From Eternity to Here: The Quest for the Ultimate Theory of Time - Sean Carroll (2010)
Part II. TIME IN EINSTEIN’S UNIVERSE
Chapter 4. TIME IS PERSONAL
Time travels in divers paces with divers persons.
—William Shakespeare, As You Like It
When most people hear “scientist,” they think “Einstein.” Albert Einstein is an iconic figure; not many theoretical physicists attain a level of celebrity in which their likeness appears regularly on T-shirts. But it’s an intimidating, distant celebrity. Unlike, say, Tiger Woods, the precise achievements Einstein is actually famous for remain somewhat mysterious to many people who would easily recognize his name.53 His image as the rumpled, absentminded professor, with unruly hair and baggy sweaters, contributes to the impression of someone who embodied the life of the mind, disdainful of the mundane realities around him. And to the extent that the substance of his contributions is understood—equivalence of mass and energy, warping of space and time, a search for the ultimate theory—it seems to be the pinnacle of abstraction, far removed from everyday concerns.
The real Einstein is more interesting than the icon. For one thing, the rumpled look with the Don King hair attained in his later years bore little resemblance to the sharply dressed, well-groomed young man with the penetrating stare who was responsible for overturning physics more than once in the early decades of the twentieth century.54 For another, the origins of the theory of relativity go beyond armchair speculations about the nature of space and time; they can be traced to resolutely practical concerns of getting persons and cargo to the right place at the right time.
Figure 10: Albert Einstein in 1912. His “miraculous year” was 1905, while his work on general relativity came to fruition in 1915.
Special relativity, which explains how the speed of light can have the same value for all observers, was put together by a number of researchers over the early years of the twentieth century. (Its successor, general relativity, which interpreted gravity as an effect of the curvature of spacetime, was due almost exclusively to Einstein.) One of the major contributors to special relativity was the French mathematician and physicist Henri Poincaré. While Einstein was the one who took the final bold leap into asserting that the “time” as measured by any moving observer was as good as the “time” measured by any other, both he and Poincaré developed very similar formalisms in their research on relativity.55
Historian Peter Galison, in his book Einstein’s Clocks, Poincaré’s Maps: Empires of Time, makes the case that Einstein and Poincaré were as influenced by their earthbound day jobs as they were by esoteric considerations of the architecture of physics.56 Einstein was working at the time as a patent clerk in Bern, Switzer land, where a major concern was the construction of accurate clocks. Railroads had begun to connect cities across Europe, and the problem of synchronizing time across great distances was of pressing commercial interest. The more senior Poincaré, meanwhile, was serving as president of France’s Bureau of Longitude. The growth of sea traffic and trade led to a demand for more accurate methods of determining longitude while at sea, both for the navigation of individual ships and for the construction of more accurate maps.
And there you have it: maps and clocks. Space and time. In particular, an appreciation that what matters is not questions of the form “Where are you really?” or “What time is it actually?” but “Where are you with respect to other things?” and “What time does your clock measure?” The rigid, absolute space and time of Newtonian mechanics accords pretty well with our intuitive understanding of the world; relativity, in contrast, requires a certain leap into abstraction. Physicists at the turn of the century were able to replace the former with the latter only by understanding that we should not impose structures on the world because they suit our intuition, but that we should take seriously what can be measured by real devices.
Special relativity and general relativity form the basic framework for our modern understanding of space and time, and in this part of the book we’re going to see what the implications of “spacetime” are for the concept of “time.”57 We’ll be putting aside, to a large extent, worries about entropy and the Second Law and the arrow of time, and taking refuge in the clean, precise world of fundamentally reversible laws of physics. But the ramifications of relativity and spacetime will turn out to be crucial to our program of providing an explanation for the arrow of time.
LOST IN SPACE
Zen Buddhism teaches the concept of “beginner’s mind”: a state in which one is free of all preconceptions, ready to apprehend the world on its own terms. One could debate how realistic the ambition of attaining such a state might be, but the concept is certainly appropriate when it comes to thinking about relativity. So let’s put aside what we think we know about how time works in the universe, and turn to some thought experiments (for which we know the answers from real experiments) to figure out what relativity has to say about time.
