### Light bending

Whether or not General Relativity is the correct theory of gravity on very large scales, it has passed all tests (many pulsar-based) when applied to planetary and solar system scales. One important feature of the theory is that in a gravitational field, light follows curved trajectories (technically geodesics are "as straight as possible", but they are curved in the practical sense). In familiar settings, this means tiny but detectable effects in laboratory experiments, or small but measurable deflection of stars near the Sun. But it turns out that pulsars are small enough and massive enough that light near their surfaces is bent a great deal - enough that you can actually see almost all of the surface of the pulsar at the same time.

This may sound bizarre, and to some degree it is, but it produces potentially measurable effects. A pulsar's radio emission is produced by plasma somewhere in its magnetosphere, and in fact we're not at all sure just where the emitting plasma is. But for many pulsars, their X-ray emission comes from "hot spots" on the surface, at the magnetic poles. For young pulsars, these hot spots arise because the magnetic field in the crust makes it much easier for heat to flow out where the field is vertical than where it's at an angle. For the very old millisecond pulsars, these hot spots arise from gigantic sparks in the magnetosphere blasting the surface with high-energy particles, heating it. In either case, we sometimes see X-ray pulsations with a thermal spectrum, and light-bending can explain some of the properties of these pulsations.

I did a few very simple simulations of the light-bending, and made some illustrative videos.

I made a set of three videos illustrating this effect. For a rotating pulsar, there are a number of geometric parameters, including the angle between the line of sight and the rotation axis, the angle between the rotation axis and the magnetic axis, and the size of the hot polar cap. I fixed values for all these. There is also the question of the physical size of the pulsar: the more compact and dense it is, the more light-bending we will see. I have generated three videos. The first shows the geometry with no light bending:

Below the actual animation I have a little plot showing a pulse profile, based on a very simple model (blackbody emission from the polar cap). If I make the pulsar more compact (R=3M), I get:

And more compact still (R=2.1M):

The size of each pulsar model is given in "geometrized units", where R=2M is the size of a black hole, the most extreme possible light bending. For a 1.4 solar mass neutron star, this is 4.1 km. Realistic neutron star models vary quite a lot in radius, from ~6 km to ~24 km (~3M to ~12M), so light bending will probably not be as strong as the neutron stars depicted here.

There are other effects to consider as well; neutron stars are expected to have atmospheres, and in fact their spectra do not look like black-body spectra. The atmospheres affect these results by "limb darkening", that is, the radiation that emerges is directed more vertically than simple black-body radiation, so these pulse profiles are not really right. This can be done better, but I just wanted to write a quick hack (using the usual suspects, python, numpy, scipy, and matplotlib) and get a feel for the effect.

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