I am confused. Your hackdrive moves your ship, but you do not gain momentum. In traveling from a distant point to a point close to the sun, you travel from a region of high potential energy to a region of low potential energy.
Let me rephrase that in terms of Nova Roma science.
In traveling from a distant point to a point close to the sun in Universe A, you travel in Universe A from a region of high potential energy to a region of low potential energy, but the entire time, your ship is inside a bubble called Universe B. There is a shadow of your bubble in Universe A, but it's more of a distortion of the fabric of space and it is massless. Gravity does not affect it, nor does Universe B gain momentum in Universe A.
If the ship were to turn off the bubble, it would start to gain momentum and move toward the center of the Sun's gravity well. In fact, this happens whenever the ship drops the bubble for 100 ms to get a sensor reading.
The potential energy that the ship gains must be accounted for when the bubble is turned back on. The total mass (including gravity potential and other energy) work against the machine that creates the bubble. That is, it's harder to pull the ship out of Universe A back into Universe B because of the additional energy it has picked up from the gravity well. It's still fairly small due to the short duration.
I realize now that I'm totally ignoring curved space and general relativity. If the hack drive simply translates the bubble through normal space, then it is subject to gravity because gravity is just space curvature. However, I still think that the ship would not "fall" in a gravity well like a normal ship in Universe A because it is not actually IN Universe A. If it wants to move in Universe A, though, it must travel "further" to get out of a gravity well and it travels a "shorter" distance as it goes into a gravity well.
Mass/Energy can neither be created nor destroyed. I will assume that some feature of the hackdrive conserves energy, because it must or it violates a fundamental law of the universe.
Yes, good catch. Both energy and momentum must be conserved.
A ship in the bubble has zero energy or momentum outside the bubble. The bubble itself has zero energy and zero momentum and zero mass in the outside universe. That's just how it works.
As I said above, dropping the bubble for even a short time brings the ship back into the main universe and so it starts gaining momentum as gravity forces apply to it. The increase to momentum is dependent on mass and relative velocity (p=mv). 100 ms is not a lot of time and thus velocity doesn't increase that much (remember, it starts from zero). Still, the ship has to conserve that momentum to bring the bubble back up, and that means an expenditure of reactor drive energy at the time of bubble activation.
Maybe activating the bubble causes a huge flash of energy equal to the total energy of the system being negated. That is, only the momentum is being negated, and perhaps some of the light it gives off, so there's a one-time flash. The rest mass of the ship (and the energy of its reactors, batteries, etc.) inside the bubble continues to be conserved inside the bubble, so no problem there.
In other words, as you change position from a higher point in the gravity well to a lower point, what would otherwise be your kinetic energy of motion is instead being expressed as some other form of energy, possibly light: the hack drive glows in infrared as it releases energy gained from changing position in the gravity well, leaving the ship inside with no change in velocity when the hack drive flickers off.
I don't think that's necessary, since the bubble is not part of the universe where the gravity well is.
Put another way: Earth whips around the sun at about 67 thousand miles per hour. Mercury whips around the sun at almost 108 thousand miles per hour. Something closer still to the sun would have to move even faster to maintain a stable orbit. Back when you were at earth orbit, you were stable with respect to the sun, moving at 67 thousand miles per hour right along with the Earth; you didn't notice you were moving 'cause Earth was moving right along with you.
Now, at Mercury's orbit, you are still moving 67 thousand miles per hour, but it's nowhere near fast enough to maintain an orbit: the instant you turn off the field, you start falling sunward like a rock. In fact, every time you blinked the field off while on the way here, you started falling a bit, gradually accumulating more and more downward velocity. Moreover, an object in a stable orbit near Mercury is flying 41 thousand miles per hour with respect to you (his 108 thousand minus your 67 thousand), and an object in a stable orbit farther in is flying even faster. You are in essence a falling rock trying to catch a ride on a passing hypersonic jet.
No, because you've cheated the universe to move in it. You never actually moved. It's an error in the calculation logic of quantum mechanics that let you jitter your way across space. It doesn't have to conserve momentum in that way. You've merely put yourself in another universe and stayed "at rest" in your bubble the entire time, while the bubble glitches its way around.