Last week a museum visitor wondered, “What’s the volume of a black hole?” A question easier asked than answered.
Let’s look at black hole (“BH”) anatomy. If you’ve seen Interstellar, you saw those wonderful images of “Gargantua,” the enormous BH that plays an essential role in the plot. (If you haven’t seen the movie, do that. It is so cool.)
A BH isn’t just a blank spot in the Universe, it’s attractively ornamented by the effects of its gravity on the light passing by:
adapted from Dr Kip Thorne’s book, The Science of “Interstellar”
Working from the outside inward, the first decoration is a background starfield warped as though the stars beyond had moved over so they could see us past Gargantua. That’s because of gravitational lensing, the phenomenon first observed by Sir Arthur Eddington and the initial confirmation of Einstein’s Theory of General Relativity.
No star moved, of course. Each warped star’s light comes to us from an altered angle, its lightwaves bent on passing through the spatial compression Gargantua imposes on its neighborhood. (“Miles are shorter near a BH” — see Gravitational Waves Are Something Else for a diagrammatic explanation.)
Moving inward we come to the Accretion Disc, a ring of doomed particles destined to fall inward forever unless they’re jostled to smithereens or spat out along one of the BH’s two polar jets (not shown). The Disc is hot, thanks to all the jostling. Like any hot object it emits light.
Above and below the Disc we see two arcs that are actually images of the Accretion Disc, sent our way by more gravitational lensing. Very close to a BH there’s a region where passing light beams are bent so much that their photons go into orbit. The disc’s a bit further out than that so its lightwaves are only bent 90o over (arc A) and under (arc B) before they come to us.
By the way, those arcs don’t only face in our direction. Fly 360o around Gargantua’s equator and those arcs will follow you all the way. It’s as though the BH were embedded in a sphere of lensed Disclight.
Which gets us to the next layer of weirdness. Astrophysicists believe that most BHs rotate, though maybe not as fast as Gargantua’s edge-of-instability rate. Einstein’s GR equations predict a phenomenon called frame dragging — rapidly spinning massive objects must tug local space along for the ride. The deformed region is a shell called the Ergosphere.
Frame dragging is why the two arcs are asymmetrical and don’t match up. We see space as even more compressed on the right-hand side where Gargantua is spinning away from us. Because the effect is strongest at the equator, the shell should really be called the Ergospheroid, but what can you do?
Inside the Ergosphere we find the defining characteristic of a BH, its Event Horizon, the innermost bright ring around the central blackness in the diagram. Barely outside the EH there may or may not be a Firewall, a “seething maelstrom of particles” that some physicists suggest must exist to neutralize the BH Information Paradox. Last I heard, theoreticians are still fighting that battle.
The EH forms a nearly spherical boundary where gravity becomes so intense that the escape velocity exceeds the speed of light. No light or matter or information can break out. At the EH, the geometry of spacetime becomes so twisted that the direction of time is In. Inside the EH and outside of the movies it’s impossible for us to know what goes on.
Finally, the mathematical models say that at the center of the EH there’s a point, the Singularity, where spacetime’s curvature and gravity’s strength must be Infinite. As we’ve seen elsewhere, Infinity in a calculation is Nature’s was of saying, “You’ve got it wrong, make a better model.”
So we’re finally down to the volume question. We could simply measure the EH’s external diameter d and plug that into V=(πd3)/6. Unfortunately, that forthright approach misses all the spatial twisting and compression — it’s a long way in to the Singularity. Include those effects and you’ve probably got another Infinity.
Gargantua’s surface area is finite, but its volume may not be.
~~ Rich Olcott
Of all the wave varieties we’re familiar with, gravitational waves are most similar to (NOT identical with!!) sound waves. A sound wave consists of cycles of compression and expansion like you see in this graphic. Those dots could be particles in a gas (classic “sound waves”) or in a liquid (sonar) or neighboring atoms in a solid (a xylophone or marimba).
Einstein noticed that implication of his Theory of General Relativity and in 1916 predicted that the path of starlight would be bent when it passed close to a heavy object like the Sun. The graphic shows a wave front passing through a static gravitational structure. Two points on the front each progress at one graph-paper increment per step. But the increments don’t match so the front as a whole changes direction. Sure enough, three years after Einstein’s prediction, Eddington observed just that effect while watching a total solar eclipse in the South Atlantic.
