We cannot define "the curved metric of spacetime" for a complex system?

UPDATE October 12, 2021: We added that one can define pressure by varying the spatial metric, and toned down our claims.

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Consider again the pressurized vessel example from yesterday. We have a test particle inside the pressurized vessel. This time the particle carries both an electric charge q and a mass M.

              _______________

             |                            |

             |    ● M, q             |

             |______________|

              pressurized vessel

        charge q ---- electromagnetic interaction ---- atoms in the liquid

        mass M ---- gravity ---- atoms in the liquid

The particle interacts with the atoms in the liquid.

We assume that both the electric charge q and the mass M affect the pressure in the vessel.

We can determine the force F on the particle by varying the position of the particle and calculating the total energy of the system at different positions. There is an effective potential for the particle inside the vessel.

Can we determine which part of the force is due to gravity and which part due to the charge q? 

Usually not. The potential typically is not linear on both M and q. If it is not linear, there is no meaningful way to divide what part of F is due to gravity and what part due to electromagnetism.

One could claim that the contribution of gravity to F is what the Einstein field equations calculate from the pressure, and the rest is due to electromagnetism. This claim sets electromagnetism inferior to gravity.

A complex system under gravity and electromagnetism

         •       •       •          system S

         •       •       •          of many particles

         •       •       •

                    ● test particle M, q

Let us then consider another experiment. There are many particles bound to each other with various forces. Let us call this the system S.

We define that in a volume element, the pressure in the x direction is obtained by varying the spatial metric in the x direction, and calculating the change in the energy of the system.

If we move a test particle with a mass M and a charge q close to the system, then S is deformed, and the energy of the system changes. There is an effective potential on the test particle, and a force F.

Can we this time determine what part of the force is due to gravity?

Again the potential probably is not linear on M and q, and there is no meaningful way to divide the force F between the effects of gravity of M and the charge q.

The Einstein field equations calculate the metric from the pressure, and other parameters. If the potential from pressure varies with q, then the equations say that the gravitational force is different for particles with the same mass M, but a different electric charge q. This breaks an equivalence principle which says that the gravitational force is the same for all particles which have the same mass.

What about setting q to zero but keeping the other system as is, varying the position of the test particle, and claiming that the calculated force on the particle is the force of gravity? This is unsatisfactory because then we ignore possible gravitational effects of the electric field of q.

What if we remove pressure and shear stresses from the stress-energy tensor?

Then the metric of spacetime would be defined solely based on the location and the movement of mass-energy. It is like the electromagnetic field, which is determined by the charges and their movement.

Could this work?

The metric is then different from the metric of general relativity, though, since pressure and shear stresses no longer have an influence on the metric.

Does this affect the Schwarzschild solution outside the central mass? Yes. The metric must be matched with the interior solution, and the interior solution is affected by pressure.

We conclude that it is not a good idea to drop pressure and shear stresses from the stress-energy tensor.

Conclusions

For a complex system we cannot define "the curved metric of spacetime" in a good way. The hamiltonian or lagrangian of the system couples gravity and other forces, and we cannot split the total energy between different fields. We cannot say what is the separate effect of gravity alone.

If we treat gravity as an ordinary field under the flat Minkowski metric, this problem does not arise. There is no need to split the combined effect between different fields.

Under most conditions, gravity successfully imitates a curved metric of spacetime. This is why people in the last century were mislead to believe that gravity is a curved metric of spacetime. The example of the system S shows that the geometric interpretation is not very good for a complex system.

Pressure does not produce gravity: the force is the combined effect of gravity and the pressure force field

UPDATE January 19, 2022: We estimated today the radial speed. Karl Schwarzschild was right.

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In this blog we have been wondering why pressure appears in the energy-momentum tensor in general relativity, while from the interior Schwarzschild solution we see that pressure does not change the spatial metric at all. Pressure does slow down time, though.

It is as if the spatial geometry of spacetime would have infinite rigidity against forces which try to stretch it.

            _____________

          |                          |

          |     ● M              |

          |_____________|

           pressurized vessel

We have been trying to devise a perpetuum mobile which would take advantage of the infinite rigidity of the spatial metric.

