Exploring "The Mathematical Theory of Black Holes" by S. Chandrasekhar

[kparchevsky]

In page 67 of book "The mathematical theory of black holes" by S. Chandrasekhar in chapter 2 "Space-Time of sufficient generality" there is a theorem that metric of a 2-dimensional space
$$ds^2 = g_{11} (dx^1)^2 + 2g_{12} dx^1 dx^2 + g_{22} (dx^2)^2$$
can be brought to a diagonal form.

I would do this in the following way: introduce new contravariant coordinates $x'$ (how $x$ depend on $x'$) $x^1 = p(x'^1, x'^2), x^2= q(x'^1, x'^2)$, differentiate them, plug $dx^1$ and $dx^2$ into the metric above, and equate factor at $dx'^1 dx'^2$ to zero.

That is not how it is done in the book. First they introduce new contravariant coordinates $x'$ such that the inverse functions are defined (how $x'$ depend on $x$) Eq.(6)
$$x'^1=\phi(x^1,x^2),\qquad x'^2=\psi(x^1,x^2)$$
then they try to reduce to the diagonal form the contravariant form of the metric ($g^{\mu\nu}$ with up indexes, $dx_\mu$ with low indexes) Eq.(7),
$$ds^2=g^{11}(dx_1)^2+2g^{12}dx_1dx_2+g^{22}(dx_2)^2$$
though coordinate transformations are defined for contravariant coordinates (up indexes).

I cannot follow the logic of the derivation.

Could you help me to understand how it is derived in the book?

Thank you.

[Mentor Note -- New user has been PM'd about posting math using LaTeX and has been pointed to the "LaTeX Guide" link, especially for threads with the "A" prefix]

 

[strangerep]

[...] they introduce new contravariant coordinates $x'$ such that the inverse functions are defined (how $x'$ depend on $x$) Eq.(6)$$x'^1=\phi(x^1,x^2),\qquad x'^2=\psi(x^1,x^2)$$ then they try to reduce to the diagonal form the contravariant form of the metric ($g^{\mu\nu}$ with up indexes, $dx_\mu$ with low indexes) Eq.(7),

$$ ds^2=g^{11}(dx_1)^2+2g^{12}dx_1dx_2+g^{22}(dx_2)^2$$ though coordinate transformations are defined for contravariant coordinates (up indexes). I cannot follow the logic of the derivation.

You didn't say at which equation in the book you get stuck. Do you understand eqs (8) and (9)? They're basically just specific cases of the general transformation formula$$g'^{\mu\nu} ~=~ \frac{\partial x'^\mu}{\partial x^\alpha} \; \frac{\partial x'^\nu}{\partial x^\beta} \; g^{\alpha\beta} ~,$$ although Chandrasekhar uses a notation convention of putting primes on the indices rather than the main symbol as I've done above.

 

[kparchevsky]

You didn't say at which equation in the book you get stuck. Do you understand eqs (8) and (9)? They're basically just specific cases of the general transformation formula$$g'^{\mu\nu} ~=~ \frac{\partial x'^\mu}{\partial x^\alpha} \; \frac{\partial x'^\nu}{\partial x^\beta} \; g^{\alpha\beta} ~,$$ although Chandrasekhar uses a notation convention of putting primes on the indices rather than the main symbol as I've done above.

Thank you. Trying to prove Eq.(8) I took differential from both sides of Eq.(6), solved it for $dx^i$, converted it to $dx_i$, plugged into Eq.(7) and zeroed term at $dx'^i dx'^j$, but I just had to use the definition of a tensor! The rest of derivation in the book is clear.

 

[haushofer]

Can't you just see this by counting? In 2 dimensions the metric has 1/2×2×3=3 independent components. With 2 general coordinate transformations (gct's) you have enough freedom to put one component to zero. Explicitly you can write down the transformed compononent for g_12, put it to zero, and see what constraints you get for the gct. This partially gauge fixes the gct's.

I don't get why you use "covariant coordinates" in the first place. For a similar calculation, see any book on string theory how to gauge fix the worldsheet metric and why this implicates that string theory is a CFT.

 

[kparchevsky]

Can't you just see this by counting? In 2 dimensions the metric has 1/2×2×3=3 independent components. With 2 general coordinate transformations (gct's) you have enough freedom to put one component to zero. Explicitly you can write down the transformed compononent for g_12, put it to zero, and see what constraints you get for the gct. This partially gauge fixes the gct's.

I don't get why you use "covariant coordinates" in the first place. For a similar calculation, see any book on string theory how to gauge fix the worldsheet metric and why this implicates that string theory is a CFT.

>I don't get why you use "covariant coordinates" in the first place
The goal was to prove the specific formula in the specific book, and this formula was written in covariant coordinates.