Tuesday, February 7, 2012

Resolving the “Twin Paradox”

Fans of science-fiction space travel know that if someone goes on a rocket at close to the speed of light, he will age more slowly than someone back home. Returning to Earth and reuniting with a twin, he would find that the twin had aged more, perhaps by years. If the trip is long enough and gets really close to the speed of light, the returning traveler could find an Earth that’s millions of years in the future, maybe even devoid of human life. For many people, that’s the “twin paradox” — how could something this strange possibly happen?

Actually, twins aging differently isn’t the issue; no paradox there. The “paradox” lies in the fact that according to relativity (the very effect that causes the twins to age differently), the ideas of motion and rest are relative. Suppose rather than leaving from Earth, the experiment is done in deep space. Twin A takes off and leaves Twin B behind. But, once the twins are separated, who’s to say which twin is moving and which is at rest? After all, if you consider the picture from the perspective of either twin, it’s the other one that’s moving. Twin A sees Twin B receding rapidly in the rear-view mirror, just as if Twin B was the one who had taken off. In relativity, the question of moving vs. stationary depends upon the perspective of the observer. Shouldn’t this mean that the aging will be equal for both twins when they finally reunite? That’s the “twin paradox.”

The twin paradox isn’t really a paradox, because it can be resolved in several ways. For one thing, the situation isn’t symmetrical. One of the twins, Twin A, has to turn around at some point, whereas Twin B can just cool his heels. Physicists say that Twin B remains in an inertial reference frame — a state of constant motion (or rest), without any change in speed. Twin A, though, spends portions of his trip in two different inertial reference frames, one on the way out, and one on the way back, and in between he has to slow down, stop, and accelerate in the other direction. (Or, he could make a circular U-turn, but that still counts as an acceleration.) It’s clear how changing velocity affects the local passage of time, from the formula known as the Lorentz transformation: Higher velocities mean slower clocks, so as Twin A slows down, his onboard clock begins speeding up. Twin B’s velocity never changes, though. Therefore, it’s wrong to say the twins are moving in identical ways relative to the other.

Thinking this through, the situation is a bit peculiar. How does Twin A and his mechanical onboard clock “know” that they had turned around? Suppose Twin B, rather than staying put, secretly takes off in the other direction, goes even faster, turns around, and races back to the starting point, just in time to meet the returning Twin A. In that case, Twin B would age slower than Twin A — who, having just gone on a fast trip that included turning around, expects to meet an older Twin B. Instead he finds a younger Twin B. How does Twin B (and his clock) know that he had gone farther and faster than Twin A?

The solution all comes down to the turning-around part: acceleration. When you’re in an inertial reference frame, you can’t tell whether you’re moving or not. You could be sitting in your living room, or you could be on a rocket ship going 99% the speed of light. Close your eyes and the situations are indistinguishable. However, when you’re slowing down or speeding up, or making a turn, you can definitely tell that something’s going on. You feel a pull toward one direction, just as you do in a car with the brakes applied. This acceleration* is what causes a clock (biological or otherwise) to slow down. For Twin B who secretly went on a really fast trip, his larger accelerations slowed down his biological clock even more than Twin A’s, who now needs a facelift to look as young as his twin again.

When you add gravity into the mix, it gets even stranger. Twin A — rather than turning on the reverse-thrust engine to slow down and turn around — could instead have a close call with a massive star, and like Halley’s comet, take a tight orbit around and be fired back toward his starting point. In such a situation, the ship is in free fall with respect to the star, and counter-intuitively, doesn’t feel the acceleration as it gets slung back toward Twin B. Our traveler could continue his game of three-dimensional billiards in the weightless environment of his ship, even as it hooks sharply around the star. If there were no windows, he might not even know when he was passing behind and starting to head back. Yet incredibly, the curvature of space from the star’s gravity would slow down time aboard the ship. While the traveling twin works on his weightless billiards game, the stay-at-home twin would have the time to master not only billiards but also croquet and miniature golf, much to Twin A’s later envy.

The “twin paradox” is one of those cases where the universe just works out perfectly right, out of sheer mathematical consistency. Relative velocity determines the rate at which clocks tick; Einstein showed that this malleability of time is necessary in a universe where the speed of light is measured the same by all observers. Therefore, changing velocity (accelerating) changes the local rate at which time passes. All of this can be calculated from the Lorentz transformation. But a star’s gravity will also slow down time for Twin A and his ship, by exactly the same amount as if he had used his engines to slow down and turn around. The curvature of space due to gravity — by having the ability to sling a spaceship around and back in the other direction, on momentum alone — simply has to make an adjustment to the ship’s onboard clocks. Otherwise the math wouldn’t work out. And then we’d have a real paradox.

Anything in our universe with mass slows down clocks in its wake. For the Earth, the effect is not only measurable, it needs to be built into your GPS to avoid large, cumulative errors that would render it useless. Thanks to Einstein’s discovery, you can reliably arrive at your destination. Isn’t it nice when things work out?

* The term “acceleration” refers to both slowing down and speeding up. Slowing down is simply accelerating in the opposite direction.

Monday, February 6, 2012

The Zero Universe

The world is like a great theater where we watch history unfold. This colossal story features a cast of billions, who not only witness the arc of its epic plot, but also actively take part in its creation. Look around you; it’s raucous and noisy inside this theater we call “the world.” But, if it were possible for us to step outside the theater and take a look, there would be nothing.

