# Metric Variations

*xTras* automatically defines 'proper' variations with respect to the metric via

VarD and

VarL whenever you define a metric.

VarD[g[-a,-b],cd][L] | returns while integrating by parts with respect to the covariant derivative cd. |

VarL[g[-a,-b],cd][L] | returns while integrating by parts with respect to the covariant derivative cd. |

Computing variations w.r.t. the metric.

Let's begin with defining a manifold.

DefMetric has a new option,

DefMetricPerturbation. It defaults to True, and so by default a metric perturbation is defined whenever you define a metric.

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In xTras, defining a metric perturbation automatically defines proper metric variations. All in all, defining a metric automatically defines proper metric variations.

We can now perform variations with respect to the metric:

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Using

VarL automatically takes care of factors of

:

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Varying with respect to the inverse metric gives an overall minus sign:

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More complicated expressions can also be varied easily:

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This can be simplified further with the help of

FullSimplification:

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