Talk:Metric tensor (general relativity)

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edit·history·watch·refresh Stock post message.svg To-do list for Metric tensor (general relativity):
  • Discussion of metric as gravitational potential (+ link to weak-field approximation).
  • Raising and lowering indices.
  • Cut down the 'volume' section (and shift that derivation to another article - or just get rid of it).

Cut material[edit]

I cut out the material on the volume form which properly belongs at volume form. Here it is for reference:

Let [g] be the matrix of elements g_{\mu\nu}. Matrix [g] is symmetric, so due to a corollary of the spectral theorem, there exists an orthogonal transformation matrix Λ which diagonalizes [g], e.g.

 D = \Lambda^\top [g] \Lambda

where D is a diagonal matrix whose diagonal elements are eigenvalues of [g]:  D_{\alpha\alpha} = \lambda_\alpha . (Note that Λ can be chosen so that the eigenvalues are in numerical order, D00 being the smallest.) Then there is a diagonal matrix V which "unitizes" D, i.e. which applies the mapping  \lambda_\alpha \mapsto \mbox{sgn} (\lambda_\alpha) to the diagonal elements of D. Such matrix V has diagonal elements

 V_{\alpha\alpha} = \left\{ \begin{matrix}  {1 \over \sqrt{| \lambda_\alpha |}} & \quad \mbox{if} \, \lambda_\alpha \ne 0 \\
 0 & \quad \mbox{if} \, \lambda_\alpha = 0 \end{matrix} \right.

Then

 [\eta] = V^\top \Lambda^\top [g] \Lambda V

and for a given manifold, the trace of [η] will be the same for all points and is referred to as the signature of the metric. (A signature of +2 is synonymous with a signature of (− + + +). ) This matrix [η] has the components of the Minkowski metric, which means that the manifold is, at each one of its points, locally smooth.

The matrix (V \Lambda)^\top is a Jacobian (a multivariate differential, or push forward) which transforms [η] to [g],

 [g] = V \Lambda [\eta] \Lambda^\top V^\top

and taking determinants

 g := \mbox{det}([g]) = \mbox{det}\,(V \Lambda) \,\mbox{det}([\eta]) \,\mbox{det}(\Lambda^\top V^\top)
 = \mbox{det}^2 (V \Lambda) \, \mbox{det}([\eta]), \
 g = -\mbox{det}^2 (V \Lambda), \
 \mbox{det}(V \Lambda) = \sqrt{-g},

but due to a property of diffeomorphisms, a volume element  dx^0 dx^1 dx^2 dx^3 whose factors are components of an orthonormal basis (locally), when transformed to components  dx^{\bar\mu} , has the determinant of the Jacobian matrix J as conversion factor:

 G = dx^0 dx^1 dx^2 dx^3 = \mbox{det}(J) \, dx^{\bar 0} dx^{\bar 1} dx^{\bar 2} dx^{\bar 3}.

See also volume form.

-- Fropuff 18:02, 22 February 2006 (UTC)

Amusing Veblen/Einstein anecdote[edit]

See Sign convention ---CH 01:54, 25 May 2006 (UTC)


Question on Metric Equation[edit]

g_{\bar \mu \bar \nu} = \frac{\partial x^\rho}{\partial x^{\bar \mu}}\frac{\partial x^\sigma}{\partial x^{\bar \nu}} g_{\rho\sigma} = \Lambda^\rho {}_{\bar \mu} \, \Lambda^\sigma {}_{\bar \nu} \, g_{\rho \sigma} .

should perhaps be:

g_{\bar \mu \bar \nu} = \frac{\partial x^\rho}{\partial x^{\bar \mu}}\frac{\partial x^\sigma}{\partial x^{\bar \nu}} g_{\rho\sigma}\overset ? =
\Lambda_{\bar \mu} { }^ \rho \, \Lambda_{\bar \nu}{ }^\sigma   \, g_{\rho \sigma}.

as per discussion

http://www.physicsforums.com/showthread.php?p=3398120#post3398120 (especially post #36) JDoolin (talk) 14:51, 11 July 2011 (UTC)

I withdraw the question based on post #39 in the same thread. Thanks. (JDoolin (talk) 15:11, 12 July 2011 (UTC))

Conflicting definitions[edit]

First, in the section "Definition", g is defined as a 4 x 4 metric tensor, but in the section "Local coordinates and matrix representations" g is defined as a scalar valued bilinear form (?)

g = g_{\mu\nu}dx^\mu dx^\nu.\,

There is also, parenthetically, a third definition of g as a tensor field.

Finally, there is a definition of ds² as the line element and as the "metric", but the line element is ds, not ds².

I suggest separate, clear, correct and unambiguous definitions of the metric tensor, the metric, the tensor field, and the line element.

Then there should be a statement regarding the informal conflation of these by physicists who know what they are doing despite appearances to the contrary.

Perhaps there should also be an explanation of the relation of the element of proper time to the line element, i.e., dtau = ds/c.

200.83.113.147 (talk) 02:37, 19 February 2015 (UTC)

There is only one definition. These are merely showing the relationship between different notational schemes applied to the same notion of a metric. JRSpriggs (talk) 16:23, 20 February 2015 (UTC)