The sound of gravity as dreamed by Johannes Kepler
- From: Albertito <albertito1992@xxxxxxxxx>
- Date: Sun, 1 Jun 2008 04:29:11 -0700 (PDT)
I'm collecting evidences for speed of gravity and speed
of light near massive bodies. I can guess both speeds,
c_g and c are naturally linked in a beautiful form. But firstly,
I must define what speed of gravity, c_g, is. Suppose there
is a central massive body, with mass M, and a test body in
circular orbit at radius |r|. What will happen to that test body
if the central body magically and instantaneously vanishes?
Obviously, that test body will follow a tangent straight path with
a constant speed equals to its tanget speed v = sqrt(GM/|r|).
The question is, when will that event happen?. We know where
it will happen, it will happen at a distance |r| from the position of
the center of the massive body. Suppose now, the central body
vanishes instantaneously when the test body vector position, r,
has an angle phi_0 in a given inertial frame of reference (say wrt
fixed stars). Then, if the test body starts to move in a straight
path
when its vector position angle has that angle phi_0, at first glance,
we'd claim the speed of gravity from M to the test body would have
been infinite. But, if its angle is phi > phi_0, then the speed of
gravity
would have been finite. How can we quantify that speed of gravity
c_g?. In order to answer that question, we need to assume there is
a finite c_g. Suppose that value c_g and the radius |r| are such
that the test body starts to moves in a straight line at angular
position phi_0, but after it had performed a complete circular
revolution. As the angular speed of the test body is w = sqrt(GM/|r|
^3),
its period is T = 2pi/w, so the speed of gravity would be
c_g = |r|/T,
c_g = sqrt(GM / |r|) / (2 pi),
In general, we would always deduce the angular position
phi_0, for any
c_g = sqrt(GM / |r|) / (2 n pi),
with integer n = 0,1,2,3, ...
We clearly see an infinite speed of gravity, c_g = oo, is
deduced for n = 0. We can even replace the integer n,
by any positive real k
c_g = sqrt(GM / |r|) / (2 k pi).
Some questions arise naturally, is k a universal constant?
Can we claim conclusively that k=0? For constant fixed |r|
and k, we see c_g increases as M increases. For constant
fixed M and k, we see c_g decreases as |r| increases.
A wonderful conclusion would be the speed of gravity c_g is
actually the speed of sound! The denser the medium, the higher
c_g. Now, we see that the speed of light, c, behaves in an
opposite manner as that c_g. The denser the medium, the
lower the speed of light c. So, both speeds must be tightly
related.
.
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