Hyperbolicity of scalar-tensor theories of gravity

Physical Review D - Particles, Fields, Gravitation and Cosmology

Volumn 77, Issue 10, 2008, Pages

  Salgado, Marcelo ^a   Río, David Martínez Del ^a   Alcubierre, Miguel ^a   Núñez, Darío ^a,b  

^a INSTITUTO DE CIENCIAS NUCLEARES   (Mexico)

^b ALBERT EINSTEIN INSTITUT   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 43949124853     PISSN: 15507998     EISSN: 15502368     Source Type: Journal    
DOI: 10.1103/PhysRevD.77.104010     Document Type: Article

References (40)

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30. In fact, as suggested in Ref., when one works with coordinates other that Cartesian-type, it turns out to be better to define ψ so that det γ∼ij=detf∼ij, where fij represents a background Riemannian metric. For instance if Σt is asymptotically flat one can take fij to be the flat metric in some coordinates adapted to the symmetry of the problem. In spherical symmetry, say, fij=diag(1,r2,r2sin 2θ) and detγ∼ij=r4sin 2θ. This approach has the advantage that ψ becomes a true scalar while all the tensorial quantities defined using γ∼ij and ψ become true tensors instead of tensor densities.
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39. For comparison purposes with the original BSSN equations, a typo in sign has to be taken into account in the second line of Eq. (24) of.
40. If the function c(t,x) is globally null, the equation for u2 decouples and u2 is itself an eigenfunction propagating with speed d; the other eigenvalue and eigenfunction can be obtained by proceeding in the same way, taking c=0 in Eq. 3. This results in w2=u1+bu2/(a-d) (with a≠d) and λ2=a. However, if c=0, a=d, and b≠0 then the system degenerates and becomes weakly hyperbolic. In such a case the characteristic matrix is a Jordan block [cf. Eq. 4], which implies that it cannot be diagonalized. A similar situation happens if b=0, a=d, and c≠0. On the other hand, if c=0=b then both Eqs. 1 2 decouple and therefore u1 and u2 are themselves eigenfunctions (or any linear combination of them is an eigenfunction as well if in addition a=d).