Let us have an infinitely rigid vessel filled with incompressible fluid. By moving a mass M in and out of the vessel we can change the volume of the vessel, and potentially create an infinite pressure. That might make a perpetuum mobile possible.

Now we realize that the "gravity" produced by pressure is not gravity at all, but the combined effect of the gravity of the mass M and the force field which creates the pressure.

The mass M has an approximate Schwarzschild metric around it. Radial lengthts have grown a little, but horizontal lengths are as in the surrounding space. The spatial volume around M is a little larger. Moving M into the vessel frees energy from the force field which creates the pressure, because the volume of the vessel increases slightly.

Why does light slow down in the vessel? It is because a photon entering the vessel acquires some extra energy from the pressure force field, and carries it to the other side of the vessel before leaving the vessel. The photon carried some extra baggage, which caused extra inertia and slowed down its progress.

Let us assume that the extra baggage on a photon is 1%.

We can explain the slowdown of clocks inside the vessel just as in the case of the Schwarzschild metric: clock parts have 1% larger inertia than in outer space because moving them causes energy to be displaced in the pressure force field. The electromagnetic field, which moves the parts of the clock, has grown 1% weaker inside the vessel because the gravity of the electromagnetic field did some work when we lowered the clock into the vessel.

We still have to explain why the radial speed of light in the vessel is the same as the horizontal speed. In the Schwarzschild metric, the radial speed is slower. The potential inside the vessel is the lowest at the center, just like in the Schwarzschild metric.

The reason might be that the field energy in the Schwarzschild case is spread over a large volume of space. Moving a photon radially in the field can cause a displacement of field energy over large distances.

But in the vessel, the extra volume generated by the mass M is (?) close to the mass. Maybe that is the reason why the energy in the pressure field has to be displaced the same distance s that we move the mass M.

The Einstein-Hilbert action

https://en.wikipedia.org/wiki/Einstein%E2%80%93Hilbert_action

The Einstein field equations are derived from the Einstein-Hilbert action through varying the metric. The action contains a lagrangian for the matter fields, as well as a term for gravity.

It is no wonder that the "gravity" which is generated by pressure really is interplay of the pressure force field with the Schwarzschild geometry which is around a test mass M.

The stress-energy tensor contains terms for "simple" interactions of the metric with matter fields. We can imagine a more complicated interaction which cannot be reduced to pressure or shear stresses.

If we have "pure" gravity between point masses, with no other interactions, then the stress-energy tensor has the pressure and shear stress elements zero. We could say that this is the "real" gravity.

A photon inside glass versus a photon in the vessel

       photon ---- electromagnetism ---- charges in atoms in glass

       photon ---- gravity ---- charges in atoms in liquid

A photon slows down in any interaction. Inside a glass pane, the electromagnetic field of the photon interacts with the charges of the atoms in glass.

In the pressurized vessel experiment, the photon interacts through its gravity with the charges in the atoms of the liquid which fills the vessel.

General relativity claims that in the vessel case, it is the geometry of spacetime which guides the photon, while inside glass it is an ordinary force. Why is gravity special?

Philosophical aspects - is spacetime really curved?

In newtonian gravity, only mass-energy produces gravity - pressure or shear stresses do not produce gravity. It looks like the same holds for general relativity, once we recognize that pressure and shear stress elements in the stress-energy tensor only describe indirect effects which are caused by gravity along with other fields.

Does pressure alter the metric of spacetime?Pressure certainly attracts matter, but our analysis did not require us to assume that pressure itself causes any change in the spacetime metric. We did assume that the Schwarzschild metric surrounds our test mass M, but we showed in our October 7, 2021 blog post that the Schwarzschild geometry may be just an illusion caused by the interplay of a newtonian gravitational field with matter and other fields.

Let us again look at the philosophical question if gravity changes the spacetime metric at all. If we work in special relativity and newtonian gravity, we may be able to derive all the effects of general relativity. If that happens to be the case, why should we not do newtonian mechanics, this time correctly, taking into account all the effects of various fields?

General relativity is supposed to be a better description of natural phenomena than newtonian mechanics. That is the reason why people in the 20th century adopted the claim that matter changes the Minkowski metric and creates a curved spacetime. If newtonian mechanics, done correctly, produces the same phenomena, why should we accept the superfluous claim that the metric of spacetime has changed?

