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2.4.7 Limits to Gravity

The presence of a gravitational structure centered on the origin of the coordinate system results in a spherically symmetrical displacement of the continuum. In the above analysis, the continuum was assumed to be flat, however, its clear that at large values of radius the structural curvature of the continuum from flatness should be taken into account. Since this curvature is universally present, even in the gravitational structure itself, it can be included simply by adding it to local curvature. In addition, local structural gradient, \(-r/R_o\), can be included with gravitational gradient, so that total gradient is given by \(A/r^2 - r/R_o\). The value of \(R_o\) is of the order \(10^{26}\) meters, so that structural gradient near the central mass, \(r<<R_o\), can be neglected.

However, these components of gradient are of opposite sign, at some radius, \(r_x\), they will cancel out, that is when, \(A/r_x^2 - r_x/R_o =0\). This yields a cross-over radius, as follows. \begin{align}
\frac{A}{r_x^2} = \frac{r_x}{R} \nonumber \\
AR=r_x^3 \nonumber \\
and\ r_x = (AR)^{\frac{1}{3}} \nonumber
\end{align}

This is effectively a limit to gravitational attraction. The same result can be obtained via Hubble's law. It is not generally understood that Hubble's law which describes recession velocity also describes recession acceleration.
\begin{align}
recession\ velocity\ v_r = \frac{\partial r}{\partial t} = Hr \nonumber \\
acceleration\ = \frac{\partial v_r}{\partial t} = \frac{\partial Hr}{\partial t} =H \frac{\partial r}{\partial t} \nonumber = H^2r \nonumber
\end{align}

In Newton's theory of gravity, two masses, \(M\) and \(m\), separated by a distance, \(r\), will accelerate towards each other with an acceleration, \((M+m)G/r^2\). For a single mass, \(M\), its gravitational attraction at radius, \(r\), is \(MG/r^2\). The logical result of this is that, at some value of \(r\), say \(r_x\), Newtonian gravitational attraction and Hubble expansion will cancel each other. This cross-over radius, \(r_x\), can be calculated by setting, \(H^2 r_x = MG/r_x^2\), yielding \(r_x^3 = MG/H^2\).

Given that, \(MG=AHc\), and \(c=HR\), we find that, \(MG/H^2=(AH)(H R)/H^2 = AR\) where \(R\) is the radius of the continuum. This is the same result obtained via gradients, where, \(r_x^3 = AR\).

In general terms, we note that gradient due to the inherent curvature of the universe and the gradient due to gravitational curvature interact to limit the range of gravitational attraction. This simple fact explains why galactic rotation does not obey Newton's laws, and why neighboring galaxies have not coalesced into one huge galaxy. This phenomena is consistent with the mass of the Milky Way galaxy and its effective diameter. A discussion of this phenomenon is given below.

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