Extra inertia of a test charge is NOT directly proportional to its potential

Suppose that the extra inertia of a test charge in an electric field is caused by the the energy flows it causes in the electric field. We have been hypothesizing that the inertia comes from the potential energy of the test charge in the field. But the following is a simple counterexample.

       ^

       |

                                      ____

        • -                        /         \

        c                          \____/  +

                                        C

Let the large charge C be a hollow spherical shell of positive charge. There is no electric field inside the shell. When we move the test charge c, there is less energy flow in the electric field than there would be if C were a point charge. The potential energy of the test charge c does not depend on the form of C, though.

We have been assuming that the test charge immediately feels the entire inertia caused by the large charge. But how would the test charge know if the large charge is a point charge or a shell?

The answer might be that the electric field of the large charge somehow, besides the potential, also knows the inertia of the field. We may imagine that the field contains a mechanism to test how much energy flow is associated with an infinitesimal movement of a test charge. Is this realistic? Such mechanism would be complicated. The potential can be summed from the potential of each elementary charge. The inertia is much more complicated.

Implications for gravity

The analogous setup in gravity involves a significant pressure along the walls of the spherical shell C. Could it be that the pressure makes up for the inertia which is lost when we transform C from a point mass into a hollow shell?

Suppose that we have an electrically charged oscillator on the surface of Earth. In this blog, our hypothesis has been that the inertia of the mass in the oscillator is larger exactly by the amount which is the absolute value of the negative potential energy of the oscillator. That is why the oscillation is slower. When the oscillator produces a photon, it works against the gravity field of Earth and loses energy. It is redshifted by an amount which exactly matches the slower oscillation.

However, if the potential energy does not match the extra inertia, things get more complicated. There might be more photons when their number is measured on the surface of Earth than their number in faraway space. This is not a contradiction. The number of photons sent by an accelerating oscillator in space is not exactly determined, either.

We could test this experimentally. Is the redshift of a known natural process exactly determined by the negative gravitational potential it is in?

Let us assume that the inertia close to Earth is surprisingly small compared to the negative potential.

Then any atomic process runs surprisingly fast. Also, a satellite orbits Earth surprisingly swiftly.

Measuring the gravitational potential of Earth is not easy if we cannot trust a redshift measurement.

Inertia of the electron in the hydrogen atom is different in an external electric potential versus a gravity potential

The Stark effect alters the hydrogen spectrum in an external electric field, but there is no known effect of an external electric potential.

We firmly believe that a free electron in an external potential must have extra inertia. We have to figure out why a bound electron does not show this property.

A low gravity potential causes a redshift to the spectrum. This is a major difference in the inertia caused by the electric field and the gravity field. The extra inertia from a gravity potential does show up, while the extra inertia from an electric potential does not.

Conclusions

If the extra inertia of a test charge or a test mass is not directly proportional to its potential, things get complicated.

In a dynamic system, there is no well defined potential at all. We were bound to encounter this problem, anyway.

This requires a lot of new research.

Inertia of an expanding and contracting shell of charge in an external electric field

In the previous blog post we studied a collapsing shell of charge. Let us now look at a related problem of a pulsating spherical shell of charge in an external electric field.

                     ___

                  /        \

                  \____/  -                               ● +

  expanding/contracting            external

         shell of charge                     charge

Is there extra inertia in the movement of the shell? The field outside the shell does not change at all. There is no flow of field energy there. But what about the field at the surface of the shell?

                       shell              shell

                       wall               wall

   . . . . . . . . . . . .| -----------------  |  =======  ● +

                          -                        -

                         -->                   <--

   weak field     medium field     strong field

Schematically, the electric field looks like the one in the diagram. The arrows tell that the shell is contracting. The left wall gains energy from the field since it makes the field weaker. The right wall consumes energy because it makes the field stronger.

Our previous blog post claims that the walls gain/lose energy from the adjacent electric field. There is no shipping of energy over large distances. We conclude that there is no extra inertia in the contracting/expanding movement of the spherical shell.

If we would be moving a pointlike test charge, then there would be extra inertia. There would be an energy flow in the combined field of the charges.

Actually, here we have a strong argument for the existence of extra inertia in an electric interaction!

Claim. In an external electric field, the inertia of a test charge is larger when it is a single pointlike charge than when the test charge is a part of a pulsating spherical shell of charges.

Heuristic proof. Let us have a large external charge C and a test charge c.

          <--------  E

                 • -                           ● +

                 c              r             C

             

When a pointlike test charge c moves, there is an energy flow in the combined electromagnetic field of c and C. This adds some inertia H, which depends on the electric field E at c, as well as the distance r. That is, the inertia is proportional to the negative potential of c in the field of C.

Suppose then that c is a part of a pulsating shell of charge. Could there be some mechanism which adds the same inertia H to the test charge c when it is a part of a pulsating shell?

That is implausible. Why would the shell be interested in the distance r and put just the right amount of extra inertia H, which depends on r?

Q.E.D.

Our claim refutes the "geodesic hypothesis" for electromagnetism. The inertia of a test charge depends on the configuration of test charges. In general relativity, the inertia does not depend on the configuration of test masses, but it is determined by the external gravity field and its metric.

Inertia of a collapsing electrically charged spherical shell

Let us analyze once again what is the inertia of a collapsing spherical shell of electric charge. The electric field of a sphere outside the sphere is constant. If the extra inertia comes from a flow of field energy, then there should be less inertia in the collapse of a spherical shell than when a pointlike test charge falls.

Ordinary, electrically neutral matter

Suppose that an electron would feel individually the inertia caused by the fields of electrons and nuclei in a nearby lump of matter, and one could sum these amounts of inertia to get the total inertia. We calculated the inertia in our March 31, 2023 post.

Then that lump of matter would increase the inertia of an electron enormously. This does not happen.

Rule 1. If we have many charges of opposite signs, then we can calculate the sum of the fields, and the inertia of a test charge is determined by the sum.

If we use neutral matter as a test object, then the following probably holds:

Rule 2. The inertia of a group of opposite charges in an electric field can be calculated from the sum of the fields of individual charges.

