EINSTEIN’S MOST FAMOUS EQUATION
Einstein’s major 1905 paper in which he laid out the principles of special relativity, “On the Electrodynamics of Moving Bodies,” took up thirty pages in Annalen der Physik, the leading German scientific journal of the time. Soon thereafter, he published a two-page paper entitled “Does the Inertia of a Body Depend upon Its Energy Content?”69 The purpose of this paper was to point out a straightforward but interesting consequence of his longer work: The energy of an object at rest is proportional to its mass. (Mass and inertia are here being used interchangeably.) That’s the idea behind what is surely the most famous equation in history,
E = mc2.
Let’s think about this equation carefully, as it is often misunderstood. The factor c2 is of course the speed of light squared. Physicists learn to think, Aha, relativity must be involved, whenever they see the speed of light in an equation. The factor m is the mass of the object under consideration. In some places you might read about the “relativistic mass,” which increases when an object is in motion. That’s not really the most useful way of thinking about things; it’s better to consider m as the once-and-for-all mass that an object has when it is at rest. Finally, E is not exactly “the energy”; in this equation, it specifically plays the role of the energy of an object at rest. If an object is moving, its energy will certainly be higher.
So Einstein’s famous equation tells us that the energy of an object when it is at rest is equal to its mass times the speed of light squared. Note the importance of the innocuous phrase an object. Not everything in the world is an object! For example, we’ve already spoken of dark energy, which is responsible for the acceleration of the universe. Dark energy doesn’t seem to be a collection of particles or other objects; it pervades spacetime smoothly. So as far as dark energy is concerned, E = mc2 simply doesn’t apply. Likewise, some objects (such as a photon) can never be at rest, since they are always moving at the speed of light. In those cases, again, the equation isn’t applicable.
Everyone knows the practical implication of this equation: Even a small amount of mass is equivalent to a huge amount of energy. (The speed of light, in everyday units, is a really big number.) There are many forms of energy, and what special relativity is telling us is that mass is one form that energy can take. But the various forms can be converted back and forth into one another, which happens all the time. The domain of validity of E = mc2 isn’t limited to esoteric realms of nuclear physics or cosmology; it’s applicable to every kind of object at rest, on Mars or in your living room. If we take a piece of paper and burn it, letting the photons produced escape along with their energy, the resulting ashes will have a slightly lower mass (no matter how careful we are to capture all of them) than the combination of the original paper plus the oxygen it used to burn. E = mc2 isn’t just about atomic bombs; it’s a profound feature of the dynamics of energy all around us.
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TIME IS FLEXIBLE
The reason why the universe is eternal is that it does not live for itself; it gives life to others as it transforms.
—Lao Tzu, Tao Te Ching
The original impetus behind special relativity was not a puzzling experimental result (although the Michelson-Morley experiment certainly was that); it was an apparent conflict between two preexisting theoretical frameworks.70 On the one hand you had Newtonian mechanics, the gleaming edifice of physics on which all subsequent theories had been based. On the other hand you had James Clerk Maxwell’s unification of electricity and magnetism, which came about in the middle of the nineteenth century and had explained an impressive variety of experimental phenomena. The problem was that these two marvelously successful theories didn’t fit together. Newtonian mechanics implied that the relative velocity of two objects moving past each other was simply the sum of their two velocities; Maxwellian electromagnetism implied that the speed of light was an exception to this rule. Special relativity managed to bring the two theories together into a single whole, by providing a framework for mechanics in which the speed of light did play a special role, but which reduced to Newton’s model when particles were moving slowly.
Like many dramatic changes of worldview, the triumph of special relativity came at a cost. In this case, the greatest single success of Newtonian physics—his theory of gravity, which accounted for the motions of the planets with exquisite precision—was left out of the happy reconciliation. Along with electromagnetism, gravity is the most obvious force in the universe, and Einstein was determined to fit it in to the language of relativity. You might expect that this would involve modifying a few equations here and there to make Newton’s equations consistent with invariance under boosts, but attempts along those lines fell frustratingly short.
Eventually Einstein hit on a brilliant insight, essentially by employing the spaceship thought experiment we’ve been considering. (He thought of it first.) In describing our travels in this hypothetical sealed spaceship, I was careful to note that we are far away from any gravitational fields, so we wouldn’t have to worry about falling into a star or having our robot probes deflected by the pull of a nearby planet. But what if we were near a prominent gravitational field? Imagine our ship was, for example, in orbit around the Earth. How would that affect the experiments we were doing inside the ship?
Einstein’s answer was: They wouldn’t affect them at all, as long as we confined our attention to relatively small regions of space and brief intervals of time. We can do whatever kinds of experiments we like—measuring the rates of chemical reactions, dropping balls and watching how they fall, observing weights on springs—and we would get exactly the same answer zipping around in low-Earth orbit as we would in the far reaches of interstellar space. Of course if we wait long enough we can tell we are in orbit; if we let a fork and a spoon freely float in front of our noses, with the fork just slightly closer to the Earth, the fork will then feel just a slightly larger gravitational pull, and therefore move just ever so slightly away from the spoon. But effects like that take time to accumulate; if we confine our attention to sufficiently small regions of space and time, there isn’t any experiment we can imagine doing that could reveal the presence of the gravitational pull keeping us in orbit around the Earth.
