Mathematics Asked by MSDG on November 2, 2020

I am looking at the derivation of the Einstein field equations as the Euler-Lagrange equations of the Hilbert functional.

To do this one starts with a variation

$$ g(t) = g+th $$

of the metric, where $h$ is a symmetric 2-covariant tensor.

For small $t$, $g(t)$ will be invertible (if we interpret it as a matrix), so it makes sense to consider the components $g(t)^{ij}$ of the inverse. We have

$$ 0 = frac{d}{dt}Big|_0 (g(t)_{ij} , g(t)^{jk}) = h_{ij} g^{jk} + g_{ij}h^{jk} quadRightarrow quad h^{lk} = – g^{il} g^{jk} h_{ij}. $$

**My question:** What exactly do the coefficients $h^{lk}$ represent?

I always thought that if one has a, say, 1-covariant tensor $A = A_i dx^i$, then $A^i$ denote the components of the 1-contravariant tensor $A^#$ (cf. musical isomorphisms), given by $A^i = g^{ij}A_j$. But in the formula for $h^{lk}$ above we also have a minus sign in front, so as far as I can see the coefficients $h^{lk}$ are not obtained by raising the indices of $h_{lk}$. That being said, the metric is not really fixed in this case so the musical isomorphisms aren’t either, which may be the source of confusion for me.

Your mistake is that $$h^{jk} neq frac{d}{dt}bigg|_{t=0} g(t)^{jk}.$$In fact, $h^{jk}$ *is* obtained from raising both indices of $h_{jk}$. What you have shown is that $$frac{d}{dt}bigg|_{t=0} g(t)^{ell k} = -g^{iell}g^{jk}h_{ij} = - h^{ell k}.$$The philosophy behind it is: you know that the initial velocity of the curve $tmapsto g(t)$ is $h$, and you want to compute the initial velocity of other geometric objects associated to $g(t)$ in terms of $h$. The initial velocity of the curve of inverse coefficients is $-h$ (up to musical isomorphisms).

Dealing with coefficients, you're doing computations on matrix Lie groups. There, the formula for the derivative of the inversion $iota(A)=A^{-1}$ is given by $diota_A(H) = - A^{-1}HA^{-1}$, so everything is fine (compare with $(1/x)'=-1/x^2$ and make it non-commutative; each $A^{-1}$ corresponds to raising one index).

Correct answer by Ivo Terek on November 2, 2020

1 Asked on January 29, 2021 by mk7

curves differential geometry riemannian geometry submanifold surfaces

3 Asked on January 29, 2021 by ramez-hindi

discrete mathematics elementary number theory factorial inequality

1 Asked on January 29, 2021 by siddharth-bhat

2 Asked on January 29, 2021

1 Asked on January 28, 2021

1 Asked on January 28, 2021 by inoc

integration lebesgue integral real analysis trigonometric integrals

1 Asked on January 28, 2021 by chris-steinbeck-bell

0 Asked on January 28, 2021 by ziqi-fan

1 Asked on January 28, 2021 by fady

1 Asked on January 28, 2021 by t-i

convergence divergence examples counterexamples general topology sequences and series

2 Asked on January 28, 2021

2 Asked on January 28, 2021

3 Asked on January 27, 2021 by questmath

1 Asked on January 27, 2021 by user-492177

1 Asked on January 27, 2021 by user21

0 Asked on January 27, 2021 by rivers-mcforge

collatz conjecture convergence divergence elementary number theory prime numbers recursion

1 Asked on January 27, 2021 by almhz

2 Asked on January 27, 2021 by sicmath

Get help from others!

Recent Answers

- Joshua Engel on Why fry rice before boiling?
- haakon.io on Why fry rice before boiling?
- Jon Church on Why fry rice before boiling?
- Peter Machado on Why fry rice before boiling?
- Lex on Does Google Analytics track 404 page responses as valid page views?

Recent Questions

- Iv’e designed a space elevator using a series of lasers. do you know anybody i could submit the designs too that could manufacture the concept and put it to use
- Need help finding a book. Female OP protagonist, magic
- Why is the WWF pending games (“Your turn”) area replaced w/ a column of “Bonus & Reward”gift boxes?
- Does Google Analytics track 404 page responses as valid page views?
- Why fry rice before boiling?

© 2022 AnswerBun.com. All rights reserved. Sites we Love: PCI Database, MenuIva, UKBizDB, Menu Kuliner, Sharing RPP