B. Linet

The static metrics with cylindrical symmetry describing a model of cosmic strings

We discuss a set of static metrics with cylindrical symmetry describing the interior and exterior space-time of a model of cosmic strings considered recently in cosmology. The interior metric depends on one arbitrary function and the exterior on one constant. We find the relation between this constant and the linear mass density of the cosmic string. A cosmic string can be also treated as a line source in the framework of the distribution-valued curvature formalism which allows us to obtain again the same relation.

Electrostatics and magnetostatics in the Schwarzschild metric

The electrostatic field of a point charge at rest in the Schwarzschild metric is given in algebraic form using some results of Copson (1928). It is possible also to determine the magnetostatics. As an example, the magnetostatic field of a current loop surrounding a black hole is given in integral form.

Time transfer and frequency shift to the order 1/c**4 in the field of an axisymmetric rotating body

Within the weak-field, post-Newtonian approximation of the metric theories of gravity, we determine the one-way time transfer up to the order 1/c^4, the unperturbed term being of order 1/c, and the frequency shift up to the order 1/c^4. We adapt the method of the world-function developed by Synge to the Nordtvedt-Will PPN formalism. We get an integral expression for the world-function up to the order 1/c^3 and we apply this result to the field of an isolated, axisymmetric rotating body. We give a new procedure enabling to calculate the influence of the mass and spin multipole moments of the body on the time transfer and the frequency shift up to the order 1/c^4. We obtain explicit formulas for the contributions of the mass, of the quadrupole moment and of the intrinsic angular momentum. In the case where the only PPN parameters different from zero are beta and gamma, we deduce from these results the complete expression of the frequency shift up to the order 1/c^4. We briefly discuss the influence of the quadrupole moment and of the rotation of the Earth on the frequency shifts in the ACES mission.

World function and time transfer: general post-Minkowskian expansions

In suitably chosen domains of spacetime, the world function may be a powerful tool for modelling the deflection of light and the time/frequency transfer. In this paper, we work out a recursive procedure for expanding the world function into a perturbative series of ascending powers of the Newtonian gravitational constant G. We show rigorously that each perturbation term is given by a line integral taken along the unperturbed geodesic between two points. Once the world function is known, it becomes possible to determine the time transfer functions giving the propagation time of a photon between its emission and its reception. We establish that the direction of a light ray as measured in the 3-space relative to an observer can be derived from these time transfer functions, even if the metric is not stationary. We show how to derive these functions up to any given order in G from the perturbative expansion of the world function. To illustrate the method, we carry out the calculation of the world function and of the time transfer function outside a static, spherically symmetric body up to the order G2, the metric containing three arbitrary parameters β, γ, δ.

On the supermassive U(1) gauge cosmic strings

Within the framework of the U(1) gauge cosmic strings, the supermassive case where the linear mass density is 1/4 in units for which G=c=1 is analysed. Making use of a system of first-order differential equations, the form of the solutions is given explicitly. Two possibilities are found: (i) the solution represents a vortex, or an antivortex, and the spacetime is the product of R2 by a 2-surface which is asymptotically a cylinder; (ii) the solution represents a system of a vortex and an antivortex at equilibrium and the spacetime is the product of R2 by a 2-surface having the topology of a sphere.

A vortex-line model for a system of cosmic strings in equilibrium

We show that the field equations of a scalar-gauge theory in general relativity can admit vortex-type solutions describingN parallel vortex lines that we interpret asN infinite straight cosmic strings remaining in equilibrium.

Principle of equivalence and electromagnetism

We show that the local expression of the electrostatic potential of a point charge suggested from the equivalence principle is different of the one resulting from the global consideration in the Schwarzschild space-time.

A Universal Tool for Determining the Time Delay and the Frequency Shift of Light: Synge's World function

In almost all of the studies devoted to the time delay and the frequency shift of light, the calculations are based on the integration of the null geodesic equations. However, the above-mentioned effects can be calculated without integrating the geodesic equations if one is able to determine the bifunction

Dynamics of a self-gravitating thin cosmic string

\Omega(x_A, x_B)

Equation of motion including the reaction of gravitational radiation

giving half the squared geodesic distance between two points