## 3. General theory of the gravitational scintillationIn a region of spacetime free of electric charge, the propagation
equations for the electromagnetic vector potential
are ( when is chosen to obey the Lorentz gauge condition It is convenient here to treat as a complex vector. Hence the electromagnetic field tensor is given by The corresponding electromagnetic energy-momentum tensor is defined by where . The components of this tensor satisfy the conservation equations as a consequence of Eqs. (3). For an observer located at the spacetime point and the density of electromagnetic energy as measured by the observer is In this paper, we use the geometrical optics approximation. So we assume that there exist wave solutions to Eqs. (3) which admit a development of the form where is a slowly varying complex vector
amplitude, is a real function and
a dimensionless parameter which tends to zero
as the typical wavelength of the wave becomes shorter and shorter. A
solution like (9) represents a quasi plane, locally monochromatic wave
of high frequency (Misner Let us define the phase and Inserting (9) into Eqs. (3) and (4), then retaining only the leading terms of order and , yield the fundamental equations of geometrical optics with the gauge condition Light rays are defined to be the curves whose tangent vector field is . So the parametric equations of the light rays are solutions to the differential equations where follows from (11), it is easily seen that satisfies the propagation equations These equations, together with (12), show that the light rays are null geodesics. Inserting (9) into (5) and (6) gives the approximate expression for and for averaged over a period where From (7) and (19), it is easily seen that the Poynting vector is
proportional to the null tangent vector . This
means that the energy of the wave is transported along each ray with
the speed of light. Let us denote by the energy
flux received by an observer located at This formula enables us to determine the photon flux
received by the observer located at The spectral shift Consequently, the photon flux may be written as The scalar amplitude where denotes the total differentiation of a scalar function along . Then, integrating (25) gives where is an arbitrary point on the light ray . In the following, we consider that the light source is at spatial infinity. We suppose the existence of coordinate systems such that on any hypersurface , when , with . We require that in such coordinate systems the quantities , and respectively fulfill the asymptotic conditions when , with . Moreover, we assume that the scalar amplitude in Eq. (26) remains bounded when and we put It results from these assumptions that may be written as Now, let us differentiate with respect to
As a consequence, we can write The convergence of the integrals is ensured by conditions (27). Eqs. (29) and (31) allow to determine the factor in from the energy content of the regions crossed by the light rays and from the geometry of the rays themselves. It is well known that (or
) can also be obtained in the form of an
integral along the light ray (see In fact, the scintillation phenomenon consists in a variation of
with respect to time. For this reason, it is
more convenient to calculate the total derivative of
along the world-line of
a given observer, moving at the point Given a scalar or tensorial quantity where is the line element between two events and on . In Eq. (24), the quantity is the energy
of a photon emitted by an atom of the light source as measured by an
observer comoving with this atom. So is a
constant which depends only on the nature of the atom (this constant
characterizes the emitted spectral line). Consequently, the change in
the photon flux with respect to time is simply due to the change in
the scalar amplitude Henceforth, we shall call the contribution
in Eq. (33) the geometrical scintillation because the variations
in Let us now try to find expressions for and in the form of integrals along . In what follows, we assume that the ray hits at each of its points a vector field which satisfies the boundary condition Let us emphasize that can be chosen
arbitrarily at any point It results from the boundary conditions (27) and (34) that may be written as Thus we have to transform the expression taken along . Of course, we must take into account the propagation equation (25) which could be rewritten as Noting that then using the relation which holds for any scalar where the bracket of and is the vector defined by Taking (37) into account, it is easily seen that Now, using the identity (1) and the definition (2) yields Let us try to write the term in the form of an integral along . In agreement with (27), we have at any point of : A tedious but straightforward calculation using (1), (2) and (17) leads to the following result In the above formulae is an arbitrary
vector. So we can choose so that the transport
equations
are satisfied along the ray . Since (46) is a system of first order partial differential equations in , there exists one and only one solution satisfying the boundary conditions (34). With this choice, is given by the integral formula: Now we look for an integral form for the total derivative
along . Henceforth, we
suppose for the sake of simplicity that the observer is freely
falling, Since is a constant characterizing the observed spectral line (see above), it follows from (23) and (48) that Given an arbitrary vector field fulfilling the boundary condition (34), Eq. (49) may be written as Using (1), (17) and (41), a straightforward calculation gives the general formula which holds for any freely falling observer. Now let us choose for the vector field defined by (46) and (34). We obtain © European Southern Observatory (ESO) 1998 Online publication: October 22, 1998 |