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Hi guys. This is regarding section 13.6 (p.327) in MTW. Here the authors consider an arbitrary accelerated observer in any space-time and construct a set of local coordinates carried along the entire worldline of the oberver with the origin of the coordinates comoving with the observer; they term this the "proper reference frame" of the accelerated observer. Note that they do not assume the observer's spatial basis vectors are Fermi-Walker transported along his/her worldline.

In ex.(13.14) they consider an accelerated observer and a freely falling particle that at some event is coincident with the origin of the observer's "proper reference frame". They say to show that the 3-acceleration of the freely falling particle relative to the "proper reference frame" at that event is given by ##\frac{\mathrm{d} ^{2}x^{j}}{\mathrm{d} x^{0^2}}e_j = -a - 2\omega \times v + 2(a\cdot v)v## where ##j = 1,2,3##, ##(e_j)## are the spatial basis vectors carried by the observer, ##\omega## is the angular velocity of rotation of the spatial basis vectors, ##v## is the 3-velocity of the freely falling particle, and ##a## is the acceleration of the observer him/herself.

Because this event is on the worldline of the observer (i.e. the origin ##x^j = 0## of the observer's "proper reference frame"), the non-zero christoffel symbols are all given in (13.69a) and (13.69b). Now the freely falling particle satisfies the equations of motion ##\nabla_u u = 0## as usual. Taking again ##j = 1,2,3## in the above coordinates, we simply calculate ##\frac{\mathrm{d} ^2 x^{j}}{\mathrm{d} \tau^2} = -\Gamma ^{j}_{\mu\nu}\frac{\mathrm{d} x^{\mu}}{\mathrm{d} \tau}\frac{\mathrm{d} x^{\nu}}{\mathrm{d} \tau}\\ = -\Gamma ^{j}_{00}(\frac{\mathrm{d} x^0}{\mathrm{d} \tau})^2 - 2\Gamma ^{j}_{k0}\frac{\mathrm{d} x^{k}}{\mathrm{d} \tau}\frac{\mathrm{d} x^0}{\mathrm{d} \tau} \\= -a^j(\frac{\mathrm{d} x^0}{\mathrm{d} \tau})^2 + 2\omega^{i}\epsilon_{0ijk}\frac{\mathrm{d} x^{k}}{\mathrm{d} x^0}(\frac{\mathrm{d} x^0}{\mathrm{d} \tau})^2\\ = (-a^j - 2(\omega \times v)^j)(\frac{\mathrm{d} x^0}{\mathrm{d} \tau})^2##

because all other Christoffel symbols vanish as per (13.69a) and (13.69b). I don't get where that third term ##2(a\cdot v)v## is coming from. Could someone point it out? Thanks in advance!

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# Proper Reference Frame -Accelerated observer

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