There are some weaknesses in general relativity. The most prominent weakness is the appearance of singularities. In newtonian mechanics we seem to be able to avoid the singularity at the center of a black hole.

There is still one good reason to claim that the geometry of spacetime can be changed from the Minkowski metric. The reason is the equivalence principle. If we claim that in a freely falling laboratory, the "real physics" is what would happen in an inertial laboratory far away in space, then we are tempted to claim that the geometry of spacetime itself has changed. The claim that the "real physics" in the freely falling case is the same as far away in space is strange, though. The laboratory is not floating freely. It is close to a huge mass. Moreover, there are tidal forces which are not present far away.

There is an equivalence principle in the old gravity model introduced by Isaac Newton in 1687. Physics in a freely falling laboratory looks the same as in a laboratory which floats freely in space. The simple explanation is that gravity attracts everything in the laboratory in the same way. There is no need to claim that the metric of spacetime has changed.

If we do newtonian mechanics more precisely, and observe that light moves slower close to a large mass, why should we then start claiming that the metric of spacetime has changed?

Is general relativity completely equivalent with special relativity & newtonian mechanics?

We probably can always claim that the curved geometry of spacetime in general relativity is just an illusion which is caused by the interplay between the fields and matter. It might be that it is only a philosophical question which interpretation we adopt.

A brave physicist could do an experiment and jump into a black hole. If he freezes for an eternity at the event horizon, then the newtonian model might be the right. If he continues unharmed through the horizon, then general relativity might be correct.

But we need to think carefully. Is there a more practical way to test the different interpretations?

Gravitational waves break general relativity - they can create a new source of gravity?

UPDATE October 8, 2021: The book Gravitation of Misner, Thorne, and Wheeler solved this mystery. Section 35.12, formula (35.54) says that the ripples of gravitational waves do alter the background geometry like mass-energy does. Using the rubber membrane analogy of gravity, a wave stretches the membrane a little bit, just like a weight placed on the membrane.

----

In our October 7, 2021 blog post we asked if gravitational waves break general relativity by giving mass-energy to ordinary matter and creating a new "source" of gravity.

It looks like that exactly is the case.

A photograph of Albert Einstein by Oren Jack Turner, 1947. https://en.wikipedia.org/wiki/File:Albert_Einstein_1947a.jpg

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

https://physics.stackexchange.com/questions/41662/why-does-no-physical-energy-momentum-tensor-exist-for-the-gravitational-field

A basic principle of the Einstein field equations is that only matter can act as a "source" of the field. Matter contains mass-energy, momentum, shear stress, and pressure. All these appear as sources of the Einstein gravity field in the equations. The gravity field itself is not a source.

A gravitational wave does not contain a source. It is a wave of the homogeneous (vacuum) Einstein equations.

But the wave can interact with matter, make the matter to vibrate and to heat up. Thus, the wave can create a new source of gravity.

             )   )   )   )                         ●

         gravitational         lump of matter M

         wave packet

Assume that we have a lump of matter in an asymptotically Minkowski space. Its mass-energy is M. We let a gravitational wave packet to hit the matter and increase its mass-energy to M'.

Assumption 1. We assume that a "reasonable" solution for the Einstein equations exists for this configuration. That is, there is a solution where the mass M is surrounded by an approximate Schwarzschild geometry and the wave packet looks like wave packets in electromagnetics.

If there is no solution, then the Einstein equations are too strict - there probably are only solutions for artificial symmetric configurations, like the Schwarzschild solution.

Let us look at the development of the gravitational field after the wave hits the matter and increases its mass-energy.

Far away, the field

       G M / r²

suddenly changes to a stronger field

       G M' / r².

If we look at the lines of force of the field, it looks like this:

                                 --------            --------

       M' ●                 --------

                                 --------            --------

                             stronger          weaker

                             field M'            field M

Assumption 2. We assume that there exists a solution where which describes the temporal behavior of the system, and where the mass M' is surrounded by an approximate Schwarzschild geometry, except that far away, the geometry is still for M.

