UPDATE October 9, 2021: We changed the claim that the inertia of a clock part is the same up in space and 1% down the potential. Now we claim that the inertia is 1% larger down. The assumptions: the electromagnetic force is 1% less and the inertia is 1% more cause the clock to run 1% slower. Alternatively, we could use as a clock a photon which bounces horizontally between two mirrors 1 meter apart.
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Hypothesis. Let us have a model which consists of special relativity and newtonian gravity. We claim that if we add complex effects of fields, we arrive at general relativity. That is, there really exists no distortion of the flat Minkowski spacetime geometry. We were fooled by the complex interplay of fields and particles to believe that the geometry of spacetime itself is altered.
Our new hypothesis immediately has a challenge in cosmological models. What is the Big Bang if the geometry is the flat Minkowski?
The donut and the photon - the Shapiro delay
photon
~~~~~~ O ~~~~~~
donut
According to general relativity, the gravitational potential at the hole is low, and the speed of light there is slightly less than in empty space. The photon is delayed when it flies through. This is called the Shapiro delay. Since the speed of the center of the mass of the system has to be conserved, it has to be that the donut moved a little bit to the flight direction of the photon.
Conservation of energy and momentum dictates that the donut did not steal any energy or momentum from the photon.
Let us analyze the process in newtonian mechanics in a flat Minkowski space. The photon and the donut interacted, which gave the photon a small effective mass, and consequently, delayed its flight. This is equivalent to the photon flying through a pane of glass. The analysis is intuitive.
The analysis in general relativity is not intuitive. The donut alters the geometry of spacetime around it. The photon travels in that altered geometry and is delayed because of the geometry. And somehow the photon manages to move the donut a little bit forward as it flies through.
Interaction with matter always seems to give the photon a small mass. The photon makes matter to move, which increases the inertia of the photon. If we have a tight string, then obviously, most interactions with an external object delay the propagation of a wave. A part of the energy of the wave is temporarily stored into the external object and the interaction.
If we send a slow massive particle through the donut, gravity speeds up its travel. Does the donut steal momentum from the particle? No. Conservation of energy prohibits momentum transfer to the donut. The particle makes the donut to move a little to the opposite direction. Interaction seems to reduce the inertia of the particle in this case.
The donut experiment is a particularly simple collision experiment which conserves the momenta of the colliding bodies. It does change the position of the bodies.
donut
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+
e- -------------------------------->
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+
donut
Instead of gravity, we may think of an experiment where the donut is positively charged and the photon is replaced with an electron which moves at almost the speed of light.
Calculating the Shapiro delay from newtonian mechanics
Let us model the potential V of the donut with a simple potential well. We have an electron which moves at almost the speed of light.
v ≈ c
e- ---------------->
------- ------- V
\___________/
When the electron descends down to the potential well, it draws energy from the field. When it climbs up, it returns the energy to the field. The electron transported, besides itself, some energy of the field from one location to another.
The electron is carrying extra inertia with it. That is why it slows down.
(The picture is by AllenMcC. See https://commons.wikimedia.org/wiki/File:Flamm.jpg#mw-jump-to-license about licensing.)
In the Schwarzschild metric of a spherical mass, horizontal (normal to the radius) lengths are not contracted or dilated. When a photon flies horizontally, it carries an extra baggage: the energy it drew from the field when it descended down. If the potential is -1% of m c², the extra baggage is 1% on top of the photon energy. The photon consequently flies 1% slower.
The Schwarzschild solution tells us that time progresses 1% slower at that depth in the potential well. The horizontal speed of light is 1% slower in the Schwarzschild solution.
The slowdown of light is explained by newtonian physics. There is no need to appeal to altered spacetime geometry.
The Shapiro delay exists in the experiment with the electron and the electrically charged donut, too. The effect is not specific to gravity.
Why does light slow down to a crawl close to the event horizon?
We saw that when light descends down in the potential by 1% of its current mass-energy, its horizontal speed slows down by 1%. The descent down to the potential -m c² involves an infinite number of such 1% reductions: the horizontal speed of light drops to zero.
