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 E is the photon energy, h is Plank’s constant and ν is frequency. The photon mass is related to its energy by way of the formula m=E/c², a formula that is surprisingly easy to derive, and often shown as E= mc². The version that’s relevant to photons and black holes is:

m =  hν/c².

Now consider that gravity affects ν by affecting the energy of the photon. As a photon goes up, the energy and frequency goes down as energy is lost. The gravitational force between a star, mass M, and this photon, mass m, is described as follows:

F = -GMm/r²

where F is force, G is the gravitational constant, and r is the distance of the photon from the center of the star and M is the mass of the star. The amount of photon energy lost to gravity as it rises from the surface is the integral of the force.

∆E = – ∫Fdr = ∫GMm/r² dr = -GMm/r

Lets consider a photon of original energy E° and original mass m°= E°/c². If ∆E = m°c², all the energy of the original photon is lost and the photon disappears. Now, lets figure out the height, r° such that all of the original energy, E° is lost in rising away from the center of a star, mass M. That is let calculate the r for which ∆E = -E°. We’ll assume, for now, that the photon mass remains constant at m°.

E° = GMm°/r° = GME°/c²r°.

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

r° =  GM/c².

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/c²

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 × 10³⁰ 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 a small asteroid. 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 an asteroid and is not a black hole: we can see light from the sun’s surface with minimal space-time deformation (there is some seen in the orbit of Mercury). Still, if the mass were a lot bigger, the radius would be a lot bigger and the density would be less. Consider a black hole the same mass as our galaxy, about 1 x10¹² solar masses, or 2 x 10⁴²  kg. This number is ten times what you might expect since our galaxy is 90% dark matter. The Schwarzschild radius with the mass of our galaxy would be 3 x 10¹² km, or 0.3 light years. That’s far bigger than our solar system, and about 1/20 the distance to the nearest star, Alpha Centauri. This is a very big black hole, though it is far smaller than our galaxy, 5 x 10¹⁷ km, or 50,000 light years. The density, though is not all that high.

Now let’s consider a black hole comprising 15 billion galaxies, the mass of the known universe. The folks at Cornell estimate the sum of dark and luminous matter in the universe to be 3 x 10⁵² kg, about 15 billion times the mass of our galaxy. This does not include the mass hidden in the form of dark energy, but no one’s sure what dark energy is, or even if it really exists. A black hole encompassing this, known mass would have a Schwarzschild radius about 4.5 billion light years, or about 1/3 the actual size of the universe when size is calculated based on its Hubble-constant age, 14 billion years. The universe may be 2-3 times bigger than this on the inside because space is curved and, 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 actual size of the known universe is remarkably similar to its Schwarzschild radius, and this is without considering the mass its dark energy must have if it exists.

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 source of the energy might be some zero-point effect, but wherever it comes from, the significant amount of energy must have significant mass, E = mc². 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 and aesthetically pleasing. Einstein originally didn’t consider the possibility that the hole could be expanding, but a reviewer of one of his articles convinced him it was possible.

Based on the above, we now know how to calculate the size of a black hole of any mass, and we now know what a black hole the size of the universe would look 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 the tidal forces around black holes, they are horrific only for the small black holes that most people write about. If the black hole is big, the tidal forces are small.

 Dr. µß Buxbaum Nov 17, 2014. The idea for this post came from an essay by Isaac Asimov that I read in a collection called “Buy Jupiter.” You can drink to the Schwarzchild radius with my new R° cocktail.