In this blog we have argued that time, as seen by a faraway observer, cannot stop completely inside a black hole of a mass M, because the inertia of an arbitrary particle cannot grow larger than M. If we have a photon inside a solar mass black hole, the inertia of the photon can grow at most
M / m
-fold, where m is the mass-energy of the photon. For a 1 eV photon inside a solar mass black hole, the ratio is ~ 10⁶⁶.
How do we know that M is the maximum possible inertia? We do not know. There might exist a "gear system" which raises the inertia to be even larger than M.
What is the lowest possible inertia? We do not know that either. Let us try to find out.
The increase of the Schwarzschild radius from a falling particle
The Schwarzschild radius of a black hole of a mass M is
R = 2 G / c² * M
= 1.5 * 10⁻²⁷ m/kg * M.
If let a particle of a mass m falls toward a Schwarzschild black hole, we may expect the movement of the particle to differ from the idealized case of a zero mass particle when it is at the distance of
R' = 2 G / c² * m
from the horizon in the Schwarzschild standard coordinates. This is because the radius of the black hole grows from R to R + R' as the particle is absorbed.
How slow does time progress at that distance, if we use the Schwarzschild metric for the mass M?
The metric is:
We have
dτ² = (1 - R / (R + R')) dt²
<=>
dτ = sqrt(R' / R) dt.
This figure might give us an estimate on how slowly a particle will sink into the black hole. The particle moves essentially at the local speed of light. The speed of the particle as seen by a faraway observer is
sqrt(m / M) c,
if it moves horizontally. If it moves vertically, then the speed is
m / M * c,
because the radial metric is stretched. Here c is the speed of light in faraway space.
The Schwarzschild metric is probably correct if the particle is "far" from the horizon. Very close to the horizon, and inside the horizon, we expect the particle to move even slower, relative to a faraway observer.
For a solar mass black hole, the vertical speed of a falling proton would be
10⁻⁵⁷ c,
as seen by a faraway observer. That is
3 * 10⁻⁴⁹ m/s.
The journey to the center would last 10⁵² s, or 3 * 10⁴⁴ years.
Filling a black hole with black body radiation: the Hawking temperature is much too high
If the speed of light is as slow as calculated above, then the black hole will appear to have an immense volume for a photon inside it. Furthermore, a smaller energy photon will see the volume even larger.
The mass-energy density of black body radiation is
σ T⁴ / c³,
where σ is the Stefan-Boltzmann constant
σ = 6 * 10⁻⁸ W / (m² K⁴).
The Hawking temperature of a solar mass black hole is 60 nanokelvins. Let us calculate what is the mass of the black body radiation if we fill a solar mass black hole with it.
The wavelength at 6000 K is 0.5 micrometers. The wavelength at 60 nK is 0.5 * 10⁵ m or 50 km.
The mass of a photon is
m = h c / λ * 1 / c²
= h / (λ c).
For 60 nK, the mass of a photon is m ~ 10⁻⁴⁷ kg.
The mass of the Sun is M = 2 * 10³⁰ kg.
The mass-energy density of black body radiation at 60 nK is
~ 10⁻⁶⁰ kg/m³.
Let us assume that the speed of light inside a solar mass black hole is
~ sqrt(m / M) c
~ 10⁻³⁸ c.
The volume of the black hole is then
V ~ (10³ / 10⁻³⁸)³ m³
= 10¹²³ m³.
The mass-energy of the black body radiation is
~ 10⁶³ kg,
or much larger than the mass of the Sun.
We conclude that the Hawking temperature is much too high, at least by a factor 10¹⁰.
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horizon
"hole"
outside black hole
space interior
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If we lower the temperature by such a huge factor, then the photons will have a wavelength which is too long to let them escape through the "hole" which is the black hole horizon. The diameter of the hole is ~ the Schwarzschild radius.
The time to thermalization, or "scrambling"
Yasuhiro Sekino and Leonard Susskind (2008) claim that a black hole attains a thermodynamic equilibrium phenomenally fast, in a fraction of a second.
Our analysis suggests that it is the exact opposite: it will take an immense time for a black hole to thermalize. This is because the speed of light is so slow inside a black hole, as seen by an external observer. It can take 10⁴⁴ years for infalling particles to collide at the center.
Let us analyze this with an object which is "almost a black hole". Let us have very massive thin spherical shell of matter which is immensely strong so that it can withstand the gravitational pull without breaking. Alternatively, we can assume that the shell is kept from collapsing by filling its interior with very lightweight, incompressible matter.
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/ \ heavy shell of matter
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\______/
We can put the entire shell into a very low potential, without it collapsing to form a black hole.
Now if we pour some gas (or photons) down on the shell, it will take a long time to thermalize because the heat and vibrations propagate at most at the speed of light.
The closer the shell is to be inside its Schwarzschild radius, the slower the thermalization.
If we let the shell to collapse into a black hole, an external observer will see the matter in the shell falling slower and slower toward the horizon. Thermalization in this case probably takes an immense time.
Question. The slow speed of light is due to the photon adopting a lot of inertia from its interaction with the heavy mass in the system. Could it be that this interaction quickly thermalizes the photon?
Let us analyze this for a photon on the surface of Earth. If the photon would lose some of its energy to the 6 * 10²⁴ kg of atomic matter in Earth, we would observe scattering or some other optical phenomenon. We might also see a gravitational lens of a galaxy be somewhat opaque. We conjecture that the extra inertia does not help a photon in thermalization.
Conclusions
An arbitrary physical system, like a neutron star, does not usually have any one temperature. The temperature varies depending on the part of the system, and varies with time. The temperature hypothesis of Stephen Hawking is at odds with this principle of physics. Hawking's hypothesis leads directly to the information paradox of black holes.
The information paradox constitutes strong evidence against the hypothesis of Hawking.
The speed of light inside a black hole seems to be extremely slow, as measured by a faraway observer. If a photon of a certain frequency falls freely into the black hole, its wavelength is extremely short relative to the Schwarzschild radius. Thus, the volume of a black hole is immense from the point of view of an individual photon.
Since speed of light is very slow, it takes a very long time for a black hole to thermalize. Billions of times longer than the age of the universe.
For photons, the black hole appears to have a volume much larger than the visible universe, connected to the outside space through a hole whose size is ~ the Schwarzschild radius.
If we let photons to thermalize in the large volume, their temperature will be 10⁻¹⁰ the Hawking temperature, or less. The photons have such a long wavelength that they cannot escape from the black hole through the hole which is the horizon. Also, the temperature is so low that essentially zero photons will escape. The black hole is truly black.
However, in principle it is possible for a photon to escape. The horizon is not a one-way membrane. We have argued earlier that a physical system cannot have a one-way membrane.
