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Signature of the Metric

  1. Apr 2, 2013 #1
    Looking for clarification on something.

    Take the metric in this form

    ds^2 = dt^2 - dx^2

    Brian Green likes to say a null path is one where motion through space is shared equally with motion through time, or (ds=0).

    Motion through space is not allowed that is "faster" than motion through time because that would make the root of ds negative. (i.e. the Lorentzian signature of the metric does not allow motion through space faster than c.)

    However, seems like this is the more popular convention

    ds^2 = -dt^2 + dx^2

    which throws all that out the window, doesn't it?

    Or was it incomplete/wrong reasoning in the first place?

  2. jcsd
  3. Apr 2, 2013 #2

    Ben Niehoff

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    It's wrong reasoning to begin with, and Brian Green should know better...
  4. Apr 2, 2013 #3


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    This viewpoint still works if you flip the signs in the metric.

    This also still works if you flip the signs in the metric; you just also have to flip the interpretation of the signs of ds^2 (so a *positive* ds^2 is now a spacelike interval--"motion" along such a curve would mean moving through space "faster" than moving through time).

    The former convention (where a positive ds^2 means a timelike interval and the sign of dt^2 in the metric is positive) is called a "timelike convention"; the latter convention (where a positive ds^2 means a spacelike interval and the sign of dt^2 in the metric is negative) is called a "spacelike convention". As the names indicate, it's just a convention, and different texts use different ones. It doesn't change the physics at all.
  5. Apr 2, 2013 #4


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    Since Ben commented on this, I should too, to clarify my previous post. What I said about the two possible conventions for the sign of ds^2 is still valid even if you think (as Ben does, and as I tend to as well) that the stuff Brian Greene says is wrong reasoning to begin with. The reasons for thinking that what Greene says is wrong have nothing to do with metric sign conventions.
  6. Apr 2, 2013 #5

    Just curious, but why do you guys think his reasoning is wrong. Here's more of what Greene says.

    (1) Since Einstein (and others) have put space and time on the same footing, we must now think of motion through spacetime (4-velocity). It doesn't end there - we deal with many 4-vectors now.

    (2) Everything moves through spacetime at the same rate - the speed of light. It's just a matter of how much your motion is split between motion through space and motion through time. A photo shares it equally - a null path. Mortals like ourselves have most of our motion in the time direction.

    (3) Lastly this explains the twin paradox. The path with all it's motion just in the time direction is the path with the maximum proper time. The twin that puts some of his motion into motion through space, his clock ticks slower in comparison

    Where is Green going wrong specifically?
  7. Apr 2, 2013 #6
    Thanks Peter. I was mostly familar with that, but it makes me uneasy for some reason! Some confirmation is good, though
  8. Apr 2, 2013 #7


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    The standard meaning of "4-velocity" is a unit 4-vector that is tangent to an object's worldline. However, that only works for timelike objects; it doesn't work for light. See below.

    Or 1 in units where the speed of light is 1. But again, that only works for timelike objects. See below.

    This is not actually what Greene says. He says a photon moves entirely through space and not at all through time, because a photon's worldline has zero length, which is sometimes (misleadingly, IMO) described as "time doesn't pass for a photon". Conversely, a timelike object at rest moves entirely through time and not at all through space (according to Greene). Timelike objects moving at some speed less than that of light move partly through time and partly through space--as they move faster through space, they move more slowly through time (according to Greene).

    This is not wrong, exactly; you can rewrite the math of relativity so it does look something like this. However, I think it is misleading because:

    (1) Saying "everything moves through spacetime at c" implies that there is some 4-vector describing every object's motion that has length c--or a unit vector, with length 1, in units where c = 1 (which are the natural units to use in relativity). But a null path has a tangent vector with zero length. Which means there is *no* unit vector tangent to the photon's worldline; there can't be, since you can't have a unit vector with zero length. So saying that a photon "moves through spacetime at c" is misleading because it obscures the fundamental difference between timelike vectors with unit length and null vectors with zero length.

    (2) Spacetime is a 4-dimensional manifold in which each point represents a unique event, such as the intersection of two worldlines. But Greene's descriptions, though they claim to be descriptions of "spacetime", are actually based on looking at space vs. *proper* time. Space vs. proper time is *not* a 4-dimensional manifold in the above sense; it does not assign unique coordinates to unique events. It's perfectly possible for the same event (such as the twins meeting up again in the twin paradox--see below) to have two *different* sets of space vs. proper time values assigned to it. This completely obscures the geometry of spacetime, which means you basically have to unlearn what Greene says if you want to learn more about the subject.

