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Re: Tidal Gravity And Spacetime Curvature

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Posted by Zephram Cochrane/">Zephram Cochrane on February 8, 2000 10:13:38 UTC

This message was mainly intended for another physics board, but it doesn't accept html tags so I'd rather post here. Though there are a few interesting discussions in that forum there is too much fighting so I doubt I'll post there in the future, but I thought I would point out why the core arguments going on are over nothing but a misunderstanding of words. Probably the biggest argument going on is over the relationship of curvature with uniform gravitational fields and tidal forces. The people involved don't realize that the problem arises from diverse, but valid definitions being used. For instance; One person here is stuck on defining the state of curvature in terms of what is usually called spacetime curvature. Another person here is stuck on defining curvature as what is usually called Gaussian curvature. I'll demonstrate the differences.

First lets consider the definition for curvature given by Gaussian curvature. Here's a classic thought experiment. Imagine a two dimensional creature living on the surface of a sphere. It will start at a point we'll call the north pole and move directly away staying on the spheres surface for some distance r. Then it will orbit the pole once staying a distance r along the ball away from the pole and measure the circumference C. It expects to find the circumference to be C = 2pr, But to its surprise measures that C is less than 2pr. After careful thought it realized that its two dimensional world must be a negatively curved space imbedded in a third dimension. Because of that the R that it should have used to get C = 2pR was not the distance traveled long the surface, but was actually a distance that cut through the third dimension going through the ball. Limiting ourselves to the consideration of space curvature the Gaussian curvature K of a two dimensional cross section of space is defined as K = (3/p)(Lim r - 0)(( 2pr - C)/r3) From here we can make a more general definition of curvature in terms of excess radius. We will say that whenever a space-time is represented in someone's Cartesian coordinate frame and there remains a non-zero affine connection then the space-time has space and/or time curvature in the coordinate frame of that observer. Here is an example of a space-time with space and time curvature that has zero spacetime curvature(i.e. Rlmsn = 0) ds2 = (1 + 2az/c2)dct2 - dx2 - dy2 - dz2/(1 + 2az/c2) Here is an example of a space-time with time curvature, no space curvature, and no spacetime curvature. ds2 = (1 + az/c2)2dct2 - dx2 - dy2 - dz2 (these two space-times are examples of globally uniform gravitational fields, the gravitational field itself is also being argued over due to a difference in definitions being used)

Next lets consider spacetime curvature. Lets say the two dimensional creature on the sphere carries an arrow pointing parallel to the surface in the direction of its motion as it moves from the north pole to the equator. Once it reaches the equator it moves along the equator one quarter the way around without twisting the arrow so that it stayed pointing away from the pole. The creature then returns to the pole without twisting the arrow. Upon the creatures return the creature realizes the arrow is now pointing 90 degrees away from where it initially pointed. From this the creature again infers that it is living in a universe curved within a higher dimensional space. From here we can generalize to define spacetime curvature in terms of the parallel transport of four vectors. Lets say we parallel transport a four vector along an infinitesimal path da then along db then along -da and then -db to close a little path then the total infinitesimal change in a four vector dvl = vmRlmsndandbs Rlmsn is called the Reimann tensor. Going by this definition of curvature, spacetime curvature is the same thing as having any non-zero elements of Rlmsn. An example of a space-time with spacetime curvature would be ds2 = (1 - 2GM/rc2)dct2 - dr2/(1 - 2GM/rc2) -r2(dq2 + sin2qdf2) This space-time also has Gaussian curvature in every global frame.

Four other words causing major arguments over nothing more than diverse but valid definitions are gravitational field tidal force mass uniform but I'm tired of typing so hopefully that was enough to convey the point.


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