Mathematical Recreations : Gravity of a Black Hole

Part of being a good scientist - in any discipline - is learning how to recognize the unconscious assumptions that others are making in their reasoning and then setting about examining those assumptions. I recently came across a video describing the gravity at the center of a black hole as infinite.

This didn't match my intuition, so I spent some time thinking about what might be the assumptions behind it.

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My first step was to back away from the extreme case of a black hole and look at something we know more about, Earth. The force of gravity at the surface of the Earth is described by the formula...

\( g = -\frac{GM}{r^2} \).

where M is the mass of the Earth (5.97219x10²⁴ kg), r is the radius of the Earth (6,378.1 km), and G is the gravitational constant (6.67384x10^-11 m³kg^-1s^-2). This calculates to ~9.8 ms^-2.

If you start going beneath the surface of the Earth, the equation for calculating the force of gravity becomes more complicated. Any mass below you is pulling you downwards, while any mass above you is pulling you upwards.

Wikipedia has a nice article going into detail about the force of gravity as you go further into the Earth. Because of variations in how density increases at different depths, the force of gravity increases until you reach the transition between the lower mantel and the outer core of the planet. As you go deeper than this transition, the gravity decreases rapidly.

At the very center of the Earth, the calculation is much simpler. The mass pulling you each direction is balanced by an equal amount of mass pulling at you in the the other direction. This cancelation results in the force of gravity at the center being 0.0 ms^-2.

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What is the gravity at the center of a black hole? The standard answer is that the gravity at the center of a black hole is infinite, because the matter within the object is infinitely compressed.

Nobody knows exactly what goes on within a black hole, because we can't go there and we can't look at it from afar. However, people spend a great deal of time thinking about them because they're interesting cases where our knowledge is limited.

The calculations which show the gravity at the center of a black hole to be infinite rely on the assumption that the mass within a black hole will collapse inwards forever, with all of its mass collected to an infinitesimal point at the center with infinite density.

I think infinite density is something that can't exist in our universe, so a point of infinite gravity also can't exist. If infinite density can't exist, there must be a universal maximum density constant. This would be similar to how there is a maximum speed, the speed of light.

The only certain calculation we can make about the density of mass within a black hole is that all of the matter is compressed to within the Schwarzschild radius of the object. This calculation gives us a lower bound for the maximum density constant. The constant could be much higher, without being anywhere near infinite.

A maximum density constant implies some consequences.

1. As you pass towards the center of a black hole, eventually you will enter the physical object. Once you are inside the mass at the core of the black hole, the force of gravity will start decreasing until you reach the center. At the center, the force of gravity would equal 0.0 ms^-2.
2. There would be a minimum size for black holes. According to the wikipedia page on the Schwarzschild radius, the average density of a black hole with the mass of Earth would be 2.04x10²⁷gcm^-3. If the maximum density constant were lower than this, there could not be a black hole with mass as low as the Earth. For certain values of the constant, this would explicitly rule out the microscopic black holes which some people were worried about being created by the Large Hadron Collider.
3. Since the Big Bang was an expansion of spacetime itself, a finite maximum density in that spacetime isn't a problem.
4. As a black hole evaporates (via Hawking Radiation), its Schwarzschid radius will eventually shrink to fall inside the physical surface of its core (because a maximum density implies a minimum size for a black hole). At this point, radiation from the surface of the object will be able to escape. At this time or some point later, the surface itself will begin to decompress (explode). The result is that this form of black hole would end with dramatic incandescence and explosion. The detailed output spectrum from this would probably differ from that produced by the infinite density black hole model decaying away to nothing.
5. The central singularity of a black hole doesn't exist, so quantum mechanics and general relativity can be unified into a fuller understanding of what gravity is.

There are probably other important consequences, but I haven't worked through the math fully. I'm still working on the thesis for my Genetics PhD, so it will be some time before I can being to examine the math in detail.

The remaining problem is of estimating what the constant of maximum density is. I've conceived of three general approaches:

1. Analyze of the sizes of black holes (which we have a hard time finding) and note the lower limit to their mass/size range. There is a very large black hole at the center of our galaxy, but this provides for a very limited estimate of the constant because it is at the wrong end of the size range.
2. Calculate the differences in emission spectra from the different models of black holes and look out into the universe for detonating black holes to test between the models.
3. Calculate the maximum density from other - more knowable - physical measures. I have no notion of how one would go about doing this, but the inter-connectedness of physics suggests there may be a path.

Can you think of any other approaches to determine what the constant of maximum density is?

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So, how does this connect with (or how can I connect it with) biology?

Living things depend upon extracting energy from naturally occurring energy (electrical, chemical, light, etc.) gradients. Around a black hole, there is a profound gradient in gravitational energy. This leads me to the idea of complex self-organizing, self-reproducing structures (life) that feed off of those intense gravitational gradients.

The author Robert L. Forward published the book "Dragon's Egg" in 1980, about intelligent lifeforms (the Cheela) living on the surface of a neutron star.

I'll leave it to you to ponder on what the physical nature of life forms feeding off the surface of the more extreme case of a black hole might be.