The above video shows how marbles do elliptical orbits on lycra that is depressed by a massive steel ball. It's analogous to satellites orbiting stars and planets. In this post we show how real spacetime is dented or curved and work out the mathematics of orbiting satellites. So let's define some variables:
Below is diagram D-1. It's a bit crude, but it shall be our guide as we do the math.
Imagine two adjacent volumes of space. Both have volume V. Neither has any significant energy or mass--just empty space. That brings us to equation 1:
Suppose we add a star to one the volumes (see D-1). The star's matter and energy take up some of the space. That leaves a net volume V'. V' is only slightly less voluminous than V, since the star is made of atoms that are mostly space. Suppose the star collapses into a black hole. Surely a black hole takes up no space and V' should equal V. But the black hole and the star have common ground: both have the same energy density within volume V.
At the top of D-1 is the Compton wave formula. When energy (or mass) is added, wavelengths decrease. Shorter wavelengths take up less space and are equivalent to a smaller volume. At D-1, the squiggly lines represent the wavelengths. Note that the top portion (V) has longer wavelengths than the bottom portion (V') If we think of volume in terms of wavelengths, V' is definitely less than V.
The V minus V' average wavelength difference is equivalent to the volume taken up by the star. We can add that volume to V' to get V:
Thus the variables of equation 2 have the same values whether we have energy density in the form of a giant star or a black hole.
Now, let's divide by the z axis (see D-1) to get areas A and A'. Equation 4 gives us the net area (the broken-line rectangle at the center of D-1).
At 5 we divide the areas by the square of the average relative time it takes for a satellite to go along the x and y axes. At 6 through 8 we find the squared velocity of the satellite. It is small compared to light speed--due to gravity being weak ... and ... gravity is weak due to the minor difference in V and V'.
Equation 7 is illustrated at D-2 and D-3 below. Due to increasing energy density (shorter wavelengths), when particle-waves move toward the star, they gain momentum (or lose less momentum). Due to decreasing energy density (longer wavelengths), when particle-waves move away from the star, they lose momentum (or gain less momentum). Thus, on average, the particle waves that make up our satellite, spacetime, etc., are attracted to the star.
At D-3, the long arrows represent the increasing momentum of stuff coming in. The short arrows represent the decreasing momentum of stuff going out.
The converging arrows at D-4 represent the net momentum, the dent in the lycra--the gravitational field. Its intensity increases as the satellite falls due to increasing energy density.
At 10 through 12 we work the star's mass and momentum into the equation. Whenever mass is included in the field, so is it's inertia. Inertia cancels mass and thus different masses fall at the same rate, so we divide by mass as well as multiply.
At 13 through 16 we convert the mass and momentum into n units of Planck mass and Planck momentum.
At 17 we borrow from Heisenberg's uncertainty relations and write the Planck mass in terms of Planck's reduced constant. We make a substitution at 18. At 19 we convert the reciprocal of the Planck momentum squared and make another substitution at 20. Some variables cancel each other.
At 20 we have something close to what we need. All that's left to do is to put the variables back in that we took out earlier (for ease of computation):
At last! We have equation 24. It gives us the satellite's instant velocity at any point during its orbit. Equation 23 allows us to change the ellipse coefficient so we can have a variety of elliptical orbits (see D-5 below) on the dented lycra of spacetime.
Update: Below is a formal proof showing that equation 2 is a solution of Einstein's field equations.
Update: How much actual space does a black hole occupy? It has a singularity with a zero-limit radius (r), but a physical extent of radius (rs), the Scharzschild radius. The answer appears to be a volume with the Schwarzschild radius. Below is the mathematics showing how a black hole takes up space:
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