is the Hubble constant, and D is the distance to the source of radiation.  Hawking⁽⁴⁾ has made the following comment about this rate of expansion:

 

“Why did the universe start out with so nearly the critical rate of expansion that separates models (of the universe) that re-collapse from those that go on expanding forever, so that even now, ten thousand million years later, it is still expanding at nearly the critical rate?  If the rate of expansion one second after the Big Bang had been smaller by even one part in a hundred thousand million million, the universe would have re-collapsed before it ever reached its present state.”

 

Rees⁽⁵⁾ has discussed at length the critical balance between gravitation and unknown forces causing the expansion.  He presents the case for “dark matter” as a potential cause for the repulsion.  He says, in addition,

 

“We should keep our minds open (or at least ajar) to the possibility that our ideas on gravity need reappraisal.”   

 

It could be that these questions arise because it is assumed that the forces of gravitational attraction are distinct from the forces causing the observed expansion.   It is the purpose of this paper to propose that these forces are not separate, but instead the accelerating rate of expansion is the very source of what we call gravitational attraction.  Expansive and attractive forces are equal and opposite according to Newton’s third law. All observers, regardless of their location in the universe, will experience the effect of this continual, accelerating stretch of dimensions.

 

                                                Analysis

                                                 

Consider the dimensions of G, the universal gravitational constant. It has the units newton  m² per kg², which may also be expressed as m³ per kg per sec².  That is, G may be expressed in terms of H² and a volume mass density,r.

 

We have

 

            G = bH² /r  m³ per kg per sec² ,                                                                     (3)

 

where b is a constant to be determined.  If G is to be constant, independent of position in space, then it would be appropriate to take r as the average mass density of space.  Using equation (3) in equation (1), we obtain an alternative expression for Newton’s law of gravitation.

 

a = F / m₂ = (bH² M) / (r R² ) m/sec² ,                                                                      (4)                       

 

which is the acceleration of gravity at a distance R from mass m₁ = M..

 

Referring to Figure 1, where the radius R, with origin at the center of mass, is expanding with velocity HR.  The kinetic energy associated with this expansion is obtained by integrating the energy associated with each ring dR, with the integration extended from R = 0 to R = R₀.

 

            E = ò0.5 r 4p H²R⁴ dR.  joules

 

The result of this integration is

            E = 0.4 p r H² (R₀)⁵ = 0.3 H² M (R₀)²                                                 (5)

 

The kinetic energy per unit volume is then

 

            e = 1.8 M H² / (8 p R₀)   joules/unit volume                                                      (6)

 

As the radius of observation extends beyond the mass, M, there is still an enclosed energy per unit volume.  The gradient of this energy is a force per unit volume, given by

 

            f = de/dR  =  -1.8 M H² / 8 p R²  newton / m³                 (R ³ R₀)                 (7)                 

  

The negative sign shows that the force is directed inwardly along the radius.  Dividing equation (7) by the average density of space, r, yields the acceleration relative to the free-space average.

 

Comparing equations (7) and (4), we can write

 

a = -1.8 M H² / (8 p r R²) newton/kg  (or m / sec²).                 (8)                                                                                            (7)

                                                                                               

We now find that                                                                                                                                            

b = 0.225 / p,

and

                                    G = b H² / r                                                                                                      (9)

 

Newton’s law for the acceleration of gravity associated with mass M and radius R may now be expressed as

                      a = G M /R²  = 0.225 H² M / (p r R² ) newton / kg                              (10)

 

Equation (9) implies that there is relationship between the average mass density of space and the Hubble constant.  This relation is plotted in Figure 2.

 

The range of values shown in Figure 2 for the mass density may be high or low depending upon the researcher being quoted.  Rees⁽⁶⁾ contends that the critical density (for re-collapse of the universe) is on the order of 5 atoms per cubic meter, and the observed density is on the order of 0.2 atoms per cubic meter. Using the mass of the hydrogen atom as approximately 10^-27 kg/cubic meter, this is equivalent to a mass density of 2 x 10^-28 kg per cubic meter.  On the other hand, Chaisson and McMillan⁽⁷⁾ report the average is roughly a million atoms per cubic meter (10^-21 kg per cubic meter).  An example from an encyclopedia⁽⁸⁾ is 10⁵ atoms per cubic meter (10^-22 kg per cubic meter).

 

Using the Hubble Space Telescope, Freedman⁽⁹⁾ has obtained a value of 72 km/sec/Megaparsec. Equation (9) then results in a density of about 6 x 10^-27 kg per cubic meter, a value toward the lower range of other data.

