The theory of gravitational lensing is reviewed from a spacetime perspective, without quasi-Newtonian approximations More precisely, the review covers all aspects of gravitational lensing where light propagation is described in terms of lightlike geodesics of a metric of Lorentzian signature It includes the basic equations and the relevant techniques for calculating the position, the shape, and the brightness of images in an arbitrary general-relativistic spacetime It also includes general theorems on the classification of caustics, on criteria for multiple imaging, and on the possible number of images The general results are illustrated with examples of spacetimes where the lensing features can be explicitly calculated, including the Schwarzschild spacetime, the Kerr spacetime, the spacetime of a straight string, plane gravitational waves, and others

https://link.springer.com/content/pdf/10.12942%2Flrr-2004-9.pdf

Gravitational Lensing from a Spacetime Perspective

The theory of gravitational lensing is reviewed from a spacetime perspective, without quasi-Newtonian approximations. More precisely, the review covers all aspects of gravitational lensing where light propagation is described in terms of lightlike geodesics of a metric of Lorentzian signature. It includes the basic equations and the relevant techniques for calculating the position, the shape, and the brightness of images in an arbitrary general-relativistic spacetime. It also includes general theorems on the classification of caustics, on criteria for multiple imaging, and on the possible number of images. The general results are illustrated with examples of spacetimes where the lensing features can be explicitly calculated, including the Schwarzschild spacetime, the Kerr spacetime, the spacetime of a straight string, plane gravitational waves, and others.

We have studied the science rationale, goals and requirements for a mission aimed at using the gravitational lensing from the Sun as a way of achieving high angular resolution and high signal amplification. We find that such a mission concept is compromised by several practical problems. Most severe are the effects due to the plasma in the solar atmosphere which cause refraction and scattering of the propagating rays. These effects either limit the frequencies that can be observed to those above ∼1 THz, or they move the optical point outwards beyond the vacuum value of ≥550 au. (Thus for observing frequency of 300 GHz the optical point is moved outwards to ∼ 680 au.) Density fluctuations in the inner solar atmosphere will further cause random pathlength differences for different rays. The corrections for the radiation from the Sun itself will also be a major challenge at any wavelength used, but could be mitigated with coronographic techniques. Given reasonable constraints on the spacecraft (particularly in terms of size and propulsion), source selection as well as severe navigational constraints further add to the difficulties for a potential mission. Nevertheless, unbiased surveys of small-scale structure on the sky at short wavelengths might be the most promising application of such a mission.

The 550‐au Mission: a critical discussion

This paper gives the Poynting vector of a plane electromagnetic wave diffracted by the gravitational field of a large spherical body (large compared to its Schwarzschild radius) and shows in detail how this body works as a gravitational lens. The most interesting results are (1) an extreme amplification of intensity near to the axis of symmetry in the far field behind the body, with a factor of 10 times the Schwarzschild radius divided by the wavelength of the light, and (2) the appearance of double images, differing in shape and position from the predictions of geometrical optics.

Wave Optics of the Spherical Gravitational Lens. Part I: Diffraction of a Plane Electromagnetic Wave by a Large Star

We study the optical properties of the solar gravitational lens (SGL) under the combined influence of the static spherically symmetric gravitational field of the Sun---modeled within the first post-Newtonian approximation of the general theory of relativity---and of the solar corona---modeled as a generic, steady-state, spherically symmetric free electron plasma. For this, we consider the propagation of monochromatic electromagnetic (EM) waves near the Sun and develop a Mie theory that accounts for the refractive properties of the gravitational field of the Sun and that of the free electron plasma in the extended Solar System. We establish a compact, closed-form solution to the boundary value problem, which extends previously known results into the new regime where gravity and plasma are both present. Relying on the wave-optical approach, we consider three different regions of practical importance for the SGL, including the shadow region directly behind the Sun, the region of geometrical optics and the interference region. We demonstrate that the presence of the solar plasma affects all characteristics of an incident unpolarized light, including the direction of the EM wave propagation, its amplitude, and its phase. We show that the presence of the solar plasma leads to a reduction of the light amplification of the SGL and to a broadening of its point spread function. We also show that the wavelength-dependent plasma effect is important at radio frequencies, where it drastically reduces both the amplification factor of the SGL and also its angular resolution. However, for optical and shorter wavelengths, the plasma's contribution to the EM wave is negligibly small, leaving the plasma-free optical properties of the SGL practically unaffected.

Diffraction of light by the gravitational field of the Sun and the solar corona

Using the technique of Debye potentials a rigorous solution of the diffraction problem is given as a superposition of an incident wave, strongly connected with the Coulomb scattering wave function, and a scattered wave, which is purely outgoing for large distances. The solution fulfils the boundary conditions to be the light of a very distant star and to be purely ingoing at the Schwarzschild horizon. The phase shifts of the partial waves are evaluated in the WBK approximation.

Diffraction of a Plane Electromagnetic Wave at a Schwarzschild Black Hole

This paper gives (in a suitable approximation) the Poynting vector of a plane electromagnetic wave diffracted by the gravitational field of a Schwarzschild black hole. The relation between the approximation procedure and the concept of rays is established. The main results are as follows: (1) On the focal line an extreme amplification of intensity takes place. (2) In the whole space off this focal line a double image is to be seen. (3) Rays having revolved around the black hole repeatedly give small corrections only. (4) The phase differences (transit-time differences) along the different rays are computed.

Wave optics of the spherical gravitational lens. II. Diffraction of a plane electromagnetic wave by a black hole

If the two invariants of an arbitrary non-static electromagnetic vacuum field are finite at the Schwarzschild radiusr=M, the field behaves atr=M+ either as a purely ingoing or as a purely outgoing wave.

E. Herlt

Papers

Wave Optics of the Spherical Gravitational Lens. Part I: Diffraction of a Plane Electromagnetic Wave by a Large Star

Diffraction of a Plane Electromagnetic Wave at a Schwarzschild Black Hole

Wave optics of the spherical gravitational lens. II. Diffraction of a plane electromagnetic wave by a black hole

Electromagnetic Test Fields around a Schwarzschild Singularity