And We Have A Smooth Take Off

From the post "Hollow Earth",

\(F=-\cfrac { GM }{ 4Rr^{ 2 } } \int _{ s=R-r }^{ s=R+r }{ \left( 1+\cfrac { r^{ 2 }-R^{ 2 } }{ s^{ 2 } }  \right)  } ds.m\)

\( g(s)=\cfrac { F }{ m } =-\cfrac { GM }{ 4Rr^{ 2 } } \int { \left( 1+\cfrac { r^{ 2 }-R^{ 2 } }{ s^{ 2 } }  \right)  } ds\)

On the outer surface of hollow earth,

\( s=r-R\),  \(s\gt0\)

\( \cfrac { d\, g(s) }{ d\, s } =-\cfrac { GM }{ 4Rr^{ 2 } } \left( 1+\cfrac { r^{ 2 }-R^{ 2 } }{ s^{ 2 } }  \right) +\cfrac { GM }{ 2Rr^{ 3 } } \int { \left( 1+\cfrac { r^{ 2 }-R^{ 2 } }{ s^{ 2 } }  \right)  } ds\)

\( =-\cfrac { GM }{ 4Rr^{ 2 } } \left\{ 1+\cfrac { r^{ 2 }-R^{ 2 } }{ s^{ 2 } } -\cfrac { 2 }{ r } \int { \left( 1+\cfrac { r^{ 2 }-R^{ 2 } }{ s^{ 2 } }  \right)  } ds \right\} \)

\(=-\cfrac { GM }{ 4R(s+R)^{ 2 } } \left\{ 2+\cfrac { 2R }{ s } -\cfrac { 2 }{ s+R } \int { \left( 2+\cfrac { 2R }{ s }  \right)  } ds \right\} \)

\(=-\cfrac { GM }{ 2R(s+R)^{ 2 } } \left\{ 1+\cfrac { R }{ s } -\cfrac { 2 }{ s+R } \int { \left( 1+\cfrac { R }{ s }  \right)  } ds \right\} \)

The gradient of gravity \(g^{'}(s)\) has a discontinuity at \(s=0\) due to the \(\cfrac{R}{s}\) term.  This high value in gradient results in a sharp increase in gravity \(g\rightarrow 2g\) at \(s=0^{+}\), on the outer surface of hollow earth.

In reality, gravity doubles only after some distance from sea level. The difference between \(s=0\) and \(s=0^{+}\) is significant.  And the next rocket launch should be less shaky.

Double Vision

One way to generate a high frequency pulse is to use a detector in the path of a travelling wave

where a stationary detector at D, intersects the wave fronts at,

\(f=\cfrac{v}{\lambda_s}\)

For a piece of metal with photoelectric threshold frequency of \(f_T\), embedded in a low resistance semiconductor, \(v\) is the speed of light in the semiconductor, then

\(\lambda_s=\lambda_T\)

where \(\lambda_T\) is the corresponding threshold wavelength.  We have,

\(f=f_T\)

The problem with this is being totally confused with particle wave duality.

However;

Given a small enough metal surface area, are the electrons emitted/detected at \(f\)?  Only if the light is pulsed at \(f\). So, given a wave train pulsed at \(f_p\),

\(f_p=\cfrac{v}{\lambda_p}\)

\(n_s=\cfrac{\lambda_p}{\lambda_s}\)

where \(n_s\) is the refractive index at the detector.

\(f=f_pn_s\)

\(f\) can then be modulated by changing \(n_s\).

If \(n_s\) changes with gravity as in an aerogel or the like, we then have an instantaneous gravity detector.  No more swinging pendulum, LPPL indeed.

And Gravity Doubles

On moving away from the shell outer surface of hollow earth, like going up a flight of stairs in a flats or by the side of a mountain, gravity doubles at sufficient distance from ground level then it decreases  by the inverse square law.

This can actually be verified on Earth.

So, Earth is hollow!