From Eternity to Book Club: Chapters Four and Five

By Sean Carroll | February 9, 2010 8:05 am

Welcome to this week’s installment of the From Eternity to Here book club. This week we’re tackling two chapters at once: Chapter Four, “Time is Personal,” and Chapter Five, “Time is Flexible.” That’s just because these chapters are relatively short; next time we’ll return to one chapter per week.


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 the future.

It’s that last bit that really gets people. In our reflexively Newtonian way of thinking about the world, we insist that some far away event either happened in the past, 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.

These two chapters take on a task that is part of the responsibility of any good book on modern cosmology or gravity: explaining Einstein’s theory of relativity. Both special relativity and general relativity, hence two chapters. In retrospect they are pretty short, so an argument could be made that I should have just combined them into a single chapter.

The special challenge of these chapters is precisely that many readers — but not all — will already have read numerous other popular-level expositions of relativity. But you have to do it. Fortunately, my favorite way of talking about relativity is a little bit different from the standard one, and lines up well with the overarching goal of understanding the meaning of “time.” In particular, I try to make the point that the secret to relativity is to think locally — to compare things happening right next to each other in spacetime, not events that are widely separated. You’re allowed to compare separated events, of course, but the answers are necessarily dependent on arbitrary choices of coordinates, and that leads to endless confusion. So you won’t read a lot about “length contraction” or “time dilation,” but you will read a lot about the actual amount of time measured along a trajectory.

Unfortunately, a search for vivid examples of the maxim “freely-falling paths through spacetime experience the longest amount of proper time” led me directly to the most embarrassing mistake in the book. (At least, “most embarrassing mistake so far uncovered.”) Sordid details below the fold!

The mistake is the claim that a clock that sits stationary on a tower will experience less proper time than a clock that orbits the Earth at the same height above ground. That’s wrong: the orbiting clock will measure less time. This appears in the paragraph at the bottom of page 85 and top of 86, and is elevated from “unfortunate” to “a real doozy” by being illustrated in graphic detail by the figure on page 86. Not really any way I can claim it was just a typo.

sphere.two.geodesics The subtle issue underlying the mistake is illustrated in this figure, which shows two paths connecting two points on a sphere. Both paths are great circles. The shortest distance between two points on a sphere is a great circle; but it certainly doesn’t follow that any path following a great circle gives us the shortest distance between two points. If you go more than half the way around the sphere, you end up with a pretty long path!

The same kind of thing happens in spacetime. The trajectory of longest proper time between two events will always be a freely-falling trajectory (a geodesic). But not every freely-falling path gives us the longest time, and that’s exactly the case in this example. Given two events at the same position above the Earth, the actual path of longest time is a radial freely-falling orbit. If you want your clock to experience the longest time it can, you throw it straight up in the air to where the gravitational field is weaker (and clocks run more quickly with respect to time measured at infinity) and let it fall back down. A circular orbit actually loses time by staying at the same altitude but zipping around the Earth. I relied on my affection for the general underlying principle, and didn’t bother to sit down and work out the actual numbers in this case, so I never found the mistake. Pretty sure my membership in the general relativists’ guild is going to be permanently revoked for this one.

If you’re still not convinced of the wrongness of my example, here’s an equation, the line element along a circular trajectory in the equatorial plane in the Schwarzschild metric:

$latex dtau^2 = left(1-frac{2GM}{r}right) dt^2 – r^2 dphi^2,.$
On the left we have a small interval (squared) of the proper time τ, what a clock would measure along some path. The first term on the right is the contribution from our motion with respect to t, the time measured at infinity; for any given amount of t, we experience less proper time τ as our height r decreases and the coefficient (1-2GM/r) becomes smaller. The second term on the right is the contribution from our angular motion φ. Taking the square root of the whole thing and integrating along a path gives you the proper time.

We don’t have to go through the entire calculation to convince ourselves that staying stationary on the tower has a longer proper time than the circular orbit does. Both trajectories get the same contribution from the first term on the right side, while the second term is zero for the clock on the tower (it’s not moving, so =0), but it’s negative for the orbit. So the orbit is definitely less time. To be corrected in the next printing.

