Metric tensor in General Relativity or otherwise

Question

1. What is the metric tensor?

2. How can this be a covariant and contravariant tensor, or a mixed tensor, by raising and lowering indices?

3. How it relates to distance function (metric) and angles?

4. How does it transport basis vectors from one coordinate system to another?

5. How is it different from the field tensor, Riemann curvature tensor and Ricci curvature tensor?

This post imported from StackExchange Physics at 2014-03-30 15:46 (UCT), posted by SE-user narayanadash

Comments

Have you looked at a book? Seriously there are many comprehensive books and websites on this subject. Googling for "mathematical foundations of general relativity" and "differential geometry" will go a long way.

This post imported from StackExchange Physics at 2014-03-30 15:46 (UCT), posted by SE-user Michael Brown

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I also suggest Ch 14, called "Calculus on Manifolds" of Roger Penrose's "The Road to Reality" as a pithy summary of the foundations of Riemannian geometry. He does an awful lot with very little, gives a good explanation of the notions of curvature before introducing the metric. Exercises in the book are well worth doing, and there is a website where readers have put their solutions. One I also like for its pithiness, thoroughness and clarity is Wulf Rossmann's "Differential Geometry" - and its free!

This post imported from StackExchange Physics at 2014-03-30 15:46 (UCT), posted by SE-user WetSavannaAnimal aka Rod Vance

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You might want to break this up into several questions.

This post imported from StackExchange Physics at 2014-03-30 15:46 (UCT), posted by SE-user Dan

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"Road to Reality" is great if you want a heavy book on just about everything foundational in modern physics. If you would like a modern, breezy, "physics first" approach devoted to general relativity with plenty of interesting historical footnotes I'd recommend Zee's book "Einstein Gravity in a Nutshell". Then there is the classic tome by Misner, Thorne and Wheeler. The advanced theory and experimental sections are a little out of date, but the book is truly great if you want geometric intuition with pictures... oh so many pictures...

This post imported from StackExchange Physics at 2014-03-30 15:46 (UCT), posted by SE-user Michael Brown

Answer 1

1. It's a tensor that describes the geometry of spacetime (or any manifold, for that matter.). It's components give the dot products of the unit vectors in spacetime (or "vierbeins", or generally, "vielbeins".). I.e.

$$g_{\mu\nu}=\hat e_\mu\cdot \hat e_\nu$$

2. Like any other tensor. I.e. $g_{\mu\nu}$ is the covariant form, $g^{\mu\nu}$ is the contravariant form, and $g_\mu^\nu=g_\nu^\mu$ is the mixed form.

3. As I said earlier, it gives the dot product between the unit vectors. I.e.

$$g_{\mu\nu}=\hat e_\mu\cdot\hat e_\nu=\|\hat e_\mu\|\cos\theta\|\hat e_\nu\|$$

Note: in flat spacetime, this term would simplify a bit, obviously, to just $\cos\theta$.

4. As I said, the metric tensor can be expressed in terms of the vielbeins, so that answers the question. But speaking of coordiante systems, Christoffel symbols can map between say, a minkowski coordinate system with metric tensor $\eta_{\mu\nu}$ and a curved coordinate system $g_{\mu\nu}$.

5. Just' like how $x$ is different from $5x^{2+\cos\sin\ln x}+i\cos x+2\sin\left(x^2\right)$. Riemann Curvature, Ricci Curvature, etc. can be written completely in terms of the metric tensor. E.g.

$$\Gamma^i_{k\ell} = \frac{1}{2}g^{im} \left(\frac{\partial g_{mk}}{\partial x^\ell} + \frac{\partial g_{m\ell}}{\partial x^k} - \frac{\partial g_{k\ell}}{\partial x^m} \right) = {1 \over 2} g^{im} (\partial_\ell g_{mk} + \partial_k g_{m\ell} - \partial_m g_{k\ell}) = {1 \over 2} g^{im} (g_{mk,\ell} + g_{m\ell,k} - g_{k\ell,m}), $$ (christoffel symbols)

$$R^\rho{}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho{}_{\nu\sigma} - \partial_\nu\Gamma^\rho{}_{\mu\sigma} + \Gamma^\rho{}_{\mu\lambda}\Gamma^\lambda{}_{\nu\sigma} - \Gamma^\rho{}_{\nu\lambda}\Gamma^\lambda{}_{\mu\sigma}$$ (riemann curvature tensor)

$$R_{\mu\nu}=g^{\rho\sigma}R_{\mu\nu\rho\sigma} $$ (ricci scalar)

$$R=g^{\mu\nu}R_{\mu\nu}$$

For a more detailed discussion, you may want to get a good GR text - book, like Ludvigsen General Relativity: A geometric approach.

Comments

Small quibble: it's vier / vielbein

This post imported from StackExchange Physics at 2014-03-30 15:46 (UCT), posted by SE-user WetSavannaAnimal aka Rod Vance

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@WetSavannaAnimalakaRodVance: Done .

This post imported from StackExchange Physics at 2014-03-30 15:46 (UCT), posted by SE-user Dimensio1n0

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Sorry - the word is a favourite of mine - quite cute really because it's also the word a German zoologist would use for "tetrapod", so I imagine little animals - probably ones found in a small child's book - scurrying all over the manifold. :)

This post imported from StackExchange Physics at 2014-03-30 15:46 (UCT), posted by SE-user WetSavannaAnimal aka Rod Vance