Black holes are regions of the universe where gravity is so strong that light can not emerge. And, since the motion of light is related to the fundamental structure of space and time, they must also be regions where space curves on itself, and where time appears to stop — at least as seen by us, from outside the black hole. But what does space-time look like inside the black hole.

NASA’s semi-useless depiction of a black hole — one they created for educators. Though it’s sort of true, I’m not sure what you’re supposed to understand from this. I hope to present a better version.

From our outside perspective, an object tossed into a black hole will appear to move slower as it approaches the hole, and at the hole horizon it will appear to have stopped. From the inside of the hole, the object appears to just fall right in. Some claim that tidal force will rip it apart, but I think that’s a mistake. Here’s a simple, classical way to calculate the size of a black hole, and to understand why things look like they do and do what they do.

Lets begin with light, and accept, for now, that light travels in particle form. We call these particles photons; they have both an energy and a mass, and mostly move in straight lines. The energy of a photon is related to its frequency by way of Plank’s constant. E= hν, where h is Plank’s constant and ν is frequency. Their mass is related to their energy by way of the formula m=E/c^{2}. This formulate is surprisingly easy to derive, and is often shown as E= mc^{2}. In classical form, the gravitational force between a star, mass M, and this photon or any other object of mass m described as follows:

F = GMm/r^{2}

where F is force, G is the gravitational constant, and r is the distance of the photon from the center of the star. The potential energy of a photon of the mass increases as it rises from the star surface, but the internal energy (proportional to frequency) decreases — the photon gets redder. The amount of internal energy lost to gravity as it rises from the surface is the integral of the force, and is thus related to the mass of the object and of the star.

∆E = ∫Fdr = ∫GMm/r^{2} dr = -GMm/r

Lets consider a photon of original energy E° and original mass m°. Lets figure out the radius of the star r° such that all of the original energy, E° is lost in rising away from the star. That is let calculate the r for which ∆E = -E° as the photon rises to freedom. Lets assume, for now, that the photon mass remains constant at m°.

E° = GMm°/r° = GME°/c^{2}r°.

We now eliminate E° from the equation and solve for this special radius, r°:

r° = GM/c^{2}.

This would be the radius of a black hole if space didn’t curve and if the mass of the photon didn’t decrease as it rose. While neither of these assumptions is true, the errors nearly cancel, and the true value for r° is double the size calculated this way.

r° = 2GM/c^{2}

r° = 2.95 km (M/M_{sun}).

Karl Schwarzschild 1873-1916.

The first person to do this calculation was Karl Schwarzschild and r° is called the Schwarzschild radius. This is the minimal radius for a star of mass M to produce closed space-time; a black hole. M_{sun} is the mass of our sun, sol, 2 × 10^{30} kg. To make a black hole one would have to compress the mass of our sun into a ball of 2.95 km radius, about the size of an SUV. Space-time would close around it, and light starting from the surface would not be able to escape.

As it happens, our sun is far bigger than a SUV and is not a black hole: we can see light from the sun’s surface with minimal space-time deformation. Still, if the mass were a lot bigger, the radius would be a lot bigger and the needed density less. Consider a black hole the same mass as our galaxy, about 1 x10^{12} solar masses (mostly dark matter), or 2 x 10^{42 }kg. The Schwarzschild radius of a star with the mass of our galaxy would be 3 x 10^{12} km, or 0.3 light years, about 1/20 the distance to Alpha Centauri. This is far bigger than the six of our solar system, but far smaller than the actual size of the galaxy, 5 x 10^{17} km, or 50,000 light years. Still, the difference between 0.3 light years and 50,000 light years isn’t that great on the cosmic scale, and it’s worthwhile to consider a black hole comprising something 10 to 100 Billion times more massive than our galaxy — the universe as a whole.

The folks at Cornell estimate the sum of dark and luminous matter in the universe to be about 15 billion times the mass of our galaxy, or 3 x 10^{52} kg. This does not include the mass of the dark energy, but no one’s quite sure what dark energy is. Considering only this physical mass, the Schwarzschild radius for the universe would be about 4.5 billion light years, or about 1/3 the size of our universe based on its age. The universe appears to be 14 billion years old, so if it’s expanding at the speed of light, the radius should be 14 billion light years. The universe may be 2-3 times bigger than this on the inside (rather like Dr. Who’s Tardis it’s bigger on the inside) but in astronomical terms a factor of 3 or 10 is nothing: the size of the universe is remarkably similar to its Schwarzschild radius, and this is without considering the mass its dark energy must have.

Standard picture of the big bang theory. Dark energy causes the latter-stage expansion.

The evidence for dark energy is that the universe is expanding faster and faster instead of slowing. See figure. There is no visible reason for the acceleration, but it’s there. The significant amount of energy must have significant mass, E = mc^{2}. If the mass of this energy is 3 to 10 times the physical mass, as seems possible, we are living inside a large black hole, something many physicists, including Einstein considered extremely likely. Einstein originally didn’t consider the possibility that the hole could be expanding. But now we know how to calculate the size of a black hole, and we know what a large black hole looks like from the inside. It looks just like home.

Wait for further posts on curved space-time. For some reason, no religion seems to embrace science’s 14 billion year old, black-hole universe (expanding or not). As for tidal forces, they are horrific only for the small black holes that most people write about. If the black hole is big enough, the tidal forces are small.

Dr. µß Buxbaum Nov 17, 2014. You can drink to the Schwarzchild radius with my new R° cocktail.

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