Which brings to mind my biggest question about the game: conservation of energy. An object gains potential energy as it moves away from the source of a gravitational field. The energy (U) that goes into lifting a mass (m) to a given height (h) is stored in its position in the gravitational field as potential energy
U = mgh
where U is the potential energy in joules of the object relative to the Earth's surface, m is the mass of the object in kilos, and g is the acceleration due to gravity in meters/sec2. As an object moves toward the source of a gravitational field, the object loses its potential energy-of-position and gains kinetic energy
EK = mv2/2
where v = final velocity of the moving object.
Now assume your IN portal is higher than your OUT portal -- that is, that your exit portal is closer to the Earth's center of mass than the one you enter. As you step through the IN portal, your body possesses a potential energy-of-position (U) equal to its mass times the distance in height between the two portals times 9.8 m/sec2. As you emerge, you have changed position relative to the Earth -- yet your net change in velocity has not changed ( v=0). What happens to the potential energy in your body as you step through? You have not gained velocity, yet you have somehow lost potential energy without gaining kinetic energy. If conservation of energy holds, the potential energy has to go somewhere. But where?
Secondly: it seems to me that by arranging the IN and OUT portals in the proper way one could create a perpetual motion machine. If an OUT portal is placed directly above its own IN portal, a man falling through the IN portal will instantaneously exit the OUT portal directly above and fall through the IN portal, thence to emerge instantaneously from the OUT portal above, and so ad infinitum.
So far so good -- the man just keeps falling -- but once again we see a conservation of energy problem. The man's body possesses a kinetic energy
EK = mv2/2
as he moves. But since he keeps falling, his v increases by 9.8 m/sec for every second he falls. And since his falling isn't going to stop (assuming nobody moves the portals), his v (and this the kinetic energy in his body) isn't going to stop increasing. He'll just keep falling faster and faster.
And therein lies the problem. As he falls, his velocity (v) increases from its initial value (vO) as time (t) passes
t = v - vO / a
Solving for t, we discover that the falling guy's velocity will increase dramatically. The speed of light (c) is 299,792,458 meters per second; if the guy is allowed to fall for 30,591,067 seconds (a little over 354 days) he will asymptotically approach the speed of light. Imagine an 80 kg man tearing through the air at nearly the speed of light...
Only he won't get there - because of special relativity. As the falling man's velocity begins to approach the speed of light, his length along the acceleration axis will approach zero, and his mass (and therefore his EK) will approach infinity. The longer he falls, the greater his relativistic mass will become. Eventually he will come to have the same mass as the Earth, then the mass of the sun, then...
I could go on, but I think I've made my point. Either
a) conservation of energy does not hold, or
b) any given IN portal/OUT portal pair must be at the same level of potential energy relative to the local gravitational field.
In other words: great video!
@mumblix: Thanks, but that's a handwave on the game designers' part. "Speed" (velocity) must be conserved between the IN and OUT portals because the average velocity (v) of an object moving a given distance through space ( = displacement, d) over time interval (t)is
v = Δd/Δt
and since there is no distance between the IN and OUT portals (Δd = 0) then velocity along the axis through the portal must equal zero as well (0 times anything = 0).
The problem is not conservation of velocity but conservation of energy. An object that is raised to a higher point in a gravity field gains potential energy relative to any lower point. For example, as a guy climbs the stairs to an upper floor of a building, he gains potential energy relative to a guy on the ground floor. If he returns to the ground floor, say by stairs or by jumping, that potential energy will be converted to kinetic energy, either gradually (stairs) or all at once (jumping). No matter how he descends, no matter how fast he does it, his potential energy relative to the ground floor will be zero once he is at ground level.
Now let's say the upper floor guy steps through the IN portal upstairs and emerges instantaneously from the OUT portal on the ground floor. His height in the gravity field of Earth relative to the ground floor has changed yet he has gained no kinetic energy. His potential energy relative to the ground floor suddenly disappears! But where has the potential energy of his body gone? It can't have simply vanished -- not if conservation of energy holds.
His "speed" may be conserved, but his energy isn't. It doesn't matter if he steps through the portal or runs through at at a full sprint -- in either case, he has gone from a higher energy state to a lower energy state. Energy can't be created or destroyed. His potential energy has to be somewhere. But where?
It just doesn't make sense -- which of course, has nothing to do with the game's entertainnment value! Thanks for the reply.