orbital mechanic

If this thing always produces constant thrust per constant power input without propellant and with higher efficiency than a photon drive then it seems unavoidable that it can be turned into a free energy machine, as there will be some speed (less than c) after which it will be gaining more kinetic energy than the energy you put in.

Are you trying to claim that using virtual particles in this manner is consistent with quantum mechanics as it is currently understood? Because it seems that the people NASA hired to investigate the theoretical validity of the work done by Eagelworks reached the opposite conclusion: White and his lab believe current QM to be wrong about how the quantum vacuum works but even they are not claiming that their work is supported by current quantum mechanics.

I think it is of the order of h/(mc^2), where h is the planck constant and m is the mass. Quite a few orders of magnitude smaller than that.

Let's say I design my lottery machine such that the winning numbers are produced by a random number generator that uses a geiger counter and ensures that each bit has equal probability. How is your computer going to predict the numbers?

Yes, it has. Particle accelerators move particles at speeds very close to the speed of light and the dilation observed in these particles agrees with the predictions of relativity perfectly. If you are talking about special relativity, then no. Quantum Mechanics and Special Relativity are perfectly compatible. Time dilation is indeed not linear: https://en.wikipedia.org/wiki/Time_dilation#Time_dilation_due_to_relative_velocity

Not true and not what relativity says. Very wrong, as stated by others already. Time dilation has been experimentally observed many times and works exactly as predicted by the Theory of Relativity.

It is not antimatter, but matter with negative mass/energy. There is a big difference: antimatter is known to exist and we can even produce it (although it is very difficult/expensive to produce and store). Matter with negative mass is not known to exist at all. Some people (non-physicists) claim that they can use the casimir effect for a proof of concept. Last time I looked at this, mainstream physicists consider this nonsense. This does not necessarily mean that it can't be done, but we should be clear that it is not supported by mainstream physics. Also, it is important to note that even in the best case this is only useful to detect if spacetime can be warped in such a way. It is very clear that this particular method, even if it works, cannot be used for actual propulsion, so you still need to find actual negative mass/energy if you want to build an Alcubierre drive capable of propelling a ship. Another point, that is not usually brought up is that if you can actually build an Alcubierre drive (and assuming General Relativity is correct) then it immediately follows that you can also build a time machine. This does not necessarily mean that the Alcubierre drive is impossible but it is important to note and in fact if it were possible it would be far more significant than actual faster-than-light travel.