Working through intro GR at the moment and I'm a little unclear on how tangent spaces are used to carry four-vectors over from SR to GR.(adsbygoogle = window.adsbygoogle || []).push({});

So, at every point in spacetime we construct a tangent space. We can form a basis for this space with the tangent vectors (i.e. the four-velocities) of one time-like and three space-like worldlines passing through this point (am I doing OK so far?) Thus, we can construct various vectors and one-forms in this tangent space and, if a metric is provided, turn one into the other. Extend it to tensors, and we're done!

Except...what about four-vectors that (as far as I can tell) can't be expressed as a tangent vector times some scalar? Where do they "live"? For instance: the four-potential. It's a four-vector because its components transformlikethe spacetime coordinates, not because theyarespacetime coordinates. With something like four-velocity and four-momentum, I can imagine a vector in a tangent space. Since the components of four-momentum are just [itex]p^\alpha=mU^\alpha[/itex], I can actually think of the four-momentum as [itex]p=p^\alpha e_{(\alpha)}[/itex], where the [itex]e_{(\alpha)}[/itex] are the tangents to four worldlines. Can I do that with something like the four-potential?

I guess I can rephrase my question like this, if that's not clear: if the components of four-potential at some point in spacetime are [itex](\phi,\vec{A})[/itex], what is the basis that those components refer to? Is it a basis of tangent vectors at that point? If that's not the case, and not every four-vector lives in a tangent space, then how do such non-tangent vectors carry over into GR?

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# Do all four-vectors live in a tangent space?

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