Gravity as a Force related to the Energy Density of the Vacuum and the Critical Mass Density

We know that the force of gravity is proportional to the mass of the object in question, and inversely proportional to the square of the distance between the center of the mass-carrying objects, but the problem with gravitational force is that, in absence of a second mass carrying body, the force does not exist except as a potential force. An analysis of G needs to take into account the context within which the gravitational constant is used, so we will first look at the Newtonian equation used to calculate the gravitational force between two mass carrying objects:

F = GM₁M₂ / r²

My first observation here is that the gravitational constant is only used once; it does not need to be used a second time in conjunction with the second mass (M₂). This would seem to imply that the gravitational constant, G, must be a proportional constant that describes the properties of both gravitational fields (of both masses) simultaneously, since when we add the second mass into the equation the constant does not change or need adjusting. What I would like to suggest here is that we substitute G and replace it with a constant that purports only to explain the properties of one gravitational field.

Given that the units of G are Nm² kg², it may be better to represent the equation by stating that F = (√G)² M₁M₂ / r² . This gives a value of -8.168^-06 N(1/2) m kg for the square root of G (√G). This also allows us to split the equation into two parts and allows us to show exactly how the gravitational constant should be related to each mass. For the ease of notation, we shall say that AF = √G (thus AF = -8.168^-06 N(1/2) m kg), where AF is the constant of proportionality of the force of the vacuum (F_V) exerted upon any mass, and can be accurately represented by stating that the force of the vacuum (F_V) is proportional to the mass of the object, and inversely proportional to the distance from the center of the mass:

F_V = A_FM / r

F = (A_FM₁ / r) x (A_FM₂ / r)

or

F = (A_FM₁) x (A_FM₂)
r²

If we substitute the values into the above equation and use two familiar masses for M₁ and M₂, then we should get exactly the same result as when using G. We shall say M₁ is the Earth, and M₂ is a 1kg object at the earth's surface. Thus:

(A_FM₁) = (8.168^-06 N(1/2) m kg x 5.98²⁴ kg) = 4.884464¹⁹ N(1/2) m

(A^FM²) = (8.168^-06 N(1/2) m kg x 1kg) = 8.168^-06 N(1/2) m

The force of gravity is essentially the product of the force exerted by the vacuum by each of the two masses. In this respect, the force exerted upon one mass by the vacuum is multiplied by the force exerted upon the second mass. This results in an attractive force (gravity) between the two masses, which is strong enough to give rise to motion where one of the masses is substantially large enough.

Therefore, the force of gravity between the two masses must be:

F = (4.884464¹⁹ N(1/2) m) x (8.168^-06 N(1/2) m kg)
r²

or

F = (4.884464¹⁹ N(1/2) m / r) x (8.168^-06 N(1/2) m / r)

The distance, r, between the center of the earth and the 1kg mass (essentially the radius of the earth) is 6376500m, so if we input this final information we should get:

Certainly using the value A_F demonstrates why the dimensions of the gravitational constant G are Nm² kg², but its primary advantage is that it shows us from a classical mechanics point of view how gravity is propagated, and indeed what it is; gravity is a resultant force (F_R) of the product of the two partial forces exerted on two mass carrying bodies by the vacuum, thus:

The question now is, how does a vacuum exert a force on a mass carrying body? What I would like to propose here is that the vacuum of space is a "negative energy plenum", where each cubic meter of space can accommodate a maximum amount of energy of 8.168^-06J, termed the Constant of Accommodation (A). This essentially means that the vacuum of space has a vacuum energy density (A_V) of -8.168^-06J/m³, and that space is inclined to be accommodated with the energy held within the matter in order to cancel out these energies and give a zero value. This inclination manifests itself as a partial force (measured in N(1/2)) determined by the constant of proportionality A_F.

The important point to note here is that this approach requires that mass carrying bodies do not occupy space, but rather exist within their own spatially extended dimensions. This is something that Einstein thought significant when he stated in Relativity:

"Physical objects are not in space, but these objects are spatially extended. In this way, the concept of empty space loses its meaning" ^[1]

In response to this I would say that empty space is the volume of space that was previously occupied by energy but which, due to the propagation of matter, has now been vacated. From this, we should deduce that there is a corresponding amount of energy (A) in the observable universe for every cubic meter of space (or conversely, every cubic meter of space can only accommodate 8.168^-06J of energy (A), which in turn suggests a vacuum energy density (AV) of -8.168^-06J/m³. If this is true, then we should be able to calculate a rough estimate of the Critical Mass Density (CMD) simply by using the constant A (derived form the gravitational constant), and the approximate volume of the universe.

According to WMAP^[2], the radius of the observable universe is 13.7 billion light years, giving an approximate radius of 1.296120075²⁶m. Given that the universe is uniform in all directions, an accurate way to calculate its volume would be to calculate the volume of a sphere with radius R_u:

V_u = (4/3Ï€) x R_u³

V_u = 4.188790205 x R_u³ (where R_u = 1.296120075²⁶ m)

V_u = 4.188790205 x (1.296120075²⁶m)³ = 9.120619146⁷⁸ m³

From here, two equations are needed to find the mass of the universe. The first shows the equation for finding the energy in the universe (E_u), while the second converts the units of measurement from joules to kilograms, and thus from energy into matter.

This calculation assumes that all energy within the universe exists in mass form, which is clearly not the case, but the value 8.28978902⁵⁶kg does give a good indication of the absolute maximum amount of matter in our 13.7 billion year old observable universe. From this information, it should also be possible to ascertain the critical density of the universe. If we accept that the constant of accommodation, A, sets the energy density of the universe at 8.168843247^-06J/m³ then the calculation for the critical density of the universe should be:

A/C² = 8.168843247^-06J/m³ / 8.987551787¹⁶m/s = 9.089063897^-23kg/m³

or

9.086063897^-28g/cm³

This value is slightly lower that the 1 x 10^-29g/cm³ that seems to be the general consensus for the value of the critical mass density (CMD) of the universe. The most important point to take from this rudimentary calculation is that G, A, A_V and the CMD of the universe are inextricably linked. The value given for the CMD of 9.086063897^-28g/cm³ should also be viewed as a maximum value, given the fact that it is based upon the notion of all the energy in the universe existing in matter form.

In practice, the only way to measure the values of A, A_F, A_V and the critical density is with more accurate measurements of G, but with the knowledge of what G actually is (G is the square of the constant of proportionality of the force of the vacuum, A_F), the accommodation constants may provide, within a classical framework, a deeper understanding when it comes to grand unified theories and theories of quantum gravity.