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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