These hold for neutral matter. But we are not sure if they hold for a group of charges which have the same sign.

              ||                                    ● +

              ||                                    ● +

              ||                                    ● +

             + -    --->            

      layers of negative          positive charges

      and positive charge

      attached to each other

Suppose then that we have thin layers of positive and negative charge moving in an electric field, as in the diagram. The positive charges pull the negative layer, that is, give energy to its movement.

If the energy to the negative layer would be shipped from far away, then there would be significant extra inertia. But the double layer is quite similar to neutral matter: the only difference is that the charges are ordered in layers. We do not believe that there is much extra inertia.

Rule 3. We conclude that the energy to the negative layer must be shipped from very close, from the adjacent electric field just right from the layer.

We have suspected that there might be extra inertia associated with the low potential of the negative layer in the field of the positive charges. Our reasoning with the double layer suggests that it is not the case. There is no extra inertia when we move the double layer, even though the negative layer is in a low potential and the positive layer in a high potential.

Rule 4. The extra inertia seems to be associated only with energy flow in the electromagnetic field.

A spherical shell of negative charge collapses on a positive charge

We can now analyze our perennial problem of the inertia of a collapsing shell of charge.

                    -

                _____

              /           \

       -    |      ●+     |   -

              \______/

                    

                    -

The shell is negatively charged. We have suspected that when the shell contracts, there is extra inertia from two sources:

1. The potential of the shell is low. The shell might ship "negative energy" from one place to another.

2. The shell harvests energy from the field of the positive charge and gains kinetic energy. The harvested energy might not come from the adjacent field but from remote locations. Then there would be a lot of energy shipping and extra inertia.

Rule 4 says that item 1 does not contribute extra inertia. No energy flows in the electric field when we lower the shell.

Rule 3 says that the kinetic energy is shipped from the adjacent field. There is no extra inertia.

Conclusions

We used examples consisting of neutral matter to analyze extra inertia within an electric field. We believe that neutral matter does not feel any extra inertia. Our conclusion is that energy flow in the electromagnetic field determines the extra inertia.

Furthermore, if a layer of charge moves, the layer harvests its kinetic energy from the adjacent field. There is no extra inertia in that process.

When a poinlike test charge moves within an electric field, the situation is wholly different. The Poynting vector says that energy flows in the electromagnetic field.

Spherical shells, in a sense, are electromagnetism in one spatial dimension. It is quite different from a pointlike test charge in three spatial dimensions.

Einstein-Hilbert action and extra inertia: a case study of the Sun

Our April 14, 2023 blog post called for a detailed analysis of the inertia as well as action/reaction in the case where we have pressure inside a body M, and a test mass m moves inside or close to that body. An example is a star in a hydrostatic equilibrium and a small test mass.

It is best to abandon the use of the geodesic hypothesis entirely. The Einstein-Hilbert action is the fundamental formula.

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

The basic idea in the Einstein-Hilbert action: we can reduce the contribution of the mass M in the integral by slowing down time in the metric g, but we must pay the price of increasing Ricci scalar curvature R. The mass M is considered potential energy in the action, as well as the Ricci scalar curvature R.

Moving a test mass m inside or close to the Sun

Let us calculate an example with the Sun,

       M = 2 * 10³⁰ kg,

and m = 1 kg. In our blog post on November 10, 2021 we calculated that if the 1 kg test mass moves at a velocity which is comparable to the escape velocity of the star, then the mass in the star has enough time to adjust to the changed metric.

The Schwarzschild radius of a 1 kg mass is

       r_s = 2 GM / c²

              = 1.3 * 10⁻²⁷ m.

Let us assume that the Sun has enough time to adjust. If we move the mass m = 1 kg some 1 million kilometers, then we may estimate that some 10³⁰ kg of the mass of the Sun moves ~ 10⁻²⁷ m. The contribution to the inertia of the test mass m is something like 10⁻⁶.

In the Komar mass formula, we integrate

       ρ + 3 p

over the volume, where ρ is the energy density. Inside the Sun,

       ρ ~ 10²⁰ J/m³,

       p ~ 10¹⁶ Pa.

We see that the contribution of pressure to "gravity" is ~ 3 * 10⁻⁴.

The gravity potential of the Sun at the surface of the Sun per kilogram of mass is

       1/2 v²,

where v is the escape velocity

       v = 600 km/s.

The energy of mass per kilogram is

       c²,

where c is the speed of light. The ratio

        1/2 v² / c² ~ 2 * 10⁻⁶.

At the center the potential is 3/2 times the potential at the surface. We conclude that the gravity potential contributes 3 * 10⁻⁶ to the inertia.

The conclusion: at the center

1. gravity adds 3 * 10⁻⁶,

2. mass flows add 10⁻⁶,

3. gravity caused by the pressure adds 9 * 10⁻¹⁰

to the inertia of a test mass.

Suppose that the Sun does not have time to adjust to the movement of the 1 kg test mass. If we move the test mass a distance of 1 million kilometers outside the Sun, then some pressure energy moves ~ 1 million km. How large is this energy?

The 1 kg mass stretches the radial metric by a ratio ~ 10⁻³⁶ inside the Sun. The volume change is

       ~ 10⁻³⁶ * 10²⁷ m³

       = 10⁻⁹ m³.

The pressure energy is

       ~ 10¹⁶ Pa * 10⁻⁹ m³

       = 10⁷ J.

A reasonable part of the pressure energy moves ~ 1 million kilometers.

The mass-energy of 1 kg is m c² = 10¹⁷ J.

We see that the flow of pressure energy contributes some

       ~ 10⁻¹¹

to the inertia of the test mass. If the Sun has time to adjust, then mass flows contribute much more, ~ 10⁻⁷.

What is the magnitude of the ordinary tidal effect on the Sun? How much does that add to the inertia of the 1 kg test mass? The shape of the Sun becomes elongated in the field of that 1 kg. The elongation is something like

       m / M ~ 10⁻³¹.

That corresponds to a volume 10⁻¹³ m³, or 10⁻¹⁰ kg. We conclude that the ordinary tidal effect adds

       10⁻¹⁰

to the inertia of the test mass.

Is the Einstein-Hilbert action aware of the extra inertia?

Is the Einstein-Hilbert action aware of these various amounts of extra inertia?