The deep point, of course, remains true: the time measured by clocks in general relativity depends on their path through spacetime, and the way to maximize that time is to take a freely-falling path. Just not that one.

  • Joseph Smidt

    “These two chapters take on a task that is part of the responsibility of any good book on modern cosmology or gravity: explaining Einstein’s theory of relativity.”

    Isn’t that the truth. When I was in high school trying to read various popular physics books, like The Elegant Universe, it took a while before I found a good one to help me understand the basic ideas of GR.

    And more than that. Later, when I formally took GR, the intuition I built from those initial “popular physics” books helped a lot.

    I guess what I am trying to say is it may be that your popular science books may lay the foundation for the intuition of future physicists, as well as the public at large.

  • Rohan


    I’m absolutely itching to read this book! Any word on a paperback edition?
    (my ONLY reading time is on my crowded train commute where hardbacks are impractical)

  • Lawrence Kuklinski

    Hello Sean,
    I understand how gravity is explained to work on a macroscopic level by changing the “spacetime” around it. What I do not know is the same concept true on a subatomic level? Is spacetime warped between an electron and the proton?

  • Sean

    Joseph– Couldn’t agree more. It’s the hope of anyone who writes a popular book that it will fall into the hands of a young person who will be inspired to learn more.

    Rohan– There will be a paperback, I can’t tell you when. Probably later this year, but we’re not sure.

  • Sean

    Lawrence– We think so, yes. Of course, once you’re getting down to subatomic scales, the particles are described by quantum mechanics, so the curvature of spacetime should be, too. But we don’t know how to do that yet, at least not within a complete theory. (Semiclassical methods work pretty well for questions like that.)

  • Lawrence Kuklinski

    Einstein had a thought experiment where he wanted to ride alonside a beam of light. The answer I was told is that you cannot do it. If possible what do you think a photon would look like? Also when a photon stops say on my retina does it translate to energy?

  • uhmmm

    A stab at answering 6…

    A massive observer accelerating to ultrarelativistic speeds on a trajectory parallel to a perfectly collimated beam of light will eventually see the beam’s photons converge on him with increasing per-photon energy (i.e., it will blueshift as the observer accelerates).

    An unaccelerating observer at a great distance would see objects being attracted to the accelerating massive object (“amo”, Einstein in your question). If this observer could see the photons in the beam and the beam “itself”, the latter would increasingly bend towards the amo, and this observer also would see the photons falling on the amo as losing energy (redshifting).

    (If the light beam were monochromatic and passing through a uniform scattering medium, photons scattered towards the distant observer’s retina would look the same colour to her until the amo’s mass-energy becomes sufficient to bend the freefalling paths of the medium and beam at which point beam photons scattered from near (and also ahead of) the ultrarelativistic amo would be seen to have a redder spectrum as the gravitational potential gradient near the amo steepens.)

  • Limmo

    Hi Sean, so in the experiment where they’ve put atomic clocks on two planes and sent them around the earth in and counter the direction of earth’s rotation, respectively, is it that the clock that traveled with the rotation of the earth experiences less proper time than the other one? Your above explanation lets me think it may be twice the term r^2 dphi^2. Does the rotation of the earth play any role?

  • Doug

    I think I side with your original example (though I don’t want to stake my guild card on it either). If the first satellite that comes to the reader’s mind is one in geosynchronous orbit, then its dphi^2 is the same as for the clock in the tower, with only r much different. In this case the satellite wins, yes?

  • Sean

    Lawrence– The answer really is that you can’t travel at the speed of light. For one thing, the proper time along a speed-of-light trajectory is exactly zero, so you don’t experience any time passing.

    Limmo– In an airplane, you’re pushing against the surrounding air, which is moving with the Earth. So you are effectively moving faster if you’re traveling in the direction of the Earth’s motion. All that is just special relativity, not general relativity, since the planes are in the same kind of gravitational field.

