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Question on Curvature of Space

  1. Nov 16, 2007 #1

    Art

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    Would it be possible to adjust for the curvature of space between 2 points and so by taking the shortcut (a true straight line) beat a light source in a race between the 2 points whilst travelling at less than light speed?
     
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  3. Nov 16, 2007 #2

    Chris Hillman

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    You mean curvature of spacetime, not curvature of "space". I highly recommend the excellent popular book by Geroch, General Relativity from A to B (the author is a leading expert on gtr).

    After you have read this book, which has almost no mathematical prerequisites, you should be able to sketch and understand the following spacetime diagram: draw the world line of a massive object such as a neutron star, and the world line of two nearby test particles A, B on roughly opposite sides of the neutron star and not traveling very quickly with respect to the star (so that all three world lines are "almost vertical"). Draw two null geodesics, starting from the same event on the world line of A and passing on either side of the world line of the star, one closer than the other, and ending at different events on the world line of B. (Which signal is received later? Recall that the path of one passes closer to the star.)

    This sketch shows that while according to gtr electromagnetic signals always travel at the speed of light (in vacuo) it is possible for two signals which traverse different paths to wind up "at the same destination" but "at different times". (Roughly speaking: more properly one must say exactly what I said: two signals sent from the same event on one world line can wind up at different events on a second world line.) This is an example of the effects that curvature of spacetime can have on the propagation of signals in some region which is strongly curved.

    Now you can imagine an extreme case in which the first signal is actually sent along a fiber optic cable or something like that, but still beats a second signal sent through a vacuum via a path which passes much closer to a massive object. (In this case, the first signal would have a world line which is a timelike geodesic, not a null geodesic.)

    But I stress that you won't understand correctly what I am trying to explain until you have had a chance to learn from the book by Geroch (or perhaps from a similar book) the essential elements of spacetime geometry. In particular, you won't understand why the local vs. global distinction in the theory of manifolds shows that nothing I have said contradicts any principles of relativistic physics.
     
    Last edited: Nov 16, 2007
  4. Nov 16, 2007 #3
    Do you really mean the curvature of space? Or do you mean the curvature of space-time? They are different things, each having a particular meaning.

    In GR it is possible that there could be two worldlines between the same two events, which are widely spaced in space-time (i.e. two events which are located a places which are "far", i.e. in the Euclidean sense, from each other). Each of which might possibly correspond to two different (coordinate) times of travel.

    Pete
     
  5. Nov 16, 2007 #4

    Art

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    As the choice of a time co-ordinate would seem to be largely arbitary in this example I thought the underlying space curvature was the significant aspect but I stand open to correction.

    The essential point of my question was is it possible for a sub-lightspeed particle to reach a destination sooner than light by taking a 'shorter' route by by plotting a course which compensated for curvature?
     
    Last edited: Nov 16, 2007
  6. Nov 16, 2007 #5

    A.T.

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    Since light can orbit a massive object it is certainly possible to construct such a case: Send a photon on an orbit, and move a bit in the opposite direction. If the photon hits you after one round, you can say: "I was here first!"
     
  7. Nov 16, 2007 #6

    robphy

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    You don't need local curvature....
    Consider a cylindrical universe (with periodicity in the spatial x-direction).
     
  8. Nov 16, 2007 #7
    Excellant point Rob, excellant! :smile:

    A.T. - Great idea about the photon sphere too A.T.! :smile: I had misread the OPs post. Had I not misread it I woiuld have suggested that myself.

    Pete
     
  9. Nov 16, 2007 #8

    Art

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    I meant more in the sense of is it possible to send information faster to a point than light can carry the same information whilst obeying the rule of not traveling faster than light can traveling at it's max speed whilst not deliberately constraining the speed of light.
     
    Last edited: Nov 16, 2007
  10. Nov 18, 2007 #9

    A.T.

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    In my example the light is sent on a detour, while you take a shortcut. If you assume that light takes the quickest way too, then it will be always faster than sub-c information.
     
  11. Nov 18, 2007 #10

    pervect

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    Whew. Are you trying to ask "Are shortcuts like wormholes possible"? The answer to whether or not wormholes are possible is a qualified yes - there is currently no proof they are impossible, which is not quite the same thing as demonstrating that they are possible.
     
  12. Nov 20, 2007 #11

    Art

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    I'm just curious if one could theoretically pilot a sub-LS craft along an arc rather than follow the grand circle light has to follow. A little like how a sailing boat can sail slightly into the wind to maintain a straight heading.
     