                                            ■■■■■

              • ----------------------------   ■■■     M

                                            ■■■■■

             m       lever

Let us consider a general lagrangian and a mechanical system where moving a test mass m also moves a very large mass M slightly through a lever. Does the lagrangian understand that the inertia of m is greatly increased by the lever?

The action does not depend on time, and is invariant under spatial translations. By Noether's theorem it conserves energy and momentum. The action certainly understand the mechanics of the lever system, and understands that the inertia of the test mass m is effectively larger.

We conclude that the Einstein-Hilbert action is aware of the various items of extra inertia which we calculated in the previous section.

The action has to be written very precisely, so that it includes the pressure mass-energy and the kinetic energy of that mass-energy if it is shipped around.

Does a test mass feel immediately the full extra inertia from the flow of pressure energy? Probably not

In our April 20, 2023 blog post we argued that a test charge feels immediately the full extra inertia from its interaction with a static electric field. If that would not be the case, then we should see an oscillation of a static electric field - which has not been observed.

What about a test mass and the flow of pressure energy? We assume that the pressure field is static. Is the test mass aware of the flow of pressure energy which will happen if we move the test mass?

                     ___

                  /         \ 

                   \____/   large mass M with pressure

                       •  --->   test mass m

Suppose that there is pressure in the large mass M, and the system is initially static.

In principle, it is possible that the test mass m would "know" how much pressure energy it will move inside M, if m moves. The system is static. There is plenty of time to communicate information around about the state of the pressure field.

Let us assume that the test mass m feels the inertia of future pressure energy flows immediately. Let us move the test mass m and input some momentum p against the extra inertia.

But then we suddenly remove the pressure inside M through some mechanism. M may be supported by some mechanical structures and we let these structures to come loose.

Now we face a problem: where are we going to put the extra inertia p which we input? There is no pressure energy to move around.

It looks plausible that the test mass m does not feel the inertia of pressure energy flows inside M in advance.

Even less likely is that the test mass would feel in advance the inertia from the ordinary tidal effect, or from the adjustment of the mass M.

Does the test mass m feel the inertia from the direct gravity force of M immediately? Probably yes

In our April 20, 2023 blog post we argued that an electric test charge c does feel the inertia from the Coulomb interaction with a large charge C immediately.

For the analogous problem in gravity we do have empirical data. Photons passing the sun seem to feel the extra inertia from the gravity of the Sun immediately.

Also, if it would take a few milliseconds for atomic clocks to feel the inertia from the gravity of Earth, then someone probably would have noticed a weird behavior of clocks.

(Traditional) general relativity assumes that there is a metric which the large body M creates around itself, and the test mass m must obey this metric: move along a geodesic. That is, in general relativity the test mass m must feel the inertia from the field M immediately. If that were not the case, then the concept of a geodesic would not make sense.

General relativity also assumes that the gravity from the pressure inside M is a part of the metric. Thus, the inertia associated with the pressure is felt immediately by m. This might be wrong.

Let us calculate the presumed inertia from pressure inside Earth. Our test mass is 1 kg.

The ratio of stretching in the Schwarzschild metric at the distance of 10,000 km is 10⁻³⁴. The volume change is

       ~ 10⁻³⁴ * 10²¹ m³

       = 10⁻¹³ m³.

The pressure is ~ 10¹¹ Pa. The pressure energy

       ~ 0.01 J.

That is only 10⁻¹⁹ of m c² for the 1 kg mass. We conclude that it is impossible to measure the contribution of the pressure to the inertia.

Conclusions

We believe that a test mass m feels the inertia from the direct (newtonian) gravity field of M immediately. There is empirical evidence of this.

We believe that the inertia caused by pressure inside M is not felt immediately. There is a "self-force" from the gravity field of m on m itself, which determines how m behaves. The extra inertia deforms the gravity field of m and the deformation causes a force on m.

The ordinary tidal effect, and mass flows inside M, when M adapts, cause more substantial extra inertia for m. We firmly believe that these are not immediately felt by the test mass m. The extra inertia only unfolds when M changes its shape.

The self-force on an electric charge is a mystery which is well over a century old. In our blog post on October 1, 2021 we wrote about an electron in a periodic motion and the self-force from its own electric field.

For gravity, the self-force might be derivable if we knew the energy of a deformed gravity field. The Schwarzschild metric is seen as the "natural field" of a free test mass. If we deform it in a way or another, we must use energy. Gravitational waves are born from deformations to the field of an accelerating mass.

The Infeld and Schild 1949 proof of the geodesic hypothesis

Leopold Infeld and Alfred Schild tackled the geodesic hypothesis in 1949 through studying a point particle which is a singularity in the gravity field. The particle moves in an external gravity field, and its neighborhood does not contain any other fields than the gravity field.

https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.21.408

The point particle moves superluminally relative to a static coordinate system

We remarked in the blog post on March 17, 2023 that one cannot model the movement of a very heavy neutron star in a static coordinate system because the speed of light is very slow close to the surface of the neutron star, as seen by a faraway observer. If we would use a static coordinate system, then the neutron star would move superluminally relative to the coordinate system. One cannot define the stress-energy tensor T in such a case because matter can move superluminally relative to the coordinates.

The point mass of Infeld and Schild is a singularity, that is, a black hole. Any movement against a static coordinate system is superluminal.

An analysis of the proof

Infeld and Schild consider a timelike worldline L in a fixed background metric g. They choose coordinates such that at every point of L the metric g is equal to the flat Minkowski metric:

       g = η.

Enrico Fermi proved the existence of such coordinates in 1922. The coordinates are "freely falling" on the line L relative to the background metric.

The authors then introduce comoving coordinates

       t, z¹, z², z³

along L, in section 3 of their paper.

The authors let a point mass singularity move along the worldline L. They aim to show that L has to be a geodesic. They will prove that the approximate Schwarzschild metric around the point mass cannot be accelerating relative to a freely falling frame in the background metric.

There is an obvious problem: the singularity cannot truly move along the line L because the speed of light is zero close to the singularity. If we imagine a physical thread laid along the spatial path of L, the singularity will push the entire thread in front of it. However, the singularity can appear to move along L for outside observers.

