How does spacetime get bent?S

In Einstein's universe, spacetime is supposed to be some crazy rubber sheet full of folds and bends. But the idea of curved space is not the most intuitive in the world. And what does light have to do with any of this? In this week's Ask a Physicist, we'll find out.

For the last few weeks, we've been running a contest to answer some of the most interesting questions about the cosmos. This week's winner is RT who should shoot me an email to get a free signed copy of my new book. He/she asks:

If light has no mass, then how does it interact with massive objects like the sun through gravity causing it to bend? Also, on the extreme side, what does the edge of the event horizon look like on a black hole as the light is bent around?

If you want to understand gravity, you need to understand General Relativity – with no disrespect to Isaac Newton intended. To understand Relativity, we need to understand "spacetime" – which, as just like it sounds like, is just space and time smashed together.

The great John Archibald Wheeler had a ready description for how all of this fits together. "Spacetime tells matter how to move;" he said, "matter tells spacetime how to curve."

This curvature of spacetime has caused a lot of confusion over the years, and while bowling balls on rubber sheets may give a vague conceptual idea of how gravity works, let's get back to basics and figure out where this curvature comes from. It will also give me a really good opportunity to riff on one of Einstein's more awesome thought experiments.

Before we get into how gravity works, I need to say a few words about space and time.

How does spacetime get bent?S

Image Credit: NASA, Andrew Fruchter and the ERO Team

A Relativity Primer

Einstein came up with his famous Theory of Special Relativity in 1905. The idea, as you may recall, is that the speed of light should be the same for everybody, and so long as you're traveling at a constant speed and in a constant direction, you shouldn't be able to tell that you are moving.

Those were his assumptions (which turned out, as it happens, to be perfectly in accord with the actual physics of the universe), and from them, he found some incredibly surprising things:

1) A clock on a moving spaceship will run slow compared to stationary observers outside. This is also true for heartbeats, pendulums, digital watches, and so on.

2) A moving spaceship will appear to be compressed along the direction of motion.

In both cases, the effect becomes ridiculously huge as you approach the speed of light, but are small enough to be easily ignored under normal terrestrial conditions, which is why nobody noticed them until Einstein came along.

In order to extend those predictions to gravity, starting in 1907 (and over many iterations) , Einstein devised what he referred to as the Equivalence Principle, which says (roughly):

[We] assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system.

He argued that there is no measurable distinction between real gravity and acceleration, which is why there's no functional difference between being thrown the back of a rocketship because it's accelerating, and being thrown to the bottom of a refrigerator carton with a picture of a rocket ship drawn on it; it's sitting on earth, and you tend to fall in the downward direction.

Even without working out any of the details of General Relativity, a process that took Einstein almost an entire additional decade post–Special Relativity, he quickly had an inkling of what the final theory should look like. By using the Equivalence Principle, Einstein came up with a scenario for relating artificial to real gravity, one that I’m going to shamelessly steal.

Life in Antworld

Imagine life on top of a large spinning disk.

In this universe, there are a bunch of superintelligent ants slowly crawling around on the surface. The queen sits perfectly still at the center of Antworld. Her royal court surrounds her in close proximity. To an outsider (you), her courtiers slowly rotate about the queen. They don’t know any of this, of course. They just grip so as to not be thrown outward by the tug of the rotating disk. As far as they’re concerned, “out” is “down.”

The farther the ants are from the Queen, the faster they move and the stronger they are tugged outward. From the perspective of the ants, their Antworld feels very much like a hill with the queen at the top, a hill that gets steeper the farther they go out. An ant that loses its grip will roll outward—down the hill—at an ever-accelerating rate.

How does spacetime get bent?S
Credit: Herb Thornby (one of his awesome illustrations from my new book!)

There’s at least one sense in which this analogy isn’t perfect. If you fall down a hill on earth, you’ll simply roll down in a radially outward path. An ant falling down the hill in Antworld will start rolling straight down, but will then slowly start rolling around the hill as well. This is the famous Coriolis Effect. It’s the same thing that causes cyclones to spin counterclockwise in the Northern Hemisphere and clockwise in the Southern.

But supposing they don't move around too much, we can pretty much ignore the Coriolis Effect entirely. Your toilet water also doesn't move around too much, which is why, whatever else it does, the Coriolis Effect doesn't determine the direction of your toilet water funnel.

