Proper Reference Frame -Accelerated observer

[WannabeNewton]

"Proper Reference Frame"-Accelerated observer

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!

 

[Bill_K]

I believe the point is that the desired equation contains a d²/dx0², whereas the geodesic equation contains d²/ds², so you have to use the metric (13.71) to convert one to the other, and this produces the additional term. For example d/dx0 on the a·x in the metric will give you an a·v.

 

[WannabeNewton]

Hi Bill, thanks for the reply! I don't know how but I totally forgot that they wanted $\frac{\mathrm{d} ^{2}x^{j}}{\mathrm{d} x^{0^{2}}}$ and not $\frac{\mathrm{d} ^{2}x^{j}}{\mathrm{d} \tau^2}$ where, as before, $\tau$ is the proper time along the worldline of the freely falling particle. That certainly helped but I'm still a bit confused.

For example, we have $\frac{\mathrm{d} ^{2}x^{j}}{\mathrm{d} x^{0^{2}}}e_j = \frac{\mathrm{d} \tau}{\mathrm{d} x^{0}}\frac{\mathrm{d} }{\mathrm{d} \tau}(\frac{\mathrm{d} \tau}{\mathrm{d} x^{0}}\frac{\mathrm{d} x^{j}}{\mathrm{d} \tau})e_j\\ = (\frac{\mathrm{d} \tau}{\mathrm{d} x^0})^{2}\frac{\mathrm{d} ^{2}x^{j}}{\mathrm{d} \tau^2}e_j + \frac{\mathrm{d} \tau}{\mathrm{d} x^0}\frac{\mathrm{d} x^{j}}{\mathrm{d} \tau}e_j(\frac{\mathrm{d} }{\mathrm{d} \tau}\frac{\mathrm{d} \tau}{\mathrm{d} x^0})\\ = -a - 2\omega \times v + v(\frac{\mathrm{d} }{\mathrm{d} \tau}\frac{\mathrm{d} \tau}{\mathrm{d} x^0})$

I just don't get how $\frac{\mathrm{d} }{\mathrm{d} \tau}\frac{\mathrm{d} \tau}{\mathrm{d} x^0}$ equals $2a_i v^i$ using (13.71) or otherwise. The freely falling particle isn't at rest in the "proper reference frame" of the accelerating observer so I don't get how the equality comes about.

 

[Bill_K]

I'd say to start from the d/ds end, and use the fact that d/ds is a directional derivative along the world line:

d/ds = dt/ds ∂/∂t + dx/ds · ∂/∂x

When the dust clears, use the fact that this is evaluated at the origin, and set x = 0. So

d²/ds² = (dt/ds ∂/∂t + dx/ds · ∂/∂x)(dt/ds ∂/∂t + dx/ds · ∂/∂x)
= (dt/ds)² ∂²/∂t² + (∂/∂t)(dt/ds) ∂/∂t + dx/ds · ∂/∂x (dt/ds) ∂x/∂t + (dx/ds)² ∂²/∂x²

Then (∂/∂t)(dt/ds) = a·v so the second term is a·v v, and dx/ds · ∂/∂x (dt/ds) = v·a so the third term is a·v v also.

EDIT: Good illustration of why I don't like MTW! They seem to make up their notation ad hoc as they go along.

 

[WannabeNewton]

Ah ok. That makes perfect sense, thank you again Bill! I appreciate the help

And yeah, I'm relatively new to MTW and their notation is making my head spin like crazy haha; I've gotten too used to the notation in Wald it would seem.