In section 4 the authors assume that the metric with the particle is

       g = η + a + m b,

where η + a is the background metric, m b is the correction from the existence of the particle, and m is the mass of the particle. Recall that a is zero on the line L - we chose the coordinates in such a way.

The argument of the authors is based on the asymptotic behavior of the metric close to the singularity. In (5.04) they write the metric in a form

       β₋₁ + β₀ + ...

where β₋₁ varies roughly as 1 / r, β₀ is roughly a constant, and so on. Here r is the spatial distance from the singularity.

Here we see a problem in the proof: the β₋₁, β₀, ... are not uniquely determined. The authors claim that the terms of the same order in r must match. That would be true if the terms would be of the form

       constant * 1 / rⁿ,

but they are not of that precise form.

An example: let

       f(r) / r + h(r) = 0

for all r. That does not imply that f(r) and h(r) are zero for all r. If f(r) and h(r) were constants, then the implication would hold.

The authors continue to write equations believing that β₋₁, β₀, ... are uniquely determined. They proceed to show that the second time derivative of the position of the point mass ξ must be zero. That is, the point mass does not accelerate relative to a freely falling frame.

The proof may be erroneous.

Another problem in the proof: if the singularity moves on the worldline L, then close to the singularity, the static coordinate system x¹, x², x³ moves superluminally relative to the comoving coordinates z¹, z², z³. The authors base their argument on the asymptotic behavior close to the singularity. But one cannot use static coordinates there at all.

The singularity does not need to move along a geodesic of the background metric

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

A system should follow a path where the action has a local extreme value. That is, any small change to the metric or the position of particles increases the value of the integral if the extreme value is a minimum, and vice versa for a maximum.

Can we assume that there is a local extreme value of the Einstein-Hilbert action for the development of this system? It could happen that the value of the action explodes and there is no local extreme value at all. Let us assume that there is a local minimum.

                      Schwarzschild

                             metric

               • ----------------------             #######

   point mass m                           rigid object

Let us design the lagrangian L in such a way that a neighboring object repels the curved spatial Schwarzschild metric around a point mass. The object is very rigid and resists any deformation of its form away from the spatial metric which its own gravity field created.

If we remove the point mass, then the stress-energy tensor is not aware of the rigidity. That is, the rigidity does not affect the background metric created by the object. Let the background metric be

       g.

We can tune the rigidity of the object without changing g.

Let us assume that the force resisting a deformation of the object is constant regardless of the compression or stretching ratio of the object.

The spatial stretching (strain) is approximately linearly dependent on m in the Schwarzschild solution. If we move the point mass m close to the object, we must do work worth

       ~ m.

Let us have a freely falling frame in the metric g, at the point mass m. The frame does not depend on the rigidity of the object.

We have a point mass accelerating relative to a freely falling frame of the background metric g.

The solution does satisfy the Einstein field equations, because the Einstein equations simply state that the path is a local extreme value of the Einstein-Hilbert action. Specifically, close to the point mass, the Ricci curvature tensor is zero. It is empty space.

In our example, the backreaction of the metric inside the object is very strong. Any small mass m causes a large pressures in the object. Can we find an example where the backreaction is small?

Yes. Our November 10, 2021 blog post discusses "tidal" effects on a test mass. The inertia of a test mass of a weight m grows near a pressurized sphere because it moves pressure energy around inside the sphere. The increase in the inertia is linearly proportional to m. Setting m very small sets the metric of the whole system very close to the metric of the sphere alone. Still, the inertia of m increases with some fixed ratio I > 1. That affects its path considerably.

Birkhoff's theorem states that the metric around a spherically symmetric system is the same regardless of what we do with the pressure inside the sphere. But the pressure affects how much pressure energy our test mass moves around, that is, the inertia of the test mass. Therefore, the background metric cannot determine the path of the test mass, even if m is very small.

We assume that the Einstein-Hilbert action understands inertia and the whole process. It can be used to determine the path of the system.

This is a counterexample to the claim of Infeld and Schild. We have a system where:

1. The path of a test mass differs considerably from a geodesic of the background metric.

2. The backreaction of the background metric is proportional to m and can be set arbitrarily small.

3. The system satisfies the Einstein equations for empty space around the test mass. This is because the path is an extreme value of the Einstein-Hilbert action, and the Einstein equations are derived through varying the metric around the path. The Einstein equations must be satisfied by the path.

Conclusions

Wikipedia states that proofs of the geodesic hypothesis are "controversial". We found two possible problems in the proof by Infeld and Schild. They do not address the superluminal movement problem in any way. Their decomposition of the β metric seems to be nonunique.

We also presented a counterexample to the claim of Infeld and Schild: a sphere where pressure modulates the inertia of a test mass close to the sphere. The metric of general relativity does not understand "tidal" processes, though the Einstein-Hilbert action presumably understands them.

We are not sure if the geodesic hypothesis holds for a set of pure point masses, either. There may be tidal forces also in that case, and the background metric does not understand them. Einstein, Infeld, and Hoffman in a 1938 paper claim that the geodesic hypothesis holds for slowly moving point masses. We should check their proof.

The geodesic hypothesis in general relativity can be said to fail

We wrote about this problem also in our November 12, 2021 blog post.

Albert Einstein, Leopold Infeld, and Banesh Hoffmann (1938) tried to derive the geodesic equation for an infinitely light point particle from the Einstein field equations:

https://edition-open-sources.org/media/sources/10/17/sources10chap15.pdf

Later, Leopold Infeld and Alfred Schild (1949) wrote another paper on the problem:

https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.21.408

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

Wikipedia says that the papers are "controversial".

http://philsci-archive.pitt.edu/5072/

David B. Malament (2012) writes that various proofs require energy conditions. They do not follow from the Einstein equations alone.

Steven Weinberg's derivation of the geodesic equation in Wikipedia

The Wikipedia page presents a derivation of the geodesic equation by Steven Weinberg (1972).

Let us assume that we have coordinates and a metric for spacetime.

Weinberg assumes the following equivalence principle for a freely falling observer: the observer can define local coordinates such that the local metric then looks cartesian.

If we have a stationary observer in the Schwarzschild solution, then the proper time in his neighborhood runs at a rate which depends linearly on the radial distance r. He cannot define cartesian coordinates in his immediate neighborhood.

