Full Einstein Equations Research Articles

Maxwell's equations, linear gravity, and twistors

A detailed outline is presented of several convergent points of view connecting the self-dual and anti-self-dual fields with their free data. This is done for the Maxwell and for linearized gravity as exemplifying the approaches. The Sparling equation provides one tool of great power and characterizes one approach. The twistor theory of Penrose yields another equally powerful point of view. The links between these two basic approaches given in this paper provide a unification that allows workers and others with interest in this area to proceed more readily toward the goal of understanding the full nonlinear Einstein equations.

Linearized quadrupole waves in general relativity and the motion of test particles

We write out the explicit form of the metric for a linearized quadrupole gravitational wave in the transverse-traceless gauge. We give a collection of formulas which are useful for testing numerical codes to integrate the full nonlinear Einstein equations. We also show that a test particle initially at rest acquires no energy from a linearized gravitational wave.

A reformulation of the post-Newtonian approximation to general relativity

A new form of a first post-Newtonian approximation to general relativity is presented, according to an iteration scheme, for solving Einstein’s equation for isolated, self-gravitating systems, recently formulated by Ehlers. The post-Newtonian metric is constructed and the corresponding equations of motion are set up in two iteration steps. The computation is based on a reduced field equation which is obtained from the full equation by dropping some terms which vanish in harmonic co-ordinates. The equivalence of the harmonicity condition for the co-ordinates to the local equations of motion is proved in the post-Newtonian approximation. It is shown that the full Einstein equation is approximately satisfied provided the local equations of motion are. Finally, a method with rigorous error estimates is developed, in order to obtain the components of the metric tensorg ab, oncek ab : =η ab − √−gg ab = (g = det (g ab )) are known.

Stationary solutions, energy, and the Bel-Robinson tensor

The absence of stationary source-free solutions and the positive energy problem in general relativity are discussed, at the linearized level, in terms of the Bel-Robinson tensor. The possibility is raised that there may exist stationary solutions to the full Einstein equations in five dimensions.