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Xcts::Tags::ShiftStrain< DataType, Dim, Frame > Struct Template Reference

The symmetric "strain" of the shift vector \(B_{ij} = \bar{D}_{(i}\bar{\gamma}_{j)k}\beta^k = \left(\partial_{(i}\bar{\gamma}_{j)k} - \bar{\Gamma}_{kij}\right)\beta^k\). More...

#include <Tags.hpp>

Public Types

using type = tnsr::ii< DataType, Dim, Frame >

Detailed Description

template<typename DataType, size_t Dim, typename Frame>
struct Xcts::Tags::ShiftStrain< DataType, Dim, Frame >

The symmetric "strain" of the shift vector \(B_{ij} = \bar{D}_{(i}\bar{\gamma}_{j)k}\beta^k = \left(\partial_{(i}\bar{\gamma}_{j)k} - \bar{\Gamma}_{kij}\right)\beta^k\).

This quantity is used in our formulations of the XCTS equations.

Note that the shift is not a conformal quantity, so its index is generally raised and lowered with the spatial metric, not with the conformal metric. However, to compute this "strain" we use the conformal metric as defined above. The conformal longitudinal shift in terms of this quantity is then:

\begin{equation} (\bar{L}\beta)^{ij} = 2\left(\bar{\gamma}^{ik}\bar{\gamma}^{jl} - \frac{1}{3}\bar{\gamma}^{ij}\bar{\gamma}^{kl}\right) B_{kl} \end{equation}

Note that the conformal longitudinal shift is (minus) the "stress" quantity of a linear elasticity system in which the shift takes the role of the displacement vector and the definition of its "strain" remains the same. This auxiliary elasticity system is formulated on an isotropic constitutive relation based on the conformal metric with vanishing bulk modulus \(K=0\) (not to be confused with the extrinsic curvature trace \(K\) in this context) and unit shear modulus \(\mu=1\). See the Elasticity::FirstOrderSystem and the Elasticity::ConstitutiveRelations::IsotropicHomogeneous for details.

The documentation for this struct was generated from the following file: