# Variation of Metric Tensor Under Coord Transf | 65 chars

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• QipshaqUli
In summary, the variation of the metric is -\frac{\partial}{\partial x^{\alpha}}\varepsilon^{\alpha}+g^{\mu\beta}\frac{\partial \varepsilon^{\nu}}{\partial x^{\beta}}+g^{\alpha\nu}\frac{\partial \varepsilon^{\mu}}{\partial x^{\alpha}}.
QipshaqUli
TL;DR Summary
Under the coordinate transformation $\bar x=x+\varepsilon$, the variation of the metric $g^{\mu\nu}$ is:
$$\delta g^{\mu\nu}(x)=\bar g^{\mu\nu}(x)-g^{\mu\nu}(x)=-\frac{\partial{ g^{\mu\nu}}}{\partial x^{\alpha}}\varepsilon^{\alpha}+ g^{\mu\beta}\frac{\partial \varepsilon^{\nu}}{\partial x^{\beta}}+g^{\alpha\nu}\frac{\partial \varepsilon^{\mu}}{\partial x^{\alpha}}$$
the right hand side is equal to $$- {g^{\mu\nu}}_{,\alpha}\varepsilon^{\alpha}+ {\varepsilon^{\mu,\nu}}+{\varepsilon^{\nu,\mu}} Under the coordinate transformation \bar x=x+\varepsilon, the variation of the metric g^{\mu\nu} is:$$
\delta g^{\mu\nu}(x)=\bar g^{\mu\nu}(x)-g^{\mu\nu}(x)=-\frac{\partial{ g^{\mu\nu}}}{\partial x^{\alpha}}\varepsilon^{\alpha}+ g^{\mu\beta}\frac{\partial \varepsilon^{\nu}}{\partial x^{\beta}}+g^{\alpha\nu}\frac{\partial \varepsilon^{\mu}}{\partial x^{\alpha}}
$$the right hand side is equal to$$- {g^{\mu\nu}}_{,\alpha}\varepsilon^{\alpha}+ {\varepsilon^{\mu,\nu}}+{\varepsilon^{\nu,\mu}}=\varepsilon^{\mu;\nu}+\varepsilon^{\nu;\mu}$$I have problem with the proof of the last equality.$$
\varepsilon^{\mu;\nu}+\varepsilon^{\nu;\mu}=g^{\alpha\nu}{\varepsilon^{\mu}}_{;\alpha}+g^{\alpha\mu}{\varepsilon^{\nu}}_{;\alpha}=

g^{\alpha\nu}({\varepsilon^{\mu}}_{,\alpha}+\Gamma_{\beta\alpha}^{\mu}\varepsilon^{\beta})+g^{\alpha\mu}({\varepsilon^{\nu}}_{,\alpha}+\Gamma_{\beta\alpha}^{\nu}\varepsilon^{\beta})=

\varepsilon^{\mu,\nu}+g^{\alpha\nu}\frac{1}{2}g^{\mu\gamma}(g_{\gamma\beta,\alpha}+g_{\gamma\alpha,\beta}-g_{\beta\alpha,\gamma})\varepsilon^{\beta}+
\varepsilon^{\nu,\mu}+g^{\alpha\mu}\frac{1}{2}g^{\nu\gamma}(g_{\gamma\beta,\alpha}+g_{\gamma\alpha,\beta}-g_{\beta\alpha,\gamma})\varepsilon^{\beta}=
$$Considering the summation over the repeated indeces each of the three items in both brackets gives the same quantity coupling with the respective indeces as: A(B+C-D)E, ABE=ACE=ADE, then A(B+C-D)E=ACE. I chose ACE$$
\varepsilon^{\mu,\nu}+\varepsilon^{\nu,\mu}+g^{\alpha\mu}g^{\nu\gamma}g_{\gamma\alpha,\beta}\varepsilon^{\beta}={g^{\mu\nu}}_{,\beta}\varepsilon^{\beta}+{\varepsilon^{\mu}}^{,\nu}+{\varepsilon^{\nu}}^{,\mu}
$$I have the first term with plus sign, opposite to the original one. What I did wrong? You do not need to use the expression for the Christoffel symbols. All that is needed is the metric compatibility of the connection$$
\nabla_\mu g^{\nu\rho} = \partial_\mu g^{\nu\rho} + \Gamma^\nu_{\mu\sigma} g^{\sigma\rho} + \Gamma^\rho_{\mu\sigma} g^{\nu\sigma} = 0.


QipshaqUli

## 1. What is the metric tensor?

The metric tensor is a mathematical object that describes the geometry of a space. It is used to measure distances and angles in a coordinate system.

## 2. How does the metric tensor change under coordinate transformations?

The metric tensor changes under coordinate transformations according to a specific rule, known as the transformation law. This law ensures that the metric tensor remains consistent regardless of the choice of coordinates.

## 3. Why is it important to consider the variation of the metric tensor under coordinate transformations?

Considering the variation of the metric tensor is important because it allows us to properly account for the effects of changing coordinate systems on the geometry of a space. This is crucial in many fields, such as general relativity and differential geometry.

## 4. Can the metric tensor be used to measure distances in curved spaces?

Yes, the metric tensor can be used to measure distances in both flat and curved spaces. In curved spaces, the metric tensor takes into account the curvature of the space and allows for accurate measurements of distances between points.

## 5. How does the metric tensor relate to the concept of a metric space?

The metric tensor is a mathematical tool used to define a metric space. In a metric space, the metric tensor provides a way to measure distances between points and define the geometry of the space. This is important in many areas of mathematics and physics.

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