Likewise, as far as the the ants are concerned, they live on a hill and aren’t spinning at all. Outside the Antworld, we know better. The queen isn’t moving. Nearby ants are moving slowly. Ants farther out move faster. The ants out in the hinterlands are moving fastest of all. This is where all of our training in Special Relativity begins to really pay off. We know something about the flow of time of moving ants. The faster they move, the slower time will appear to pass compared to the queen. The farther out an ant is, the slower that ant will appear to age.

The ants don’t know they’re moving so they don’t know that Special Relativity should come into play at all. The ants, so far as they can tell, are living in a gravitational field. They've discovered that the farther you go “down” the slower time runs.

The ant physicists are absolutely correct—about their universe and ours. Time runs slower the closer you get to a massive body, and the more massive the body, the more dramatic the effect. These effects are quite real, but normally ridiculously tiny. Time runs slower on the surface of the earth by less than 1 part in a billion compared to time out in deep space. Over the surface of the earth, the effect is even smaller. Time runs slower at the bottom of Mount Everest than at the top by about 1 part in a trillion. Given that we’ve been confined to the surface of the earth for most of our existence, it’s not that surprising that nobody before Einstein noticed that the flow of time changes based on where you are.

There are much more extreme environments out there. You could hang out on the surface of a neutron star where time runs slower by 20 percent or more. After a decade, two extra years will have passed far away. What you’ve done here is built a (pretty crappy) time machine into the future. But because the gravity on a neutron star is so strong that you’d be squashed like a pancake, traveling to the future is probably the least of your concerns.

But what about the curvature? Here's where things get even weirder. Remember when I said that space appears contracted along the direction of motion? Well, since the ants are moving around and around, distances seem compressed. Suppose an outlying ant decides to take a trip around the entire world, a circle. His trip would seem shorter than the queen might have guessed using simple Euclidean geometry. As seen by its residents, Antworld is curved. What’s true for the ants is true for us.

"Straight" Lines

Even though we "know" that Antworld really is a flat disk, it doesn't seem so to the ants. An astronaut flying around in a rocket will take what she perceives to be a straight line, but it's anything but. In other words, even in a 2-d world, there's a difference between what the inhabitants see, and what we might see in a higher dimensional space.

A straight line is normally considered to be the shortest distance between two points, but from our perspective, the rocketship curves outward (as you can see in the Antworld drawing above).

Because there's no difference between "real" gravity and a spinning disk or an accelerated rocketship, everything falls at the same rate (Thanks, Galileo!), whether they're ordinary matter, particles with negative mass, or particles with no mass at all.

Which brings us to light.

If you've got an old school physics education (the kind where you played with pendulums, pulleys, and did force diagrams without end), you probably learned that only particles with mass experience gravitational acceleration. Wrong, wrong, wrong. Light beams bend in gravitational fields as well, because they are, after all, only following the shortest route. This effect is known as "Gravitational Lensing," and was, in fact, the first successful test of General Relativity.

Time runs slow near a massive source; that's what the ants told us at any rate. Light, which wants to travel in as fast a route as possible will tend to avoid the slow time parts of the universe, and as a result, will get deflected. Because there are several possible "fastest" routes, this can also mean that we get multiple images of the same object.

How does spacetime get bent?S

G2237+305. Credit: NASA, ESA, and STSCI.

But back to RT's original question. We've gotten the "why?" and "how?" light gets bent, but we still haven't touched on the "how much?" Normally, the effects of lensing are quite small because, compared to black holes, most places in the galaxy have very, very weak gravitational fields.

However, something interesting happens near a black hole. As most good io9 readers know, black holes are bounded by a region of no return known as the "Event Horizon." For black hole the mass of our sun, for instance, the event horizon is at a radius of about 3km.

Interestingly, it turns out that if you fire a laser pointed at 1 1/2 times the event horizon, you can (if you aim absolutely perfectly) get your laser to orbit the black hole. Any less, and it spirals inwards, never to be seen again. Any more, and your laser will boomerang back at you.

There's a moral to all of this: If you're going to be firing lasers near black holes, you're going to shoot your eye out.

Dave Goldberg is a Physics Professor at Drexel University. His newest book, The Universe in the Rearview Mirror is coming out July 11, but you can totally pre-order it now. You should definitely become a fan on facebook, or better yet, send a (non-contest) question about the universe.