Einstein–æther models III: conformally static metrics, perfect fluid and scalar fields

The asymptotic properties of conformally static metrics in Einstein–æther theory with a perfect fluid source and a scalar field are analyzed. In case of perfect fluid, some relativistic solutions are recovered such as: Minkowski spacetime, the Kasner solution, a flat FLRW space and static orbits depending on the barotropic parameter $$\gamma $$ γ . To analyze locally the behavior of the solutions near a sonic line $$v^2=\gamma -1$$ v 2 = γ - 1 , where v is the tilt, a new “shock” variable is used. Two new equilibrium points on this line are found. These points do not exist in General Relativity when $$1<\gamma <2 $$ 1 < γ < 2 . In the limiting case of General Relativity these points represent stiff solutions with extreme tilt. Lines of equilibrium points associated with a change of causality of the homothetic vector field are found in the limit of general relativity. For non-homogeneous scalar field $$\phi (t,x)$$ ϕ ( t , x ) with potential $$V(\phi (t,x))$$ V ( ϕ ( t , x ) ) the symmetry of the conformally static metric restrict the scalar fields to be considered to $$ \phi (t,x)=\psi (x)-\lambda t, V(\phi (t,x))= e^{-2 t} U(\psi (x))$$ ϕ ( t , x ) = ψ ( x ) - λ t , V ( ϕ ( t , x ) ) = e - 2 t U ( ψ ( x ) ) , $$U(\psi )=U_0 e^{-\frac{2 \psi }{\lambda }}$$ U ( ψ ) = U 0 e - 2 ψ λ . An exhaustive analysis (analytical or numerical) of the stability conditions is provided for some particular cases.

DIMENSIONAL REDUCTION AND THE TCP THEOREM IN THE SUPERSTRING THEORY

Previously, we have shown that the Kasner solution [Formula: see text] to the D = (M+N+1)-dimensional vacuum Einstein theory [Formula: see text], where dx2 and dy2 are flat spaces of dimensionality M and N, respectively, exists only for (M - N)2 = M + N, that is D = 2+4K(K + 1), 1 + 4L2. The first three quantum numbers K = 0, 1, 2 for even D correspond to the three superstring dimensionalities D = 2, 10, 26 and are also given by the condition D = 2 mod 8 for the existence of Majorana–Weyl fermions. Here, we explain this result in terms of world-sheet symmetry and the TCP theorem applied to the superstring theory, the no-scale metric exhibiting T non-invariance which compensates the CP non-invariance due to the fermions, whose chirality thus gives the cosmological arrow of time its sense of direction.