 

It should be emphasized that there is nothing in this proposed model of gravitation that requires G or H to be constant in time or space.  Much additional study is required.

 

                                    Implications for the Earth

 

tc \l1 "Implications for the earthIf the earth radius is expanding, what are the implications for measuring this effect?

The velocity of expansion at the earth’s surface is negligible and not measurable by

presently used methods.  For an earth radius of 6.4 x 10⁶ meters it is only about

1.5 x 10^--11 meter/second.

 

The expression for the radius as a function of time is

 

R = R₀ exp(Ht),                                                            (11)

                                                              

where R₀ is the initial radius at time t = 0.  Table I shows this growth versus time for H = 72 km/sec/Megaparsec and an initial radius of 6.4 million meters. The value of H, expressed in years^-1 is approximately

 

H = 72/(3.09 x 10¹⁹) per second

   = 7.35 x 10^-11 per year

 

                        It is likely this growth, as represented by equation (11), is not detectable by known methods for earth ages well below one million years.  Opportunities for indirect, if not direct, observations may be more likely as the age of the earth increases.  These observations may include changes in the earth’s gravitational field, as well as the sun’s, and consequences for the location of the earth’s orbit around the sun.                      

 

                       Also, there are implications for the measured speed of light.  If this speed is absolutely unchanging over the passage of time, and if all of our so-called “measuring sticks” are expanding with time, as described by equation (11), then we would falsely conclude that the speed of light is decreasing. Using equation (11) and a value of 72 km/sec per Megaparsec for H, it is a straightforward calculation to show the expected, apparent reduction in the speed of light over a 100-year period.  The result is that the decrease would be 100c₀H, where c₀ is the initial speed of light.  With H given by 7.35 x 10^-11 per year, and a light speed of 3x10⁸ meters per second, the result is that the apparent speed will decrease on the order of 2.2 m/sec per 100 years of observation.

 

                                Figure 3 shows the results of measurements of the speed of light observed between 1972 and 1983.  These data were chosen with the assumption that more recent measurements, particularly since the advent of laser and improved optical technology, may be more accurate. There is also considerably less scatter in that data when compared with earlier results. The eight data points are taken from the list provided by Setterfield⁽¹⁰⁾.  The best-fit trend-line has a slope of -5 m/sec per 100 years, which is about a factor of 2 larger than the prediction. It is realized that this result may or may not support the proposed theory of gravitation.  It is offered for consideration as a future research topic.

                                                            The Cause

                        The word of God contains answers for our questioning minds.  The ancient Hebrews were given insights that could relate to our search for the cause of stretching.  Turning to Isaiah 40:22, we read:

                        It is He who sits above the circle of the earth, and its inhabitants are like grasshoppers, who stretches out the heavens like a curtain and spreads them out like a tent to dwell in. (NASB)

                        And in Isaiah 44:24-25,

I, the Lord, am the maker of all things, stretching out the heavens by Myself, and spreading out the earth all alone, causing the signs of boasters to fail, making fools out of diviners, causing wise men to draw back, and turning their knowledge into foolishness. (NASB)

 

And again in Zechariah 12:1 we read:

Thus declares the Lord who stretches out the heavens, lays the foundation of the earth, and forms the spirit of man within him. (NASB)

 

The stretching is always under control by the Lord Jesus Christ, for we read in Colossians 1:16-17:

For by Him all things were created, both in the heavens and on earth, visible and invisible, whether thrones or dominions or rulers or authorities, all things have been created by Him and for Him.  And He is before all things, and in Him all things hold together. (NASB)

 

The proposed model for the source of gravity is the expansion of all dimensions and objects in space in accordance with Hubble’s Law.  The gravitational constant, G, can be defined in terms of the square of the Hubble Constant and the average mass density of space.  There are consequences for the measured speed of light.  Observed trends toward an apparent, decreasing speed of light can be approximated by this model. There is nothing within the model that shows how accurate are the current estimates of the Hubble Constant, nor is there any attempt made to determine if the Gravitational constant is the same everywhere in space or time.  Considerable research is needed to further investigate these topics.

                                    Table I.  Growth of Earth Radius versus Time

Elapsed time (years)	Radius ( meters)	Growth (meters)	Percent growth
1000	6400000.5	0.5	7.35E-06
10,000	6400004.7	4.7	7.35E-05
100.000	6400047	47	7.35E-04
1 million	6400470	470	0.00735
10 million	6404700	4700	0.0735
100 million	6447149	47149	0.738
1 billion	6887430	487430	7.63
10 billion	13333744	6933744	108.55

Figure 1.  Mass M Observed at Radius R