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Does it make sense to talk about Christoffel symbols in flat space time? Do they have non-zero values? I understand that the Christoffel symbols appear as an indication of curvature in space. So, are they non-existent in flat space-time?

nihal
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Yes, it makes sense to talk about Christoffel symbols in flat spacetime. Every coordinate system has associated Christoffel symbols. On Minkowski spacetime in the standard coordinates, the Christoffel symbols are all zero. But in different coordinates (e.g., spherical coordinates), they will not be zero. The Christoffel symbols contain information about the intrinsic curvature of the spacetime and about the "curvature of the coordinates".

frakbak
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  • Then, in the Einstein Field Equations, when you try to reduce them to Newton's equation( Gauss's Law for Gravity), how come the R00 component of the Ricci Curvature is a non-zero quantity when the Ricci Curvature is expressed in terms of Christoffel Symbols? – nihal Aug 29 '15 at 07:08
  • Are you talking about Minkowski spacetime or a perturbation of Minkowski spacetime? – frakbak Aug 29 '15 at 15:27
  • Sorry, I was talking about the perturbation of Minkowski spacetime. The Ricci component(R00) does have a non-zero value of 8 pi times Gravitational Constant times density of the mass. My doubt is how that component of the Ricci Curvature alone is left out as a non-zero quantity when reducing the Einstein field equations to Gauss's Law for Gravity. – nihal Aug 29 '15 at 18:23
  • Without more context I am not sure what you are asking. – frakbak Aug 29 '15 at 18:29
  • @frakbak Nice answer, I want to ask you, is there anything concrete (mathematical or physical) that proves as you said "The Christoffel symbols contain information about the intrinsic curvature of the spacetime and about the "curvature of the coordinates"."? – PhilosophicalPhysics Oct 13 '15 at 00:14
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    @PhilosophicalPhysics It's a mathematical observation. The Riemann curvature tensor (which is an intrinsic, coordinate-independent object) can be computed from the Christoffel symbols, so the Christoffel symbols definitely contain information about the intrinsic curvature. On the other hand, as I explained in my answer, the spacetime can be flat (Riemann curvature tensor is identically zero) and yet the Christoffel symbols can be nonzero in certain coordinates. This means that the coordinates themselves are "curved" and the Christoffel symbols are detecting that. – frakbak Oct 13 '15 at 19:52
  • If the Riemann curvature tensor is zero but Christoffel symbols are not in certain coordinates, how would this imply that the coordinates are curved? If you can give an example, but if not, how does one conclude that coordinates are curved if the Riemann tensor is zero? – PhilosophicalPhysics Oct 13 '15 at 20:11
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    It's a matter of definition. By definition, flat coordinates are coordinates in which all the Christoffel symbols are zero. Then we have the result that coordinates are flat if and only if the open set covered by the coordinates is locally isometrically Euclidean. I think making this definition makes good sense, but you are free to disagree. Now if I have coordinates on a space which I know is flat (Riemann tensor is zero) but the Christoffel symbols are nonzero, I am forced to conclude that the coordinates are curved, by which I mean, not flat according to the definition I just gave. – frakbak Oct 13 '15 at 20:37
  • Thanks, this was very helpful. – PhilosophicalPhysics Oct 13 '15 at 23:37
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As frakbak explained, one has a notion of Christoffel symbols in flat spacetime, as they basically record information about derivatives of the metric tensor with respect to different indices, each. It makes kind of sense, since changes of the metric tensor describe local changes of a scalar product which encodes information about changes of projections to a new coordinate basis, featureing a new tangent plane. While such changes of the projection to a new coordinate basis are inevitable in an arbritrarily curved spacetime, you can just deliberately provide the covariant derivative with such unnatural changes in flat spacetime by changing to an "inappropriate", curved coordinate system.

Victor Caran
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