Piotr P. Goldstein
A quadric of kinetic energy in the role of phase diagrams - application to the BKL scenario
The phase diagrams turn into intractable objects if a Hamiltonian system is more than one dimensional. However, it is convenient to describe the evolution of the bound states, whose potential part $E_p<0$, by the sole kinetic part, especially if it is quadratic in the momenta. In the space of momenta the evolution takes place inside or outside of a quadric $E_k-H<0$ for each value of $H$. Moreover, it provides complete information on the dynamics of the system like phase diagrams do in 1 dimension.
We apply this tool to the Belinski-Khalatnikov-Lifshitz (BKL) scenario, which describes behaviour of an anisotropic Universe near the cosmic singularity. It consists of 3 equations and a constraint (which arise as asymptotics of the Einstein equations in the synchronous frame of reference) \begin{equation} \frac{d^2 \ln a }{d t^2} = \frac{b}{a}- a^2,\qquad\frac{d^2 \ln b }{d t^2} = a^2 - \frac{b}{a} + \frac{c}{b},\qquad\frac{d^2 \ln c }{d t^2} = a^2 - \frac{c}{b}, \end{equation} subject to the constraint \begin{equation} \frac{d\ln a}{dt}\;\frac{d\ln b}{dt} + \frac{d\ln a}{dt}\;\frac{d\ln c}{dt} + \frac{d\ln b}{dt}\;\frac{d\ln c}{dt} = a^2 + \frac{b}{a} + \frac{c}{b} \, , \end{equation} where $t$ is the time parameter while $\,a=a(t),\, b=b(t)$ and $\,c=c(t)$ are the directional scale factors, whose evolution defines the dynamics of the characteristic lengths in three principal directions. The limit $t\to\infty$ corresponds to the cosmological singularity.
A transformation of variables leads to a Hamiltonian model with the constraint $H=0$, while $E_p<0$. The boundary of the quadric, $E_k=0$, which is a cone, corresponds to the limit $t\to\infty$. A comprehensive description of the asymptotics for large $t$ is provided by this method. For systems like BKL, it seems to be more natural and simpler than the diagrams of Misner Thorne and Wheeler [2].
Bibliography:
- V. A. Belinskii, I. M. Khalatnikov, and E. M. Lifshitz, Oscillatory approach to a singular point in the relativistic cosmology, \textit{ Adv. Phys.}, 1970, {\bf 19} (80), 525--573.
- C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation, W.H. Freeman & Co., San Francisco 1973, p. 811.
