The original question was: I was reading an article about tachyons in Wikipedia and stumbled upon this sentence: “Because the total energy must be real then the numerator [mc^2] must also be imaginary”. I’m confused by the fact that in the article they discuss imaginary mass, but don’t even consider imaginary energy.
My question is why energy is bound to be real? Is there any law that precludes energy from having an imaginary value? Perhaps this somehow follows from the law of conservation of energy?
Also, if you don’t mind, could you please discuss negative energy. Is there a law that prevents it from existing? What would be the implications if imaginary or negative energy has existed?
Physicist: This’ll be a little disappointing.
All physical “laws” are just observed patterns. In every case energy has always been conserved and real.
However, we’ve made observations that imply that energy can be a little negative for a very short time (negative energy virtual particles), but actual negative energy has never been directly observed. Virtual particles (by definition) can never be observed, so the same is likely to be true of negative energy.
So the physical law that forces energy to be real is: “energy is always real”.
You can re-write/change/make-up the laws of the universe to permit energy to have any complex values without running into any particularly nasty problems, although the universe would be extremely, incomprehensibly different. That’s the context that the wikipedia article is written in: made-up physics (There is no reason to think that tachyons exist). And sadly, imagining a thing just doesn’t make it so (I’m talking to you, “The Secret“).
More to the point: when you write down the equation of energy for most systems you find that “the energy is quadratic”. For example:
Where m and k (mass and spring constant) are positive. Since everything else is squared (quadratic) the energy must be positive. The non-quadratic parts (in these cases m and k) always seem to be positive.
With regard to negative energy: if you could get a lot of it, and condense it into negative matter, you could make some serious money. Negative matter, often called “exotic matter” and not to be confused with anti-matter, does exactly the opposite of what ordinary matter does. For example, it repels normal matter and it radiates coldness (as opposed to a lack of heat). But what’s really exciting about it is that it twists up spacetime in weird ways.
The equations that describe the curvature of spacetime are dependent on the distribution of matter and energy in space. You can turn those equations on their head and ask “what is the distribution of matter that would lead to a spacetime shaped like ____?”. Sometimes the result is a distribution of positive mass (and so is possible), and other times the solution requires negative matter (which is a no go).
In this way we’ve managed to figure out: how to arrange matter to force a region of space to move quickly through time (possible with normal matter), stabilize a wormhole (requires lots negative matter), and even build a warp drive (negative matter again).
There are plenty of people excited about negative energy (so explore around), but don’t expect any of it to pan out.