ME: I just sought the Relativistic math behind the acquisition of a particle of negative energy with respect to infinity, resulting in it depleting the black hole.
AI: The pure relativistic mechanism behind a particle possessing negative energy relative to infinity is determined by the projection of its four-momentum onto a time translation Killing vector field. [1, 2]
In General Relativity, energy is not an absolute scalar value. It is an observer-dependent quantity calculated by taking a vector dot product. When a particle crosses a boundary where the geometric flow of time changes directions relative to a distant observer, its mathematically calculated energy flips from positive to negative. [1, 2]
The exact relativistic math of how this happens and depletes the black hole breaks down into four sequential stages:
- Define Energy Relative to Infinity [1]
To define the energy (E) of any particle in a stationary spacetime, we map a mathematical vector field called a Killing vector field (\xi ^{\mu }), which represents the symmetry of time translation (the fact that the geometry of spacetime does not change from one moment to the next). [1, 2, 3]
For an observer sitting infinitely far away from the black hole (at âasymptotic infinityâ), this Killing vector points directly forward along their timeline. The energy of a particle with a four-momentum vector (p^{\mu }) as measured by that observer at infinity is defined by the covariant dot product: [1, 2, 3]
E=-g_{\mu \nu }p^{\mu }\xi ^{\nu }=-p_{\mu }\xi ^{\mu }
In normal, flat space, \xi ^{\mu } is strictly timelike (g_{\mu\nu}\xi^\mu\xi^\nu < 0). Because a physical particleâs four-momentum p^{\mu } must also be future-directed and timelike, this dot product is mathematically forced to always yield a positive energy (E > 0). [1, 2, 3]
- Locate the Causal Flip (The Killing Horizon)
As you move closer to a black hole, the metric components (g_{\mu \nu }) change. In a standard Schwarzschild (non-rotating) black hole, the metric component associated with time translation is: [1, 2, 3]
g_{tt}=-\left(1-\frac{2GM}{c^{2}r}\right)
When a particle crosses the event horizon (r = r_g = \frac{2GM}{c^2}), the sign of g_{tt} flips from negative to positive. [1, 2]
This means that past the horizon, the time translation Killing vector \xi ^{\mu } changes its physical character entirely: it transitions from a timelike vector to a spacelike vector. Inside the horizon, moving along the direction of \xi ^{\mu } no longer means moving forward in timeâit means moving through space. [1, 2, 3]
- Mathematical Realization of Negative States
Because \xi ^{\mu } is now a spacelike vector inside the horizon, the mathematical restriction that forced the dot product (-p_\mu \xi^\mu) to remain positive is completely removed. [1]
Inside this region, a perfectly real, physical particle can exist with a valid, future-directed timelike momentum vector (p^{\mu }) whose alignment relative to the geometry causes the dot product to flip signs: [1]
E=-p_{\mu }\xi ^{\mu }<0
Crucially, this particle does not feel anything ânegativeâ in its own local frame of reference. To a local astronaut falling into the black hole alongside it, the particle has standard, positive energy. The energy is only globally negative when mathematically evaluated against the time standard (\xi ^{\mu }) of the observer sitting at infinity. [1, 2, 3]
- Quantify the Depletion of the Black Hole
The depletion of the black holeâs mass is governed by the conservation of the total energy-momentum tensor (T^{\mu \nu }), which satisfies the continuity equation \nabla_\mu T^{\mu\nu} = 0.
When quantum field fluctuations spontaneously create a wave state near the horizon, the state is split. [1]
- One component carries a positive energy flux (\Delta E_{out} > 0) and escapes to infinity.
- The entangled partner component carries a negative energy flux (\Delta E_{in} < 0) and crosses the horizon, tunneling into the interior region where such states are geometrically permitted. [1, 2]
When the black hole absorbs this inward-directed negative energy flux, its total mass (M) changes according to the relativistic mass-energy equivalence: [1]
\Delta M=\frac{\Delta E_{in}}{c^{2}}
Because \Delta E_{in} is a negative value relative to infinity, the addition of this particle directly forces the black holeâs mass to decrease (\Delta M < 0