Sachs and Wu's General Relativity for Mathematicians

[Avogadro Number]

I am trying to study "religiously" the book by Sachs and Wu, but I am finding the Exercises very much of a challenge. Does anyone know if there exists a source for solutions one can consult when stuck?

 

[Ibix]

We don't really "do" solution manuals here, but if you post the question and your working (you may need to read the LaTeX guide linked below the reply box if you don't know LaTeX) we'll be happy to help. That's one reason we're here. Technically, you should probably post in the Advanced Physics Homework Help forum, but mentors seem to be a bit more relaxed about graduate level exercises in the technical forums and they probably won't disintegrate you if you put it in the wrong place.

 

[Avogadro Number]

Apologies about posting this in the wrong channel. Thanks for the tip.

 

[MathematicalPhysicist]

I tried several years ago to read from it.
Got stuck on one question, and didn't proceed from there.
https://physics.stackexchange.com/questions/61298/ex-0-2-1-in-sachs-and-wus-textbook

I see it's from an old computer my brother always promised to fix it (and didn't).

 

[Avogadro Number]

I have made an attempt at this exercise: Is the following alright?:

If we take $\mathcal{E}=\mathbb{R}$, and $\gamma:\mathcal{E}\rightarrow \mathbb{R}^2$ defined by $\gamma u= (x(au),au),$ for $u\in \mathcal{E}=\mathbb{R}$,
then $\gamma_*u=a\frac{dx}{du}(au)\partial_{u^1}+a\partial_{u^2}$. This gives:
(a) $du^2(\gamma_* u)=(a\frac{dx}{du}(au)\partial_{u^1}+a\partial_{u^2})(u^2)=0+a\cdot 1=a,$
(b) $\gamma^1(u)=x(au)=x(\gamma^2 u)=(x\circ \gamma^2)$ for all real $u$, and
(c) $g(\gamma_* u,\gamma_* u)=(a\frac{dx}{du}(au))^2-(a)^2=a^2[(\frac{dx}{du}(au))^2-1^2]=a^2\cdot 0=0.$
$\phantom{(c)}$Here we have used $v=|\frac{dx}{du}|=1$ everywhere.

I haven't yet tried showing uniqueness of $\mathcal{E}$ and $\gamma$.

 

[Avogadro Number]

I tried several years ago to read from it.
Got stuck on one question, and didn't proceed from there.
https://physics.stackexchange.com/questions/61298/ex-0-2-1-in-sachs-and-wus-textbook

I see it's from an old computer my brother always promised to fix it (and didn't).

I have made an attempt at this exercise: Is the following alright?:

If we take $\mathcal{E}=\mathbb{R}$, and $\gamma:\mathcal{E}\rightarrow \mathbb{R}^2$ defined by $\gamma u=(x(au),au)$, for $u\in \mathcal{E}=\mathbb{R}$, then $\gamma_∗ u=a\frac{dx}{du}(au)\partial_{u^1}+a\partial_{u^2}$. This gives:

(a) $du^2(\gamma_∗u)=(a\frac{dx}{du}(au)\partial_{u^1}+a\partial_{u^2})(u^2)=0+a⋅1=a$,

(b) $\gamma^1(u)=x(au)=x(\gamma^2 u)=(x\circ \gamma^2)(u)$ for all real $u$, and

(c) $g(\gamma_∗u,\gamma_∗u)=(a\frac{dx}{du}(au))^2−(a)^2=a^2[(\frac{dx}{du}(au))^2−1^2]=a^2⋅0=0$. Here we have used $v=|\frac{dx}{du}|=1$ everywhere.

I haven't yet tried showing uniqueness of $\mathcal{E}$ and $\gamma$.