Gravitational interaction of two photons, initially separating Imagine two photons are flying in a void of empty space and pass each other such that one is moving forward and the other upward.
Since energy creates gravity and photons are pulled by gravity, wouldn't the small energy and therefore small gravitational attraction of photons bend their trajectory such that at some time after their passing they will meet again despite the fact that they initial flew in different directions at 90 degree angle?
 A: The path of a photon is a geodesic, which is similar to a line in Euclidean geometry or a great circle on the surface of a sphere. In a spacetime with curvature, it is possible to have two geodesics that intersect in more than one place. This is different from Euclidean geometry, where lines can intersect at most once. This can indeed happen, and it's called gravitational lensing, but it requires a third (usually fairly massive) object to provide enough curvature.
To see that this can't happen with just the two photons, note that it's possible to switch frames of reference to a frame called the center of mass frame, in which the total momentum is zero. In your example, if one photon is going in the positive x direction and the other in the positive y direction, in the original frame $\text{F}_1$, then the center of mass frame $\text{F}_2$ will be defined by an observer who is moving, relative to $\text{F}_1$, along a line at a 45-degree angle between the x and y axes. The speed of $\text{F}_2$ relative to $\text{F}_1$ will be less than $c$. An observer in $\text{F}_2$ will see the two photons as moving off in opposite directions. As they separate, their gravitational attraction may cause them to be red-shifted, but this redshift will approach some finite limit. The photons will not stop and come back together. Therefore, back in $\text{F}_1$, the photons will be deflected, but not enough to reunite.
All of the above assumes that the photons have energies small enough so that the curvature they create can be treated as a perturbation on a background of flat spacetime. Only under this condition does it make sense to talk about things like global frames of reference. When the gravitational fields are very strong, we can get qualitatively different phenomena, such as geons or a kugelblitz.
Related: Are there bound states from light-light or gravity-gravity scattering in general relativity?
A: Photons are elementary particles, they do have stress-energy, and do bend spacetime. I actually asked a question about this:
Do photons bend spacetime or not?

The question of whether photons cause spacetime curvature is a question about quantum gravity, and we have no accepted theory of quantum gravity. However, we have standard ways of quantizing linear perturbations to a metric, and reputable journals such as Physical Review D have published papers on graviton-mediated photon-photon scattering, such as this one from 2006. If such calculations are no longer mainstream, it is news to me. Given that photons have energy and momentum, it would surprise me if they do not induce curvature.

If you accept that photons do bend spacetime, then your question can be answered in theory. In a empty universe, where there are only two photons, initially coming from the same position, in 90 angle, then the answer is only up to the energy levels of the photons.


*

*If the photons' energy levels (stress-energy) are so big (comparable to that of a black hole), then the two photons will bend spacetime so much, that the curvature will be so that the photons will not be able to escape the gravitational field of each other (geodesics will be bent so much), and they will meet again. This is because the escape velocity (from that gravitational field) will exceed the speed of light.



The local speed of light is always c, but if you use the Gullstrand–Painlevé coordinates to analyse what happens at the event horizon you find that:
  at the horizon you are falling inwards at the speed of light
  relative to you the light is travelling outwards at the speed of light
  so the net speed of the light away from the event horizon is zero
  And that's why the light can't escape from the black hole.

If the speed of light is constant, why can't it escape a black hole?


*But in any other case, when the photons' energy levels are smaller, the two photons' effect on spacetime curvature is so little, that they will affect each other very little (and will not meet again).

A: While it is true that photons add to the stress-energy tensor, and therefore exert some gravitational pull, it would hardly affect the two photons in your thought experiment, and their geodesic paths would just be straight lines. Think of how little deviaton results from a massive object like the earth, surely the inter-photon attraction will be negligible.
A: The actual question here is whether two photons can be in a gravitationally bound state. This can be seen by transforming to the rest frame, in which the sum of the photon momenta vanishes.
For photons to form a gravitationally bound state they would have to be incredibly close, much closer than the Planck length for optical photons. Any statements on what happens in this regime is pure speculation. Just my 2 cts...
