Does spacetime curvature require a fifth dimension?

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daniel_i_l
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In 2D for example, warping it only makes sense 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 sense 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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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
 
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daniel_i_l said:
In a 2D world, bending 2D would be meaningless.

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].
 
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")?
 
daniel_i_l said:
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")?

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.