But a freely falling observer can define locally cartesian coordinates. Weinberg shows that the geodesic equation holds for the path of the freely falling observer.

Weinberg's argument proves the following. Let us assume:

1. the Einstein equations determine a metric which correctly describes time intervals and distances as they are measured by observers;

2. there is a "natural" way to match the clock times and measuring rods in any local coordinates of a moving observer to the metric;

3. a freely falling observer can define locally cartesian co-moving coordinates.

Then the path of the freely falling observer must obey the geodesic equation.

But is the derivation circular? If rays of light would not obey the geodesic equation, could it be that the stationary observer in the Schwarzschild metric could see clocks running at the same rate regardless of the radial distance r of the observer? We must assume that rays of light cannot propagate to the past. Let us add:

4. rays of light cannot go to the past in the metric. This is included in the word "natural" in item 2.

The dominant energy condition fails? Probably not

Let us have two people pulling on a very rigid, tense, almost massless rope.

               tense rope

              ------------------

        o/                         \o

         |                            |     

        /\                          /\

There are also other people holding on the rope, placed at every meter of the rope. At some time t, the people let the rope to move to the right some distance. The person at the left end can harvest some energy E from the tension of the rope, and the person on the right loses that same energy E.

Did the energy E travel superluminally from the right end to the left end? If the rope is almost massless, then that is the only conclusion that we can make. The dominant energy condition fails. But we cannot construct almost massless ropes. This saves the condition.

https://arxiv.org/abs/2006.07125

Erik Lentz (2020) has constructed a superluminal soliton. The basic idea in his soliton might be similar to our rope example.

If there is interference of waves, constructive interference may appear to move faster than light. That may be yet another violation of the dominant energy condition.

Let us assume that we have two circularly polarized waves A and B moving to the same direction. The rotation of the waves is clockwise. B has a somewhat lower frequency than A. Let both waves move at the speed of light. If A and B have a full constructive interference at some point P, then P moves at the speed of light to the same direction as A and B. Let P be at a location x at t = 0 s, and at y at t = 1 s.

Let us now make B a little slower than light. We assume a constructive interference at x at 0 s. 

       B lagging

                     \     |  A not lagging

                       \   |

                         \ |

          location y at t = 1 s

Then at y, after 1 second, the phase of B is "lagging behind" a little, compared to what it would have been were it moving at the speed of light. The wave A already had a constructive interference with B before t = 1 s. We conclude that the point of constructive interference moves at a superluminal speed. Energy appears to be transported superluminally.

For B to propagate slower than A, they have to be in a medium which is not a vacuum. This may save the dominant energy condition: the apparent superluminal speed may be a result of a small part of the total system energy concentrating at the location of a constructive interference. The total energy pool moves much slower than light.

The 2012 proof by David Malament

http://philsci-archive.pitt.edu/5072/

Let us check the proof of the geodesic hypothesis by David Malament in his 2012 paper. It is proposition 3.1 in his paper.

The proposition assumes that an arbitrary small "body" moves along a path I, and its stress-energy tensor T satisfies the conservation condition, such that the energy and the momentum of the body is conserved. That is supposed to mean that there are no external forces on the body.

Malament also assumes that the effect of the small body on the "background metric" is "negligible".

Our counterexamples to the geodesic hypothesis versus Malament's and Weinberg's proofs

Our counterexamples in this blog are such that the test mass does not affect the "background metric" at all.

In order to capture what people usually understand with the geodesic hypothesis, Malament's proposition should contain the following part: for an arbitrary (neutral) body there exists a path where the conservation condition for T holds.

Our counterexamples do not refute Malament's proof because in our examples, complex interactions, which act through the gravity field of the body, change T along its path.

Our counterexamples show the following:

Theorem. If we have a test mass m which does not affect the background metric at all, and if m has no other charges than the gravity charge, then it can still happen that m does not move along a geodesic path.

Our Theorem shows that the metric of general relativity does not really capture the movement of test masses. However, the Einstein-Hilbert action might capture it.

Weinberg's proof assumes that the metric correctly describes time intervals and distances measured by an observer. Customarily, it is assumed that the observer uses rays of light and clocks to make measurements. In our counterexamples, rays of light do not obey the metric. Weinberg's assumption is not correct in the counterexamples.

Conclusions

The proofs of the geodesic principle seem to ignore complex interactions which are mediated through the gravity field of a test mass.

General relativity, if we define it through the Einstein-Hilbert action, probably captures these complex interactions. Proofs of the geodesic hypothesis assume things which are not true when general relativity is defined this way.

The proofs are, in a sense, circular. They make assumptions which are quite close to assuming the geodesic hypothesis itself.

Retardation: a test charge "sees" the entire static electric field and the entire extra inertia

Suppose that we have an electric charge C which is suddenly moved at a time t. In electromagnetism it is assumed that a test charge c only sees a change in the field of C when light carries the information of the movement of C to the location where c resides.

                                              • c  test charge

             C  ● -->          ----------------------

        charge                   field of C

The test charge sees the field of C as if C would have been moving at a constant velocity. The velocity in this example is 0. This is the usual assumption about retardation.

In our blog we assume that there is extra inertia associated with c when it moves in the field of C.

Hypothesis. The test charge c feels the extra inertia as if C would have been moving at a constant velocity.

How to prevent a perpetuum mobile?

Retardation seems to open the doors for a perpetuum mobile.

              + ● --->                     <--- ● +

Suppose that we have two charges with the same sign. We suddenly move them closer to each other. Each charge sees the "old" field of each other, which is weaker than the field would be if it would be updated infinitely fast.

The force when we move the charges is less than the force which will repel them later? Do we have a perpetuum mobile?

No. The fields of each charge "bend" when we move them. We have to do work to cause the bending. We cannot recover all that work after the process is completed. That spoils the perpetuum mobile.

                  /    \        field lines = "wires"

                /        \  

          + ●           ● +

This can be understood through a wire model of the fields. The repulsion between the charges come from stresses of the wires. We can cheat with the repulsion for a moment by moving the charges quickly, but we have to pay the price of bending the wires. No extra energy can be gained.

