Gravity Exponential Form Again

Consider s space density function of the form,

${ d }_{ s }(x)=A{ e }^{ -bx }+B$  where A, B and b are constant to be determined.

At  ${ d }_{ s }(0)={ d }_{ e }$  space is compressed, its space density is  ${d}_{s}$,  on surface of earth. And  ${ d }_{ s }(x\rightarrow \infty )={ d }_{ n }$  where at long distance, space is relaxed and the corresponding space density is ${d}_{n}$.

We have,

${ d }_{ s }(x)={ e }^{ -bx }({ d }_{ e }{ -d }_{ n })+{ d }_{ n }$ ----(1)

Then we let the inverse relationship between time speed squared  ${v}^{2}_{t}$  and space density,  ${ d }_{ s }(x)$   to be,

${v}_{t}^{2}=C-D{ d }_{ s }(x)$   where  $C, D$   are to be determined.

We know that as  $x\rightarrow \infty, { d }_{ s }(x)\rightarrow{ d }_{ n }$  where space is relaxed and time speed is $c$,

${v}_{t}^{2}={c}^{2}$,  ${ d }_{ s }(x)={ d }_{ n }$

So,

$C={c}^{2}+D{d}_{n}$    then,

${v}_{t}^{2}={c}^{2}-D({ d }_{ s }(x)-{d}_{n})$

Differentiating with respect to time,

$2{v}_{t}\cfrac{d{v}_{t}}{dt}=-D\cfrac{d({d}_{s}(x))}{dx}\cfrac{dx}{dt}=-D\cfrac{d({d}_{s}(x))}{dx}v_s$ ---(2)

But from the energy equation,

$2{v}^{2}_{t}+{v}^{2}_{s}={c}^{2}$ differentiating it

$4{v}_{t}\cfrac{d{v}_{t}}{dt} + 2{v}_{s}\cfrac{d{v}_{s}}{dt}=0$, since $\cfrac{d{v}_{s}}{dt}=g$

$2{v}_{t}\cfrac{d{v}_{t}}{dt} = -g.{v}_{s}$    substitute into (2)

and so,

$g=D.\cfrac{d({d}_{s}(x))}{dx}$

From (1), differentiating with respect to x,

$\cfrac{d({d}_{s}(x))}{dx}=-b{e}^{-bx}({d}_{e}-{d}_{n})$ subsitute into the above we have,

$g=-D({d}_{e}-{d}_{n}).b{e}^{-bx}$

when  $x=0$,    $g=-g_o$

$g_o=\cfrac{G_o}{r^2_e}=D({d}_{e}-{d}_{n}).b$

$g=-g_o{e}^{-bx}=-\cfrac{G_o}{r^2_e}{e}^{-bx}$

And if  we consider

$\int^{\infty}_0{g}dx=\cfrac{G_o}{r_e}=\int^{\infty}_0{\cfrac{G_o}{r^2_e}{e}^{-bx}}dx$

$\cfrac{G_o}{r_e}=\cfrac{G_o}{r^2_e}[^{\infty}_0-{e}^{-bx}\cfrac{1}{b}]$

$r_e=\cfrac{1}{b}$

$b=\cfrac{1}{r_e}$

So,

$g=-g_oe^{-\cfrac{x}{r_e}}=-\cfrac{G_o}{r^2_e}{e}^{-\cfrac{x}{r_e}}$

This expression is based on compression of free space, in a exponential manner.