Surviving the Plunge


“The Conversation” answered this question: Could a human enter a black hole to study it?

I’m adding some figures to the question. Let’s see what that tells us.

Falling into a Black Hole can be unhealthy. However it’s not gravity that’ll kill you – it’s the first derivative of gravitational acceleration with respect to the radial distance i.e. the tidal forces that will be experienced when swinging past (or into) a compact object.

Black holes are the pointy end of a spectrum of objects. Stars are a balance between the outward pressure from their fusion energy production and the inward pressure from their self gravity. When fusion energy production ends, the cores of stars begin collapsing, held above the Abyss by firstly successive fusion energy reactions, then electron degeneracy pressure from squeezing free electrons too close together (via the Pauli Exclusion Principle) and when that isn’t enough, neutron degeneracy pressure and beyond.

Pressure is a measure of the ‘expansive’ energy packed in a volume. Dimensionally we can see that F/m2 (Pressure) = E/m3 (Energy per unit volume) and as the inward gravitational squeeze pushes inwards on a mass of particles pushing back against each other thanks to the Pauli Exclusion Principle for fermions (electrons, protons, neutrons etc) that pressure increases and increases… too much and equilibrium is never achieved. Thanks to Special Relativity we know that energy has mass, thus adding to the inward squeeze of gravity, the harder particles are squeezed together. When the Gravity Squeeze- Push-Back Pressure process self-amplifies and runs away, the mass collapses ‘infinitely’ inwards forming a Singularity. Such a Singularity cuts itself of from the rest of the Universe when it squeezes inwards past the mass’s Schwarzschild Radius:

$$r_s = \frac{2GM}{c^2}$$

This is the so-called Event Horizon which defines a Black Hole. It’s the ‘surface of no return’ for everything, including light. Nothing escapes from within the Event Horizon. At least when pointing outwards.

The minimum mass to cause such an inwards collapse and form a Black Hole for a mass of fermions in the present day Universe is about 3 solar masses, squished into a volume smaller than 18 kilometres across.

Before we get to that point there are White Dwarfs and Neutron Stars – objects defined by Electron Degeneracy Pressure and Neutron Degeneracy Pressure. White Dwarfs are typically composed of carbon and oxygen – the ashes of hydrogen & helium fusion – and mass anywhere between 0.1 and 1.4 Solar masses. Their radius is proportional to the inverse 1/3 power of the mass:

$$R(M) = \frac{R*}{M^\frac{1}{3}}$$

R* is a reference radius. For a cool white dwarf of 1 solar mass, the radius is about 0.8 Earth’s – 5,600 km. A space vehicle falling from infinity, on a flyby very close to such a star’s surface will rush past the lowest point of its orbit at 6,900 km/s, experiencing over 430,000 gee acceleration. In free-fall however it feels only the first derivative of that acceleration:

$$\dot g(R) = -\frac{2GM}{R^3}$$

Which in this example is 0.15 gee per metre of lateral structure squeezing inwards from the sides and a pulling force of half that is directed radially away from and towards the white dwarf. Easily protected from for small structures.

Neutron stars are smaller again – typically 20 km wide for a 1.3 solar mass neutron star. A near surface flyby isn’t recommended, since the tidal forces are thus almost a million fold stronger. However close proximity to a magnetic neutron star is probably lethal due to the intense magnetic fields long before the tidal forces rip you to shreds. Heavier neutron stars get smaller – just like white dwarfs – until they totally collapse as a black hole.

But… black holes reverse the trend. The event horizon gets bigger linearly with the mass. And there’s no limit to their mass. Take the black hole (call it Chandra) at the core of our Galaxy – 27,000 light years away and massing 4.3 million solar masses. The Event Horizon radius, for masses in “Suns” is simply:

$$r_{EH} = 2.95325 \cdot \frac{M*}{M_\odot}$$ kilometres

Thus Chandra is 12.7 million kilometres in radius. Substituting the Event Horizon radius equation into the Tidal force equation gives us:

$$\dot g(R) = -\left(\frac{c}{r_{EH}}\right)^2$$

Or in our chosen quantities:

$$\dot g(R) = -\left(\frac{299,792,458 m/s}{12,700,000,000 m}\right)^2$$ = -5.5E-4 m/s2 per kg per metre. (NB: I left a few units out, to focus on the key numerical factors, until this point.)

How far into Chandra can we then fall? If we can brace ourselves against 1,000 gees per metre of squeezing and stretching, then quite a long way…

$$R = \left(\frac{2GM}{\dot g(R)}\right)^{\frac{1}{3}}$$

Which gives a distance of 48,822 kilometres from the centre. In other words 99.6% of the way to the central Singularity. What wonders might we see? Quantum Gravity is yet to give a clear answer. One suggestion, based on modified General Relativity which has non-zero Torsion (space-time ‘twistiness’), is that the imploding Star forms the Genesis of a whole other Universe.

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