    Doug– In the example in the book, I explicitly ignored the rotation of the Earth. And the point of the exercise was to keep the two satellites at the same altitude, so you would have to imagine a tower that reached up to geosynchronous orbit. In which case, the satellite on the tower would actually be freely floating! (Assuming your tower was on the equator.)


  • ftt377q

    You use the term “free falling” several times in the book and I am wondering what the definition of the term is. In particular, does it mean falling in a gravitational field or can it also mean just moving through space at a constant velocity.

    For example, in one sentence on page 84 you seem to use it both ways. You write, “Just as you can’t tell the difference between freely falling in interstellar space and freely falling in low-Earth orbit . . . .” And if you mean for free-falling to require gravity, is there enough gravity in interstellar space to cause much free falling?

    Also, on page 85, did you mean to write “surface of a three-dimensional sphere” when you wrote “surface of a two-dimensional sphere,” or did you mean the two-dimensional surface of a three-dimensional sphere?

  • Sean

    “Free fall” is the state you are in when you don’t feel any forces acting on you. That could be in orbit, or it could be far away from any gravitational field.

    In that quote I was using conventional mathematical language, not everyday language. To a mathematician, a what most people might call “the surface of a sphere” is simply “a sphere,” while what most people might call “a sphere” is called “a ball.” The interior does not count as part of the sphere. The surface of the Earth, for example, is (approximately) a two-dimensional sphere.

  • Lawrence Kuklinski

    If I cannot travel at the speed of light how about if I capture an image of a photon or electron in action thru “Attosecond Physics”?

    Thank you for the opportunity to ask these questions and to you for answering them.

  • paul valletta

    Sean, so locally the Earth can constrain Relativity? ..example take the Earth at the north pole with a decending shaft though to the south pole. If I jump in at the north pole feet first, will I emerge at the south pole feet first? Being there is distance between the two poles, if one reduces the distance, ie the surface area thus the Earths core meets poles, will I be retro-flipped the instant I jump in? Freefalling here would of course be unnatural?

    I know this is slightly idealized, but I read somewhere that at quantum levels particles have a “central core”..that connects to string theory, actually more of a bead theory, particles the beads move according to the string?

    P.S I no longer have to wait until april for your book, it arrives this weekend, pv.

  • Sean

    Lawrence– What if you do? There’s nothing to stop you from taking images of photons, even though you can’t travel at the same speed as them. Every camera image ever is an example, not to mention your eyes.

    Paul– Sorry, not sure what you are asking. There are plenty of tests of relativity here on Earth, but you need to find something that would distinguish from Newtonian gravity.

  • NicoleS

    I’m a little behind and haven’t yet read Chapter 5, so if my question is better explained in the next Chapter, please let me know to hold my horses.

    Okay. You had me and then you lost me. Figure 12 and its explanation is what had me most at a loss. Particularly why an object if it bounces around all willy nilly in time will elapse less time than if you traveled straight through spacetime.

    There are parts of this section that I understood, or at least some glimmer of understanding flashed through my brain – I thought I’d just quickly cover those just in case the answer lies in what I do understand but I’m just not piecing it together correctly:

    – I get that if you were to travel more slowly than the speed of light, you’d be traveling mostly through time and than if you were traveling faster, it’d be mostly through space. Time would still exist, but at that speed it’s not so much of a factor. Hopefully I have that right.
    – So that I also understand how because we move more slowly, time to us is a feature of the universe rather than a measure of spacetime intervals. Or at least I’m grasping at that one.

    Still immensely enjoying the read!

  • Will

    Now I’m confused: I thought dtau was the invariant interval between two events, measured to be the same by all observers, and that dt was the difference in time measured by each different observer. Have I got this wrong, then?

  • Cartesian

    Error is human, the problem is only if there are too much of them. So do not lose motivation.

  • Plato

    Finding examples is probably a good thing.

    For example before John Glenn, we speculated historically about how the world looked and then he looked from space. A Mother of Pearl.

