*Post by SEKI*Hawking radiation postulates that particles with negative energy

fall into a black hole.

That is only *one* possible, the *alternative* conceptualization of Hawking

radiation (HR), and it only applies for an observer far away from the event

horizon (EH) of the black hole (BH).

Usually HR is *locally* conceptualized as the result of pair production (PP)

from vacuum fluctuations (VF) near but outside the EH, where due to the BH

the virtual particles become real, one particle falls beyond the EH (“into

the black hole”) and the other one does not. So considering that the vacuum

energy from which the pair was created formerly contributed to the energy

density (ρ = dE∕dV) of the BH, the BH loses mass–energy (density) equivalent

to that of the escaping particle’s total energy (E) (density). As the

(assumed non-rotating, uncharged) BH’s Schwarzschild radius is rₛ = 2 G M∕c²

and E₀ = M c², if dE₀∕dt < 0, then dM∕dt < 0 → drₛ∕dt < 0: the BH shrinks.

It can also be conceptualized as PP from VF beyond the EH where one of the

particles tunnels through the EH (see “quantum tunneling effect”); the same

as above applies.

<https://en.wikipedia.org/wiki/Hawking_radiation>

*Post by SEKI*Einstein's gravitational equation is presented in terms of linear

expression of energy-momentum tensor.

That is an oversimplification. Instead, the _Einstein field equation*s*_

(EFEs) state that the curvature in a region of spacetime (ST) is equivalent

to the density and flux of energy and momentum in that region.

The EFEs are:

G_µν ≡ R_µν − (½ R − Λ) g_µν = 8πG∕c⁴ T_µν

“ST curvature” = “energy–momentum”

G_µν – Einstein tensor

R_µν – Ricci curvature tensor

R – Ricci curvature scalar R = tr_g R_µν

Λ – cosmological constant

g_µν – metric tensor

G – Newton’s gravitational constant

c − speed of light in vacuum

T_µν – (stress–)energy–momentum tensor

<https://en.wikipedia.org/wiki/Einstein_field_equations>

*Post by SEKI*So, I suppose that particles with negative energy, if actually

present, are to be repelled by the black hole.

No, I do not think so. Contrary to popular belief, a BH is _not_ a massive

*body* (with the Schwarzschild radius as its radius) that attracts

everything, but a *region* of *ST* where the ST curvature (STC) is that

large that “space falls faster than light” (Andrew J.S. Hamilton, see also [1]).

The total STC in the ST neighborhood of the negative-energy particle (NEP)

is the result of the STC equivalent to the energy–momentum of the NEP *and*

that of the BH *combined*. The energy−momentum density and flux, and

therefore the STC in the BH’s singularity is *infinitely* *large* (that is

why it is a *singularity*: all spacetime geodesics *end* there).

That is, even if the ρ_NEP of the NEP is *a little* negative, the ρ_BH in

the region beyond the EH is *infinitely more positive*. So the total scalar

STC where the NEP is is very much *positive*, and *all* geodesics for the

NEP *still* lead *towards* the singularity, “*into* the BH”.

[That is also why the evaporation of large BHs due to HR would take a

very long time, and does not even happen if the BH is large enough and

there are enough positive–energy particles in the vicinity to fall into

it: HR would still be emitted, but in total the BH will still *stay the

same* or *grow*.]

IOW, for us distant observers to observe the NEP to be repelled by the BH,

i.e. for the total scalar STC to be negative, the absolute of its negative

ρ_NEP would have to be *greater than* the absolute of the (positive) energy

density of the BH:

|ρ_NEP| > |ρ_BH|

IOW, the NEP would have to be a white hole that is greater than the BH.

Obviously, that is NOT so.

If that helps, you can think of the NEP near the EH like a canoe near a

waterfall: Sitting in the canoe, you can use all your strength (negative

energy) paddling against the downward current, but eventually you will still

go down the waterfall because the speed relative to the abyss that you can

achieve by paddling against the current is still less than the speed of the

downward current:

At the EH, space is falling at the speed of light in vacuum, c.[1] A NEP

with non-zero mass m cannot even move away *at* c, much less *more* than that:

T = E − E₀ = m c² (1∕√(1 − u²∕c²) − 1)

lim T = +∞,

|u|→c

where T is the particle’s local kinetic energy and u is its local relative

(3-)velocity.

[1] <http://jila.colorado.edu/~ajsh/insidebh/waterfall.html>

*Post by SEKI*Then, isn't Hawking radiation unrealistic?

Yes, it is NOT. You are proceeding from false assumptions.

However, note that as long as we have no theory of quantum gravity¹, HR is

really only a semi-classical hypothesis: it is based on the *assumption*

that the laws of quantum mechanics apply *unchanged* in curved ST. AFAIK,

to date there is no evidence of HR, no evidence of evaporating BHs.

You should have asked this in <news:sci.physics.relativity> as it pertains

more to BHs and general relativity (GR) than particle physics.

_______

¹ <https://en.wikipedia.org/wiki/Quantum_gravity>

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