Contrast the difficulty of detecting a gravitational field with, for example, the ease of detecting an electric field, which is a piece of cake. Just take your same fork and spoon, but now give the fork some positive charge, and the spoon some negative charge. In the presence of an electric field, the opposing charges would be pushed in opposite directions, so it’s pretty easy to check whether there are any electric fields in the vicinity.
The difference with gravity is that there is no such thing as a “negative gravitational charge.” Gravity is universal—everything responds to it in the same way. Consequently, it can’t be detected in a small region of spacetime, only in the difference between its effects on objects at different events in spacetime. Einstein elevated this observation to the status of a law of nature, the Principle of Equivalence: No local experiment can detect the existence of a gravitational field.
Figure 16: The gravitational field on a planet is locally indistinguishable from the acceleration of a rocket.
I know what you’re thinking: “I have no trouble detecting gravity at all. Here I am sitting in my chair, and it’s gravity that’s keeping me from floating up into the room.” But how do you know it’s gravity? Only by looking outside and checking that you’re on the surface of the Earth. If you were in a spaceship that was accelerating, you would also be pushed down into your chair. Just as you can’t tell the difference between freely falling in interstellar space and freely falling in low-Earth orbit, you also can’t tell the difference between constant acceleration in a spaceship and sitting comfortably in a gravitational field. That’s the “equivalence” in the Principle of Equivalence: The apparent effects of the force of gravity are equivalent to those of living in an accelerating reference frame. It’s not the for
ce of gravity that you feel when you are sitting in a chair; it’s the force of the chair pushing up on your posterior. According to general relativity, free fall is the natural, unforced state of motion, and it’s only the push from the surface of the Earth that deflects us from our appointed path.
CURVING STRAIGHT LINES
You or I, having come up with the bright idea of the Principle of Equivalence while musing over the nature of gravity, would have nodded sagely and moved on with our lives. But Einstein was smarter than that, and he appreciated what this insight really meant. If gravity isn’t detectable by doing local experiments, then it’s not really a “force” at all, in the same way that electricity or magnetism are forces. Because gravity is universal, it makes more sense to think of it as a feature of spacetime itself, rather than some force field stretching through spacetime.
In particular, realized Einstein, gravity can be thought of as a manifestation of the curvature of spacetime. We’ve talked quite a bit about spacetime as a generalization of space, and how the time elapsed along a trajectory is a measure of the distance traveled through spacetime. But space isn’t necessarily rigid, flat, and rectilinear; it can be warped, stretched, and deformed. Einstein says that spacetime is the same way.
It’s easiest to visualize two-dimensional space, modeled, for example, by a piece of paper. A flat piece of paper is not curved, and the reason we know that is that it obeys the principles of good old-fashioned Euclidean geometry. Two initially parallel lines, for example, never intersect, nor do they grow apart.
In contrast, consider the two-dimensional surface of a sphere. First we have to generalize the notion of a “straight line,” which on a sphere isn’t an obvious concept. In Euclidean geometry, as we were taught in high school, a straight line is the shortest distance between two points. So let’s declare an analogous definition: A “straight line” on a curved geometry is the shortest curve connecting two points, which on a sphere would be a portion of a great circle. If we take two paths on a sphere that are initially parallel, and extend them along great circles, they will eventually intersect. That proves that the principles of Euclidean geometry are no longer at work, which is one way of seeing that the geometry of a sphere is curved.
Figure 17: Flat geometry, with parallel lines extending forever; curved geometry, where initially parallel lines eventually intersect.
Einstein proposed that four-dimensional spacetime can be curved, just like the surface of a two-dimensional sphere. The curvature need not be uniform like a sphere, the same from point to point; it can vary in magnitude and in shape from place to place. And here is the kicker: When we see a planet being “deflected by the force of gravity,” Einstein says it is really just traveling in a straight line. At least, as straight as a line can be in the curved spacetime through which the planet is moving. Following the insight that an unaccelerated trajectory yields the greatest possible time a clock could measure between two events, a straight line through spacetime is one that does its best to maximize the time on a clock, just like a straight line through space does its best to minimize the distance read by an odometer.
Let’s bring this down to Earth, in a manner of speaking. Consider a satellite in orbit, carrying a clock. And consider another clock, this one on a tower that reaches to the same altitude as the orbiting satellite. The clocks are synchronized at a moment when the satellite passes by the tower; what will they read when the satellite completes one orbit? (We can ignore the rotation of the Earth for the purposes of this utterly impractical thought experiment.) According to the viewpoint of general relativity, the orbiting clock is not accelerating; it’s in free fall, doing its best to move in a straight line through spacetime. The tower clock, meanwhile, is accelerating—it’s being prevented from freely falling by the force of the tower keeping it up. Therefore, the orbiting clock will experience more elapsed time per orbit than the tower clock—compared to the accelerated clock on the tower, the freely falling one in orbit appears to run more quickly.
