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GRAVITATIONAL LENSE

Part V

A very interesting set of Einstein rings are the ``self" Einstein rings, where observers can see themselves. The most well known of these can be seen when the observer is located at the photon sphere. There observers can simply look along the photon sphere, where light travels in a circle, and see the backs of their heads! All observers in the presence of a sufficiently compact lens, however, can see themselves. Here, light can leave the observer, travel around the lens and return to the observer to be viewed. Observers would see themselves as a series of Einstein rings. The more times light can circle the lens and return to the observer, the more "self" images the observer can see. For a lens compact enough to have a photon sphere, observers can, theoretically, see themselves in every self Einstein ring: an infinite number of times.

Amusingly, there is only a single case where observers can see only a single image of themselves - and this is the case that is well known - when observers are at the photon sphere! Here all the self Einstein rings actually merge with the photon sphere to form a single observer image.

Observers who see themselves would be viewing themselves with high amplification. This is because the self images observers would see would be on or near Einstein rings - which carry the highest amplifications. Therefore gravity has become a powerful microscope! When at the photon sphere observers can microscopically view the backs of their heads, and when far away observers can microscopically view their own eyes. This is because the light that returns to the observer has left on a nearly radial trajectory - and the part of the observer most nearly radial is the observer's own eye. When close to and inside the photon sphere, observers can inspect annular rings on their heads (or spacecrafts).

A fairly detailed description of the distortion effects a space-traveler (or camera) would see on a visit to a high gravity star is now possible. The case that will be described first will be a trip to a "normal" neutron star: one with a currently popular equation for the interior structure of the star. This star is not dense enough to have an event horizon or photon sphere.

The second case that will be described is that of visiting a black hole. This case is more complex in that many bound and unbound photon orbits exist near the black hole. There is, however, a somewhat simpler aspect to describing this case than the previous one in that one does not have to track surface feature distortions for a black hole.

The third and last case that will be described is that of visiting a ultracompact neutron star - one with an extreme equation for its interior structure that allows a mean density so high the star has a photon sphere. This is the most complicated case of all to describe as it involves all three types of photon orbits described above as well as requiring a description of both the sky and surface feature distortions.

To more clearly delineate what the viewer would see, a set of computer generated figures were created that document the distortion effects in terms of familiar icons. In these illustrations, the sky in the background behind the high gravity star was taken to be the night sky as viewed from present-day earth. More specifically, the background sky is taken from the Bright Star Catalogue, allowing all stellar images as dim as 5th magnitude to be seen, and stellar images as dim as 7th magnitude may be amplified into visibility. In the two cases of neutron stars, a map of the earth was projected onto the surfaces of the stars and allowed to distort. These figures are, in many aspects, fully general relativistically correct. The resolution of the figures is about 3 arcminutes (0.05 degrees).

Stellar image brightnesses are shown by the area the stellar image takes on the plots: the area is directly proportional to the flux the observer would receive from the image. It was impossible to change the pixel brightness, so many of the single pixel images would actually be seen dimmer than shown in the figures. Stellar images were allowed to get brighter or dimmer by angular amplification effects, but time induced amplification effects have been suppressed.

Note that for A_angular > 1 the stellar image flux would actually be seen as an increase in angular area of the image, so that the amplified angular area of the stellar images in the computer generated plots are, in this sense, realistic. However, the distortions in the amplified images would not be readily observable, as these background images would be unresolved by the viewer and hence indistinguishable from point sources. A small amplification would not cause the image to be resolved. Stellar images will therefore always be depicted as circles, even when they undergo angular amplification, as these convey best the idea of an unresolved point sources.

Only the two brightest images of all sources were tracked by the computer programs used. All stars originally 5th magnitude or brighter are plotted as secondary images, no matter their magnitude after gravitational distortion. Stars originally 5th magnitude are only plotted as primary images, however, if their final post-lensed magnitude was 5 or brighter. Higher order images undergoing larger angular amplification could potentially be visible but one would need significantly better angular resolution so see them (the only exception to this will be Fig. 2p), so they will be suppressed. An angular amplification limit of a factor of 100 was placed on all images for plotting purposes.

The hypothetical "camera" used in the simulations is somewhat fanciful but has several defining characteristics. First of all the camera is asymptotically small so that no general relativistic light bending effects are important over the length of the camera. The camera's field of view is 90 degrees across the middle of the picture. Lastly, the illustrations that follow, produced by the "camera," have been "flat-fielded" so that angular area on the spherical sky is directly proportional to spatial area on the flat page.