Inside, the theater is a mind-boggling swirl of information, actions, and reactions; from the outside, as seen by a cosmic Google Earth beyond space and time, it’s completely empty. I am not just making a cute metaphor here — this is a real feature of our universe, with profound implications.

It seems that in the final analysis, when all things are considered, the universe adds up to exactly zero. For one example (there are others at the bottom), consider the relationship between space and time. Bear with me as I review an idea from high-school geometry. If we have a triangle that includes a 90° angle, we can use the Pythagorean theorem to determine the length of the diagonal from the length of the right-angle sides:

a2 + b2 = c2

where a and b are the lengths of the right-angle sides, and c is the length of the diagonal. A nifty geometrical diagram proves why this is true. If you make a square box along each of the three sides, then each box has an area of the length of that side, squared. The area of the smaller squares adds up to the area of the largest square: If a = 3 and b = 4, then c = 5.

The Pythagorean theorem works in three dimensions, too. If you see a blimp in the sky, you can calculate the exact straight-line distance to the blimp by knowing its altitude above the ground (the value z), as well as how far east-west (x) and north-south (y) you'd have to go to get right under the blimp:

x2 + y2 + z2 = d2

where d is the distance to the blimp. I don't have a 3D diagram, but you can prove it for yourself with a little effort.

Now it gets interesting. This trick extends to four dimensions. Time is typically cited as the fourth dimension. Does the decidedly non-geometric idea of time work into the Pythagorean theorem? Incredibly, it does — but first, you have to convert the time measurement into a distance-like measurement. Then, the total distance you’re calculating is the spacetime distance in the bizarre four-dimensional world where east-west, north-south, up-down, and earlier-later mean the same thing, only in eight different directions. Represented by the letter s, spacetime distance (also known as a Minkowski interval) is determined by an amazing formula. Let’s break it down:

x2 + y2 + z2 – (ct)2 = s2

As before, x is the distance (for example) to the east, y is north, and z is up, but we’ve added a fourth term for time (t), which gets multiplied by a constant, c. Notice the minus sign before the term for time. When it comes to distance through spacetime, elapsed time counteracts spatial distance, and vice versa: If we travel a distance through space, and do it in a very short interval of time,* the distance traversed is effectively reduced. This is why a space traveler could reach stars across the galaxy within their lifetime if they got close enough to the speed of light. Time goes in the opposite “direction” of space!

That constant, represented by c? It’s the same c that represents the speed of light in equations such as E = mc2. What better number to convert units of time (seconds) into a distance-like measurement — after all, we know that for light, there are 186,000 miles per second. See what Einstein did there? The speed of light is more than just a speed; it’s a universal conversion factor that turns time into a distance-like measurement. By treating time as a negative and multiplying it by c, we can exchange time and space in our formulas as readily as nature exchanges them. That’s what special relativity is all about.

This extra meaning of the speed of light has interesting consequences. If you could look out the window of a rocket going at the speed of light, you wouldn’t “c” a thing — the entire universe would vanish to a point through Lorentz contraction. In order for space and time to enter into what we call “reality,” they must be measured by an observer, something not traveling at the speed c. In the real world, that’s anything with mass. For any observer with mass (make your own couch-potato joke), zero spacetime distance separates into the components familiar to us: a measurable amount of space and a measurable amount of time.

It’s as if the presence of mass causes “zero” to pull apart into the familiar ideas of spatial distance and temporal duration, like taffy. But since the universe is by definition everything there is, you have to be inside the universe to witness this incredible stretching apart of zero, to experience space and time as different things. If you were taking in the all-seeing “God’s-eye view” from a timeless, spaceless, massless perspective outside, you would see the same thing the speed-of-light traveler sees — nothing. To witness the action, you have to be inside the theater, in your seat.

Space and time cancel out to exactly zero for the universe as a whole. But that’s just one example of the zero-sum nature of the physical world. A few others:
• The mass–energy of everything in the universe is exactly balanced by the universe’s gravitational energy. The latter is expressed as a negative number, just as time is in the spacetime formula. A while back Alex Filippenko, who’s a familiar smiling face to science-TV geeks, co-wrote an essay about how this means the universe may have come from “nothing at all.” Like the pulling apart of space and time, mass–energy and gravitational energy were also pulled apart in the Big Bang.
• For similar reasons, the net charge of the universe is generally believed to be zero, with the number of positively charged particles equaling the number of negative. (This is unproven.)
• Certain pairs of phenomena, like electricity and magnetism or mass and the curvature of space, are linked such that they seem to keep each other in check. The great physicist John Wheeler was fascinated by these “automatic” connections, pointing out how they are constrained together by zero sums, the way the ends of a see-saw are always the same total distance from horizontal. “That this principle should pervade physics, as it does,” he asked in 1986, “is that the only way that nature has to signal to us a construction without a plan, a blueprint for physics that is the very epitome of austerity?”

On the one hand, it’s surprising that quantities totaling zero show up again and again in nature. But on the other it makes sense, if the universe is a closed system incorporating everything there is. As a teen I remember being into the Taoist idea of Yin and Yang — I thought that in the final analysis, the universe as a whole couldn’t be anything but perfectly balanced. On a level deeper than I imagined, I may have been right.

* Slow speeds (which mean long elapsed times) cause the time part of the formula to overwhelm the space part, resulting in large spacetime distances. Spacetime distances only get small when you approach the speed of light, for example, covering 186,000 miles in 1.1 seconds — then the (negative) time part almost cancels the space part.