Tensor Variation with Respect to Metric in First Order Formalism

In summary, the conversation is discussing a calculation and whether it is being done correctly. The equations being used involve tensors and a tensor expansion. It is mentioned that the calculation may be correct if using a 1st order formalism with a flat metric expansion, but may not be correct otherwise.
  • #1
Chris Harrison
3
0

Homework Statement


I'm just wondering if I'm doing this calculation correct?
eta and f are both tensors

Homework Equations

The Attempt at a Solution


[tex]\frac{\delta \left ( \gamma_{3}f{_{\lambda}}^{k}f{_{k}}^{\sigma}f{_{\sigma}}^{\lambda} \right )}{\delta f^{\mu\nu}}=\frac{\delta\left (\gamma_{3} f^{\epsilon k}\eta_{\lambda\epsilon}f^{\rho\sigma}\eta_{k\rho}f^{\omega\lambda}\eta_{\sigma\omega} \right ) }{\delta f^{\mu\nu}}\\
=\gamma_{3}\left ( \delta_{\mu}^{\epsilon}\delta_{\nu}^{k}f^{\rho\sigma}f^{\omega\lambda}+\delta_{\mu}^{\rho}\delta_{\nu}^{\sigma}f^{\epsilon k}f^{\omega\lambda}+\delta_{\mu}^{\omega}\delta_{\nu}^{\lambda}f^{\epsilon k}f^{\rho\sigma} \right )\times\left ( \eta_{\lambda\epsilon}\eta_{k\rho}\eta_{\sigma\omega} \right )\\
=\gamma_{3}\left ( f{_{\nu}}^{\sigma}f{_{\sigma}}^{\lambda}\eta_{\lambda\mu}+f{_{\nu}}^{\lambda}f{_{\lambda}}^{k}\eta_{k\mu}+f{_{\nu}}^{k}f{_{k}}^{\sigma}\eta_{\sigma\mu} \right )\\
=3\gamma_{3} f{_{\nu}}^{\sigma}f{_{\sigma}}^{\lambda}\eta_{\lambda\mu}[/tex]
 
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  • #2
It looks ok if this is a 1st order formalism where the metric is being expanded around the flat metric: ##g_{\mu\nu}=\eta_{\mu\nu} + f_{\mu\nu}##. If it is something else, it may or may not be correct.
 

1. What is a tensor?

A tensor is a mathematical object that represents the linear relation between different coordinate systems in a multi-dimensional space. It is a generalization of vectors and matrices and can have multiple components in different dimensions.

2. What is meant by "variation wrt metric"?

Variation with respect to metric refers to the change in a tensor when the underlying metric, which defines the distance between points in a space, is varied. This variation is used to study the properties of the tensor in that particular space.

3. How does the metric affect a tensor?

The metric affects a tensor by changing its components or values in different coordinate systems. This is because the metric defines the distance between points, and the tensor represents the relation between these points in the space.

4. What are the applications of tensor variation wrt metric?

Tensor variation wrt metric is used extensively in the field of general relativity, where the metric tensor represents the gravitational field. It is also used in differential geometry and other areas of mathematics to study the properties of tensors in different spaces.

5. Is tensor variation wrt metric the same as tensor differentiation?

No, tensor variation wrt metric is not the same as tensor differentiation. While tensor differentiation involves finding the derivative of a tensor with respect to its components, tensor variation wrt metric involves studying the change in a tensor when the metric is varied. However, both concepts are closely related and are used in different contexts in mathematics and physics.

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