Does a "Photon Box" have gravitational mass? The photon box is a thought experiment involving a massless, perfectly reflective box containing light.
When the box is accelerated in any direction it results in a difference of force being applied to the walls of the box that are in line with the acceleration. The "front" being pushed less and the "back" being pushed more. This results in a resistance to acceleration or inertial mass and this mass can even be given a value in Kilograms regardless of the fact that no individual components are massive.
While I understand that this box is "massive" I do not know whether the box would have "weight".
Would the box fall in a gravity field?
If so, why?
 A: It would also have weight. Due to gravitational red-shift the momentum of a photon hitting the bottom is higher than the same photon hitting the ceiling of the box. If you calculate this I think you end up with a 'mass' of 1/3 E/c^2.
A: The box will indeed fall, for the same reason that light is lensed by the gravitational field of the Sun (for example). The gravitational field curves spacetime, and the photons will follow the curved paths. The photons will also in general exert their own gravitational field, though a very negligible one in all realistic scenarios, and the details are a bit complicated.
A: Two points:


*

*even massless particles (such as photons) "fall in a gravitational field";

*the amount of mass actually doesn't change how it falls, either in Newtonian gravitation (where gravitational and inertial masses coincide), or in General Relativity (where free objects follow space-time geodesics), provided the box mass is small enough not to change the gravitational force/curvature.


But, rigorously, any amount of energy in the box will influence the space-time curvature. What brings us to a possibly more interesting question: "would objects with mass fall towards the box?". And the answer is: "yes". A box with enough light inside would create the same gravity well as, e.g., the Earth.
A: to understand gravity well, you must wrap your head around the stress-energy tensor. basically, how much something interacts with gravity is equal to how much matter and energy it has.  that explains both why moving objects have a stronger gravitational field (because they have more kinetic energy), and why a photon box interacts with gravity (both falling and pulling). honestly, the only reason laymen think light doesn't have mass is because ordinary matter can be converted into massive amounts of light. for example, your fingernail could provide enough energy to generate all the light you will ever see (because e=mc squared). unfortunately, we do not yet have the technology to turn fingernails into photons efficiently.
A: There are many good answers available above, however, I'm adding a more simple and conclusive way to tackle the problem:
It does't matter to gravity whether there is mass or energy or both: Mass-energy equivalence, $E = mc^2$, tells us that the total energy $E$ contained in the given space can be visualized equivalently as the mass of value $m$. And for low $m$ value, you can simply solve the problem with Newtonian gravity or otherwise, for large $m$ values, you have to use Einstein's field equations instead.  