To that end, imagine we are isolated in a sealed spaceship, floating freely in space, far away from the influence of any stars or planets. We have all of the food and air and basic necessities we might wish, and some high school-level science equipment in the form of pulleys and scales and so forth. What we’re not able to do is to look outside at things far away. As we go, we’ll consider what we can learn from various sensors aboard or outside the ship.
But first, let’s see what we can learn just inside the spaceship. We have access to the ship’s controls; we can rotate the vessel around any axis we choose, and we can fire our engines to move in whatever direction we like. So we idle away the hours by alternating between moving the ship around in various ways, not really knowing or caring where we are going, and playing a bit with our experiments.
Figure 11: An isolated spaceship. From left to right: freely falling, accelerating, and spinning.
What do we learn? Most obviously, we can tell when we’re accelerating the ship. When we’re not accelerating, a fork from our dinner table would float freely in front of us, weightless; when we fire the rockets, it falls down, where “down” is defined as “away from the direction in which the ship is accelerating.”58 If we play a bit more, we might figure out that we can also tell when the ship is spinning around some axis. In that case, a piece of cutlery perfectly positioned on the rotational axis could remain there, freely floating; but anything at the periphery would be “pulled” to the hull of the ship and stay there.
So there are some things about the state of our ship we can determine observa tionally, just by doing simple experiments inside. But there are also things that we can’t determine. For example, we don’t know where we are. Say we do a bunch of experiments at one location in our unaccelerated, non-spinning ship. Then we fire the rockets for a bit, zip off somewhere else, kill the rockets so that we are once again unaccelerated and non-spinning, and do the same experiments again. If we have any skill at all as experimental physicists, we’re going to get the same results. Had we been very good record keepers about the amount and duration of our acceleration, we could possibly calculate the distance we had traveled; but just by doing local experiments, there doesn’t seem to be any way to distinguish one location from another.
Likewise, we can’t seem to distinguish one velocity from another. Once we turn off the rockets, we are once again freely floating, no matter what velocity we have attained; there is no need to decelerate in the opposite direction. Nor can we distinguish any particular orientation of the ship from any other orientation, here in the lonely reaches of interstellar space. We can tell whether we are spinning or not spinning; but if we fire the appropriate guidance rockets (or manipulate some onboard gyroscopes) to stop whatever spin we gave the ship, there is no local experiment we can do that would reveal the angle by which the ship had rotated.
These simple conclusions reflect deep features of how reality works. Whenever we can do something to our apparatus without changing any experimental outcomes—shift its position, rotate it, set it moving at a constant velocity—this reflects a symmetry of the laws of nature. Principles of symmetry are extraordinarily powerful in physics, as they place stringent restrictions on what form the laws of nature can take, and what kind of experimental results can be obtained.
Naturally, there are names for the symmetries we have uncovered. Changing one’s location in space is known as a “translation”; changing one’s orientation in space is known as a “rotation”; and changing one’s velocity through space is known as a “boost.” In the context of special relativity, the collection of rotations and boosts are known as “Lorentz transformations,” while the entire set including translations are known as “Poincaré transformations.”
The basic idea behind these symmetries far predates special relativity. Galileo himself was the first to argue that the laws of nature should be invariant under what we now call translations, rotations, and boosts. Even without relativity, if Galileo and Newton had turned out to be right about mechanics, we would not be able to determine our position, orientation, or velocity if we were floating freely in an isolated spaceship. The difference between relativity and the Galilean perspective resides in what actually happens when we switch to the reference frame of a moving observer. The miracle of relativity, in fact, is that changes in velocity are seen to be close relatives of changes in spatial orientation; a boost is simply the spacetime version of a rotation.