Contradiction. Lines of force seem to end in empty space. There is a source (or rather, a sink) of the field in empty space which does not contain matter - a contradiction.

We know that gravitational waves exist and that they contain energy. Waves remove huge amounts of mass-energy from binary pulsars and merging black holes. The process is the reverse to what we described above: a source of gravity M suddenly becomes a lot smaller source M'.

If gravitational waves themselves would be a source of gravity, then we would have no contradiction. Then our thought experiments would only involve moving mass-energy to a different place.

There are no longitudinal waves in electrodynamics. If charge were not conserved, then we could create a longitudinal wave by letting a charge q vary in time. We believe that there are no longitudinal waves in gravity, either. Then we must have conservation of the source of gravity.

Is it possible to save Einstein's idea that gravity is described as the geometry of spacetime? Can geometry itself act as a source of geometry?

How general relativity might be saved (and is saved)

In our diagram, the gravitational wave does not contain any source of gravity, but it may still be the case that far away it bends spacetime in the same way as the mass-energy difference M' - M would do. Is this possible?

                 ___

          ___/      \         __________     drum skin

                         \___/

                wave ---->

                                            

         ----------__           __-----------      drum skin

                          \    /

                            ● weight

If we have a tight drum skin, waves in it do not bend the skin in the same way as it would bend if we put a weight on the drum skin.

Gravitational waves are derived from linearized Einstein equations. In the case of electromagnetics, the wave does not behave like an electric charge.

Weak gravity has little nonlinearity. Electromagnetic waves presumably are an excellent approximation of gravitational waves.

Looking at the literature, it turns out that gravitational waves do bend background geometry just like mass-energy does. Thus, general relativity is saved from the contradiction.

The Schwarzschild geometry follows from newtonian mechanics without general relativity?

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

Assume that we have a very heavy donut in space. We let a photon to fly through its hole. We assume that it flies exactly through the center of a symmetric gravitational field, and the photon is not deflected.

          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

                            +

                            +

        e-   -------------------------------->

                            +

                            +

                        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.)

https://en.wikipedia.org/wiki/Schwarzschild_metric

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.

Does vacuum polarization turn Newton's gravity into Einstein's gravity?

UPDATE October 7, 2021: We were not able to find a reasonable model where vacuum polarization makes the gravitational field infinitely strong at the event horizon. Instead we found out that newtonian mechanics explains the effect. See our blog post today.

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There is a similarity in the gravitational force in general relativity and the Coulomb force in QED.

In both cases, the force at long distances is from a pure 1 / r potential, but at short distances the force grows stronger.

In QED we can explain the growth with vacuum polarization.

                e-  •    ----------->

                     +     vacuum

                     -      polarization

                    ●

                proton

                  

The acceleration of the electron causes a virtual pair to appear between the electron and the proton. The pair is a dipole which attracts both the electron and the proton.

In gravity, the analogous diagram is the following:

          

                     -    vacuum

    Mercury  •     ----------->

                     +    polarization

                    ●  Sun

The acceleration of Mercury causes virtual pairs of photons to appear. The positive energy photon flies between Mercury and the Sun, while the negative energy one flies behind Mercury. The photons must have extremely small positive/negative energy because they have to move over large distances.

In recent weeks we have claimed that in QED, the large-distance physics is the "true" physics, and it is modified at short distances through vacuum polarization and other processes.

Maybe newtonian gravity is the "true" gravity, and it is modified at short distances through vacuum polarization and other processes. General relativity would have its origin in quantum physics.

In QED we have single particles while in gravity we have lumps of 10⁶⁰ particles. We need to figure out how to handle a large number of particles in quantum field theory.

In QED, the electromagnetic field couples to electrons and positrons. The interaction is a source for the Dirac field. Conversely, an electron couples to the electromagnetic field and acts as a source.

Gravity couples to all matter fields. All matter particles act as a source of gravity. Conversely, gravity should act as a source for all kinds of matter fields.

There is a problem: if all matter fields contribute to making gravity stronger, why are the Einstein equations independent of quarks and so on? The reason might be that only photons can have low enough energy so that they can stay off-shell long enough to contribute to vacuum polarization. The lifetime of a virtual pair has to be minutes if it has to travel over planetary distances.