A test charge c close to a large stationary charge C; a mass-energy shipment

This is a common configuration. The test charge feels a force field which is static. The Coulomb force is equivalent to the one which we could calculate from the energy of the combined field of c and C.

If we think in terms of the field energy, the static field magically "knows" how much the field energy would change if we moved c.

Our hypothesis above claims that the static field magically knows the inertia of such a movement.

          more field energy      

                       • +  c        -->

                     less

                       ● +  C

  

                    more

We have marked in the configuration how a test charge modifies the field energy density of the large charge C. Suppose that we suddenly move c right. The energy flows in the field. How could the field know beforehand how much energy flow there will be, and be able to tell to c how much inertia there should be?

The test charge c changes the energy density by

       ~ E E',

where E is the field of C and E' is the much weaker field of c.

The test charge c has to emit a "mass-energy shipment" to the the field, so that the energy can be shipped around the right distances.

A mass-energy shipment might be defined:

       s W,

where s is the distance and W is the amount of energy. Alternatively, as

       s m,

where m is the mass shipped.

Conservation of the center of mass is the associated property. A shipment must be balanced by an opposite shipment.

Is there some analogy in mechanics which could simulate a shipment in the electric field? Any electromagnetic wave which c creates is very weak, since the field of c is very weak. The process can only leak minimal energy and momentum into space.

Suppose that we have a tapering rope which becomes very narrow far away.

             ^  movement

             |

              ####====----- .  .  .  .   rope

It might be that if we input a shipment to the heavy end of the rope, that makes the entire rope to move a fixed distance up in the diagram. That is, the heavy end "feels" the entire inertia of the rope immediately.

1. Only minimal energy can escape as radiation when we move the test charge c.

2. The common field of C and c cannot remain oscillating. It becomes static very quickly.

Items 1 and 2 suggest that the field does behave like the tapering rope. The field can relay shipments but it cannot oscillate.

Compare the tapering rope model to a simple rod:

         ^

         |

          ================  rod

If we move the end of the rod suddenly upward, we do not feel the entire inertia of the rod immediately. The rod will start to oscillate. But such oscillation has not been observed in static electric fields. This has to be a wrong model.

Why does the test charge feel the field energy instantaneously? The Coulomb force

Let us ask the question: why does the test charge c in the field of a large charge C instantaneously "see" how much the combined field energy will change when we move c? That is, how can the Coulomb force predict the required energy?

We may appeal to the same argument as for the extra inertia. Let us move c quickly closer to C. If we could input less energy than is required to update the entire field, then there would be oscillation. But such oscillation has not been observed experimentally.

We must input the right amount of energy when we move c. That energy is then shipped to the appropriate locations in the combined electric field.

What would be a mechanical analogue for this energy flow process? For inertia, it was the tapering rope.

                     ●/\/\/\/\/\/\●

If we push the ball on the left quickly, the spring carries the energy to the right end. This mechanism might be analogous to the combined electric field of c and C. The energy is shipped as "pressure waves".

Conclusions

We presented a hypothesis that a test charge immediately feels fully the extra inertia caused by an external electric field.

We argued that the hypothesis is reasonable, and might be the only way that extra inertia can work. Other mechanisms would imply that the static electric field of the charges could oscillate, but such oscillation has never been observed.

Pressure causes double gravity in general relativity?

Let us have a pressurized spherical vessel, filled with liquid. 

                     ____ 

                 /            \

               |       •        |            • = test mass m

                 \______/

         pressurized vessel

We slowly move a test mass m from far away into the vessel. The test mass might be a small massless box filled with light. That is, the mass-energy is in the form of light.

How much energy is released when we move the test mass?

1. the test mass m extends the radial metric but keeps the tangential metric as is: the volume of the vessel grows and energy from the pressure is released;

2. each bouncing photon gains inertia because the mechanism of item 1 makes pressure energy to move around: more inertia means less kinetic energy, and energy is released from the photons.

Item 2 might explain the temporal metric of general relativity. The pressure in the Schwarzschild interior solution slows down clocks at the center of the the ball of incompressible liquid.

Our earlier calculations in this blog suggested that also item 1 would explain the metric. Do we have double gravity here?

The Einstein-Hilbert action knows the metric and is certainly also aware of item 1. When we move the test mass m to the center, the slow flow of time effectively releases some of the mass-energy m. But the action also calculates the energy released by the volume expansion of the vessel.

How quickly does the extra inertia affect a movement of m?

Let us move m abruptly a short distance inside the vessel. That will make pressure energy to move around in the vessel, and the inertia of m feels larger. But since changes in the gravity field only propagate at the speed of light, the extra inertia should first be less and then grow?

General relativity believes that it is the metric which "implements" the extra inertia. In general relativity, the inertia stays the same throughout the movement.

However, in general relativity, gravitational waves exist, which shows that general relativity is aware of the speed limit on the gravity field. Is there a contradiction here?

Does Einstein-Hilbert action count the extra inertia twice? First through the slowdown of the time in the metric, and after that through the flow of pressure energy in the vessel? Is the action smart enough to understand that pressure energy flows inside the vessel and adds to the inertia of m?

What takes care of action/reaction? Noether's theorem

The Einstein-Hilbert action describes the direct gravitational attraction between masses M1 and M2 . For them, the metric may implement the action/reaction law of newtonian mechanics. M1 falls in the field of M2, and vice versa.

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

What about pressure and even more indirect interactions of the matter lagrangian L and gravity? What ensures that momentum is conserved, i.e., there is an action/reaction? We have to check if the ADM formalism, which is a hamiltonian formulation of general relativity, proves conservation of momentum.

https://en.wikipedia.org/wiki/Noether%27s_theorem

Generally, Noether's theorem states that a lagrangian, which only has conservative forces, preserves linear momentum if the lagrangian is invariant with respect to spatial translations. The Einstein-Hilbert action seems to conserve energy, and it is invariant under spatial translations. Thus, it most probably also conserves linear momentum.

On the other hand, if we try to analyze the system using a metric, then it is not clear what would alter the metric around M to cause M to move. This might be yet another instance where the geodesic hypothesis fails.

Pressure in M and a test mass m: a hamiltonian seems to conserve momentum

In the case of pressure inside a large mass M, and a test mass m within, momentum conservation comes from the following process.