Before getting there, let’s pause to ask whether things could have been different. For example, we claimed that one’s absolute position is unobservable, and one’s absolute velocity is unobservable, but one’s absolute acceleration can be measured. 59 Can we imagine a world, a set of laws of physics, in which absolute position is unobservable, but absolute velocity can be objectively measured?60
Sure we can. Just imagine moving through a stationary medium, such as air or water. If we lived in an infinitely big pool of water, our position would be irrelevant, but it would be straightforward to measure our velocity with respect to the water. And it wouldn’t be crazy to think that there is such a medium pervading space.61 After all, ever since the work of Maxwell on electromagnetism we have known that light is just a kind of wave. And if you have a wave, it’s natural to think that there must be something doing the waving. For example, sound needs air to propagate; in space, no one can hear you scream. But light can travel through empty space, so (according to this logic, which will turn out not to be right) there must be some medium through which it is traveling.
So physicists in the late nineteenth century postulated that electromagnetic waves propagated through an invisible but all-important medium, which they called the “aether.” And experimentalists set out to actually detect the stuff. But they didn’t succeed—and that failure set the stage for special relativity.
THE KEY TO RELATIVITY
Imagine we’re back out in space, but this time we’ve brought along some more sophisticated experimental apparatus. In particular, we have an impressive-looking contraption, complete with state-of-the-art laser technology, that measures the speed of light. While we are freely falling (no acceleration), to calibrate the thing we check that we get the same answer for the speed of light no matter how we orient our experiment. And indeed we do. Rotational invariance is a property of the propagation of light, just as we suspected.
But now we try to measure the speed of light while moving at different velocities. That is, first we do the experiment, and then we fire our rockets a bit and turn them off so that we’ve established some constant velocity with respect to our initial motion, and then we do the experiment again. Interestingly, no matter how much velocity we picked up, the speed of light that we measure is always the same. If there really were an aether medium through which light traveled just as sound travels through air, we should get different answers depending on our speed relative to the aether. But we don’t. You might guess that the light had been given some sort of push by dint of the fact that it was created within your moving ship. To check that, we’ll allow you to remove the curtains from the windows and let some light come in from the outside world. When you measure the velocity of the light that was emitted by some outside source, once again you find that it doesn’t depend on the velocity of your own spaceship.
A real-world version of this experiment was performed in 1887 by Albert Michelson and Edward Morley. They didn’t have a spaceship with a powerful rocket, so they used the next best thing: the motion of the Earth around the Sun. The Earth’s orbital velocity is about 30 kilometers per second, so in the winter it has a net velocity of about 60 kilometers per second different from its velocity in the summer, when it’s moving in the other direction. That’s not much compared to the speed of light, which is about 300,000 kilometers per second, but Michelson designed an ingenious device known as an “interferometer” that was extremely sensitive to small changes in velocities along different directions. And the answer was: The speed of light seems to be the same, no matter how fast we are moving.
Advances in science are rarely straightforward, and the correct way to interpret the Michelson-Morley results was not obvious. Perhaps the aether is dragged along with the Earth, so that our relative velocity remains small. After some furious back-and-forth theorizing, physicists hit upon what we now regard to be the right answer: The speed of light is simply a universal invariant. Everyone measures light to be moving at the same speed, independent of the motion of the experimenter.62 Indeed, the entire content of special relativity boils down to these two principles:
• No local experiment can distinguish between observers moving at constant velocities.
• The speed of light is the same to all observers.
When we use the phrase the speed of light, we are implicitly assuming that it’s the speed of light through empty space that we’re talking about. It’s perfectly easy to make light move at some other speed, just by introducing a transparent medium—light moves more slowly through glass or water than it does through empty space, but that doesn’t tell us anything profound about the laws of physics. Indeed, “light” is not all that important in this game. What’s important is that there exists some unique preferred velocity through spacetime. It just so happens that light moves at that speed when it’s traveling through empty space—but the existence of a speed limit is what matters, not that light is able to go that fast.
