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4D warping

  1. Dec 30, 2005 #1

    daniel_i_l

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    In 2D for example, warping it only makes sence in a 3D enviorment. In a 2D world, bending 2D would be meaningless. So in GR when we talk about bending 4D ST, is a fifth dimension needed to give ST something to "bend into"? If not, in what sence can we talk about warped ST? I suspect that maybe it warps not in a physical (dimensional - into another dimension) way, but rather the cooardinite system gets messed up and those effects are noticable in 3D. Can someone clarify this please?
     
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  3. Dec 30, 2005 #2

    Garth

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    No daniel_i_l, you do not need a fifth dimension for space-time curvature. The curvature can be defined and described in terms of the intrinsic geometry of the manifold: do the internal angels of a triangle sum to 2[itex]\pi[/itex]? do parallel lines meet? does a vector, such as the axis of a Gravity Probe B gyroscope, still point in the same direction when you have parallel transported it around a closed circuit?

    However in order to visualise the situation, and Einstein himself said that he needed to be able to visualise his theories to understand them, as a mental exercise you do need the higher dimension in which to embed the 4D manifold of space-time.

    Garth
     
    Last edited: Dec 30, 2005
  4. Dec 30, 2005 #3
    Consider an ant walking on the surface of the earth. If there were no hills or mountains, to him, the existence of the 3rd spatial (height) dimension would be non-existent as he would never experience it. However, he could still experience the curvature of the earth's surface, which is purely a 2D space. As Garth has pointed out, he could walk a giant triangle (with a vertex at the North pole, and 2 at the equator, for example) and see that the triange has a total internal angle of [itex]3\pi[/itex].
     
  5. Dec 30, 2005 #4

    daniel_i_l

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    Thanks, so if I understand correctly (please correct me if I'm wrong) if for example 2D space was represented as a grid with eavenly spaced intersection points, in warped space the poits would be closer in some places and further is others ("messed up cooardinates")?
     
  6. Dec 30, 2005 #5
    Well, you could always consider whatever the co-ordinates are around you as "straight lines intersecting perpendicularly". But the point is that co-ordinate lines far away from you wouldn't necessarily look "straight and perpendicular" compared to the ones around you.
     
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