Let us have a hamiltonian which is equivalent to the Einstein-Hilbert action. If we move m and M closer to each other, then energy is freed from potential energy to kinetic energy. That is, the lagrangian T - V increases. The increase is the fastest if we move both m and M closer to each other. That is because if we free some potential energy ΔV and change it to kinetic energy of m and M, then the distance between m and M shrinks the fastest if both approach each other. The lagrangian favors the fastest conversion of potential energy to kinetic energy.

This is the mechanism of action/reaction.

Conclusions

In our own Minkowski & newtonian gravity model, pressure frees energy through the two mechanisms described in items 1 and 2.

In general relativity, it may be that the extra inertia of the test mass is double counted. We have to study the Einstein-Hilbert action in detail to figure out if that is the case.

The speed of the extra inertia remains an open problem. Does the test mass m possess the extra inertia immediately (like it does in a metric), or does it only acquire the extra inertia when pressure energy starts to flow?

Why the dynamic electromagnetic field does not affect energy levels in the hydrogen atom?

In our previous blog post we were perplexed about the ignorance of the hydrogen atom about the extra inertia which the potential of the proton causes on the electron.

It could be that there simply is no extra inertia in a potential, but we are not willing to abandon our extra inertia hypothesis yet.

Classically, the acceleration of the electron would cause it to radiate, but that does not happen if the electron is on a stationary orbit. Could it be that the absence of extra inertia is related to this? Maybe the field is not "dynamic" when the electron is stationary?

On March 13, 2021 we were able to derive the Lamb shift, assuming that the electric field of the electron is elastic (= the rubber plate model). This is a dynamic behavior of the field.

We should find an explanation why certain dynamic behavior of the field happens and the other not.

There is not enough energy to "implement" the miniature energy flow in the combined field of the electron and the proton?

In a macroscopic system we may hypothesize that the Poynting vector describes an energy flow in space. Let the electron in a hydrogen atom be at a typical distance from the proton. Then its potential energy is -27.2 eV.

If we have a photon whose energy is 27.2 eV, its wavelength is 44 nanometers, or 420 times the diameter of the hydrogen atom. We cannot describe miniature energy flows within the atom with such photons. The spatial resolution is too poor.

Could it be that the energy flow does not exist because a photon is not able to resolve it?

When the electron is very close to the proton, at 10⁻¹⁵ m, the potential is ~ -511 keV, and the corresponding photon wavelength is 2.4 * 10⁻¹² m. Again, it is not possible to resolve the miniature energy flow with the photon of the given wavelength.

What is the nature of the energy flow which is described by the Poynting vector? It arises from changes in the electromagnetic field. It does not look like ordinary, oscillating photons.

The Lamb shift and the rubber plate model of the electron electric field

On March 13, 2021 we were able to explain the Lamb shift through the assumption that the electric field of the electron is elastic, and the far field does not have time to follow the electron in abrupt movements. The end result is that the effective mass of the electron is reduced.

In this case, the interaction between the far field and the electron are primarily forces which transfer momentum.

"Virtual photons" which transfer momentum are quite different things from photons which transfer energy. It might be that the resolution problem, which we described in the preceding section, does not exist for virtual photons.

Also, in this case the process is private to the electron plus its own field, while the Poynting vector process is collective with the proton.

It might be that the resolution problem does not appear here.

Radiation of a photon when an electron falls into a lower energy state

This phenomenon falls into the category of energy flow. The process is private for the electron plus its own field.

The resolution problem does not appear, because it is the far field of the electron which is lagging behind the electron motion, and "produces" the photon. The photon is "born" in a large volume.

Conclusions

In this blog post we touched a fundamental problem: what classical phenomena with charges also appear in the miniature world of the hydrogen atom?

It sounds quite natural that the Poynting vector energy flow cannot be reproduced in the miniature world. The resolution is not good enough.

The production of an electromagnetic wave is a process which is private to the electron and its own field. We know that the classical process does happen in the miniature world, too.

We have explained the Lamb shift with the elasticity of the electric field of the electron. The process is somewhat like the production of an electromagnetic wave, but in this case no photon escapes. The process is private to the electron and its own field.

E² tells the localization of energy in an electric field?

Let us have an electric charge in empty space. Is there energy "localized" at places where the electric field is nonzero? Does it make sense to say that the local energy density of its electric field is

       1/2 ε E²,

where ε is vacuum permittivity and E is the electric field strength?

Harvesting energy of an electric field with capacitor plates

Let us devise a way to harvest that energy. We may use the following device:

                     +    +    + 

                    --------------      capacitor plate

   ^                              

   |                     V

   | E                        W

                    --------------      capacitor plate

                      -     -     -

                                 R

We can place the right amount of charge into the plates and make the device to cancel an electric field E in a given volume V of space. The electric field pulls on the plates and we can harvest the energy W of the electric field E from the volume V.

But does that mean that the energy W originally resided in that volume V? Or was the energy shipped from some other spatial location?

When the plates are pulled apart at a speed v, there is a magnetic field at the sides of the plates. The Poynting vector will differ from zero there. Let us vary the radius R of the round capacitor plates. The total charge of a plate goes as

       ~ R²,

the electric field strength at the plate edge per a unit charge goes as

       ~ 1 / R²,

and the electric and magnetic field strength of the plate at the edge is roughly constant.

The harvested energy goes as

       ~ R²,

while the length of the plate edge goes only as

       ~ R.

If R is large enough, then the Poynting vector says that any energy flow was small compared to the harvested energy. If we believe the Poynting vector, then the harvested energy W truly came from the volume V.

Practical experiments to test localization of field energy: circularly arranged pendulums do not work

The system is complex and the Poynting vector might misinterpret the location of energy. Measuring the gravity of the energy W is prohibitively difficult. Measuring the inertia of the process might be easier. If the energy flows from a distant location, then the inertia is larger.

On March 25, 2023 we wrote about circularly arranged pendulum clocks. Let us put electric charges to the pendulums. Can we use the clock cycle to determine the inertia?

                                    +  charge

                                 |

                                 |

                + ------                ------  +

                                 |    direction of

                                 |    oscillation

                                   +

The pendulums will oscillate synchronously. Energy from their combined electric field will flow to each pendulum and back. The pendulums will approximate a spherical shell of charge expanding and contracting.