    First of all, you don't need Greene's viewpoint to explain the twin paradox; it's explained perfectly well using relativity in its standard form, without resorting to Greene's misleading descriptions.

    Second, "the path with all its motion in the time direction" is frame dependent. Change frames and you change which observer is moving "only through time". So that alone isn't enough to explain why one twin is younger when they meet up again. To explain that you have to look at the actual paths the twins take through spacetime; but as I noted above, Greene's explanation misleads you about the actual geometric nature of spacetime.

    For example, consider how we would have to represent the twin's trajectories in a space vs. proper time graph. One worldline is easy, it's just a straight vertical line (assuming proper time is on the vertical axis). But the second worldline messes things up; it will move up and to the right (the outbound leg), and then up and to the left (the return leg), but when it meets up again with the stay-at-home twin's worldline, it will do so at a *different* proper time than the stay-at-home twin is at when the meetup occurs. This is, of course, because the traveling twin ages less. So you now have two different proper times at the same point of space assigned to the same event.
  9. Apr 2, 2013 #8
    Excellent point. I forgot that the 4-velocity is undefined for photons.

    So would you agree that "everything moves through spacetime at c", as long as we're not talking about massless particles?

    Gotta run, and still need to read/digest the rest of your post. May have more to say later, thanks.
  10. Apr 2, 2013 #9

    Ben Niehoff

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    "Everything moves through spacetime at c" is a silly thing to say. The problem is that "the rate at which an object moves along its own worldline" is not even a sensible physical concept. The worldlines themselves automatically contain all physical information about the location of objects. They don't "travel along" the worldline; they are the worldline. Or said another way, it doesn't matter how you choose to parametrize the worldline, because only the worldline is physical, not its parametrization. You must not think of the worldline as a trajectory.

    "Velocity" is a quantity that is equal to the slope of the worldline in some given inertial frame (hence all velocities are relative, as they need to be defined relative to some frame). "4-velocity" is just a mathematical tool to organize velocity information, so that if we know the velocity in one frame, we can find it in another frame.

    Since velocity is really just the ratio of the spacelike component to the timelike component, 4-velocity is only defined up to scale (i.e. ##u^\alpha## and ##\lambda u^\alpha## give equivalent 4-velocities for any positive constant ##\lambda##). For timelike 4-velocities, we can choose to normalize them to unit vectors. For null 4-velocities, they have length 0, so normalization is meaningless.

    But in any case, the time coordinate already tracks the motion of objects through space with respect to time. It makes no sense to talk about the motion of objects through spacetime with respect to "something else", because this has no measurable effect. For example, if two worldlines intersect, it is absolutely true that the objects collide at the intersection. But if two trajectories (in space) intersect, the two objects don't necessarily collide, because they might arrive there at different times.
  11. Apr 2, 2013 #10


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    I don't think it's wrong, exactly, but I don't think it's very useful either. You're basically saying you can find a unit tangent vector to a timelike worldline. That's true, but it doesn't really lead anywhere.
  12. Apr 2, 2013 #11


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    I would agree, but then it sounds really weird: "everything moves through spacetime at the speed of light except light which has no speed through spacetime".

    I agree with PeterDonis, at best it is useless but not wrong.
  13. Apr 3, 2013 #12


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    My problem with the phrase is that it's at best highly ambiguous.

    If you say "the proper 4-velocity" at which an object moves through space-time is "c", it becomes nearly unambiguous, and makes sense.

    However, there is a long tradition in physics that velocities are measured with two stationary clocks and some synchronization method.

    Proper velocities, sometimes known as celerities, require only one moving clock and no synchronization method.

    Conceptually, I think proper velocities are a lot simpler than velocities as they don't require worrying about synchronization methods at all. And the idea avoids obsessions with synchronization methods being an important funamental issue, a very common argument which I think is basically just a dead end, one that can waste a whole lot of time.

    Proper velocities aren't typically taught much, though one can find the occasional odd paper on the topic, for instance http://arxiv.org/abs/physics/0608040.

    I suspect the reason that proper velocities aren't taught more is the experimental difficulties in making a moving clock as experimentally precise as two stationary ones that are synchronized. Experimentally, the later procedure is preferred, even though the "proper velocity" concept is simpler.

    What tends to happen I think is that the lay audience is much more flexible in interpreting the words, and many read"velocity" as"proper 4-velocity" (how they manage to do this, I dont know, but a surprising number of them get it, intuitively. I can't really comment on how many "don't get it", I'm sure there are a lot that don't, what's a bit surprising is the number that do.)

    Meanwhile the non-lay audience expects precision, and assumes that velocity means velocity and doesn't mean proper velocity. So they find Greene's remarks a bit unfortunate.

    The jump from proper velocity to proper 4-velocity is more minor than the jump from velocity to proper velocity, but it's still a bit of a jump. Because it's relatively minor, I've focused more on the jump from velocity to proper velocity.

    Also people usually say 4-velocity, and automatically interpret it as a proper 4-velocity, another little bit of linguistic magic.

    Anyway, I'm not really clear how much trying to disambiguate the concepts of veocity, proper velocity, proper 4-velocity. and 4-velocity really helps. To me it seems key. Thus, I post remarks along this line from time to time, in the hope that it helps someone, somewhere.
    Last edited: Apr 3, 2013
  14. Apr 3, 2013 #13

    Ben Niehoff

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    According to all other uses of "proper" in relativity (i.e., measured with respect to my own reference frame), the "proper velocity" of any observer is exactly zero.

    That paper defines a quantity they call "proper velocity" which they compute by making measurements in two different frames of reference (!). It's essentially "Lab distance travelled divided by proper time of moving object", which is a very strange quantity indeed. I don't see how this is useful, either for doing relativity or teaching it...
  15. Apr 3, 2013 #14


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    But they only work for timelike objects, not for light; by the definition given, the proper velocity of a light beam is zero.

    I think they are all best viewed as derived concepts, not fundamental concepts. The fundamental concept, to me, is spacetime as a geometric object. Once you understand that, it's easy to fit in whatever definition of velocity you like, by matching it up to the appropriate geometric objects, or components of them. It also makes it a lot easier to generalize to curved spacetime without getting confused.
  16. Apr 3, 2013 #15
    I agree, it is rather odd!

    Anyhow, thanks for the clarification guys.

    I guess it's good for one thing, though ... it sounds really cool. Perhaps why Greene includes similar statements in his book, it sells more copies.
  17. Apr 3, 2013 #16
    Epstein's version:
    and >> forward from there to read successive pages

  18. Apr 3, 2013 #17
    Proper velocity or celerity is not uncommon especially in the field of high energy particle physics.
    Also proper velocity is used by the Gullstrand-Painlevé and Doran charts in the resp. Schwarzschild and Kerr solutions.

    A good read would be: http://geocalc.clas.asu.edu/pdf/SpacetimePhysics.pdf
    There is also a Wikipedia page on STA: http://en.wikipedia.org/wiki/Spacetime_algebra

    From the first article:

    From the proper point of view, the term “relativistic mechanics” is a misnomer, because the theory is less rather than more relativistic than the so-called “nonrelativistic” mechanics of Newton. The equations describing a particle in Newtonian mechanics depend on the motion of the particle relative to some observer; in Einstein’s mechanics they do not. Einstein originally formulated his mechanics in terms of “relative variables” (such as the position and velocity of a particle relative to a given observer), but he eliminated dependence of the equations on the observer’s motion by the “relativity postulate,” which requires that the form of the equations be invariant under a change of relative variables from those of one inertial observer to those of another.

    A sharp observation wouldn't you say?

    Also from the first article:

    The “proper formulation” given here takes another step to move from covariance to invariance by relating particle motion directly to Minkowski’s “absolute spacetime” without reference to any coordinate system.

    Occam's razor rules again?

    You might even be interested in: http://arxiv.org/abs/gr-qc/0405033 if you want to take STA a bit farther.
    Last edited: Apr 4, 2013
  19. Apr 4, 2013 #18


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    Yeah, with this interpretation that common verbal explanation makes more sense. Here is an animated Epstein diagram:

    It takes the metric:
    Writes it as:
    dt^2 = ds^2 + dx^2
    For time like worldlines you can interpret ds as aging, or "advancing though time". And dt is the Euclidean length of the worldline in the space-propertime manifold. The passage of t then becomes the speed of advance in that manifold for all observed object.
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