antonfire wrote:Agent_Irons wrote:The total energy of the system remains at zero. You can't harvest energy from the system because if you try and push around a paddlewheel, say, there's no oomph behind the traveling two-brick assembly. No momentum, no energy, etc. It's not useful, just cool.

The point is that you can take energy out of a bunch of exotic matter simply by accelerating it. So build a flywheel out of exotic matter, and it takes negative energy to spin it up, i.e. spinning it up gives you energy.

And that's the tricky part. Entropy would have this object spinning at impossible speeds and flying through the cosmos. (What's at the core of a neutron star...

) It works similarly to running simulations with negative drag (or running time in a system with drag backwards); objects quickly become unmaintainably fast. To show the effect of unequal masses, though:

mass of matter =

mmass of exotic matter (taken to be positive) =

epositive direction = the matter side

[math]\begin{array}{}

\mathbf{F}=\underset{away}{\mathbf{\hat u}}G\frac{me}{r^2}=\mathbf{\hat u}kme \\

\mathbf{a}_{m\to e}=\frac{\mathbf{F}}{m}=-\mathbf{\hat u}ke=ke \\

\mathbf{a}_{e\to m}=\frac{\mathbf{F}}{-e}=\mathbf{\hat u}km=km

\end{array}[/math]

At this point, we find an unusual problem in solving the mechanics. Under normal conditions, if the accelerations were found to be different, but the pieces are connected, we solve [imath]\mathbf{F}=m\mathbf{a}[/imath] for the total mass, given the net force. However, in this case, no matter what

m and

e are, the net force is zero (as is to be expected by inter-gravitational forces). So we are left with the paradox from the first post: if the masses are equal, the two have the same force (and not infinite or anything else weird either) in the same direction, and they need not even be connected. They merely cruise next to one another, each mutually propelling the other. If we analyze the system as a whole, though, we find the net force to be 0, and therefore they aren't going anywhere! The solution to this mathematical contradiction, to maintain one's sanity, is to treat the exotic matter as if it had positive mass and was being pushed from behind, rather than being negative-repelled. With this in mind, we can finish the derivation:

[math]\begin{array}{}

F_{net}=m\mathbf{a}_{m\to e}+e\mathbf{a}_{e\to m}=2F=2kme \\

\mathbf{a}_{net}=\frac{F_{net}}{m+e}=\frac{2kme}{m+e}

\end{array}[/math]

If

m and

e are equal, [imath]\mathbf{a}_{net}=\frac{2km^2}{2m}=km=\frac{Gm}{r^2}[/math], which makes sense, since each piece has this acceleration individually, so we now have a formula which can calculate the acceleration for the ship in the unequal-mass case. Unfortunately, the assumption that was made has, in a sense, "compromised" the formula. That is to say, it only works (now) on exotic-regular matter interactions, and is no longer applicable in the general case. More importantly, though, it now blatantly violates the law of conservation of momentum. This formula predicts a nonzero net force (with the term applied in the naive sense of "they're accelerating in the same direction") for a closed system for

any nonzero choices of masses. In the strong sense, if the net mass is nonzero (the masses are unequal),

they cannot accelerate on their own without violating CoM. So, to get back to the point, if "real" as applied in this thread means "not in violation of physical law", then what "really" happens in the unequal mass case? Surely any result other than that which is detailed above would be in violation of Newton's laws, so what is the justification? (Or does this paradox, in turn, imply the nonexistence of exotic matter?)