The general theory of relativity is famously complex. I know I can’t work the math. I’ve tried. I’m no spring chicken any more, and I just can’t wrap my brain around the complex math these days. You can teach an old dog new tricks, but not this dog, and not those tricks.
However…
Is all that complexity really necessary? The first version of anything is almost always overly complicated and a little wonky. General relativity is well tested to several decimal places, but is also known to be incomplete. “Incomplete” in physics means “useful but wrong”.
What if Einstein (and Hilbert, et al.) were wrong? They’ve built a really inspired edifice of complex math describing a geometry of curved space. But that doesn’t necessarily make for a true description of reality. Doesn’t make it false, either, to be honest. But I generally bet against overly complicated descriptions of reality. Perhaps, in this instance, we could sharpen Occam’s razor and see what we find under all that scruff?
Let us return, once again, to the examination of Schwarzschild’s solution for empty space around a non-rotating, stationary, massive body. His solution boils down to applying the spacetime metric to the standard gravitational potential energy and escape velocity. General relativity is, at its heart, a description of energy levels. But is this the proper energy to use? I grant that my entire theory is based upon the idea of a master field of potential energy, from which all other fields draw their energy. But is gravitational potential energy real?
Pro:
It’s useful and easy to compute. You subtract heights to compute energy levels.
Escape velocity is a useful concept, computed by relatively straightforward calculations.
Con:
Very few objects actually fall from infinity, which is an abstract concept to begin with.
It doesn’t handle the finite speed of propagation at all. Granted, the model is for a stationary object. But still.
The escape velocity at a point {√(2GM/r)} does not correspond well to the gravitational acceleration at that point {GM/r²}.
Is there a better model that gives similar results? Yes. Yes, there is.
We return to my fairly simple model of a potential energy field deformed by energy being withdrawn from it. Given an energy equivalent mass m, the field deforms as m/r. But, since the energy is being withdrawn from a maximum possible value, with a minimum of zero, we subtract it from one, giving 1 - m/r. This gives local energy gradients as -m/r². Gradients cause change.
Notice the boundary at radius one. You can’t withdraw more energy than exists. This particular graphic model only works for bodies of maximal possible energy density, better known as black holes. You can scale it up or down as appropriate. At the lower limit, you get a black hole with the diameter of the Planck length. Convenient, that. At that scale, you find that all particles have a radius of 1 (half the Planck length), with a depth dependent upon their energy content as a fraction of the Planck energy.
But how does this compare to the Schwarzschild solution? To the graph!
Blue is the results of my (1 - m/r) field. Purple is Schwarzschild’s results. Green is the differences, which approach but never quite reach zero. That’s math for you.
What is the Lorentz alpha factor doing here? Well, in the Schwarzschild equation, you use the escape velocity to determine the alpha factor. That corresponds directly to the m/r factor in my theory. Each alpha corresponds uniquely to a velocity. Even if the velocity isn’t actually present except notionally. The difference being that with my formula, that velocity then also corresponds directly to the local gravitational acceleration, which Schwarzschild’s equation doesn’t do. It’s -m/r² regardless. And I compute the “stress velocity” from alpha, not the other way around. Remember, alpha is time dilation. So time dilation is directly caused by energy level, not just velocity. This is important.
Notice the factor of two in the Schwarzschild curves. Those come directly from the use of escape velocity to determine the gravitational potential. So the radius of the black hole must be doubled using that model.
Notice how the curves of the different models start out differently, but rapidly converge. For clarity, this graph only extends to 6. That’s 3 Schwarzschild radii. That’s fairly close to a black hole’s event horizon by anyone’s measure. At 5 units on the x axis, the difference is already down to 5,047,531 meters per second (1.68% of the speed of light). But how small are the differences at practical, measurable distances, like the surface of the sun or Earth?
At the sun’s surface, the difference between the two velocities is approximately 676 micrometers (0.000676) per second. That’s on a calculated escape velocity of over 617 kilometers per second. At Earth’s surface, the difference is a mere 72.5 micrometers per second on an escape velocity of over 11 kilometers per second.
Figuring out a curve of m/r is a lot easier than deriving Einstein’s field equations or interpreting Schwarzschild’s results. It boils down to a distinction with little difference - except my model does not lead to absurd results. No infinities. No singularities. No absurdities like white holes or wormholes. Just particles of fixed diameter (the Planck length) interacting in simple ways. Of course, that fixed diameter leads to some interesting results, as I’ve mentioned before.
Oh, you wanted formulas? Remember, these are for a black hole, where mass = 1.
α = 1 - 1/r
v=√(2r-1) / r
The velocity formula is derived from the sin² θ + cos² θ = 1 identity, using
{ α = cos θ = 1 - 1/r } and { v = sin θ }. Relativity is mostly based upon high school trigonometry, with just a touch of first year calculus.
And as always, Copenhagen interpretation delenda est!
