Let’s begin with the formula. We assume there is a field of potential energy, from which all other energies are withdrawn. It has a large but finite constant value (U₀) at every point. The potential energy at a point outside a particle is the total potential energy minus the total energy of the particle (E) divided by the distance (r) from the particle in relation to the radius of the particle (r₀).
U = U₀ - E(r₀/r), where r ≥ r₀.
If we take the gradient (slope) of this curve, we get acceleration, which we call gravity. The instantaneous acceleration (over the minimum possible time, t₀) at a point is a velocity, as a fraction of the speed of light. This determines the Lorentz alpha factor at that point, which is the subjective passage of time. The faster you go, the slower you perceive time passing. Perceived time is to speed as cosine is to sine. (This is the fundamental relationship of Special Relativity. Yes, it really is just the Pythagorean Theorem.)
The energy gradient inside a particle ( a fixed amount unless altered by external gradients) is the particle’s own velocity (instantaneous acceleration), which determines its alpha factor. Below we have a stationary (red) particle of mass energy 0.25U₀ being accelerated to the left (dashed green) by an external gradient (blue).
Notice the blue curve is not affected by the particle at all. Locally, energy is not conserved. (However, our test particle creates a gradient which accelerates the particle causing the blue curve, so the total system balances.) Also notice how the acceleration is not influenced by the particle’s own mass/energy. Drop a feather and a hammer on the airless moon, and they both fall the same way.
Notice also that the particle is accelerated by the external gradient, but then retains this acceleration. This is why the Schwarzschild equation has two factors in the direction towards the attractive body: one for time (enormous, but decreases with speed), and one for space (minuscule, but increases with speed). Time flows much, much faster than distance does for relatively slow moving bodies, so the accumulated acceleration is much, much more of a factor than the instantaneous push of local acceleration. Light has double the expected curvature near the surface of the sun because it’s travelling distance r₀ in time t₀, so the internal and external accelerations are equal. Mercury orbits the sun a tiny bit faster than expected because it’s traveling so very quickly, and even more quickly nearest the sun. Also, the gradient across the diameter of a particle is always greater than at the central point. However, this difference is only really noticeable as you get close to the attractive body and the slope increases. Newton was fooled by the small angle approximation, which holds in everyday experience.
A highly observant reader might notice that these curves are somewhat different from the accepted values in General Relativity. (At the surface of the Earth and Sun, the values predicted differ at the ninth non-zero digit, a quantity several orders of magnitude smaller than current measurement errors.) Schwarzschild based his equation on the escape velocity of a particle approaching a body. At r = 1, this can be greater than the body’s own energy, which is nonsensical. However, the accelerated particle reaches the speed of light at distance r = 2 from a black hole mass, at which point the two particles impact each other (each having a radius of 1) and stop, there being no more energy to draw from. (You can’t have less than zero energy remaining in the field.) So, it’s not actually a contradiction. It’s merely incorrect to assume the energy of a body can be more than the actual energy of the body, and that the energy of a body can be more than the available energy. To give him credit, the concept of fields had yet to be invented, and energy still isn’t a well defined concept more than a century later.
Yes, every particle has the same radius r₀, with an inside and outside separated by a discontinuity. Why? Because Planck and Heisenberg said so. So does the geometry of 1/r, its derivative, and its integral. When combined with the finite total energy of U₀, this prevents infinities and singularities. A black hole is a thin region of maximal energy density surrounding a spheroid of zero potential energy.
You might have noticed the energy gradient (units: kg m²/s²) is being measured over a distance (m) in a time (s). That gives momentum (kg m/s). The mass (kg) is somewhat unimportant, as we convert it directly to energy (E = m c²). It has no gradient, but it does take up an important amount of space, defining the size and shape of a particle! And empty space has no mass, and thus no momentum. So we can redefine energy (anyrgy?) for our purposes to ignore the kilograms and be simply m²/s².
Empty space has energy gradients, and given the fixed speed of light and the fixed size of a particle, these gradients directly correlate to velocity. However, we aren't truly adding velocities. We are adding energy gradients. They're not the same thing, although they appear to be at low energies. The small angle approximation strikes again!
What are U₀, r₀, and t₀? I don’t know. A good guess is that they are based on the Planck units. If so, then U₀ ≈ 1.9561×109 J, r₀ ≈ 1.616255(18)×10−35 m, and t₀ ≈ 5.391247(60)×10−44 s.