If the energy comes locally directly from the field, then the inertia of the oscillation is less than if the energy is shipped from far away. Let us try to calculate an estimate.

A large static electric charge is typically one microcoulomb. The electric field of a charge q is

       E = k q / r²,

where k is the Coulomb constant

       9 * 10⁹ kg m³/(s⁴ A²).

The field at the distance 1 meter is

       ~ 10,000 V/m.

Its energy content is

       1/2 ε E² ~ 0.5 mJ/m³.

The mass involved is energy per c²:

       m = 0.5 * 10⁻²⁰ kg/m³.

The mass of a pendulum is ~ 1 kg, and the accuracy of a pendulum clock is 10⁻⁷. We conclude that we cannot measure the inertia this way.

The hydrogen atom and the 2s and 2p orbitals

The electron in a hydrogen atom moves in low electric potentials of the proton, ranging from -27.2 eV to perhaps -511 keV. We would expect the extra inertia to affect the hydrogen energy levels a lot, but the classic Schrödinger equation hydrogen model ignores the inertia of the electromagnetic field entirely, treating the electron as a point particle under a Coulomb attraction.

We should see the analogy of the precession of the perihelion of Mercury, if the inertia of the electron would be larger close to the proton.

How is it possible that the inertia does not change close to the proton?

The hydrogen atom states are stationary. Maybe there is no energy flow which would affect the inertia of the electron?

In a stationary state the electron does not radiate electromagnetic waves. Its behavior is fundamentally non-classical. Could this explain why the extra inertia is missing?

This is a fundamental question. Let us write a new blog post about this.

Conclusions

Measuring the inertia associated with the electric field seems to be hard in a macroscopic setting, with the pendulums.

When there is a current in an electric wire, then the inertia in the system is prominent: the energy of the magnetic field is substantial. But it is not clear how to interpret that as inertia of the electric field.

In a hydrogen atom, the inertia of the electric field should affect the energy levels a lot, but we do not see any effect. This is a fundamental question: why in a stationary state the electron behaves as if the electromagnetic field would not exist at all? There is no energy loss through radiation, and no inertia of the field is visible. However, in our blog post on March 13, 2021 we showed that the Lamb shift may be due to the loss of inertia in the private electric field of the electron.

Inertia from the Poynting vector

Let us study extra inertia in the case of electric charges.

A tube of a small negative charge sliding on a rod of a positive charge

                          E'               E

                   -   -   -   -

        ++++++++++++++++++  rod

                   -   -   -   -    R

                    1 meter

                  tube  v ---->

Let us calculate the Poynting vector. We assume that the length of the tube is one meter, its radius is R, and the electric field of the tube E' is much weaker than the electric field E of the rod.

The magnetic field of the moving tube is

       B = E' v / c².

The Poynting vector is defined as

       S = E × B / μ.

The energy flow per a unit area around the tube is

       1 / (μ c²)  E E' v.

The energy flows to the opposite direction from v.

Let us contract R a little, so that the normal area outside the tube grows by A. The combined electric field loses the energy density

       D = ε E E'

           = 1 / (μ c²)  E E'

there. Let us interpret this that along with the tube, the negative energy density D flows at the speed v to the right. Now we see that the Poynting vector correctly calculates the positive energy flow to the left.

We have claimed in this blog that the tube acquires extra inertia since field energy must move around. In this case, the Poynting vector captures our idea.

The extra inertia in this case is as if the tube would be carrying the potential energy that we harvested as we let R contract from infinite to its present value.

We should measure the extra inertia in practice, to be certain that the analysis is correct.

In some other cases, the extra inertia probably is not as nicely linked to the harvested potential energy.

The energy of a magnetic field is twice the kinetic energy of the corresponding electric field

The energy density of an electric field is

       1/2 ε E²

and of a magnetic field

       1/2 * 1 / μ * B².

If an electric field E moves at a velocity v << c normal to the field lines, it generates a magnetic field

       B = E v / c²,

where c is the speed of light.

The kinetic energy (1/2 m v²) density of a moving electric field is

       1/4 ε E² / c² * v²

       = 1/4 1 / (μ c²) E² v² / c²

       = 1/4 * 1 / μ * B².

We see that the kinetic energy density of the electric field is 1/2 of the energy density of the corresponding magnetic field.

A circular loop of positive charge surrounded by a tube with the equivalent negative charge

We let the tube slide around the loop at a velocity v. We adjust the charges in such a way that the electric field is zero outside the tube.

In this case, the Poynting vector does not show any energy flow at all because the electric field is zero outside the tube.

The magnetic field B, on the other hand, differs from zero outside the tube. The energy of the magnetic field is twice of what would be the kinetic energy of the electric field of the tube if there were no positive charge inside the tube.

What is the extra inertia in this case? Let I be the inertia of the tube in a rotating motion without the loop inside. Let us conjecture that the inertia without the electric field outside the tube would be I - m. The magnetic field B adds 2 m to the inertia. The inertia is then

        I + m.

The extra inertia m is as if the tube would be carrying the potential energy which we harvested as we let the radius R of the tube to contract from infinite to its present value.

Let us contract the tube to such R that we have recovered the entire mass-energy M of the tube while lowering it. The mass-energy M is the energy of the electric field of the rod from R outwards. Let us assume that the inertia of the tube, I, in this (strange) case would be zero without the magnetic field. The inertia from the magnetic field is then 2 M. We see that, with these assumptions, the inertia is I + m, using the notation above.

We should measure the extra inertia in practice, to be sure.

Conclusions

In the case of a tube with a very small charge, the Poynting vector might calculate the extra inertia which is associated with the interaction with the charged rod.

But in the case of the tube and the rod with equal charges, the Poynting vector is zero. In this case, the energy of the associated magnetic field may indicate the amount of extra inertia. Even though the electric field is zero outside the tube, it may still possess a lot of momentum, in the form of a magnetic field.

What did we learn? Even though an electric field is seemingly static, or even zero in the second case, it may still possess a lot of momentum. We are not sure how to determine the extra inertia. We used very different methods in these two cases.