A gravitational lens should produce a halo effect and not arcs. Given that the light travels not only to the right and left of the intervening massive object but also to the top and bottom, why do we typically see only arcs?

Gravitational lensing is a phenomenon where the path of light is bent as it passes through the gravitational field of a massive object. This bending of light causes the light from a distant source to arrive at an observer from different angles, creating multiple images or distorted images of the source.

Imagine a massive object, like a galaxy or a black hole, situated between a distant light source and an observer. The light emitted by the source will travel in all directions. When the light passes close to the massive object, its path is bent due to the object's gravitational field.

The shape of the resulting images depends on the geometry of the lensing system, including the alignment between the source, the intervening massive object, and the observer. When the light paths are perfectly symmetrical around the massive object, we would expect to see a complete circle, known as an "Einstein ring." However, in most cases, the alignment is not perfect, and the light paths are more prominent in some directions than others.

Typically, we observe arcs rather than complete halos due to the relative alignment of the source, the massive object, and the observer. The likelihood of a perfectly symmetrical alignment is very low, making Einstein rings relatively rare. Instead, the geometry generally favors arcs, which are partial segments of the idealized Einstein ring. Additionally, any deviation from perfect symmetry can lead to a distortion of the images along a particular axis, either horizontally or vertically, causing the formation of arcs rather than a full halo.

The light paths coming from the top and bottom of the intervening massive object are indeed also bent by the gravitational field. However, these light paths may not necessarily lead to the formation of observable arcs. Factors such as the specific lensing geometry, the brightness of the source, and the spatial distribution of the massive object's mass can contribute to the dominance of the arcs observed in one direction over the other. In some cases, arcs may be present, but too faint to be detected easily. In conclusion, the main reason why we typically see arcs instead of complete halos in gravitational lensing situations is due to the imperfect alignment and asymmetrical geometry of the source, massive object, and observer in most lensing systems. This imperfect alignment favors the formation of partial segments of Einstein rings (arcs) rather than perfectly formed circular halos. Additionally, light paths from the top and bottom may also contribute to arc formation, but their visibility might be limited by various factors.