Pressure inside a submerged rigid box, and how to change it I've been puzzling over the following thought experiment: Consider two versions of a perfectly rigid, sealable box. Suppose first we fill the first box with water at sea level, so the hydrostatic pressure inside the box is basically 1 atm, plus the pressure of the water (but let's say the box is very shallow). Now seal the box and push down into a deep body of water and attach it there.
Second, suppose that we leave the second box open, attach it under water, and then seal it there. In that case, the pressure inside the box is the same as the (high) ambient water pressure.
How should I imagine the difference between the two boxes? The first box experiences strong inward pressure (driving it to implode), which we assume is absorbed by the box, whereas the second one does not. (Only the material of the box itself is under pressure.) But the water inside and outside is incompressible, and there's nowhere for the force to act, no energy to spend (?).
Is there any theoretical mechanism by which one state could be transformed into the other, in either direction? It seems that displacing content is not an option. What if I slowly open the seal of the first box -- does the interior pressure immediately go up? Is there any net force on the lid?
(Aspects of this question appear in this related question about valves.)
Box 1:
  +-------+
  | Water |   ---->
--+-------+--
             \~~~~~~~          ~~~~~~~~~~~~~~~~~~~~~~~~~~
                                         .
                                         .
                                         .




                                         .
                                     +-------+
                                     | Water |
                                     +-------+
                               ##########################

Box 2:
     /
    /
   /
  +
  |  Air  |   ---->
--+-------+--
             \~~~~~~~     ~~~~~~~~~~~~~~~~~~~         ~~~~~~~~~~~~~~~
                                    .                        .
                                    .                        .
                                    .                        .


                                                ---->

                                    .                        .
                                    .                        .
                                    .                        .

                                   /
                                  /
                                 /
                                +                        +-------+
                                |       |                | Water |
                                +-------+                +--------
                          #####################      #################

 A: If the box is sealed and perfectly rigid, the fluid pressure inside it will not change even if it is submerged to the bottom of the sea. 
A real box is compressible to a variable extent, which depends on its structure and the material it is made from. The fluid pressure inside the box will only change in response to either a change in the volume of the box, or a change in the temperature of the fluid, or a change in the amount of fluid inside. If the box is sealed, not compressible, and held at constant temperature, the fluid pressure inside it will not change. 
No material is really incompressible. Incompressible simply means that the decrease in volume is very small for the typical forces you are considering. 
A: I believe that the problem as stated is indeterminate, in that you have combined two absolute properties:  an absolutely rigid box, and absolutely incompressible fluid in the box.
Assume that at the bottom of the sea, the box resists a fraction $x$ of the applied pressure (without changing its volume) and transmits $(1-x)$ of the pressure to the liquid in the box (without any change in the fluid's volume)  So the box and liquid have the same volume as before.
And this is true for any values of x, such that $0<x<1$.
If the box is rigid, the fluid could be as compressible as you like, even a gas.  And if the liquid is incompressible, the container could be as flimsy as you like...
