Twinkle, Twinkle Little Stars.

Gravity around a black hole

$g=-g_{eo}e^{-\cfrac{1}{r_{eo}}x}$

where,

$g_{eo}$ = 5.07071e18 ms^-2    and    $r_{eo}$ = 0.00886 m

If we approximate this gravity curve at $x=0$ with a straight line, we have for small $x$, $\Delta x$,

$\Delta g=\cfrac{g_{eo}}{r_{eo}}e^{-\cfrac{1}{r_{eo}}(x=0)}\Delta x=\cfrac {g_{eo}}{r_{eo}}\Delta x$

From the posts "In, Up, Out and Away..." and "A Right Turn, A Wrong Turn" we know that photons are repelled away from strong gravitational region, as such

$\Delta g=-\cfrac {g_{eo}}{r_{eo}}\Delta x$

If we compare this to Hooke's Law,

$F=ma=-kx$   or  $a=-\cfrac{k}{m} x$

We find that, for small $x$, $\Delta x$

$\cfrac{k}{m}=\cfrac{g_{eo}}{r_{eo}}$

That is to say, photons around a black hole or massive planet with high gravitational field have a resonance frequency of,

$f=\cfrac{1}{2\pi}\sqrt{\cfrac{k}{m}}=\cfrac{1}{2\pi}\sqrt{\cfrac{g_{eo}}{r_{eo}}}$

At some nominal surface gravity $g$, as $\Delta x$ increases $\Delta g$ increases in the opposite direction, some photons fall back with less acceleration,  as $\Delta x$ decreases $\Delta g$ decreases in the opposite direction and the same photons accelerate forward again.  The result is photons cluster into groups or packets.

For the case of a black hole,

$f$=1/(2pi)*(5.07071e18/0.00886)^(1/2) = 3.807e9 Hz = 3.807 GHz

For any other planet,

$f=\cfrac{1}{2\pi}\sqrt{\cfrac{g_{p}}{r_{p}}}$

where $g_p$ is the planet surface gravity and $r_p$ the planet radius.

In the case of Earth,

$f$=1/(2pi)*(9.80665/6371000)^(1/2) = 0.000197 Hz or 1 pulse every 5064.3 s

This frequency is not to be confused with light frequency but pulses of photons that escapes the pull of a strong gravitational field.  It is possible that such pulses be detectable with our naked eye given the right values for high $g_p$ and small $r_p$.

Twinkle, Twinkle Little Stars.