Maybe it is best not to interpret the Doppler shift thermodynamically.
Increase in inertia in a moving body
In addition to the Doppler shift, clocks have slowed down in the moving body. The body appears cooler because of time dilation. What does this mean in thermodynamics?
The inertia of mass in a moving body grows by the Lorentz factor
γ = 1 / sqrt(1 - v² / c²),
as seen by a static observer.
We could explain the slowing down of clocks with increased inertia.
Suppose that we have a test mass bouncing inside a box, riding a moving body.
^
|
• ● ---------->
| body
v
test mass
The inertia of the test mass grows as we speed up the body, but the vertical momentum p of the test mass stays the same, as seen by a static observer. The kinetic energy
E = p² / (2 m)
appears to become smaller, thinks the static observer. However, we know that the kinetic energy actually became larger. If an observer riding with the body stops the test mass, harvests its kinetic energy, and sends the energy to the static observer, the static observer receives more energy. We conclude that the formula for kinetic energy simply does not hold in this case.
Growing inertia by attaching weights
Suppose that we have a molecule of a gas bouncing inside a box. If we suddenly attach an extra weight to the molecule, the molecule starts to move slower. The gas has cooled down. But what absorbs the kinetic energy which the molecule lost? The momentum p stayed constant, but the kinetic energy
E = p² / m
diminished.
spring
• /\/\/\/\/\/\ ● ---->
extra weight molecule
Maybe we have a spring which we used to absorb the shock when we attached the weight to the molecule? The lost kinetic energy went to potential energy in the spring.
Growing inertia by approaching a large charge
Let us have a charged molecule q in a box. It bounces horizontally in the box, and we lower the box vertically toward a large charge Q.
The charges could as well be gravity charges, but let us imagine that they are electric charges, so that we do not confuse our thinking with general relativity.
box
| ----------------------
| | <----- • -----> |
| ----------------------
v q
●
Q
As the box approaches Q, the inertia of q increases, and it moves slower. The momentum p of q stays the same, but its kinetic energy E decreases. Where does the lost kinetic energy go?
Besides the electric potential, the moving charge feels a magnetic vector potential as it moves in the field of Q. If the charges of Q and q have the same sign, then the lost kinetic energy might go to the vector potential. But if they have opposite signs?
Our elastic field lines model offers a solution: a moving charge q has the field lines bent. Bent field lines require more energy. The lost kinetic energy probably goes to bent field lines. This is analogous to the previous section where we used a spring to attach a weight to a molecule.
A gas of charged particles appears to cool down close to a large charge
Let us have a gas of charged particles. The particles are moving and their combined kinetic (= heat) energy is E.
We let the gas approach a large charge Q in a way such that the particles do not gain or lose kinetic energy from the attraction or repulsion of Q.
The gas appears now to be cooler. The particles send longer wavelength radiation when they pass close to each other because they move slower.
The gas radiates at a lower rate because the particles move slower.
We can still harvest the original kinetic energy E entirely if we stop the particles from moving, and we can send E to a faraway observer.
How to interpret this? Maybe we can say that some of the original kinetic energy was stored as some kind of "potential energy". When we harvest kinetic energy from the system, the potential energy flows back to the particles as kinetic energy, and we can harvest more kinetic energy.
The gas appears cooler because a part of its kinetic energy was stored as potential energy.
Collision of clouds of opposite charges
Suppose that we have a cloud of electrons and a separate cloud of positrons. Electric attraction makes the clouds to collide, and all the matter may annihilate.
In the process the energy in the rest mass of the particles was entirely converted to a kind of heat, electromagnetic radiation.
The radiation can freely depart because the photon does not possess an electric charge. Faraway observers see a hot explosion.
The course of events would probably be very different if photons would possess a charge which would attract them strongly to other photons.
Collision of hot clouds of matter
Let us assume that the clouds are initially very hot, so that almost all of their mass-energy is heat, or kinetic energy of random motion. Heat is high-entropy energy.
We let gravitational attraction make the clouds to collide.
gravity
● --> <-- ●
hot cloud hot cloud
We analyze the process from the point of view of a faraway observer. The amount of energy is defined as how much energy would the faraway observer receive if the energy were transported to him.
When the clouds accelerate toward each other, some of the mass-energy of the cloud is converted to kinetic energy of ordered motion, or low-entropy energy.
At the collision, the ordered kinetic energy is converted to random kinetic energy again. That is heat.
We started with heat and ended up with heat. Naively, one might assume that the faraway observer sees the merged cloud having the same temperature as the original clouds.
However, now the particles of the clouds are under their collective gravity field. Being in the field of a large charge Q made the gas of charged particles appear cooler, as we deduced in an earlier section.
This is probably the mechanism which makes black holes extremely cold. The field is very strong. Almost all the heat energy is stored as a kind of potential energy.
The effect is some kind of a "syrup" effect. Individual particles move extremely slowly, and can only radiate energy away at a very slow pace.
Converting heat to low-entropy energy
Let us describe an alternative course of events in the collision of the two hot clouds.
spring
● /\/\/\/\/\/\/\/\/\/\ ●
hot cloud hot cloud
We put the hot clouds inside vessels. We assume that the particles only bounce "horizontally in the vessel as we lower the vessel down toward the other vessel.
We harvest their gravitational potential energy with a spring between the clouds.
We are able to convert most of the heat energy into low-entropy energy in the spring. Did we invent a perpetuum mobile of the second kind?
No, because we cannot make the process to run in a loop.
However, we did make the heat appear cooler to a faraway observer. He sees thermal radiation redshifted.
The lower temperature is fundamentally due to the fact that the energy levels of the cloud have been squeezed because of a low potential. There is less energy in the heat, and the energy levels are accordingly squeezed closer to each other.
The entropy, or the "disorder" of the system stays constant. The particles still move in a completely disorderly fashion. They just move slower, as seen by a faraway observer.
Let us imagine a black hole for which most of the ordered kinetic energy of the collapsing mass has already become disordered random kinetic energy. We could call this "the equilibrium state".
We believe that the temperature T of such a black hole is very close to zero. Consequently, if we add the energy dQ, the increase in entropy, dS, has to be huge. There must be a very large number of states at a very low energy. That is the case if the potential is very close to
-m c²
for a test mass m. The space of different energy states must be squeezed immensely.
On the surface of a neutron star, the potential is of the order
-0.1 m c².
The potential in a black hole is probably very close to -m c². Why would the potential stay at a larger value?
How fast does the kinetic energy in a collapsing black hole become disordered?
Let the two hot clouds collide. The inertia of each cloud, as a bulk object, cannot grow much because from where would it get huge inertia? From the other cloud? But the other cloud only has the same mass and cannot give huge inertia.
Since the inertia does not grow hugely, the clouds collide in a short time, measured in the global Minkowski time.
Then starts the phase of converting the ordered kinetic energy into random kinetic energy. It is some kind of turbulence.
It is like a collision of two flows of syrup. How quickly does the disorder spread to the low level?
As long as the kinetic energy is still mostly ordered, the black hole will appear even colder than in the final equilibrium state.
Another question is how quickly does the syrup collapse into the tiny quantum object which we described in a blog post on October 19, 2021.
We do not yet know the details, but we know that the syrup process obeys rules of classical physics, and at the late stage, quantum mechanics. There are no singularities, nor one-way membranes. The process does not involve the Hawking information paradox.
Conclusions
We showed that electromagnetism has an analogue for the low temperature of black holes.
We still have to analyze the syrup effect in more detail.
Also, we have to figure out how photons escape the syrup and show up as thermal radiation to a faraway observer. Is there any problem in that? Probably not.
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