How Does the Ricci Tensor Affect Tensor Equations in Wald's Problem?

[tommyj]

EDIT dw i figured it out, not sure how to remove it though!

 

[WannabeNewton]

Could you show me how you did part (b) of problem 10.2?

Some while back I did problem 10.2 and I thought I did part (b) of it right (https://www.physicsforums.com/showthread.php?t=687641#post4359286) but a few weeks later I found a mistake in my solution. Since then I forgot about the problem but you bringing it up has reminded me I have yet to still fix the mistake in my solution to part (b)!

 

[tommyj]

Sorry to get your hopes up but I made mistakes in both parts! How did you do part a) may i ask? I can't get terms to disappear.

Also, for part b) both constraints only seem to imply that $F^{ab}\nabla _an_b=0$ which don't really seem to help to solve for the time derivative of the initial conditions. I've spent ages looking and can't find anything on it!

 

[WannabeNewton]

Sorry to get your hopes up but I made mistakes in both parts! How did you do part a) may i ask? I can't get terms to disappear.

Let me start by showing that Gauss's law for electricity holds on the space-like Cauchy surface. Keep in mind that $n^{a}n_{a} = -1$, which implies that $n^{a}\nabla_{b}n_{a} = 0$; also keep in mind that $n_{[a}\nabla_{b}n_{c]} = 0$ since the unit normal field is hypersurface orthogonal to the space-like foliation $\Sigma_t$ (c.f. Theorem 8.3.14).

I will denote the derivative operator associated with the spatial metric $h_{ab}$ by $\tilde{\nabla}_{a}$.

We have $\tilde{\nabla}_{a}E^{a} = h_{a}{}{}^{b}h^{a}{}{}_{c}\nabla_{b}(F^{c}{}{}_{d}n^{d})\\ = (\delta^{bc} + n^{b}n^{c})(n^{d}\nabla_{b}F_{cd} + F_{cd}\nabla_{b}n^{d})\\ = n^{d}\nabla^{c}F_{cd} + F_{cd}\nabla^{c}n^{d} + n^{b}n^{c}n^{d}\nabla_{b}F_{cd} + n^{b}n^{c}F_{cd}\nabla_{b}n^{d}$.

Now $n^{c}n^{d}\nabla_{b}F_{cd} = n^{d}n^{c}\nabla_{b}F_{dc} = -n^{c}n^{d}\nabla_{b}F_{cd}\Rightarrow n^{c}n^{d}\nabla_{b}F_{cd} = 0$

and $n_{[a}\nabla_{b}n_{c]} = 0\Rightarrow n^{b}n^{c}F_{cd}\nabla_{b}n^{d} - n^{b}n^{d}F_{cd}\nabla_{b}n^{c}= 2n^{b}n^{c}F_{cd}\nabla_{b}n^{d}\\ = F_{cd}\nabla^{d}n^{c} - F_{cd}\nabla^{c}n^{d} = - 2F_{cd}\nabla^{c}n^{d}$

thus $\tilde{\nabla}_{a}E^{a}= n^{d}\nabla^{c}F_{cd} = -4\pi j_{d}n^{d} = 4\pi\rho$ by virtue of the inhomogeneous Maxwell equations.

Showing Gauss's law for magnetism holds on the spacelike Cauchy surface is very similar.

We have $\tilde{\nabla}_{a}B^{a} =-\frac{1}{2}\epsilon^{cdef} h_{a}{}{}^{b}h^{a}{}{}_{c}\nabla_{b}(F_{de}n_{f})\\ = -\frac{1}{2}\epsilon^{cdef} (n_{f}\nabla_{c}F_{de} + F_{de}\nabla_{c}n_{f} + n^{b}n_{c}n_{f}\nabla_{b}F_{de} + n^{b}n_{c}F_{de}\nabla_{b}n_{f})$.

Now $\epsilon^{cdef}n_{c}n_{f} = 0$ because the volume form is totally antisymmetric and just as before we have $n_{[a}\nabla_{b}n_{c]} = 0 \Rightarrow \epsilon^{cdef} n^{b}n_{c}F_{de}\nabla_{b}n_{f} - \epsilon^{cdef} n^{b}n_{f}F_{de}\nabla_{b}n_{c}= 2\epsilon^{cdef} n^{b}n_{c}F_{de}\nabla_{b}n_{f}\\ = \epsilon^{cdef} F_{de}\nabla_{f}n_{c} - \epsilon^{cdef}F_{de}\nabla_{c}n_{f} = -2 \epsilon^{cdef}F_{de}\nabla_{c}n_{f}$

so we are left with $\tilde{\nabla}_{a}B^{a} = -\frac{1}{2}\epsilon^{cdef} n_{f}\nabla_{c}F_{de}$. But $\epsilon^{cdef}\nabla_{c}F_{de} = -\epsilon^{cdef}\nabla_{d}F_{ce} = \epsilon^{cdef}\nabla_{e}F_{cd}$ hence $3\epsilon^{cdef}\nabla_{c}F_{de} = 3\epsilon^{cdef}\nabla_{[c}F_{de]} = 0$ by virtue of the homogeneous Maxwell equations thus we have the desired result $\tilde{\nabla}_{a}B^{a} = 0$.

Also, for part b) both constraints only seem to imply that $F^{ab}\nabla _an_b=0$ which don't really seem to help to solve for the time derivative of the initial conditions. I've spent ages looking and can't find anything on it!

I'm stuck on that calculation as well. I have to finish my particle physics HW but after that I'll take another jab at it.

 

[tommyj]

ah man I even noted the hypersurface orthogonal relation as I knew it would be useful but I forgot about it. You live and learn as they say. thanks a lot, I know how much effort it is to write tensor equations on here so I really appreciate it!

I cannot see how to do the second part. Its completely different from the example in the book, the Ricci Tensor term just messes everything up for the second constraint part