How to Solve MTW Gravitation Problems 6.5 and 6.8?

[IAmAZucchini]

Hello! I'm stuck on these two problems, any help is appreciated!

6.5

Homework Statement

Use radar as a distance-measuring device. The radar set measures its proper time \tau between the instant at which it emits a pulse and the later instant when it receives the reflected pulse. It then performs the simple computation L_{0}=\tau/2 and supplies as output the "distance" L_{0}. How accurate is the output reading of the radar set for measuring the actual distance L to the object, when used by a uniformly accelerated observer? (L is defined as the distance in the momentary rest frame of the observer at the instant the pulse is reflected, which is at the observer's proper time halfway between emitting and receiving the pulse.) Give a correct forumula relating L_{0} \overline{=} \tau/2 to the actual distance L. Show that the reading L_{0} becomes infinite as L approaches g^{-1}, where g is the observer's acceleration, as measured by an accelerometer he carries.

Homework Equations

gt=sinh(g \tau)
gx=cosh(g \tau)

The Attempt at a Solution

We need to find \tau in terms of L.
Let x_{1L} be the distance the light has traveled at the moment of reflection, and x_{1R} be the distance the spaceship has traveled at the moment of reflection. Then L=x_{1R} - x_{1L}. Setting \tau_{L} to be the proper time at which reflection occurs, and plugging in the "relevant equations", we get
\tau_{L} = - g^{-1} arcsinh (g x_{1L} )
so
x_{1R} = g^{-1} cosh (arcsinh(g x_{1L}) ) = g^{-1} \sqrt{g^{2} x_{1L}^{2} + 1 }
so L=x_{1R} - x_{1L} solves to
x_{1L}= (1-L^{2}g^{2}) / (2g^{2}L)

And then, at reabsorption of the radar beam, the light and the rocket are once again at the same coordinates, so
ct - 2x_{1L} = g^{-1}sinh(g \tau) - (1-L^{2}g^{2}) / (g^{2}L) = g^{-1} cosh(g \tau)
From this follows that
sinh(g \tau) - cosh(g \tau) = - (1-L^{2}g^{2}) / (g L)
=>
e^{-g\tau} = (1-L^{2}g^{2}) / (g L)
i.e.
L_{0} = \tau / 2 = g^{-1} ln(\sqrt{ gL / (1-L^{2}g^{2})})

Note that this satisfies the condition that as L->1/g, L_{0} -> infinity.

However, in the other limiting condition, as L->0, this formula also gives that L_{0} -> -infinity.
Clearly this is wrong...

Intuitively, I feel that the answer should take the form
L_{0}=L/\sqrt{1-g^{2}L^{2}}, because this is basically length contraction. I don't know where I went wrong though, and I can't find a mistake in my logic or my math.

Thanks for any suggestions...

I'll post my question about 6.8 in the next post (same thread)

 

[IAmAZucchini]

And now, problem 6.8, which is a beast that I am having a lot more trouble wrapping my head around.

the problem:
Fremi-Walker plus spatial rotation. Observer transports his orthonormal tetrad in the following fashion:

\Omega^{\mu\nu} = a^{\mu}u^{\nu} - a^{\nu}u^{\nu} + u_{\alpha}\omega_{\beta}\epsilon^{\alpha\beta\mu\nu}

where \epsilon is Levi-Civita and \omega is a vector orthogonal to 4-velocity u.

a) let e_{0'}=u. Show this is permitted by his transport law and that de_{\alpha'}/d\tau = - \Omega \cdot e_{\alpha'}. (easy)

b) show that the rotational part of the transport law produces a rotation in the plane perpendicular to u and \omega (easy)

here's where I start having trouble.

c) suppose the accelerated observer Fermi-Walker transports a second orthonormal tetrad e_{\alpha''}. Show that the space vectors of this first tetrad rotate relative to those of his second tetrad with angular velocity vector equal to \omega. i.e., show that at the moment when the tetrads coincide, show that (in three-dimensional notation, referring to the 3-space orthogonal to the observer's world line):
d(e_{j'} - e_{j''})/d\tau = \omega \times e_{j'}

attempt (?) at a solution

I'm mostly having trouble wrapping my head around what this means. In class, we showed that d(e_{j})/d\tau = \omega \times e_{j}. Does this mean that I simply have to show that de_{j''}/d\tau = 0? If the second tetrad is not rotating, then wouldn't the first tetrad obviously rotate relative to it with angular velocity \omega in the local lorentz frame? Do I need to actually transform coordinates?

I think I can figure out parts d, e, and f on my own, but I have yet to try to work them - they look straightforward, though. I may post questions about those and/or 7.1 later tonight or tomorrow ... need to study for a Math Methods exam as well...

Again, thanks for any help you can offer. I'm left a little confused by the question, although all the preceding material in the chapter more or less makes sense. Not asking for a solution, but a nudge in the right train of thought direction would certainly help!