An infrared "divergence" in QED is a result of summing overlapping classical probabilities

In our September 20, 2021 blog post we claimed that infrared "divergences" in quantum field theory are caused by summing overlapping classical probabilities. That is, Feynman integral formulas treat as separate cases processes which are not separate.

Bremsstrahlung is the prime example of this. 

Claim. The electron and its field behaves in the classical way at the scale of "large distances". That is, when an electron scatters from a proton and the minimum distance of the electron to the proton is not small, then the low-frequency (soft) spectrum of bremsstrahlung is essentially classical, though the energy is, of course, divided into individual quanta.

Consider the classical limit of the process. We have a macroscopic charge changing its course in scattering. The energies involved in the process are very large relative to the energy of an individual low-frequency radiated photon. Bohr's correspondence principle says that the bremsstrahlung spectrum is the classical spectrum. The charge will radiate an infinite number of low-frequency photons in a single scattering event.

                                      bremsstrahlung photon

                                      ~~~~~~

                                    /

        e- ---------------------------------

                             |

                             | 4-momentum q

                             |

            ---------------------------------

        proton

Feynman formulae do not put any limit on the charge and the mass of the scattering particle. The Feynman integral formulae for the simplest bremsstrahlung diagrams assume that the electron radiates only one photon of bremsstrahlung. That is an error. We claim that it radiates an infinite number of soft photons in one scattering event.

We can remove the infrared divergence simply by cutting off the low-frequency spectrum of bremsstrahlung. The probability of emitting a high-frequency photon in bremsstrahlung is low. The overlap of classical probabilities is negligible when the involved probabilities are small. This is the reason why regularization works for infrared divergences.

The number of photons in classical bremsstrahlung

https://en.wikipedia.org/wiki/Bremsstrahlung 

Wikipedia contains a semiclassical calculation "Simplified quantum description" for a nonrelativistic electron. Let the electron velocity be v. We assume that v is at least 100 km/s, which is the thermal velocity at 450 kelvins. The electron scatters from a proton.

We want to determine bremsstrahlung intensity for a frequency f. Wikipedia gives two values for the impact parameter,

       b_max = v * 1 / f

and

       b_min = h / (m_e v).

There b_min is the de Broglie wavelength of the electron. The bremsstrahlung output for the frequency f is obtained by integrating over the classical output between these impact factors b.

The limit b_max clearly is dictated by classical physics. If b is the impact parameter then the close encounter with the proton lasts for a time

       t = b / v.

The classical power output is only significant for frequencies

       f  <  1 / t  =  v / b.

We get

       b  <  v * 1 / f  =  b_max.

The limit b_min is quantum mechanical. Classically, there should be a lot of power output from much smaller impact parameters b. The total kinetic energy of the electron puts a ceiling on the classical output, but that ceiling is nowhere near for b = the de Broglie wavelength.

Let us consider the case of a mildly relativistic electron, whose wavelength is

       ~ 2.4 * 10⁻¹² m,

or the Compton wavelength of the electron. We probably can build an electron wave packet whose length and width are

       ~ 10⁻⁹ m = 1 nm.

The diameter of a typical atom is 0.1 nm. We can build a "screen" which forces electrons to fly past the proton at a distance of 1 nm or more. The wave packet presumably produces almost the classical spectrum of bremsstrahlung.

What is the classical spectrum like if our wave packet flies past the proton at a distance b ~ 1 nm?

It is a sudden change of the electron path in a distance of 1 nm. The Fourier transform of the classical electron path probably divides the radiated energy fairly evenly to frequencies 

       0 Hz ... 3 * 10¹⁷ Hz = c / 1 nm.

Let 2 f be smaller than c / 1 nm. The number of radiated photons in the interval

       f ... 2f

is some constant C. That is because the energy output in that range scales with f, and so does the energy of a photon. Let us divide the frequency range into intervals of the form f ... 2f. There is an infinite number of such intervals, and an infinite number of photons are emitted.

Conclusion. Bremsstrahlung from a single scattering event always contains an infinite number of soft photons. That is true classically as well as in quantum mechanics.

