Fun Fact - Spiraling into a Black Hole

So you want to time-travel into the future. You could stand next to a black hole for a few hours and watch a million years roll buy, but there are at least three serious problems here.

  1. We'll never build a rocket powerful enough to slow you down as you fall into the gravity of a black hole, so you don't get sucked in forever.

  2. If we could, then while you're standing next to the black hole, not being sucked in, you are crushed by a million G's of force.

  3. If you could survive the G forces, we'll never have a rocket powerful enough to push you back out of the black hole again.

But wait, maybe we can solve all three problems in one go, by zipping past a black hole in a hyperbolic orbit. Not too close, or the tidal forces will tear you to shreds. Make it a really big black hole, so you can zing past the event horizon and not be torn apart. There are other problems, like the intense heat and radiation at the event horizon, and a rocket to take you there and back home again, but let's say we could solve these. As you approach the black hole, time slows down for you, and you can look outward and see the universe racing ahead, like a video on fast-forward. You swing around in a free-return trajectory, and head home. So there is our time machine, a way to vault into the future.

Would you go? It's a huge gamble. A million years from now you might find a fantastic George Jetson world, but more likely, there won't be any humans at all, or if there are, they might be a species you don't recognize, living in a preindustrial age, perhaps even a stone age. You would land back on earth and have nothing in common with them. Thanks to evolutionary drift, you wouldn't be able to mate with them. You surely wouldn't speak their language, but maybe you could learn it. As your tools break down one by one, your technology drops away, and you too live in the stone age with these "people". You can't even explain that you are a million years in their past; they don't understand. Yeah I know, it's not an original idea - Planet of the apes - but I wouldn't expect that particular outcome; I would expect animals, including humanoids, as they were 100,000 years ago, and the 5 million years before that. I expect a recapitulation of a stable situation, which is not necessarily where we are now. Our intelligence, or at least our civilization and our technology, could be a poor player, That struts and frets his hour upon the stage, And then is heard no more. If that is the case, I don't want to know. So I wouldn't go.

Here's a more down-to-earth application. If you don't have to feel the G's to experience time dilation, as when falling into or around a black hole, then maybe a box could produce an artificial gravity well, such that time creeps along at a snail's pace inside. The obvious example is a refrigerator. But that's a misnomer, because there is no reason for the food to be cold any more. It ages only a minute for every day the box is closed. You open up the box and find room temperature milk, and fruit, and cheese, and meats, as fresh as the day you put them in. It would take a while to get use to eating room temperature perishables. Another application is a babysitter of sorts, which I wrote about in the previous article.

Returning to the skies, there's another problem with my time machine; orbital mechanics is not what you expect. How can an object spiral in to a black hole? Did you ever wonder? In 1687, Newton published the math for a two body orbit. It's always a conic section. Perhaps an ellipse, like the earth going around the sun. Perhaps a hyperbola, like an asteroid flying by the earth, it's path bent slightly by earth's gravity. Or it could be a parabola, if the orbit is right on the edge between an ellipse and a hyperbola. If an asteroid skims past the surface of the moon, where there is no air, it doesn't get caught in a spiral; it's path bends slightly as it sails on by. Never a spiral, so what gives?

It's all about general relativity, and it starts with light. Gravity can bend light, as demonstrated in 1919 during a total eclipse, when the stars at the edge of the sun shifted position, their light bent by the sun. We think of the sun as massive, but it barely bends starlight; the deflection is just enough for us to measure with 1919 technology. You need something much heavier to turn a beam of light several degrees.

Picture a massive black hole, without a lot of crap falling into it, cause that stuff emits tremendous radiation and would fry you to a crackly crunch. It bends light in a big way - and as the beam gets closer to the black hole, it bends even more. In fact, if the light is close enough, it can bend into a circle and remain in orbit. It's not a stable orbit, like the earth around the sun; slight perturbations push the light beam out of its circle, and eventually into the event horizon, never to return. But for a while, light is trapped around the black hole without falling in. This is called the photon sphere. If you stand at a distance of 3/2 × the event horizon, and look straight ahead, perpendicular to the radius, you will see the back of your head. Light bounces off the back of your head, bends around the black hole, and enters your eyes. So you can tell if you're having a bad hair day. Closer in, light can run around in elliptical orbits.

