Can We Walk on “Supermassive Black Holes”?
You Are Going to Be Surprised!
In Real Gravity Does Not Produce Singularities we discussed Newton’s law of gravity:
F = G×M×m/R²,
where F is the gravitational force between masses M and m at distance R, and G is a constant (G = 6.674×10⁻¹¹ m³/[kg×sec²]). This formula is an approximation for real gravity, and it works well when masses and distances are not extremely big or small. And there are variations of this formula from Einstein and Schwarzschild that have the same flaw:
with M -> ∞ (masses going to infinity, aka big masses) or with R -> 0 (distances going to zero, aka small distances) F -> ∞ (gravitational force goes to infinity).
Because of such flaws, “Black Holes” and “Event Horizon”, where time stops and nothing, even light, can escape from, appeared in cosmology (but they do not exist in real life, because to stop time, infinite energy is required). Images of “Black Holes”, presented by astronomers, are images of “nearly Black Holes”, where time slows down (but does not stop), and light can still escape from there. And we discussed the correct / exact gravitational formula from Dr. Vivian Robinson:
F = G×M×m/[ R²×exp[ 2G×M/(R×c²) ] ], where c is the speed of light 3×10⁸ m/sec.
This formula does not produce close-to-infinity forces. Gravitational force F is limited:
Max F(R) = F(G×M/c²) = mc⁴/(G×M×e²), where e ≈ 2.72,
which means that maximum of the force is achieved at the distance R= G×M/c², where it acts on mass m with acceleration c⁴/(G×M×e²). Usually, acceleration is denoted as g (similar to the acceleration near the Earth, which we denote as g = 9.8m/sec²):
Max g(R) = g(G×M/c²) = c⁴/(G×M×e²) — maximum is at R = G×M/c².
c⁴/(G×M×e²) is a finite cap for acceleration g, and this cap becomes more restrictive (decreases) when mass M grows. If calculate these values for a Solar mass M, which astronomers denote as M☉, we get:
R = G×M☉/c² = 6.674×10⁻¹¹ m³/[kg×sec²] × 2×10³⁰ kg / (3×10⁸ m/sec)² = 1.5×10³ m = 1.5 km;
Max g(R) = c⁴/(G×M☉×e²) = (3×10⁸ m/sec)⁴ / ( 6.674×10⁻¹¹ m³/[kg×sec²] × 2×10³⁰ kg × 2.72²) = 8.2×10¹² m/sec².
Such cap has no use for the Sun for two reasons:
1) R=1.5 km is basically at the center of the Sun, where (by Newton’s shell theorem) there is no gravity;
2) Gravity on the Sun surface (where it reaches its maximum) is estimated as 274 m/sec², which is way below 8.2×10¹² m/sec² cap.
But having these values for the Solar mass M☉, as the unit, we can just prorate them to any mass M = N × M☉:
M = N × M☉, Max g(R) = 8.2×10¹²/N m/sec², for R = N × 1.5 km.
We will use this line for the rest of the story, for so-called “Supermassive Black Holes”, which are just heavy non-ignited stars with masses close to a million masses of Sun , and later we’ll look at M☉-billionaires and M☉-trillionaires. Let’s start with a “small” one, similar to Sagittarius A, which sits at the center of our Milky Way. To simplify calculations, I will take a M☉-millionaire with R = N × 1.5 km = 1,000,000 km = 10⁶ km. Deriving N, we can calculate the rest:
N = 10⁶ / 1.5; M = N × M☉ = 10⁶ / 1.5 × M☉ = 6.66×10⁵ × M☉;
Max g(R) = 8.2×10¹²/N m/sec² = 8.2×10¹²/ 10⁶ ×1.5 m/sec² = 12.3×10⁶ m/sec².
For that supermassive object, gravity at the distance of a million kilometers from its center is more than a million times greater than 9.8 m/sec² gravity on the Earth, and 45,000 times greater than 274 m/sec² gravity on the Sun. No surprise here so far, and the answer to the “Can we walk on it?” is “No”. But results change unexpectedly when we start looking at the most supermassive giants.
For a 100-billion-M☉ stellar object:
M = 10¹¹×M☉; Max g(R) = 8.2×10¹²/10¹¹ m/sec² = 82 m/sec²; R = 1.5×10¹¹ km.