We should appreciate how astonishing all this is. Say you’re in your spaceship, and a friend in a faraway spaceship is beaming a flashlight at you. You measure the velocity of the light from the flashlight, and the answer is 300,000 kilometers per second. Now you fire your rockets and accelerate toward your friend, until your relative velocity is 200,000 kilometers per second. You again measure the speed of the light coming from the flashlight, and the answer is: 300,000 kilometers per second. That seems crazy; anyone in their right mind should have expected it to be 500,000 kilometers per second. What’s going on?
The answer, according to special relativity, is that it’s not the speed of light that depends on your reference frame—it’s your notion of a “kilometer” and a “second.” If a meterstick passes by us at high velocity, it undergoes “length contraction”—it appears shorter than the meterstick that is sitting at rest in our reference frame. Likewise, if a clock moves by us at high velocity, it undergoes “time dilation”—it appears to be ticking more slowly than the clock that is sitting at rest. Together, these phenomena precisely compensate for any relative motion, so that everyone measures exactly the same speed of light.63
The invariance of the speed of light carries with it an important corollary: Nothing can move faster than light. The proof is simple enough; imagine being in a rocket that tries to race against the light being emitted by a flashlight. At first the rocket is stationary (say, in our reference frame), and the light is passing it at 300,000 kilometers per second. But then the rocket accelerates with all its might, attaining a tremendous velocity. When the crew in the rocket checks the light from the (now distant) flashlight, they see that it is passing them by at—300,000 kilometers per second. No matter what they do, how hard they accelerate or for how long, the light is always moving faster, and always moving faster by the same amount.64 (From their point of view, that is. From the perspective of an external observer, they appear to be moving closer and closer to the speed of light, but they never reach it.)
However, while length contraction and time dilation are perfectly legitimate ways to think about special relativity, they can also get pretty confusing. When we think about the “length” of some physical object, we need to measure the distance between one end of it and the other, but implicitly we need to do so at the same time. (You can’t make yourself taller by putting a mark on the wall by your feet, climbing a ladder, putting another mark by your head, and proclaiming the distance between the marks to be your height.) But the entire spirit of special relativity tells us to avoid making statements about separated events happening at the same time. So let’s tackle the problem from a different angle: by taking “spacetime” seriously.
Back to the spaceship with us. This time, however, instead of being limited to performing experiments inside the sealed ship, we have access to a small fleet of robot probes with their own rockets and navigation computers, which we can program to go on journeys and come back as we please. And each one of them is equipped with a very accurate atomic clock. We begin by carefully synchronizing these clocks with the one on our main shipboard computer, and verifying that they all agree and keep very precise time.
Then we send out some of our probes to zip away from us for a while and eventually come back. When they return, we notice something right away: The clocks on the probe ships no longer agree with the shipboard computer. Because this is a thought experiment, we can rest assured that the difference is not due to cosmic rays or faulty programming or tampering by mischievous aliens—the probes really did experience a different amount of time than we did.
Happily, there is an explanation for this unusual phenomenon. The time that clocks experience isn’t some absolute feature of the universe, out there to be measured once and for all, like the yard lines on a football field. Instead, the time measured by a clock depends on the particular trajectory that the clock takes, much like the total distance covered by a runner depends on their path. If, instead of sending out robot probes equipped with clocks from a spaceship, we had sent out robots on wheels equipped with odometers from a base located on the ground, nobody would be surprised that different robots returned with different odometer readings. The lesson is that clocks are kind of like odometers, keeping track of some measure of distance traveled (through time or through space) along a particular path.
If clocks are kind of like odometers, then time is kind of like space. Remember that even before special relativity, if we believed in absolute space and time à la Isaac Newton, there was nothing stopping us from combining them into one entity called “spacetime.” It was still necessary to give four numbers (three to locate a position in space, and one time) to specify an event in the universe. But in a Newtonian world, space and time had completely separate identities. Given two distinct events, such as “leaving the house Monday morning” and “arriving at work later that same morning,” we could separately (and uniquely, without fear of ambiguity) talk about the distance between them and the time elapsed between them. Special relativity says that this is not right. There are not two different things, “distance in space” measured by odometers and “duration in time” measured by clocks. There is only one thing, the interval in spacetime between two events, which corresponds to an ordinary distance when it is mostly through space and to a duration measured by clocks when it is mostly through time.
