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.
Out[10]= | ![](Files/MetricVariations/O_1.gif) |
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:
Out[13]= | ![](Files/MetricVariations/O_2.gif) |
Using
VarL automatically takes care of factors of
![](Files/MetricVariations/28.gif)
:
Out[15]= | ![](Files/MetricVariations/O_3.gif) |
Varying with respect to the inverse metric gives an overall minus sign:
Out[16]= | ![](Files/MetricVariations/O_4.gif) |
Out[17]= | ![](Files/MetricVariations/O_5.gif) |
More complicated expressions can also be varied easily:
Out[18]= | ![](Files/MetricVariations/O_6.gif) |
Out[23]= | ![](Files/MetricVariations/O_7.gif) |
This can be simplified further with the help of
FullSimplification:
Out[24]= | ![](Files/MetricVariations/O_8.gif) |