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I understand that space is curved, but if one accounted for this curvature and warped the vector accordingly, could a straight line be produced?

When and how was it first proved that this can or cannot be done?

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    $\begingroup$ A ray of light is a straight line, almost by definition. $\endgroup$ – Alexandre Eremenko Jul 19 '17 at 20:52
  • $\begingroup$ @AlexandreEremenko but is the beam of light not curved by gravity? $\endgroup$ – DukeZhou Jul 19 '17 at 20:54
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    $\begingroup$ Only at large distances. Strictly speaking no mathematical abstraction can be EXACTLY produced in nature. A point, for example, what is a point? Mathematical notions only approximately describe what we see in the real world. A line segment corresponds to a ray of light (not very long one). $\endgroup$ – Alexandre Eremenko Jul 19 '17 at 20:57
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    $\begingroup$ If the geometry of the spacetime is not euclidean (in the sense of non-euclidean geometry), then a straight line is not "straight" in the euclidean sense. But what is the def of "straight line" ? See Euclid: "Definition 4. A straight line is a line which lies evenly with the points on itself." $\endgroup$ – Mauro ALLEGRANZA Jul 20 '17 at 8:46
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    $\begingroup$ I made the question less trivial by asking for the history of this topic. $\endgroup$ – Tom Au Jul 21 '17 at 1:15
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In general, if you have a curved space, the notion of a "straight" line breaks.

In order to rebuild it, or at least an approximation of it, remember that, in flat space, a straight line is the shortest distance between two points. In that sense, you can generalize the straight line to the geodesic, which is defined as the curve whose distance is either the maximum or minimum distance between two points (why you would have to consider the maximum is a bit beyond the scope of this question). For a globe, the geodesics are the paths on the surface that intersect planes through the center of the globe, but for a general curved surface, you have to solve an equation:

$${\ddot x}^{a} + \Gamma_{bc}{}^{a}{\dot x}^{b}{\dot x}^{c} = 0$$

where the path is given by $x^{a}$, the dots are derivatives along the path, and the $\Gamma_{ab}{}^{c}$ can be calculated from the geometry under consideration (all of them are zero for flat space in Cartesian coordinates, for example).

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  • $\begingroup$ Thanks for this! (I know fun, naive questions are generally frowned upon on Stack, presumably taken as undermining the value of strictly serious inquiry, but then again, doesn't all science arise out of naive inquiry into fascinating subjects?) I'm not a physicist, but I'm gauging that there is a connection between physics and combinatorics, and I'm trying to learn more about topology. $\endgroup$ – DukeZhou Jul 20 '17 at 17:23
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    $\begingroup$ @DukeZhou: combinatorics will come up in physics, but I wouldn't say it's the most common branch of math to show up. You'll be very far along, indeed, in physics before you run into topology. $\endgroup$ – Jerry Schirmer Jul 20 '17 at 18:05
  • $\begingroup$ en.wikipedia.org/wiki/Combinatorics_and_physics $\endgroup$ – K7PEH Jul 22 '17 at 18:42
  • $\begingroup$ Also note that a curved space of General Relativity is a pseudo-Riemannian manifold with a topology that is Euclidean in the local neighborhood around each point. $\endgroup$ – K7PEH Jul 22 '17 at 18:49

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