What decides “mostly”? The speed of light. Velocity is measured in kilometers per second, or in some other units of distance per time; hence, having some special speed as part of the laws of nature provides a way to translate between space and time. When you travel more slowly than the speed of light, you are moving mostly through time; if you were to travel faster than light (which you aren’t about to do), you would be moving mostly through space.
Let’s try to flesh out some of the details. Examining the clocks on our probe ships closely, we realize that all of the traveling clocks are different in a similar way: They read shorter times than the one that was stationary. That is striking, as we were comforting ourselves with the idea that time is kind of like space, and the clocks were reflecting a distance traveled through spacetime. But in the case of good old ordinary space, moving around willy-nilly always makes a journey longer; a straight line is the shortest distance between two points in space. If our clocks are telling us the truth (and they are), it would appear that unaccelerated motion—a straight line through spacetime, if you like—is the path of longest time between two events.
Figure 12: Time elapsed on trajectories that go out and come back is less than that elapsed according to clocks that stay behind.
Well, what did you expect? Time is kind of like space, but it’s obviously not completely indistinguishable from space in every way. (No one is in any danger of getting confused by some driving directions and making a left turn into yesterday.) Putting aside for the moment issues of entropy and the arrow of time, we have just uncovered the fundamental feature that distinguishes time from space: Extraneous motion decreases the time elapsed between two events in spacetime, whereas it increases the distance traveled between two points in space.
If we want to move between two points in space, we can make the distance we actually travel as long as we wish, by taking some crazy winding path (or just by walking in circles an arbitrary number of times before continuing on our way). But consider traveling between two events in spacetime—particular points in space, at particular moments in time. If we move on a “straight line”—an unaccelerated trajectory, moving at constant velocity all the while—we will experience the longest duration possible. So if we do the opposite, zipping all over the place as fast as we can, but taking care to reach our destination at the appointed time, we will experience a shorter duration. If we zipped around at precisely the speed of light, we would never experience any duration at all, no matter how we traveled. We can’t do exactly that, but we can come as close as we wish.65
That’s the precise sense in which “time is kind of like space”—spacetime is a generalization of the concept of space, with time playing the role of one of the dimensions of spacetime, albeit one with a slightly different flavor than the spatial dimensions. None of this is familiar to us from our everyday experience, because we tend to move much more slowly than the speed of light. Moving much more slowly than light is like being a running back who only marched precisely up the football field, never swerving left or right. To a player like that, “distance traveled” would be identical to “number of yards gained,” and there would be no ambiguity. That’s what time is like in our everyday experience; because we and all of our friends move much more slowly than the speed of light, we naturally assume that time is a universal feature of the universe, rather than a measure of the spacetime interval along our particular trajectories.
STAYING INSIDE YOUR LIGHT CONE
One way of coming to terms with the workings of spacetime according to special relativity is to make a map: draw a picture of space and time, indicating where we are allowed to go. Let’s warm up by drawing a picture of Newtonian spacetime. Because Newtonian space and time are absolute, we can uniquely define “moments of constant time” on our map. We can take the four dimensions of space and time and slice them into a set of three-dimensional copies of space at constant time, as shown in Figure 13. (We’re actually only able to show two-dimensional slices on the figure; use your imagination to interpret each slice as representing three-dimensional space.) Crucially, everyone agrees on the difference between space and time; we’re not making any arbitrary choices.
Figure 13: Newtonian space and time. The universe is sliced into moments of constant time, which unambiguously separate time into past and future. World lines of real objects can never double back across a moment of time more than once.