We see that it is not a coincidence that the event horizon in the Schwarzschild geometry is at the potential -m c².
What causes the time dilation and radial length dilation in the Schwarzschild geometry?
A pulse of light carries extra inertia from the field when it descends down 1% in the potential.
Let us have a mechanical clock 1% down in the potential. We assume that the parts of the clock mechanism move horizontally.
Apparently, the inertia of a clock part and its gravitational field is 1 % larger than up in empty space. The energy stored in an electromagnetic field is 1% less, and consequently, electromagnetic forces are 1% weaker. This explains why a mechanical clock 1% down in the potential runs 1% slower.
Why is a radial ruler contracted 1% when we descend 1% down in the potential? We should show that the speed of light is 2% slower in the radial direction than in the horizontal direction.
The following reasoning may explain this. Suppose that a pulse of light descended 1 meter down radially in the potential, and drew 1% more mass-energy. Now if it descends still 10 cm down, it will draw 0.1% more and that energy will come from a distance of 1 meter. We assume that the energy comes from the "surface" of the potential.
The extra 1% of energy is displaced by 10 cm, and an additional 0.1% is displaced by 1 meter. This means double displacement relative to a horizontal movement of 10 cm. The effective inertia in the radial direction is 2% higher while in the horizontal direction it is only 1% higher.
Optical gravity, the equivalence principle, and renormalization
We had the idea of "optical gravity" in this blog in 2018 but were not able to determine what is the "medium" where light travels. Isaac Newton had a similar idea of optical gravity some 300 years ago.
Now we see that interaction with matter can be abstracted into a "medium" where light propagates. In general relativity, the medium is further abstracted into spacetime geometry.
The equivalence principle of general relativity is like renormalization in quantum field theory. Physical processes in a gravitational field involve complex interaction with the gravitational field and other matter, but we can define effective mass, time, and length in a way that makes physics in a freely falling laboratory look like physics in an inertial laboratory in a flat Minkowski space.
What is the event horizon like? Is there a singularity in a newtonian black hole?
In our newtonian model, the event horizon is the place where light stands still. A pulse of light there is carrying an infinite baggage of inertia with it. The inertia is relative to the mass which formed the newtonian black hole.
In quantum mechanics all matter is described as waves. All waves have come to a standstill at the event horizon.
What about the matter within the event horizon? Can it still move?
How would physics inside the horizon know that a collapse happened and an event horizon formed? Astronomers living inside the horizon notice that they no longer can receive any signal from outer space. And if they try to send a light signal out, it will be frozen at the horizon. The fields of the matter within the horizon are frozen at the horizon. That causes some forces if astronomers try to move objects.
The infinite inertia at the horizon is caused by fields extending to outer space. Matter within the horizon does not experience such inertia if the fields are frozen at the horizon.
We conclude that life within the horizon can proceed in the normal way, but communications to the outside world are cut off.
What happens in the merger of two newtonian black holes?
It might be that matter which ends up inside the event horizon of the combined system can start moving again. The merger might progress in a fashion which resembles general relativity. We now know from LIGO that black hole mergers do look like what general relativity predicts.
Are there singularities in newtonian black holes? Inside the event horizon, a new collapse may happen, and a new event horizon is formed. The process continues until the remaining part inside the innermost event horizon no longer collapses. There is no singularity, if we do not regard the event horizon as such.
Conclusions
We showed that newtonian gravity combined with the Minkowski geometry may be able to imitate the Schwarzschild geometry.
We need to study rotating black holes and the Kerr geometry. Can we explain frame dragging with the newtonian model?
Can we somehow prove that our newtonian model is "right" and general relativity just an illusion? An old problem in general relativity is how gravitational waves can carry energy. They do not act as a "source" in the Einstein equations, like matter fields do.
Suppose that a gravitational wave gives energy to matter, for example, by making a bell to ring. Then the source of the Einstein equations suddenly grows on its own, without there being flow of energy which is visible for general relativity. Does this break general relativity?
In our October 5, 2021 blog post we asked if vacuum polarization makes the gravitational field very strong close to the horizon. The answer is no. It is newtonian mechanics which creates the strange phenomena close to the horizon.
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