The Feynman integral formula for bremsstrahlung assumes that an emitted photon in an interval I is a separate case from an emitted photon in another interval I', and that we can sum the classical probabilities of these events. That is clearly an error. The probabilities overlap.

The Feynman formula probably calculates right the probability of emitting a photon in a small frequency interval df, but there is no sense in summing these probabilities over all intervals.

A crude calculation of the number of photons C in a frequency interval

We assume that a mildly relativistic electron passes a proton.

Then b_min is roughly the Compton wavelength of the electron. We divide possible impact parameters into intervals

       b ... 2b.

Let us divide possible photon frequencies into intervals

       f ... 2f.

For an interval b ... 2b, let us calculate the number of photons in each frequency interval.

https://en.wikipedia.org/wiki/Larmor_formula

In the picture we have the Larmor formula from Wikipedia. The square of the acceleration a² scales as

       1 / b⁴.

The duration of the fly-by scales as b. The cross section of the interval scales as b². The radiated energy is evenly divided on frequencies from 0 Hz to ~ 1 / b. We see that each interval b ... 2b contributes an equal number C of photons to an interval f ... 2f, if b > b_max for the frequency f.

What is C? Let us calculate it in the case b = the Compton wavelength. The photon frequency range is then, given in photon energy, from 0 eV to m_e c² = 511 keV.

                  b

    e-  ----------------->

                  |  

                  | b

                 ● proton

We assume that the electron only radiates when it travels the distance b, roughly at the distance b away from the proton. The acceleration is

       a = k e² / (b² m_e) = 5 * 10²⁵ m/s²,

where k is the Coulomb constant, e is the electron charge, and m_e is the electron mass. The radiated power is

       P = 0.01 W,

and the radiated energy is

       E = 10⁻²² J = 0.7 meV.

In the frequency interval 256 keV ... 511 keV the energy of a single photon is roughly 400 keV. We see that only

       C = 2 * 10⁻⁹

photons are emitted in that interval.

Let us calculate how many photons whose wavelength is 1 m ... 2 m are emitted. Then b_max is 2 m and b_min is 2.4 * 10⁻¹² m. About 40 intervals of b contribute. The number of photons is only 10⁻⁷.

The wavelength 1 m corresponds to a photon whose energy is 1 μeV. Suppose that we measure bremsstrahlung of mildly relativistic electrons, and we can detect photons down to the energy of 1 μeV. There are then roughly 40 frequency intervals of photons. Each interval only contains some 10⁻⁷ photons. We conclude that the overlap of probabilities is negligible. We can ignore the infrared divergence of bremsstrahlung.

What part of the Green's function escapes as bremsstrahlung?

In our September 29, 2021 blog post we explained the "sharp hammer" model of the electric field of the electron. The electron keeps hammering the electromagnetic field to create its electric field.

The electron at every point of its path disturbs the electromagnetic field which we model with the massless Klein-Gordon wave equation. The disturbance is a Dirac delta source in the equation. The response is a Green's function of the Klein-Gordon equation.

We showed that for a free electron, there is total destructive interference for a component of the Green's function, if the component contains energy. A component which only contains spatial momentum survives.

Another interpretation is that the free electron totally reabsorbs the Green's function which it sent.

When the electron is scattered by a proton, the electron cannot absorb the Green's function completely. A part escapes as bremsstrahlung.

In this blog we have claimed that high-frequency components of the Green's function are almost completely wiped out by destructive interference. What about low-frequency components?

                                           ^  e-

                                         /

                                       /

                                     x

                    ●            /   \ 

                                /        \  k'

 e-    ------------------  

                    \

                      \  k

When a free electron at a point x creates a component k' of the Green's function, then destructive interference destroys k' along with a component k of the same 4-momentum as k', but with a 180 degree phase shift relative to k'. Alternatively, we may interpret that the electron reabsorbed k.

Total destructive interference is spoilt when the electron is scattered by the proton (marked with ●).

Looking at the diagram above, it is obvious that if the deflection angle is small, then destructive interference still wipes out most of a long-wavelength k component, but some of the component survives.

The geometry of the diagram is essentially the same for all long-wavelength k. We conclude that the "survival rate" is the same for all such k.

For a very short-wavelength k, destructive interference is almost total. The wave only "sees" a small part of the deflection curve before it is destroyed by interference.

