All “Black Holes” Are Escapable, Even from Their Black Surfaces
Escape velocity is derived in classic and relativistic mechanics as v = sqrt(2GM/R), where sqrt is square root, G is the gravitational constant, M is the mass of the attractor, R is the distance to the attractor. Here is the problem: for massive and dense attractors (like “Black Holes”), 2GM/R value can be greater than c², then v > c (where c is the speed of light ≈ 300,000 km/sec). That is impossible, because c is the universal speed limit. Thus, the mainstream’s claim is: some (massive) bodies at some (close) distances are inescapable (“Black Holes”). Is it for real, or is it just a math mistake? No worries — it is just a math mistake. Let’s start with a classic derivation and then correct it.
Check https://byjus.com/physics/derivation-of-escape-velocity/, on escape work done against Newton’s gravitational force GMm/x², where x is (variable) distance to the attractor and m is the mass of the escaping body:
By the energy conservation principle, that (escape) work should be equal to the kinetic energy mv²/2 of the escaping body (where v is its velocity when it was at the distance R to the attractor), thus
GMm/R = mv²/2 => v = sqrt(2GM/R) — end of the classic derivation.
The correction lies in Dr. Vivian Robinson’s precise formula for gravity (we discussed in Real Gravity Does Not Produce Singularities — Black Holes Myth and Math Busted):
GMm/[ R²×exp[ 2GM/(Rc²) ] ],
which we should use instead of Newton’s gravitational force GMm/R² for the escape work calculation:
Thus, mc²/2 × [ 1-exp( -2GM / (Rc²) ) ] = mv²/2 =>
v = c × sqrt[ 1-exp( -2GM / (Rc²) ) ] ❗
Now, because 2GM /(Rc²) is positive, -2GM /(Rc²) is negative, exp(-2GM / (Rc²)) is a positive value less than 1 , and 1-exp(-2GM /( Rc²)) is also is a positive value less than 1. Therefore, sqrt[ 1-exp(-2GM / (Rc²)) ] is less than 1, thus, escape velocity
v = c × sqrt[ 1-exp(-2GM / (Rc²) ) ] is less than c ❗
Now we can see that escaping any attractor, even from its surface, is possible.
The classic formula v = sqrt(2GM/R) for the escape velocity is just an approximation for the correct formula we just have derived, it comes from exp(x) ≈ 1+x + … (approximation by Taylor’s series for small values x):
v = c × sqrt[ 1-exp(-2GM / (Rc²)) ] ≈ c × sqrt[ 1-(1–2GM / (Rc²)+…) ]
≈ c × sqrt[ 2GM / (Rc²) ] = sqrt(2GM/R).
sqrt(2GM/R) is only appropriate for small GM /(Rc²) values.
