The joint NASA–ESA mission, LISA, relies crucially on the stability of the three-spacecraft constellation. Each of the spacecraft is in heliocentric orbit forming a stable triangle. In this paper we explicitly show with the help of the Clohessy–Wiltshire equations that any configuration of spacecraft lying in the planes making angles of ±60° with the ecliptic and given suitable initial velocities within the plane, can be made stable in the sense that the inter-spacecraft distances remain constant to first order in the dimensions of the configuration compared with the distance to the Sun. Such analysis would be useful in order to carry out theoretical studies on the optical links, simulators, etc.

The joint ESA–NASA mission LISA relies crucially on the stability of that three spacecraft constellation. All three spacecraft are on heliocentric and weakly eccentric orbits forming a nearly stable triangle. It has been shown that for certain spacecraft orbits, the arms keep constant distances to the first order in eccenticities. However, exact orbitography exhibits the so-called ‘breathing modes’ resulting in slow variations of the armlengths on the timescale of one year. In this paper, we analyse the breathing modes (flexing of the arms) with the help of the geodesic deviation equation up to the octupole order, which is shown to be equivalent to higher order Clohessy–Wiltshire equations. We analytically show that the flexing of the arms can be reduced to a peak-to-peak variation of about 50 000 km, and the corresponding peak-to-peak variation in the Doppler laser frequency shift to about 8 m s−1. This is achieved by slightly changing the well-known tilt of 60°. We further show that it is the minimum within the assumption of equivalent spacecraft orbits, where the orbit of each spacecraft is rotated by 120° from the preceding one.

The authors establish two general results. Firstly, every solution with the cosmological constant and self-dual Weyl tensor of Einstein's equation comes from the Samuel (1988) or Ashtekar-Renteln (1987) ansatz; and secondly, self-duality for spherically symmetric spacetimes implies conformal flatness.

We study the vacuum Maxwell theory by expressing the electric field in terms of its Faraday lines of force. This representation allows us to capture the two physical degrees of freedom of the electric field by means of two scalar fields. The corresponding classical canonical theory is constructed in terms of four scalar fields, is fully gauge invariant, has an attractive kinematics, but a rather complicated dynamics. The corresponding quantum theory can be constructed in a well-defined functional representation, which we refer to as the Euler representation. This representation turns out to be related to the loop representation. The resulting quantization scheme is, perhaps, of relevance for non-Abelian theories and for gravity. PACS number(s): 03.50.De, 03.70.+k

One of the remarkable features of Ashtekar's formulation of general relativity is that degenerate solutions of Einstein's equations are also admissible. By using the Ashtekar-Renteln ansatz the authors obtain some spherically symmetric degenerate Euclidian solutions of Einstein's equations with a cosmological constant. An appropriate choice of the gauge gives the well known (non-degenerate) de Sitter solution with Euclidian signature.