Conclusion. In a typical scattering event, a small part of each long-wavelength on-shell component of the Green's function escapes as bremsstrahlung. Most of the component is still destroyed by interference (alternatively, reabsorbed by the electron).

How to handle infrared divergences

Setting a cutoff for low frequencies probably works in all situations. The infinite tail of photons contains exceedingly low-energy photons. They have a very small impact on events and we can safely ignore them.

There are no infrared divergences in QED, once we recognize that Feynman integral formulae should not sum overlapping probabilities.

Literature contains a strange argument that the infrared divergences in bremsstrahlung and in the vertex function somehow "cancel each other out". They are separate processes with different outputs and cannot "cancel" each other. The argument has to be erroneous. We will analyze it.

The self-force on an electron in periodic motion, caused by its electrostatic field

In our September 29, 2021 blog post we claimed that the mass-energy of the electrostatic field of the electron has zero mass-energy if the electron is free.

We also claimed that when the electron is in a periodic motion, the far field of the electron stays static and causes a self-force on the electron. The self-force emulates reduction of the inertial mass of the electron.

That is, if the electron is in a circular motion, the self-force cancels a part of the centrifugal "force". If the electron swings back and forth, the self-force helps to return it to the central position.

In the Sommerfeld hydrogen atom model, an electron on an s orbital does a 180 degree turn very close to the proton. We claim that the far electrostatic field lags behind, and the self-force helps to overcome some of the centrifugal "force" in the sharp turn around the proton.

We have claimed that the vertex function (correction), and the major part of the Lamb shift, as well as the anomalous magnetic moment of the electron are all due to reduction in the inertial mass of the electron in sudden movements. We derived the electron anomalous magnetic moment from this assumption (for zitterbewegung) in our March 2, 2021 blog post. We derived the Lamb shift on March 13, 2021.

The anomalous magnetic moment of the electron: reduction in the inertial mass

In the anomalous magnetic moment, the inertial mass of the electron seems to be reduced by a factor

       1 - α / (2π),

where α = 1 / 137 is the fine structure constant.

The reduced mass makes the zitterbewegung loop 1 / 861 larger and increases the magnetic moment accordingly. In our model, the magnetism comes from the zitterbewegung loop which the electron circles at the speed of light. The length of the loop is the Compton wavelength of the electron. If the mass is reduced, the wavelength becomes longer.

The classical energy density of an electric field is

       1/2 ε₀ E²,

where ε₀ is the vacuum permittivity and E is the electric field strength. Classical physics is unsure if an electrostatic field really contains that energy. An electromagnetic wave does contain, that we know for sure. Let us temporarily assume that there is mass-energy in an electrostatic field, too.

In our blog post March 2021 we assumed that the electrostatic field farther than

       1/2 c t

from the electron does not contribute to the inertial mass of the electron. There t is the cycle time of the periodic movement and c is the speed of light.

We motivated our claim by the position of nodes in a standing wave in a string.

With our assumption we were able to explain why the inertial mass of the electron is reduced, the Compton wavelength becomes longer, the zitterbewegung loop is larger, and the magnetic moment 1 / 861 larger.

Let us calculate what is the self-force on the electron. Can it explain the reduction in the inertial mass?

A very crude calculation of the self-force on an electron

Let the electron circle a loop of a radius r at a speed

      v = a c,

where c is the speed of light. Let the cycle time be

       t = 2 π r / v.

We assume that the field farther than

       s = 1/2 c t = π r / a

is static.

               |     |     |

               |     |     |    electric field lines

                 s 

                     |

                     |   \    \    \       oblique

                     |     \    \    \     electric field lines

                     |       \    \    \

                     |      

                     |        |     |     | electric field lines

                     |        |     |     |

                     |       

                      ----------- • e-

                            r (displacement)

We have to connect the (dynamic) field lines going up from the electron to the vertical static field lines up at the distance s. The oblique field lines become slightly longer because of the displacement r. The density of oblique field lines grows a bit larger, which means a larger electric field |E|.

The total energy of the oblique electric field is larger than in the case where the displacement r would be zero.