    Now today we have a better picture of how the earth looks thanks to understand the value of measure. Grace satellite Now the earth does not look so round.


  • Plato

    More Info below

    GOCE delivering data for best gravity map ever

    Gravity is stronger closer to Earth, so GOCE was designed to orbit as low as possible while remaining stable as it flies through the fringes of our atmosphere. To help avoid drag and ensure that the gravity measurements are of true gravity, the satellite has to be kept stable in ‘free fall’. Any buffeting from residual air at this low altitude could potentially drown out the gravity data.

  • Sean

    NicoleS– When it comes to explaining the fact that a non-straight path through spacetime gives you a shorter elapsed time, rather than a longer elapsed time, I’m not sure if there is a better explanation than “it’s a brute fact about the universe.” In fact, it’s basically the major distinction between time and space. They’re certainly not exactly the same thing, after all.

    I think your second statement is basically right, although I would say that time “appears” to us like a feature of the universe rather than a measure of our individual spacetime intervals because we all move slowly compared to light. It still is a measure of our interval through spacetime. For the first statement, I’m not sure what “at that speed it’s not so much of a factor” really means. You always experience time passing at one second per second, regardless of what trajectory you are on.

    Will– dτ is the invariant interval along a path, full stop. “Invariant” means that it’s completely independent of coordinates. In particular, it is equal to the time that elapses for a clock that actually travels along that path. In contrast, dt is the change in coordinate time, which certainly depends on coordinates. It is not necessarily the time measured by any clock at all. (Although in our usual way of choosing coordinates, clocks that are at rest infinitely far away do measure that time — but that’s a choice.)

  • escher sketcher

    Hi Sean-

    I don’t have the book with me right now, but you mention something along the lines of:
    Physicists generally agree on the feasibility of extra space-like dimensions, but find it far less likely that there are additional time-like dimensions.
    This is certainly not something I know much about, but I was under the impression that Penrose had an alternate formulation of physics called twistor theory that relied on two space and two time dimensions. Is this a) the case, b) a meaningful distinction or just a mathematical formality, and c) a promising/interesting/productive line of research? (Obviously c) is a little more of a personal opinion question.)


  • Philoponus


    Thank you so much for the correction. Not to worry, we’ve all done worse in print.

    If I could impose on you to help me understand why GPS sat clocks gain 30 micro-secconds/day in orbit. It is simply less gravity? Will working through the equation you just gave us come up with this answer if we plug in their attitude (r)? Do these satellite clocks have any (small) slowing effect according to the Special Theory?

  • Sean

    escher– There have been a number (a small number, but still a number) of people who have thought seriously about multiple timelike dimensions. Itzhak Bars at USC is probably most closely identified with the idea. Right now it’s interesting, but hasn’t led to anything that captures the imagination of the rest of the community — not clear whether it’s promising physics, or just interesting math.

    Philoponus– Yes, it’s just that they’re further from the Earth’s gravitational field. You have to do a bit of work to get the right answer, because we’re comparing clocks at different altitudes, but it’s not that hard. I tried to cheat by considering clocks at the same altitude, which is what got me in trouble. And special relativity does count, but it depends on how you look at things — I just like to calculate the proper time along paths, so there’s no division into “special” and “general” relativity at all.

  • RantingNerd

    In Chapter 4, in the setup for relativity, you say that there’s no way in our universe to measure your absolute velocity. Is that a continuation of the “without looking out the window” setup? Or is that true even with things like the Cosmic Microwave Background? If you were to boost to 0.99c in a given direction, wouldn’t you see the CMB change dramatically (blueshifting in front and redshifting behind)? And wouldn’t you see the otherwise more-or-less isotropic distribution of matter get Lorentz-contracted so it looked denser around 90 degrees off of your direction of motion? Or is the expansion of the universe such that it’d look the same at all velocities?

    (I’m trying to remember back to my undergrad GR course and I’m not even sure we talked about this issue there.)

    Thanks —

  • Sean

    Yes, that’s certainly “without looking out the window.” Even if you do look out the window, any velocity you measure is still with respect to something, whether it’s the cosmic microwave background or the highway beneath your car.

