Universal Inverse-Square Laws without Singularities!

Even a high school student can understand!

We learn in school laws about forces that are evenly distributed over R-radius spherical area 4πR²:

• Newton’s law for gravitational force F = — G×M×m/R² between masses M and m (with gravitational constant G);
• Coulomb’s law for electrostatic force F = Q×q/(4πε₀×R²) between charges Q and q (with constant ε₀).

These formulas give infinite forces F ≈ ∞ at short distances R ≈ 0, since division by near 0 gives near infinite results. Infinities in math are called singularities in physics, but they do not exist in the nature (hold your thoughts about Black Holes, Event Horizon, and the Big Bang). About a century ago, Einstein found that Newton’s formula should be adjusted if time is variable, and it is: time in the Universe is not absolute, it flows somewhere faster, somewhere slower, and sometimes it slows down or speeds up. But even after these corrections, physicists still claimed that even the corrected formula for gravity produces singularities. Only recently Dr. Robinson came out with the precise formula for gravity that cleared off singularities. The purpose of my article is to show an elementary and universal path to physics without singularities.

Let’s denote by D time dilation factor in an area compared to another area, for example:

• D = 2 stands for twice-slower time — for a second passing in a slower-time area, two seconds pass in a faster-time area;
• D = 1 stands for the same time speed in both areas — no time dilation;
• D = 1.000002 — for 1 second passing near the Sun, 1.000002 seconds pass on the Earth (that is the true value for time dilation near the Sun).

For an object moving at speed v m/sec in a slower-time area, let’s denote such speed as v(D), where D is time dilation in this area, compared to another area where we stay and look at this moving object. What is the velocity of this object from our time perspective? In a second of time, in the slower-time area object moves v(D) meters, but in our (faster) time area, our clocks ticked not once, but D times (D seconds passed for us), thus velocity that we observe is v(D)/D (the same distance divided by our time passed). Let’s denote such observed velocity as v(1) (meaning v(for D=1), since there is no time dilation for us). We have v(1) = v(D)/D. Apparently, it is the mere consequence of our clocks ticking more often, thus, any object moves a shorter distance at 1 tick of the faster-ticking-clock. Velocity is measured in m/sec, and since velocity is just change in position per second, with the whole change in position being the same in our time and slower time, but in more seconds for us, change in position per our second is smaller than it is in slower time (for slower-ticking clock). Let’s go further: consider change in velocity per second — that is acceleration, it is denoted by a m/sec². Again, because time interval at which we track change in velocity is D times shorter in our time, and velocities are already D times smaller in our time, then change in velocities in our time is D×D=D² times smaller than in slower time: a(1) = a(D)/D². The same applies to forces, because force is acceleration multiplied by mass, and mass is independent of time:

What does it mean? F(D) is “local” force, in local time where the object is at the moment, F(1) is “observed” from outside (in our/observer time) force. Of course, we observe acceleration and not force usually, but acceleration is proportional to force. That changes traditional inverse-square law for “local” force from

F ~ 1/R² (“~” means proportional with a coefficient K, where F = K/R²)

to

F ~ 1/(R²×D²)

for the force, observed from a D-times faster time (the object, on which this force acts, is in D-times slower time area at the moment). Thus,

• F = — G×M×m/(R²×D²) for gravity in variable time;
• F = Q×q/(4πε₀×R²×D²) for electrostatic force in variable time.

Besides forces and acceleration, physicists often use potential — it is a function, derivative of which (by location) gives acceleration:

— Potential' = F/m = a (Potential' denotes derivative of potential).

And I showed in [1] (or even simpler in [5]) that potential for gravity is

and it causes (see in [2]) following dependency of D from distance R:

D = exp[G×M/(R×c²)], where exp(x)=eˣ.

That was a major discovery, originally done by Dr. Robinson, because such combined dependency

F = — G×M×m/(R²×D²) = — G×M×m/(R²× exp[2G×M/(R×c²)] )

makes F -> 0 when R -> 0 (because exp(1/R) goes to infinity faster than R² goes to 0). Meaning, gravitational force is finite now and does not produce singularities. But what about other forces like electrostatic force? Physics does not know massless particles that carry charge. Let’s denote by M mass (which can be very small) of the charge Q carrier, then the formula

F = Q×q/(4πε₀×R²×D²)

combined with

D = exp[G×M/(R×c²)] (even for infinitesimal mass M)

becomes

F = Q×q/(4πε₀×R²× exp[2G×M/(R×c²)] ),

for which again F -> 0 when R -> 0, because 1/(R²×exp[K/R]) -> 0 (K is a coefficient) <=> X²/exp(K×X) -> 0 when X = 1/R ->∞ (since exponent grows faster than any power of X).

Despite M can be very small (for proton it is 1.67×10⁻²⁷), values of F are finite now. And if we consider acceleration a = F/m, where m is mass of electron, which is even lighter (m = 9×10⁻³¹), then acceleration is still finite. Inverse-square laws by time dilation (1) and (2) have removed singularities from forces that correlate with 1/R² (i.e. inverse-square-by-distance forces).

P.S. D = exp[G×M/(R×c²)] shows that D is always finite for any R > 0 => there is no Event Horizon, where time stops (where D = ∞, more in [3]).
P.P.S. More about Dark Science debunked read [4].

References:
[1] Time Energy Potential = 0.5 c²/D²
[2] Einstein’s General Relativity Becomes Elementary in 2024
[3] Inside Black Hole
[4] Time Matters
[5] Simpler Derivation of Gravitational Potential = 0.5×c²/D²