It makes sense to keep the field closer than s / 2 from the electron undeformed because there the field follows more closely the movement of the electron. The extra energy is taken by the oblique lines higher than s / 2 from the electron.

The "deflection" angle of oblique field lines is

       2 r / s

radians. The separation of field lines is

       cos(2 r / s) = 1 - 2 r² / s²

relative to the undeformed case. The energy density of the oblique field is

       1 + 4 r² / s²

relative to the undeformed field.

The energy of the field between the distances 1/2 s and s from the electron is

          [1/2 r_e / (1/2 s) - 1/2 r_e / s] * m_e c²

       = 1/2 r_e / s * m_e c²,

where r_e is the classical electron radius, and m_e is the mass of the electron.

The extra energy in the oblique field is

       1/2 r_e / s * m_e c² * 4 r² / s².

Let us calculate the derivative of that relative to r, to find the self-force F:

       F = 4 r / s² * m_e c² * r_e / s

          = 4 a² / (r π²) * m_e c² * r_e / s.

The centrifugal "force" on the electron in the circling motion is

       F' = m_e v² / r

           = m_e c² a² / r

The ratio F / F' reveals by which proportion the self-force F reduces the effective inertial mass of the electron in the circling motion:

       F / F' = 8 / π² * 1/2 r_e / s.

There, 1/2 r_e / s is the ratio of the electrostatic field mass m_s outside the radius s relative to the mass-energy of the electron, m_e. We have

       F / F' = 8 / π² * m_s / m_e.

Since 8 / π² = 0.81 is roughly 1, we have the conclusion: the effective inertial mass of the electron is reduced roughly by the mass m_s, which is the mass of the electrostatic field farther than s.

Note that m_s ~ 1 / s. Maybe we should reduce s by a factor π² / 8, so that the self-force would explain the mass reduction exactly?

We do not need to assume any mass-energy in the electrostatic field of a free electron - we get the same effect from the self-force on the electron

We showed that we can explain the reduction in the electron inertial mass either by

1. claiming that the far field has mass-energy m_s, and that mass does not have time to take part in the circling motion, or

2. claiming that the electrostatic field does not have any mass-energy for a free electron, but the deformed field in the circling motion exerts a self-force on the electron.

The very rough calculation supports our claim that the electrostatic field of a free electron has zero energy. An alternative interpretation is that we have renormalized the electrostatic field to have zero energy. Only deformations of the free electron field contain mass-energy.

A precise calculation of the self-force probably requires a computer simulation. There is an open problem, too. We keep the field static farther than s. How does retardation affect the deformation of the oblique field? Does Nature try to minimize the extra energy?

What about the mass-energy in the deformation of the oblique field? It is small compared to m_s. The total mass-energy in the oblique field is m_s. The angle 2 r / s is small in most cases. Thus, the deformation energy is usually very small.

The border conditions at s make the electric field a standing wave at distances < s. Is this wave somehow quantized? Since the electric field is tightly bound to the electron, the wave is not an on-shell photon. The wave is an off-shell photon, or a virtual photon, which contains little energy compared to its wavelength. For an on-shell photon, the energy-momentum relation is

        E² = p²,

where E is the photon energy and p is its spatial momentum. If the photon is off-shell, it can contain less energy. In this blog we have stressed that interacting particles are always off-shell. The photon is in a very tight interaction with the electron.

The self-force is purely a classical phenomenon

Nowhere in our analysis did we refer to quantum mechanics. We calculated the self-force from elementary classical electrodynamics.

Many people believe that the vertex function is a quantum mechanical thing which only manifests itself in quantum mechanical calculations with Feynman diagrams. We claim that the vertex function is mostly a classical thing. However, quantum mechanics does affect the process. For example, emission of bremsstrahlung is constrained by the minimum energy of a quantum of light.

The calculation of the anomalous magnetic moment has at least two quantum mechanical features:

1. the length of the zitterbewegung circle is determined by the Planck constant h;

2. there is no bremsstrahlung because the electron is in its lowest energy state.

The calculation of the Lamb shift involves solving the Schrödinger equation for the hydrogen atom. The electron on the 2s orbital is in an almost stationary state, and bremsstrahlung is negligible. These are quantum aspects of the calculation.