What does all this light bending have to do with spiral orbits? This is where GR gets really weird! To set the stage, let's go back to Newton's first law: an object in motion stays in motion, unless acted upon by an outside force. Not just in motion, but in that particular direction. That seems obvious to us now, with our 21st century knowledge. We can hardly appreciate the genius of this idea; it was revolutionary in its day! For 2,000 years everybody thought the circle was God's perfect motion. Everything in the sky moves in circles. Ok, the planets didn't move in perfect circles, but people like Ptolemy spent his entire life trying to derive the orbits as wheels within wheels within wheels, as though they were drawn by a spirograph. A lifetime of careful astronomical observations and trig calculations wasted, because the premis was wrong. And not just Ptolemy, but others as well. So everyone "knew" that the circle was the perfect path, but Newton said no, he said an object would continue along in a straight line. And he was right. Throw a baseball and it travels in a straight line, except for two outside forces acting on it, gravity (curving it towards the ground), and air resistance (slowing its forward motion). When you're in a car, and the car turns, you feel a force, we call it a centrifugal force, pulling you to the outside; but really the car is exerting a force on you, turning you from a straight line into a curve. It's something we're all familiar with now. Newton's laws are second nature to us.

Newton's three laws were essential for his follow-on work on orbits. The moon would sail away forever, traveling in a straight line, but there is an outside force acting on it, the gravity of the earth. That constant and unrelenting force bends the orbit into an ellipse. From here, Newton invented calculus, then used it to show all orbits are conic sections. Now we're ready for Einstein.

Over and over again, Einstein found that Newton was right for any ordinary circumstance, but in extreme situations, the math changes. Let's revise newton's first law. An object will travel along the same path as a beam of light, unless acted upon by an outside force. Light beams travel in straight lines everywhere in our solar system, so you can see why Newton's first law was so successful. You can send a spacecraft to Mars using Newton's formulas, and it will reach its destination on time and on target. But if, somewhere in the galaxy, a light beam bends, an object will tend to travel along that same path. Linear momentum is defined by light paths, which is defined by the shape of spacetime.

Return to your station above the black hole, on the edge of the photon sphere. You can look forward and see the back of your head. Throw a baseball straight ahead and ask what happens. Obviously it gets sucked down by gravity - but in honor of Newton and Einstein, ask what happens if there are no outside forces acting on it. Without gravity, and without air resistance, it doesn't travel in a straight line away from you, it travels in a circle, and hits you in the back of the head. It travels as the light beam travels. The baseball is going in a circle, but there is no centrifugal force, because it is going the way it wants to go. We like to think that if we threw it hard enough, it would attain a stable, circular orbit around the black hole. That's what happens here on earth. If you stand above the atmosphere and throw a ball at 5 miles per second, centrifugal force exactly balances gravity, and the orbit is stable. Seen another way, gravity is just enough of an external force to pull the ball out of line, to curve it away from its light path, and it goes round and round. But at the black hole, the ball already traces a circle. We don't need a force to bend it into a circle. Any downward force, such as gravity, pulls the ball further down, below its circle. And here we have the spiral. As the ball moves further down, the light beam curves even faster, and even the forward motion of the ball exerts a downward force. I'll say it again, inside the photon sphere, centrifugal force is reversed. The faster you revolve around the black hole, the stronger the force pushes you *inward* towards the black hole. The beam of light curves tighter than your circle, an ellipse tangent to your circle on the inside, rather than a straight line tangent to the outside. With the beam of light curving on the inside, centrifugal force pulls you in, not out, and adds to gravity, instead of counterbalancing it. Light can orbit in the photon sphere, but nothing else.

There are a lot of variables here, but as a general rule, if you try to fly by a black hole, and you get within 3 or 4 × the event horizon, the light bends too much, and the nice hyperbola you thought you were tracing turns into a spiral. Kiss your ass goodbye! When I am flying past a black hole to leap into the future, it's not just the tides I have to worry about, it's Einstein and his light beams turning my nice hyperbolic orbit into a death spiral. Unfortunately, time dilation and light bending go hand in hand. You can't have one without the other. I want to be close, so that time slows down and I wind up far in the future. I'd graze the event horizon if I could. But I can't get too close, because I need Newtonian mechanics to whip me around the black hole in a hyperbolic orbit and send me back home. Once again, time travel is difficult.

Further Reading

Light is distorted near a black hole, and can be trapped inside the photon sphere.
The cauchy horizon