We’re being dynamic here, so the simulation has to include the fact that changes in the mass configuration aren’t felt everywhere instantaneously. Einstein showed that space transmits gravitational waves at the speed of light, so I used a scaled “speed of light” in the calculation. You can see how each of the new features expands outward at a steady rate.
Almost a century later, James Clerk Maxwell (the bearded fellow at left) wrote down his electromagnetism equations that explain how light works. Half a century later, Einstein did the same for gravity.
Gravitodynamics is completely unlike electrodynamics. Gravity’s transverse “force” doesn’t act to move a whole mass up and down like Maxwell’s picture at left. Instead, as shown by Einstein’s picture, gravitational waves stretch and compress while leaving the center of mass in place. I put “force” in quotes because what’s being stretched and compressed is space itself. See 
The experiment consists of shooting laser beams out along both arms, then comparing the returned beams.
Grammie always grimaced when Grampie lit up one of his cigars inside the house. We kids grinned though because he’d soon be blowing smoke rings for us. Great fun to try poking a finger into the center, but we quickly learned that the ring itself vanished if we touched it.
For instance, suppose Fred and Ethel collaborate on a narwhale research project. Fred is based in San Diego CA and Ethel works out of Norfolk VA. They fly to meet their research vessel at the North Pole. Fred’s plane follows the green track, Ethel’s plane follows the yellow one. At the start of the trip, they’re on parallel paths going straight north (the dotted lines). After a few hours, though, Ethel notices the two planes pulling closer together.
The line rotates as a unit — every skater completes a 360o rotation in the same time. Similarly, everywhere on Earth a day lasts for exactly 24 hours.
Now suppose our speedy skater hits a slushy patch of ice. Her end of the line is slowed down, so what happens to the rest of the line? It deforms — there’s a new center of rotation that forces the entire line to curl around towards the slow spot. Similarly, that blob near the Equator in the split-Earth diagram curls in the direction of the slower-moving air to its north, which is counter-clockwise.
Newton definitely didn’t see that one coming. He has an excuse, though. No-one in in the 17th Century even realized that electricity is a thing, much less that the electrostatic force follows the same inverse-square law that gravity does. So there’s no way poor Isaac would have come up with quantum mechanics.
If Newton loved anything (and that question has been discussed at length), he loved an argument. His battle with Leibniz is legendary. He even fought with Descartes, who was a decade dead when Newton entered Cambridge.
in his Les Miz role of Inspector Javert, 
And then there’s 
Newton was essentially a geometer. These illustrations (from Book 1 of the Principia) will give you an idea of his style. He’d set himself a problem then solve it by constructing sometimes elaborate diagrams by which he could prove that certain components were equal or in strict proportion.
For instance, in the first diagram (Proposition II, Theorem II), we see an initial glimpse of his technique of successive approximation. He defines a sequence of triangles which as they proliferate get closer and closer to the curve he wants to characterize.
The third diagram is particularly relevant to the point I’ll finally get to when I get around to it. In Prop XLIV, Theorem XIV he demonstrates something weird. Suppose two objects A and B are orbiting around attractive center C, but B is moving twice as fast as A. If C exerts an additional force on B that is inversely dependent on the cube of the B-C distance, then A‘s orbit will be a perfect circle (yawn) but B‘s will be an ellipse that rotates around C, even though no external force pushes it laterally.
It all started with Newton’s mechanics, his study of how objects affect the motion of other objects. His vocabulary list included words like force, momentum, velocity, acceleration, mass, …, all concepts that seem familiar to us but which Newton either originated or fundamentally re-defined. As time went on, other thinkers added more terms like power, energy and action.
There is another way to get the same dimension expression but things aren’t not as nice there as they look at first glance. Action is given by the amount of energy expended in a given time interval, times the length of that interval. If you take the product of energy and time the dimensions work out as (ML2/T2)*T = ML2/T, just like Heisenberg’s Area.
Hard Science Ain't Hard 