The Einstein Equations of Evolution-A Geometric Approach

Fischer, A. E. and J. E. Marsden

J. of Math. Phys., 13, No. 4, April 1972, 546-568

Abstract:

In this paper, the exterior Einstein equations are explored from a differential geometric point of view. Using methods of global analysis and infinite-dimensional geometry, we answer sharply the question: "In what sense are the Einstein equations, written as equations of evolution, a Lagrangian dynamical system?"

By using our global methods, several aspects of the lapse function and shift vector field are clarified. The geometrical significance of the shift becomes apparent when the Einstein evolution equations are written using Lie derivatives. The evolution equations are then interpreted as evolution equations as seen by an observer in space coordinates. Using the notion of body-space transitions, we then find the relationship between solutions with different shifts by finding the flow of a time-dependent vector field. The use of body and space coordinates is shown to be somewhat analogous to the use of such coordinates in Euler's equations for a rigid body and the use of Eulerian and Lagrangian coordinates in hydrodynamics.

We also explore the geometry of the lapse function, and show how one can pass from one lapse function to another by integrating ordinary differential equations. This involves integrating what we call the "intrinsic shift vector field." The essence of our method is to extend the usual configuration space $\mathfrak{M}=\mathrm{Riem(}M\mathrm{)}$ of Riemannian metrics to $\mathcal {T}$ x $\mathcal {D}$ x $\mathfrak{M}$, where $\mathcal {T}$ = C^$\infty$(M,R) is the group of relativistic time translations and $\mathcal {D}$ = Diff (M) is the group of spatial coordinate transformations of M. The lapse and shift then enter the dynamical picture naturally as the velocities canonically conjugate to the configuration fields ($\xi_{t}^{}$,$\eta_{t}^{}$) $\in$ $\mathcal {T}$ x $\mathcal {D}$.

On this extended configuration space, a degenerate Lagrangian system is constructed which allows precisely for the arbitrary specification of the lapse and shift functions. We reinterpret a metric given by DeWitt for $\mathfrak{M}$ as a degenerate metric on $\mathcal {D}$ x $\mathfrak{M}$. On $\mathcal {D}$ x $\mathfrak{M}$, however, the metric is quadratic in the velocity variables. The groups $\mathcal {T}$ and $\mathcal {D}$ also serve as symmetry groups for our dynamical system. We establish that the associated conserved quantities are just the usual "constraint equations."

A precise theorem is given for a remark of Misner that in an empty space-time we must have $\mathfrak{H}=0$. We study the relationship between the evolution equations for the time-dependent metric g_t and the Ricci flat condition of the reconstructed Lorentz metric g^L. Finally, we make some remarks about a possible "superphase space" for general relativity and how our treatment on $\mathcal {T}$ x $\mathcal {D}$ x $\mathfrak{M}$ is related to ordinary superspace and superphase space.