For a stellar body with the mass of trillion M☉:
M = 10¹²×M☉; Max g(R) = 8.2×10¹²/10¹² m/sec² = 8.2 m/sec²; R = 1.5×10¹² km.
For a mass of a hundred billion Solar masses, acceleration is not greater than 82 m/sec², which is several times greater than 9.8 m/sec² on the Earth, but several times weaker than 274 m/sec² on the Sun. For a mass of trillion Solar masses, we get force not greater than 8.2 m/sec², which is less than 9.8 m/sec² on the Earth. And if look further, one order up, at 10 trillion Solar masses, then gravity becomes miniscule 0.82 m/sec², which is twice weaker than 1.62 m/sec² gravity of the Moon. As we can see, masses of trillions M☉ generate weak gravitational force (like on our Moon or on the Earth). Apparently, M☉-trillionaires, with a planet-like small gravity, cannot gain mass and retain planets. Largest “Supermassive Black Holes” observed, like TON 618 or Phoenix A, were estimated as having mass between 30-billion-M☉ to 100-billion-M☉. (This estimation might be off, because it relies on Schwarzschild formula, which is wrong). For a 100-billion-M☉ object, maximum gravity is 82 m/sec² at R = 1.5×10¹¹ km. Let’s calculate average density of this object at this radius (that might include some of its atmosphere, if that globe has smaller actual radius) as 100-billion-M☉ mass divided by the volume of the sphere 4/3 π R³:
10¹¹×M☉ / (4/3 π R³) = 10¹¹×2×10³⁰ kg×0.75 / 3.14 / (1.5×10¹⁴m)³ = 0.014 kg/m³.
It is a very low density even for Hydrogen gas, meaning, actual radius (of a solid or liquid surface) is way below R = 1.5×10¹¹ km. Let’s compare this density with the Solar and the Earth density:
Solar density — 1,400 kg / m³, which is 100,000 greater than 0.014 kg/m³;
Earth density — 5,500 kg / m³ (the Earth is smaller, but denser than the Sun, on average).
Now we can estimate that surface radius of this object is way below 10¹⁰ km (we reduced 1.5×10¹¹ km radius 15 times to increase its average density 15³=3,375 times from 0.014 kg/m³ to 47 kg/m³ — which is still gas, because it is yet below 70 kg/m³ density of liquid Hydrogen).
Answer to the question “Is 100-billion-M☉ giant walkable?” is “No”, not because of 82 m/sec² gravity (astronauts are trained to survive 10×g = 10×9.8 m/sec² > 82 m/sec² acceleration for a short time), but because its surface is liquid hydrogen (like it is on the Sun or gas giants/planets in our Solar system). And as we have mentioned, estimations for the observed top supermassive masses as in 30–100 billion M☉ range might be off because astronomers used a wrong formula. Most likely the actual sizes are smaller. Let’s run numbers for 10–30 billion-M☉:
M = 10¹⁰×M☉; Max g(R) = 8.2×10¹²/10¹⁰ m/sec² = 820 m/sec²; R = 1.5×10¹⁰ km;
density 10¹⁰×M☉ / (4/3 π R³) = 10¹⁰×2×10³⁰ kg×0.75 / 3.14 / (1.5×10¹³m)³ = 1.4 kg/m³.
M = 3×10¹⁰×M☉; Max g(R) = 8.2×10¹²/3/10¹⁰ m/sec² = 273 m/sec²; R = 4.5×10¹⁰ km;
density 3×10¹⁰×M☉ / (4/3 π R³) = 3×10¹⁰×2×10³⁰ kg×0.75 / 3.14 / (4.5×10¹³m)³ = 0.16 kg/m³.
30-billion-M☉ has the same 273 m/sec² gravity as our Sun has!
Coming back to M☉-trillionaires unsustainability: with their next-to-nothing gravity and with next-to-nothing density, it becomes clear why M☉-trillionaires exist only as galaxies (sparse collection of stars instead of a single sparse M☉-trillionaire globe). Our neighbor Andromeda galaxy has trillion stars, and there are two known galaxies with a hundred trillion stars or more.
