Q: Why does energy have to be positive (and real)?

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:

$\begin{array}{ll}\textrm{Kinetic Energy}&\frac{1}{2}mv^2\\\textrm{Occilator Energy}&\frac{1}{2}kA^2\\\textrm{E/M Field Energy}&\frac{E^2+B^2}{8\pi}\end{array}$

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).

The warp drive in practice.

There are plenty of people excited about negative energy (so explore around), but don’t expect any of it to pan out.

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6 Responses to Q: Why does energy have to be positive (and real)?

1. Nate says:

How is this type of negative energy different from potential energy, after all isn’t potential energy negative?

2. Capt. Scarlet says:

Not all physical laws are just observable patterns. The idea that they are is antirational and comes from outdated and discredited Humean views. Some laws exist because of necessity, that is, the situation can’t be otherwise, e.g., a smaller body necessarily orbits a larger one, this is the rational view. Also, there is no such thing as the curvature of space-time because only matter can be curved. And it is difficult to see how energy could be less than 0, that is, less than the vacuum (0-point) energy, the point at which energy by definition is at its lowest point possible.

3. Xerenarcy says:

energy is, until i see evidence to the contrary, interchangeable with the arrow of time – we don’t see time go in reverse for the same reason we don’t find negative energy and negative mass. i’m quite confident you won’t see one without the other two.

4. Shaw says:

I see at least three instances of the Dunning-Kruger Effect. Hi guys

5. Angel says:

Capt. Scarlet: I have to disagree with you there. You claimed that the space-time curve cannot be curved because only matter can be curved. You are challenging Einstein’s theory of relativity. I think you misinterpreted a bit. Let me explain:

Space-time is not a mere lack of matter. As Einstein described, gravitation occurs due to the space time curve, not to matter, which was proven by various experiments. Space time is required to mediate forces at a macroscopic level in the vacuum. The presence of mass causes a distortion is space-time curvature, which complies that space-time curvatures must also be curved.

Also, it would do you good to research the Casimir effect, in which negative energy can be recreated, although it does not last. Not stable, in the same way imaginary mass could not be stable. Imaginary energy is not observable via physical laws like the ones we know. Negative energy, however, is still real energy. Negative matter can arise by compressing high amounts of negative energy into a quantum space and bonding the energy particles with bosons. That is nonetheless impractical to people of our era due to the difficulties that it arises.

A physical law that is true can never be broken. A physical law is a mere universal pattern that can be detected by observation. Such as Newton’s laws, or the Thermodynamic laws. Laws aren’t entitled to exist the way they do, they just do so. It’s called nihilism.

Just so to be clear, negative matter and imaginary matter are hypothetical thus far. Imaginary energy could exist, but it would not be observable as it would alter the current existing physical laws. Negative energy is possible, but as far as we know, it is only possible under the Casimir effects.

6. Angel says:

Update: it is a now a more modern approach for physicists to measure energy as complex numbers. Electrical charge of a particle is the imaginary component, while the mass has the real component.