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SpECTRE
v2024.09.29
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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 > |
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.