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Inverse coefficients of dt & dr in Schwarzschild

  1. Dec 22, 2011 #1
    Awhile back A.T. & PeterDonis helped me through several basic confusions on my way to an intuitive understanding of geodesics in curved spacetime. I looked at

    But now something is bothering my feeble brain again: the inverse coefficients of dt and dr in the Schwarzschild metric.
    All those diagrams seem to show space and time stretched out equally. I understand the time-coordinate lines diverging: this illustrates how proper time is shorter than coordinate time; the coefficient of dt is less than one. But the coefficient of dr is greater than one, meaning that proper length is greater than coordinate ("reduced circumference") length. So shouldn't the space-coordinate lines be getting closer together rather than further apart? (In the trumpet-shaped diagram, the trajectories being represented are straight up and down, so it is the radial distance we're talking about.)

    The third page listed above shows "length contraction" that makes sense to me when gravity=0 (SR only). But with nonzero gravity, and zero initial velocity, it still shows length contraction. The coordinate time is longer than the proper time, which is okay; but the coordinate length seems to be greater than the proper length, which is backwards. No?

    My general question, then, is whether there is a tradeoff between space curvature and time curvature. Perhaps the time factor overwhelms the space factor (due to conversion by c)?
    Also, what's the best way to think about the fact that in SR, the space and time expansion factors are the same, whereas in GR they're inverses.
  2. jcsd
  3. Dec 22, 2011 #2


    Staff: Mentor

    The diagram you're talking about has the vertical lines marked in meters, but they don't say whether the meters are radial coordinate meters or proper length meters, so they are basically ducking the issue of how proper length relates to coordinate length. But I think that they mean for the meters along their trumpet surface to represent proper length meters; and if so, an easy way to see how lengths get stretched at smaller radii is to think of a vertical line at the center of the trumpet as the scale of the "r" coordinate, with equal increments of r marked off along it. Since the trumpet spreads out as it gets lower, a given increment of r lower on the trumpet will correspond to a larger proper length (since the trumpet spreads more there) than the same increment of r higher up (where the trumpet spreads less).

    Another good visualization tool for seeing how proper lengths get "stretched" is the Flamm paraboloid:


    This tool has the virtue of illustrating explicitly how the "r" coordinate is defined: the "r" coordinate of any given circle that goes around the paraboloid "horizontally" (each such circle represents a 2-sphere centered on the black hole at that r coordinate) is just the circumference of the circle divided by 2 pi; so it's easy to see that, as you go down the throat of the paraboloid, two circles that differ in circumference by a constant increment 2 pi dr will be separated by a larger proper distance (i.e., distance "downward" along the paraboloid). The "trumpet" surface above doesn't have this property (the circles around the trumpet get larger as you go lower, not smaller).

    The disadvantage of the paraboloid is that it doesn't show anything about time, since it's a "snapshot" of a single slice of constant Schwarzschild time t. I'm not sure there is any single visualization that would combine all the good features of both the trumpet and the paraboloid.

    I think the way "space" is labeled in this diagram is misleading; it appears to me that the "space" axis is measured in *coordinate* units, while the time axis is measured in proper units. For the zero gravity case, after 1 s of coordinate time, the object has traveled 0.5 ls, indicating a speed of 0.5, which is the *coordinate* speed of the object.

    (Of course, in one sense a diagram with "proper time" on one axis and "proper distance" on the other would be meaningless for the object itself, since it never moves in its own rest frame so its "proper distance" is always zero, relative to itself. So I can see why the diagram is the way it is; but it's still misleading IMO.)

    (Btw, the web page is also at least misleading when it says that coordinate time is "the length of all worldlines in spacetime". That's *proper* time, for timelike worldlines, *not* coordinate time. So I think whoever made this page was either a bit confused themselves or didn't think through their wording very carefully.)

