The longitudinal tensor h || is also called the scalar part of h, the solenoidal part h is also called the vector part, and the transverse-traceless part h T is also called the tensor part. This classification of the spatial metric perturbations h ij was first performed by Lifshitz (1946). The purpose of this decomposition is to separate h ij

is the rate of strain tensor, and Ωij = 1 2 ∂qi ∂xj − ∂qj ∂xi! (1.6.6) is the vorticity tensor. Note also that (1.6.4) depends only on the rate of strain but not on vorticity. This is reasonable since a ﬂuid in rigid-body rotation should not experience any viscous stress. In a rigid-body rotation with angular velocity ω, the 3 Balance equations Volumetric–deviatoric decomposition in analogy to the strain tensorǫ, the stress tensorσcan be additively decomposed into a volumetric partσvoland a traceless deviatoric partσdev volumetric – deviatoric decomposition of stress tensorσ σ=σvol+σdev(3.1.21) with volumetric and deviatoric stress tensorσvolandσdev For example, if the symmetry is just rotation, then the term with the trace transforms like a scalar; the anti-symmetric part M i j − M j i of the tensor transforms like a pseudo-vector, while the traceless symmetric part (the last term) transforms like an ordinary 2-tensor. An interesting aspect of a traceless tensor is that it can be formed entirely from shear components. For example, a coordinate system transformation can be found to express the deviatoric stress tensor in the above example as shear stress exclusively. In the screenshot here, the above deviatoric stress tensor was input into the webpage, and arXiv:gr-qc/0703035v1 6 Mar 2007 3+1 Formalism and Bases of Numerical Relativity Lecture notes Eric Gourgoulhon´ Laboratoire Univers et Th´eories, UMR 8102 du C.N.R.S., Observatoire de Paris, Jul 22, 2015 · So, let us decompose it into irreducible parts. First, we split the tensor into symmetric and antisymmetric tensors: $$u^{i}v^{j}_{k} = \frac{1}{2} u^{(i}v^{j)}_{k} + \frac{1}{2} u^{[i}v^{j]}_{k} .$$ To make the symmetric part traceless, we subtract (and add) the symmetric combinations of traces

## First, we obtain the plane wave solution of the linearized massive conformal gravity field equations. It is shown that the theory has seven physical plane waves. In addition, we investigate the gravitational radiation from binary systems in massive conformal gravity. We find that the theory with large graviton mass can reproduce the orbit of binaries by the emission of gravitational waves.

the fully traceless part, the Weyl tensor C abcd Each piece possesses all the algebraic symmetries of the Riemann tensor itself, but has additional properties. The decomposition works in slightly different ways depending on the signature of the metric tensor g a b , and only makes sense if the dimension satisfies n > 2 .

### the fully traceless part, the Weyl tensor C abcd Each piece possesses all the algebraic symmetries of the Riemann tensor itself, but has additional properties. The decomposition works in slightly different ways depending on the signature of the metric tensor g a b , and only makes sense if the dimension satisfies n > 2 .

The momentum anisotropy contained in a sheared flow may be transferred to a pressure anisotropy, both gyrotropic and non-gyrotropic, via the action of the fluid strain on the pressure tensor components. In particular, it is the traceless symmetric part of the strain tensor (i.e. the so-called shear tensor) that drives the mechanism, the fluid vorticity just inducing rotations of the pressure The determinant can be understood as a conserved volume under stretching or shrinking or any other sort of smooth deformation. The trace is some scalar object or quantity associated with the tensor that doesn't change upon rotation (but not stretc In particular, the moment equation for the traceless part of the stress tensor can be written in a form that isolates the term proportional to the magnetic field – from which Bruce Liley derived an explicit expression for this important part of the stress tensor which is invariant (i.e. valid in any coordinate system) and applies for any the fully traceless part, the Weyl tensor C abcd Each piece possesses all the algebraic symmetries of the Riemann tensor itself, but has additional properties. The decomposition works in slightly different ways depending on the signature of the metric tensor g a b , and only makes sense if the dimension satisfies n > 2 .