  • RantingNerd

    Great, thanks. At least I can remember some relativity, even if Stat Mech still eludes me after all these years….

  • Clifford

    In Star Trek II: The Wrath of Khan, Kirk orders Sulu to translate the Starship Enterprise 10,000 meters in the negative z direction through a nebula which has blinded their sensors. Assuming this setup is equivalent to your rocket example, can Kirk’s command be obeyed? Is so, how?

  • Samuel A. (Sam) Cox

    “In relativity, the concept of “simultaneous faraway events” does not make sense.”

    Very important and very profound, for that true statement implies why relativity is and must be, in spite of its experimental veracity- conceptually incomplete.

    In a quantum universe, within the Planck Realm, gravity must be Newtonian or near-Newtonian…instantly or near instantly propagating. Since the universe is quantum entangled, we are reminded that the “relativity” universe results from our electromagnetically observing the relationships between energy-densities in a very special way.

    At its heart, the universe has a very “simultaneous” quality about it. We note that fact when we look at a far red-shifted galaxy and remember that folks “over there” see us just as far red-shifted as we observe them to be.

    The past and future are all “out there” to be sure, but upon conceptual reflection, that fact only adds to the certain basic simultaneous quality of the cosmos. In its essence, as Hawking said, “The universe just IS”.

  • Elias

    Sean (I hope you’re still reading this, two days after the original post — or that someone else can answer :-)) — in J.J. Sakurai’s Modern Quantum Mechanics (pp. 126), there’s an argument that gravity cannot be purely geometrical in quantum mechanics. It’s about “gravity-induced quantum interference” which produces an effect that depends on $latex (m/hbar)^2$ whereas, the argument goes, the mass should drop out if gravity where geometrical.

    In post 5 you say that gravity works by “changing the spacetime around it”. I’m interpreting that as gravity being purely geometrical. If so, what’s your answer to Sakurai’s argument?

  • Sean

    Elias– Sorry, but I don’t have Sakurai’s book with me right now, and that argument doesn’t really make sense to me. In any event, I’m only talking about classical general relativity, not about quantum mechanics.

  • Bruce Toor

    Sean, I have the hardback, the Kindle edition, and your DVD. I’m giving the book as a birthday present to my oldest grandson who will be entering college next year and is interested in math and physics. I hope your book and your work get the credit and attention they deserve! In 1979 I took a night class in “Relativity and Cosmology” (I’m just a lawyer)and asked the question of the prof, “If the universe is expanding, what is it expanding into?” (He had used the raisin and balloon analogy). “It’s a meaningless question,” he said, and cut off further questions. It took thirty-one years and your explanation for that answer to make sense to me. Thank you!

  • Philoponus

    Sean, is there any chance we could tempt you to say a little more that (sales-killing) tensor equation on page 87? Could we know the units of the variables? Is there a textbook example that would give us a sense of how physicists actually might apply that equation to calculate something? I just looked on Amazon and couldn’t find a TENSORS FOR DUMMIES volume in that learned series.

  • Igor Khavkine

    @Elias#30: There is no dichotomy between “gravity is geometric” and “mass appears in physical effects”. Gravity is said to be geometric because the effects of gravity on motion and the effects of gravity on geometry (measurements of lengths, time intervals and angles are interrelated). In fact, if you know the geometry, you know how gravity affects motion, and vice versa. Sakurai’s statement is simply not mindful of this point of view.

    @Philoponus: The academic analogs of the Dummies series are called Schaum’s Outlines. If you look for one on tensors or tensor calculus, you’ll find a couple.


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About Sean Carroll

Sean Carroll is a Senior Research Associate in the Department of Physics at the California Institute of Technology. His research interests include theoretical aspects of cosmology, field theory, and gravitation. His most recent book is The Particle at the End of the Universe, about the Large Hadron Collider and the search for the Higgs boson. Here are some of his favorite blog posts, home page, and email: carroll [at] .


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