    So what the animation is saying is that the effect of gravity (I used a "gravity" number of 1 to test this) is to shrink the proper time relative to coordinate time (from 0.87 s to 0.52 s per 1 s), and to increase the *coordinate* distance covered in 1 s of coordinate time (from 0.5 ls to 0.73 ls). But this makes me wonder: if the coordinate distance covered increases, that means the object is *falling* in the gravitational field; in other words, "to the right" in space means "downward". But the "length contraction" number decreases as the object falls, so this number must be the object's coordinate length relative to its proper length; it looks *shorter* as it falls deeper into the gravity well. So the coordinate length is getting smaller relative to proper length as the object falls, as expected; but the way things are labeled makes this a bit hard to see.

    Because they're due to two different things. SR length contraction and time dilation are due to relative motion; GR "length contraction" and "time dilation" are due to gravity's strength varying from place to place. I put the terms in scare-quotes for GR for that reason: using the same terms to describe different things often causes confusion, and it would be nicer if we had a wholly separate pair of terms to describe what happens in GR (since the terms "length contraction" and "time dilation" are way too entrenched in SR to change them there).
  4. Dec 22, 2011 #3
    Thanks, I should have thought of the trumpet's radius as a measure of proper length. That works out okay, I guess. It gives me more confidence that this could really represent how the inverse expansion/contraction factors play into the mapping of curved to straight lines (as we want to represent geodesics). I just have to remember that more distance around the trumpet represents shorter proper time, because the coordinate lines are spread out; whereas the larger radius represents larger proper length even though it too is spreading out the space-coordinate lines. Do I just have to take this as a limitation of the representation? It still leaves me with a bit of queasiness about the inverse factors.

    And thanks for bolstering my confidence as to the misleading labeling (or maybe it's the inherent oversimplification of a diagram trying to do too much) of the adamtoons "relativity" page.

    As for "because they're due to two different things", I suspected that might be the whole answer, but was fishing for some deeper SR-GR connection, such as I seemed to get from the centrifugal force example.
  5. Dec 22, 2011 #4


    Staff: Mentor

    Any representation is going to be limited, so I would answer "yes".

    I think it's probably the latter, mostly (though some of it, as I said, makes me think they either don't fully understand the underlying physics or they didn't think very carefully about how they were presenting it).
  6. Dec 23, 2011 #5


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    Science Advisor

    The spacial distortion is ignored in 1) & 3) which are very simplified.

    But IIRC in 2) proper distance vs. coordinate distance is shown:
    - radial proper distance : longitudinal distance along the surface
    - radial coordinate distance : projection onto the space axis
    As you see proper distance > coordinate distance. I'm not sure but you can check this out in more detail in Chapter 6 of this:
    It has the embedding formulas that the diagram 2) was based on.

    In GR you still have length contraction from movement.
    Only after the rocket started falling. That is length contraction from movement. Length contraction from spatial distortion is omitted here.
    No. Coordinate length is the projection onto the space dimension, which is shorter than the rotated proper length shown within the diagram
    No, if you use natural units (as diagram (2) does) the factors are in the same order of magnitude.
    Last edited: Dec 23, 2011
  7. Dec 23, 2011 #6


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    Not sure which trumpet you mean here. But in the space-eigentime diagram (2) the trumpet radius itself has no physical meaning. In fact there is a free embedding parameter (called k by Jonsson) which tells you how tight the diagram is rolled together. Jonsson chooses k such the the interior part is spherical, because that is how Epstein showed it in his book:
    But in (2) k is chosen somewhat differently to make it useful for all settings of mass.
    In (2) the larger proper length is not represented by the radius of the bulged cylinder, but rather by the larger distance along the surface than along the space axis.
    Last edited: Dec 23, 2011
  8. Dec 23, 2011 #7


    Staff: Mentor

    Hmm, looking again at the animation you are right, it only seems to be taking account of the effects of movement, not of the change in the g_rr metric coefficient. Essentially, it's assuming that everything takes place at a constant r, but it's still including the effects of g_tt on proper time vs. coordinate time. Another confusing aspect; locally, the effects of g_tt are not visible, only by exchanging light signals with some observer at a much larger r do those effects become apparent.
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