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## Main Question or Discussion Point

Hi all, this might be a silly question, but I was curious. In Carroll's book, the author says that, in a manifold [itex] M [/itex], for any vector [itex] k [/itex] in the tangent space [itex] T_p [/itex] at a point [itex] p\in M [/itex], we can find a path [itex] x^{\mu}(\lambda) [/itex] that passes through [itex] p [/itex] which corresponds to the geodesic for that vector ([itex] k [/itex] being the tangent vector to the path). Two conditions for this path are:

[tex] \lambda(p)=0 \\ \frac{dx^{\mu}}{d\lambda}(\lambda=0)=k^{\mu}

[/tex]

(And, of course, it must satisfy the geodesic equation.)

From this, we can then construct a map, call it [itex] \exp_p: T_p\to M [/itex] such that [tex] \exp_p(k)=x(\lambda=1) [/tex]

Where [itex] x(\lambda=1) [/itex] is the point in [itex] M [/itex] belonging to the parametrized path introduced earlier (the geodesic for [itex] k [/itex]) evaluated at [itex] \lambda=1 [/itex]. Now, my question is: why are we evaluating at [itex] \lambda=1 [/itex]? Not only does this seem arbitrary- it also seems completely independent of all aspects of the manifold. What I mean by this is that we could pick any parameter, big or small, for our geodesic (since we're working with an affine parameter, I think). Given this, it seems like we lose our ability to say that [itex] \exp_p [/itex] maps to the neighborhood of [itex] p [/itex]; it could map to faraway places in the manifold, given the right parameter. So given this,

1) Why was this chosen? Arbitrary convention?

2) Is what I pointed out above problematic, or is it a non-issue? (Regarding the fact that we can map to faraway places.)

3) Could something be chosen instead of [itex] \lambda=1 [/itex], that sort of characterizes a "small scale" in the manifold? This is related to something I'm not very sure about- is there a way to establish the "physical size" of a manifold? Throwing rigor out the window, what I mean is this: maybe we could say that a manifold has "size" [itex] S_M [/itex], and then redefine the map so that [itex] \exp_p(k)=x(\lambda=s) [/itex], where [itex] s<<S_M [/itex]. For an example of what I mean by this "size", maybe we could say [itex] S_M=2\pi R [/itex] for [itex] S^2 [/itex]? (The problem is I doubt this could be done in general :c ).

Many thanks in advance!

[tex] \lambda(p)=0 \\ \frac{dx^{\mu}}{d\lambda}(\lambda=0)=k^{\mu}

[/tex]

(And, of course, it must satisfy the geodesic equation.)

From this, we can then construct a map, call it [itex] \exp_p: T_p\to M [/itex] such that [tex] \exp_p(k)=x(\lambda=1) [/tex]

Where [itex] x(\lambda=1) [/itex] is the point in [itex] M [/itex] belonging to the parametrized path introduced earlier (the geodesic for [itex] k [/itex]) evaluated at [itex] \lambda=1 [/itex]. Now, my question is: why are we evaluating at [itex] \lambda=1 [/itex]? Not only does this seem arbitrary- it also seems completely independent of all aspects of the manifold. What I mean by this is that we could pick any parameter, big or small, for our geodesic (since we're working with an affine parameter, I think). Given this, it seems like we lose our ability to say that [itex] \exp_p [/itex] maps to the neighborhood of [itex] p [/itex]; it could map to faraway places in the manifold, given the right parameter. So given this,

1) Why was this chosen? Arbitrary convention?

2) Is what I pointed out above problematic, or is it a non-issue? (Regarding the fact that we can map to faraway places.)

3) Could something be chosen instead of [itex] \lambda=1 [/itex], that sort of characterizes a "small scale" in the manifold? This is related to something I'm not very sure about- is there a way to establish the "physical size" of a manifold? Throwing rigor out the window, what I mean is this: maybe we could say that a manifold has "size" [itex] S_M [/itex], and then redefine the map so that [itex] \exp_p(k)=x(\lambda=s) [/itex], where [itex] s<<S_M [/itex]. For an example of what I mean by this "size", maybe we could say [itex] S_M=2\pi R [/itex] for [itex] S^2 [/itex]? (The problem is I doubt this could be done in general :c ).

Many thanks in advance!