Particles having negative energy in the Penrose process (near the Black Hole) have negative energy in the comoving (local) coordinate system or in general, background, coordinates?
Penrose process - Wikipedia
In non-relativistic quantum mechanics, the zero point of energy is arbitrary, it's just an additive constant. Therefore, the difference between positive and negative energy is an alteration of a non-observable additive constant.
According to relativity, a particle's rest-mass energy naturally correlates with the particle's zero-point energy; because a particle at rest has E= mc^2 worth of energy, there's no additive for a constant amount of energy.
Hence, there are two equivalent interpretations::
A.) Particles with negative energy are positive energy particles traversing backwards in time. or
B.) Particles with negative energy are positive energy anti-particles traversing forwards in time.
The sign of the energy appears in 2 key locations. The first is the wave function of particles
ψ(t)∼eiEt/ℏψ(0)=e−i|E|t/ℏψ(0)=ei|E|(−t)/ℏψ(0)ψ(t)∼eiEt/ℏψ(0)=e−i|E|t/ℏψ(0)=ei|E|(−t)/ℏψ(0)
Hence, the negative energy particle would evolve in time exactly the same way that a positive energy particle would evolve as it were traversing backwards in time.
The second interpretation occurs if you conceptualize what would happen if an electron were traversing backwards in time while having its charge measured; its charge would disappear as it were passing through a detector. The same would happen with an antiparticle appearing in the detector (where antiparticles are particles with opposite charges).
The issue of how negative energy solutions are handled arises when dealing with the relativistic versions of quantum mechanics:
√E=±p2c2+m2c4
Note, that this is very different than a negatively squared mass particle, a tachyon, which behaves fundamentally different than an antiparticle.