  13. Nov 20, 2007 #12

    Chris Hillman

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    Assuming that by "follow the grand circle light has to follow" you mean that in gtr, the world line of a laser pulse (or a photon, if you prefer) is a null geodesic, did you read [post=1506991]my earlier post in this thread[/post]? Didn't I answer your question?
     
  14. Nov 20, 2007 #13

    Art

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    I was responding to other people's posts who posted after you.

    Yes I read your reply and replied to it re your question if I meant curvature of space or spacetime and if I understand you correctly the inference seems to be that you can send information from Point A to B (effectively) faster than light. My question then was could this be achieved by simply steering a course to nullify the effect of the curvature? I ask this as it would appear to avoid the necessity for exotic matter etc for wormholes whilst achieving the same result?

    Would there be any causality consequences if this were done?
     
  15. Nov 20, 2007 #14

    pervect

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    Art, I don't really understand what you might be asking that hasn't already been answered. I think you need to define an operational way (i.e. some thought experiment) that has a definite answer, one that doesn't depend on your understanding of abstract notions (like curvature), because I have a strong suspicion based on your usage that we don't attribute the same meaning to the word "curved".
     
    Last edited: Nov 20, 2007
  16. Nov 21, 2007 #15

    DrGreg

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    Let’s get back to the original question:

    Art, I suspect the picture you have in your head is that spacetime is a bit like the 2-dimensional curved surface of a 3-dimensional sphere. Are you asking if it is possible to bore a “straight line” in 3-D space through the middle of the sphere to get to a point faster than light travelling along the surface?

    The answer, in that sense, is no. In this picture, the whole known Universe lies within the 2-D surface - it’s not possible to step outside the Universe and take a short cut through a higher dimension, as far as we know.

    However, suppose we imagine a 2-D spacetime that is the shape of the surface of a doughnut with a hole, or the inner tube of a tyre (called a torus). In this case, you could have light travelling around the outside of the torus whilst you yourself take a shortcut through the hole and get there faster. However, if you zoom in to any small area of the torus, there is no way of travelling faster than light within that area. Locally, light always follows the shortest route, or the “straightest” route (or, to put it technically, follows a null geodesic).

    Note that in reality, the Universe looks like neither a sphere nor a torus - these are just illustrative pictures to get across the concept of curvature. Nobody knows if there really are any “holes” in spacetime, it’s just that the mathematics doesn’t forbid them.

    Also, there is no evidence to suggest that there actually exists anything outside of 4-D spacetime. The “curvature” of spacetime can be described mathematically within its own four dimensions without having to imagine any extra dimensions. (For example, the angles of a large triangle drawn on the 2-D surface of a sphere add up to more than 180 degrees; that fact would tell two-dimensional beings living on the surface that their world was curved without them being aware of any third dimension.)
     
  17. Nov 21, 2007 #16

    Chris Hillman

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    DrGreg may have uncovered a different source of confusion on Art's part, but FWIW:

    No, I did not say that. I tried to explain that to ask questions which can be sensibly answered, you need to refine your intuition and adopt a new and more powerful language adapted to the complexities of curved spacetime, which is in several respects somewhat unlike the elementary notions of geometry you are used to. One point I tried to stress is that "faster than light" could mean all kinds of things, some of them quite humdrum.

    One of the points I was trying to stress is that you need to distinguish between path curvature (of a world line in a spacetime) and the curvature of the spacetime itself. The first has units of reciprocal length while the second has units of reciprocal area.

    I said what I said about the scenario which I discussed. To repeat: one of the points which I tried to stress is that even in flat spacetime there are multiple distinct operationally significant notions of "distance in the large" and thus "speed in the large".

    There is nothing of that kind in the humdrum scenario I outlined.
     
    Last edited: Nov 21, 2007
  18. Nov 21, 2007 #17

    pervect

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    Let me discuss the following (actual) experiment, in the hope that it clears up some of the questions.

    Suppose we have a different supernovae, and observe it on Earth. What will we see first? Light (electromagnetic radiation), or massive particles?

    If interstellar space were actually empty, we would see the light first. Interestingly enough, because interstellar space isn't quite empty, we see the neutrino emissions first, which are technically massive particles. Normal refractive effects delay the light just long enough for the neutrinos (which travel at almost the same speed as light but do not interact with the interstellar media appreciably and hence are not delayed by it) to arrive first.

    In a general curved space-time, we might see multiple images of the supernovae explosion. But any given image would be announced first by the neutrino emissions, closely followed by the electromagnetic radiation (light), and with other massive particles coming in as a distant third.
     
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