Figure 18: Time as measured on a tower will be shorter than that measured in orbit, as the former clock is on an accelerated (non-free-falling) trajectory.
There are no towers that reach to the heights of low-Earth orbit. But there are clocks down here at the surface that regularly exchange signals with clocks on satellites. That, for example, is the basic mechanism behind the Global Positioning System (GPS) that helps modern cars give driving directions in real time. Your personal GPS receiver gets signals from a number of satellites orbiting the Earth, and determines its position by comparing the time between the different signals. That calculation would quickly go astray if the gravitational time dilation due to general relativity were not taken into account; the GPS satellites experience about 38 more microseconds per day in orbit than they would on the ground. Rather than teaching your receiver equations from general relativity, the solution actually adopted is to tune the satellite clocks so that they run a little bit more slowly than they should if they were to keep correct time down here on the surface.
EINSTEIN’S MOST IMPORTANT EQUATION
The saying goes that every equation cuts your book sales in half. I’m hoping that this page is buried sufficiently deeply in the book that nobody notices before purchasing it, because I cannot resist the temptation to display another equation: the Einstein field equation for general relativity.
Rµν - (t/2)Rgµν = 8πGTµν.
This is the equation that a physicist would think of if you said “Einstein’s equation”; that E = mc2 business is a minor thing, a special case of a broader principle. This one, in contrast, is a deep law of physics: It reveals how stuff in the universe causes spacetime to curve, and therefore causes gravity. Both sides of the equation are not simply numbers, but tensors—geometric objects that capture multiple things going on at once. (If you thought of them as 4x4 arrays of numbers, you would be pretty close to right.) The left-hand side of the equation characterizes the curvature of spacetime. The right-hand side characterizes all the various forms of stuff that make spacetime curve—energy, momentum, pressure, and so on. In one fell swoop, Einstein’s equation reveals how any particular collection of particles and fields in the universe creates a certain kind of curvature in spacetime.
According to Isaac Newton, the source of gravity was mass; heavier objects gave rise to stronger gravitational fields. In Einstein’s universe, things are more complicated. Mass gets replaced by energy, but there are also other properties that go into curving spacetime. Vacuum energy, for example, has not only energy, but also tension—a kind of negative pressure. A stretched string or rubber band has tension, pulling back rather than pushing out. It’s the combined effect of the energy plus the tension that causes the universe to accelerate in the presence of vacuum energy.71
The interplay between energy and the curvature of spacetime has a dramatic consequence: In general relativity, energy is not conserved. Not every expert in the field would agree with that statement, not because there is any controversy over what the theory predicts, but because people disagree on how to define “energy” and “conserved.” In a Newtonian absolute spacetime, there is a well-defined notion of the energy of individual objects, which we can add up to get the total energy of the universe, and that energy never changes (it’s the same at every moment in time). But in general relativity, where spacetime is dynamical, energy can be injected into matter or absorbed from it by the motions of spacetime. For example, vacuum energy remains absolutely constant in density as the universe expands. So the energy per cubic centimeter is constant, while the number of cubic centimeters is increasing—the total energy goes up. In a universe dominated by radiation, in contrast, the total energy goes down, as each photon loses energy due to the cosmological redshift.
You might think we could escape the conclusion that energy is not conserved by including “the energy of the gravitational field,” but that turns out to be much harder than you might expect—there simply is no well-defined local definition of th
e energy in the gravitational field. (That shouldn’t be completely surprising, since the gravitational field can’t even be detected locally.) It’s easier just to bite the bullet and admit that energy is not conserved in general relativity, except in certain special circumstances.72 But it’s not as if chaos has been loosed on the world; given the curvature of spacetime, we can predict precisely how any particular source of energy will evolve.
HOLES IN SPACETIME
Black holes are probably the single most interesting dramatic prediction of general relativity. They are often portrayed as something relatively mundane: “Objects where the gravitational field is so strong that light itself cannot escape.” The reality is more interesting.
Even in Newtonian gravity, there’s nothing to stop us from contemplating an object so massive and dense that the escape velocity is greater than the speed of light, rendering the body “black.” Indeed, the idea was occasionally contemplated, including by British geologist John Michell in 1783 and by Pierre-Simon Laplace in 1796.73 At the time, it wasn’t clear whether the idea quite made sense, as nobody knew whether light was even affected by gravity, and the speed of light didn’t have the fundamental importance it attains in relativity. More important, though, there is a very big distinction hidden in the seemingly minor difference between “an escape velocity greater than light” and “light cannot escape.” Escape velocity is the speed at which we would have to start an object moving upward in order for it to escape the gravitational field of a body without any further acceleration . If I throw a baseball up in the air in the hopes that it escapes into outer space, I have to throw it faster than escape velocity. But there is absolutely no reason why I couldn’t put the same baseball on a rocket and gradually accelerate it into space without ever reaching escape velocity. In other words, it’s not necessary to reach escape velocity in order to actually escape; given enough fuel, you can go as slowly as you like.
From Eternity to Here: The Quest for the Ultimate Theory of Time Page 11