Solving for integral curves- how to account for changing charts?

In summary, the discussion revolves around finding the integral curve associated with a vector field by solving a set of coupled first-order differential equations. This can be done in any coordinate system and the resulting curve will satisfy the equation in every coordinate system. The example provided further illustrates this concept, showing that the coordinates of the vector field at different points will be in accordance with the corresponding coordinate system. The integral curve can then be described as a combination of local solutions in different coordinate systems.
  • #1
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[Ref. 'Core Principles of Special and General Relativity by Luscombe]

Let ##\gamma:\mathbb{R}\supset I\to M## be a curve that we'll parameterize using ##t##, i.e. ##\gamma(t)\in M##. It's stated that:
If ##\gamma(t)## has coordinates ##x^i(t)## and [a vector field] ##X## has components ##X^i##, finding the integral curve associated with ##X## reduces to solving a set of coupled first-order differential equations, $$\frac{d}{dt}x^i=X^i(x^1(t),\ldots,x^n(t))$$
Immediately after there's an example: if ##X=x\partial_x+y\partial_y##, then ##dx/dt=x## and ##dy/dt=y##, which gives the integral curve passing through ##(a,b)## at ##t=0## as ##\gamma(t)=(ae^t,be^t)##.

  1. Now from the context, provided we're talking about only one curve ##\gamma##, shouldn't ##X## actually be the restriction of the vector field to the curve ##\gamma##, rather than the vector field itself?
  2. Referring to the phrase "If ##\gamma(t)## has coordinates ##x^i(t)##...", I'm guessing it's unlikely that all the points on the curve belong to a single chart. So how can we claim only one coordinate system ##x^i## to represent the coordinates of all the points on the curve? Won't we have to adjust the coordinates according to the chart?

    e.g. if some ##p,p'\in\gamma(I)## are covered by different charts, and if the coordinates of ##p## are ##x^i##, won't the coordinates of ##p'## have to be characterized by an entirely different coordinate system (e.g. some ##y^i##)?
 
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  • #2
1. No. You're starting with the vector field ##X## and solving for ##\gamma.## You don't have a curve to begin with.

2. Given ##X##, you're looking for a curve ##\gamma## with ##X(\gamma(t))=\gamma'(t)## for all ##t##. A solution to this equation will satisfy your quoted equation in every coordinate system. You're right that in general the image of the curve won't lie inside of a single coordinate chart.
 
  • #3
Infrared said:
1. No. You're starting with the vector field ##X## and solving for ##\gamma.## You don't have a curve to begin with.

2. Given ##X##, you're looking for a curve ##\gamma## with ##X(\gamma(t))=\gamma'(t)## for all ##t##. A solution to this equation will satisfy your quoted equation in every coordinate system. You're right that in general the image of the curve won't lie inside of a single coordinate chart.
What do you mean by ##X(\gamma(t))##? Is it that the coordinates are given by ##X((x^i\circ\gamma)(t))##? That's the only way I can think of that'll make it chart independent.
 
  • #4
A vector field assigns to each ##x\in M## a tangent vector in ##T_xM##. By ##X(\gamma(t))##, I mean the tangent vector that ##X## assigns to the point ##\gamma(t)##.
 
  • #5
Infrared said:
A vector field assigns to each ##x\in M## a tangent vector in ##T_xM##. By ##X(\gamma(t))##, I mean the tangent vector that ##X## assigns to the point ##\gamma(t)##.
Yes, that's clear to me so far.

In regards to the example in the OP, consider any point ##p_0\in M##. So can I say that whatever coordinate system ##\{x, y\}## is used at ##p_0## (in accordance with whatever chart covers it), the coordinates of ##X_{p_0}## in the corresponding coordinate basis (corresponding to the coordinate system) will be ##x,y##?

Essentially this means that if I use some other coordinate system ##\{u,v\}## at some other point ##p_1##, then the coordinates of ##X_{p_1}## will now be ##u,v##.

And what this implies for the integral curve that we calculate, i.e. ##(ae^t,be^t)##, is that if ##\gamma(t_0)=p_0## and ##\gamma(t_1)=p_1##, then ##(ae^{t_0},be^{t_0})## are the coordinates of the integral curve at ##p_0## in the coordinate basis ##(x,y)##, while ##(ae^{t_1},be^{t_1})## are the coordinates of the integral curve at ##p_1## in the coordinate basis ##(u,v)##. In essence, we're coming up with local solutions to the curve and "stitching them together".

Does that sound correct so far?
 

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