Every Newtonian object (a person, an atom, a rocket ship) defines a world line—the path the object takes through spacetime. (Even if you sit perfectly still, you still move through spacetime; you’re aging, aren’t you?66) And those world lines obey a very strict rule: Once they pass through one moment of time, they can never double backward in time to pass through the same moment again. You can move as fast as you like—you can be here one instant, and a billion light-years away 1 second later—but you have to keep moving forward in time, your world line intersecting each moment precisely once.
Relativity is different. The Newtonian rule “you must move forward in time” is replaced by a new rule: You must move more slowly than the speed of light. (Unless you are a photon or another massless particle, in which case you always move exactly at the speed of light if you are in empty space.) And the structure we were able to impose on Newtonian spacetime, in the form of a unique slicing into moments of constant time, is replaced by another kind of structure: light cones.
Figure 14: Spacetime in the vicinity of a certain event x. According to relativity, every event comes with a light cone, defined by considering all possible paths light could take to or from that point. Events outside the light cone cannot unambiguously be labeled “past” or “future.”
Light cones are conceptually pretty simple. Take an event, a single point in spacetime, and imagine all of the different paths that light could take to or from that event; these define the light cone associated with that event. Hypothetical light rays emerging from the event define a future light cone, while those converging on the event define a past light cone, and when we mean both we just say “the light cone.” The rule that you can’t move faster than the speed of light is equivalent to saying that your world line must remain inside the light cone of every event through which it passes. World lines that do this, describing slower-than-light objects, are called “timelike”; if somehow you could move faster than light, your world line would be “spacelike,” since it was covering more ground in space than in time. If you move exactly at the speed of light, your world line is imaginatively labeled “lightlike.”
Starting from a single event in Newtonian spacetime, we were able to define a surface of constant time that spread uniquely throughout the universe, splitting the set of all events into the past and the future (plus “simultaneous” events precisely on the surface). In relativity we can’t do that. Instead, the light cone associated with an event divides spacetime into the past of that event (events inside the past light cone), the future of that event (inside the future light cone), the light cone itself, and a bunch of points outside the light cone that are neither in the past nor in the future.
Figure 15: Light cones replace the moments of constant time from Newtonian spacetime. World lines of massive particles must come to an event from inside the past light cone, and leave inside the future light cone—a timelike path. Spacelike paths move faster than light and are therefore not allowed.
It’s that last bit that really gets people. In our reflexively Newtonian way of thinking about the world, we insist that some faraway event happened either in the past, or in the future, or at the same time as some event on our own world line. In relativity, for spacelike separated events (outside one another’s light cones), the answer is “none of the above.” We could choose to draw some surfaces that sliced through spacetime and label them “surfaces of constant time,” if we really wanted to. That would be taking advantage of the definition of time as a coordinate on spacetime, as discussed in Chapter One. But the result reflects our personal choice, not a real feature of the universe. In relativity, the concept of “simultaneous faraway events” does not make sense.67
There is a very strong temptation, when drawing maps of spacetime such as shown in Figure 15, to draw a vertical axis labeled “time,” and a horizontal axis (or two) labeled “space.” The absence of those axes on our version of the spacetime diagram is completely intentional. The whole point of spacetime according to relativity is that it is not fundamentally divided up into “time” and “space.” The light cones, demarcating the accessible past and future of each event, are not added on top of the straightforward Newtonian decomposition of spacetime into time and space; they replace that structure entirely. Time can be measured along each individual world line, but it’s not a built-in feature of the entire spacetime.
It would be irresponsible to move on without highlighting one other difference between time and space: There is only one dimension of time, whereas there are three dimensions of space.68 We don’t have a good understanding of why this should be so. That is, our understanding of fundamental physics isn’t sufficiently developed to state with confidence whether there is some reason why there couldn’t be more than one dimension of time, or for that matter zero dimensions of time. What we do know is that life would be very different with more than one time dimension. With only one such dimension, physical objects (which move along timelike paths) can’t help but move along that particular direction. If there were more than one, nothing would force us to move forward in time; we could move in circles, for example. Whether or not one can build a consistent theory of physics under such conditions is an open question, but at the least, things would be different.
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.