Real Gravity Does Not Produce Singularities
Black Holes Myth and Math Busted
Newton’s most popular formula, known as inverse square law for gravity, is:
F = G×M×m/R²,
where F is the gravitational force between masses M and m at distance R, and G is some constant.
As you can see, the smaller R, the greater force F, and with R approaching to 0, force F goes to infinity. That is the problem, which led to unrealistic conclusions, because this formula is just an approximation.
Einstein and Schwarzschild corrected Newton’s formula by taking into account so-called gravitational redshift Z (usually a small value, for example, for the Sun, Z=0.000002, and for the Earth, Z is even smaller, around one billionth):
Even with such corrections, gravity still goes to infinity when R approaches 0. Recently, Dr. Vivian Robinson derived the precise formula for gravity:
F = G×M×m/[ R²×exp[ 2G×M/(R×c²) ] ],
which behaves differently for R close to 0 (where other formulas give infinite results), c is the speed of light (about 3×10⁸ m/sec).
To simplify, let’s denote X=G×M/(R×c²), then R=G×M/(X×c²), and X goes to infinity when R goes to 0. Let’s rewrite the last formula for F in terms of X, replacing R with G×M/(X×c²):
F(X) = G×M×m/ [ [G×M/(X×c²)]² × exp(2X) ] = m×c⁴/(G×M) × [X² / exp(2X)] = m×c⁴/(G×M) × [ X/exp(X) ]².
m×c⁴/(G×M) is constant (does not depend on X), and since exp(X) grows much faster than X, then X/exp(X) -> 0, when X -> ∞ (aka, when X goes to infinity). That is why F(X) -> 0, when X -> ∞.
Q.E.D. When R -> 0, X -> ∞, then F -> 0 — no singularity (no F -> ∞, when R goes to 0, as with Newton’s formula). And if we calculate F(X) for X=0: F(0) = m×c⁴/(G×M) × [ 0 / exp(0) ]² = 0. Thus, F(X)=0 both for X=0 and for X=∞. Thus, it has some maximum value somewhere in between. Let’s find it.
Formula F(X) can be rewritten as:
F(X) = m×c⁴/(G×M) × X² × exp(-2X)
and we can find where it reaches its maximum by finding where its derivative F’(X)=0:
F’(X) = m×c⁴/(G×M) × [ 2X× exp(-2X) — 2X² × exp(-2X) ] = m×c⁴/(G×M)×2X×exp(-2X)×[1 — X],
F’(X)=0 for X=1 — that is where maximum is achieved. And for X=1:
F(X) = m×c⁴/(G×M) × X² × exp(-2X) = mc⁴/(G×M×e²),
that is maximum value for gravity, which is reached at X=1, which is G×M/(R×c²)=1, or R=G×M/c². There e ≈ 2.72 = exp(1).
To summarize,
Max F(R) = F(G×M/c²) = mc⁴/(G×M×e²) —no singularity / infinity!
Here is graph from Dr. Vivian Robinson presentation at https://www.youtube.com/watch?v=bcGCTuDbhjc , comparing his (no singularity / infinity) gravitational formula with the earlier (Newton’s, Einstein’s, Schwarzschild’s) formulas/approximations:
Continue with: Can We Walk on “Supermassive Black Holes” or Neutron Stars? You Are Going to Be Surprised!
