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I just have started studying special relativity and moving to general relativity

Special relativity only deals with inertial frame (non accelerated frame) but there are no inertial frame in the curved space time so we consider local inertial frame from the global (which is accelerating) like earth is accelerating but you can measure things (time, distance) by considering a laboratory on earth. In a local laboratory an observer carries along four orthogonal unit four-vectors which define a time direction and 3 spatial coordinates But then the book says "time like unit four vector will be tangent to the observer's world line since that is the direction a clock at rest in the laboratory moving in spacetime"

I don't get how Time like unit four vector is tangent to observer's world line!! The example is from gravity by James.hartle Can someone please explain

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  • $\begingroup$ If you draw the space time diagram of stationray observer, then it will be vertical line. Now vertical direction the vector is time, so yeah $\endgroup$ Jun 7, 2022 at 12:07
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    $\begingroup$ Special relativity can handle non-inertial frames (although the simple equations in introductory treatments of special relativity can't handle non-inertial frames). What special relativity can't handle are curved spacetimes. $\endgroup$
    – robphy
    Jun 7, 2022 at 14:46

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I don't get how Time like unit four vector is tangent to observer's world line!!

It is not necessary for the timelike unit four-vector to be tangent to an observer's worldline. However, since we have freedom in choosing our coordinates and therefore our coordinate basis vectors we can always choose our coordinates such that the timelike basis vector is tangent to the observer's worldline.

When the timelike basis vector is parallel to the observer's worldline it simply means that the observer is at rest with respect to the coordinates. So if the observer's worldline is only tangent to the timelike basis vector at one event then that means that it is only momentarily at rest. However, if the worldline is parallel to the vector for an extended period then the observer is at rest in those coordinates over that extended period.

So all they are saying is that we are constructing a coordinate system with the potentially useful property that the given observer is at rest in those coordinates, either momentarily or for an extended period, depending on the rest of the description.

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