Is the energy of a static field always zero? Renormalization and Maxwell

In our blog we have put forth a hypothesis that the energy of a static electric field is zero. Only a dynamic field, e.g., an electromagnetic wave, contains energy.

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

James Clerk Maxwell in his 1865 paper writes that energy is "essentially" positive. He cannot make sense of the gravity field, whose energy density seems to be negative, and he does not try to develop further an electromagnetic theory of gravity.

We recognize at least the following problems which concern the energy density of a force field:

1. The electric field of a pointlike electron contains infinite energy, if we assume that the energy density is 1/2 ε₀ E².

2. If we try to renormalize away the infinite energy by making the mass-energy of the electron particle minus infinite, then the combination the electron & its inner field has negative energy. Negative energy behaves in a strange way in newtonian mechanics.

3. Electromagnetic waves and gravitational waves contain positive energy. How are they born if a static field contains zero energy?

4. Since gravity is attractive, in a naive "electromagnetic" model, the gravity field has negative energy.

5. A point particle in gravity is a singularity and a black hole.

6. How do we implement energy and momentum conservation when particles interact?

Retardation and the Wheeler-Feynman absorber theory

https://en.wikipedia.org/wiki/Wheeler%E2%80%93Feynman_absorber_theory

If we have a proton and an electron, we can assume that the proton has an electric potential, and the electron moves in the potential with no retardation of the interaction. Then there is no need to assume the existence of any "electric field" of the proton, where the field would contain energy. It is enough to assume that there exists an abstract potential.

In quantum electrodynamics (QED), the interaction in the static field is carried by a virtual (off-shell) photon. A virtual photon may have zero energy though it carries momentum.

Conservation of momentum is a problem in the QED model. When do the proton and the electron feel the kick from the exchanged momentum? If we assume a finite speed of of a signal, where is the momentum stored when the signal travels?

The exchange of momentum is like a transaction which the proton and the electron decide together. If the proton emits a virtual photon, then the transaction guarantees that the electron will receive it. It is all or nothing. The virtual photon cannot be left dangling in space.

If two electromagnetic systems at a large distance from each other exchange energy and momentum, then we assume that they exchange a real photon. In this case it is hard to believe that the exchange is a transaction. The real photon can be left propagating in space on its own.

The Wheeler-Feynman hypothesis is that even real photons are inside a transaction. Every photon will be absorbed in the future. Somehow the universe decides which photon will be absorbed by which system. A "free electromagnetic field" does not exist. There are no independent photons flying around. All photons are virtual, or off-shell.

We in this blog believe that the Wheeler-Feynman hypothesis is too radical. We certainly observe electrons losing their kinetic energy in a radio transmitter. Why would the photon lack independent existence, while the electron can exist on its own?

However, we do not know how to decide the demarcation line between virtual photons, which exist inside a transaction, and real photons, which have independent existence.

How does retardation create waves in the field?

We have been claiming that when an electron is accelerated, its "static electric field" lags behind and extracts energy from the movement of the electron. That energy is the energy of the electromagnetic wave.

https://physics.weber.edu/schroeder/mrr/mrrtalk.html

Edward M. Purcell derived the Larmor formula of the extracted energy from:

1. the retardation of the electric field;

2. the fact that the electric lines of force cannot break;

3. the energy density of the dynamic electric field is 1/2 ε₀ E².

In the link Daniel V. Schroeder presents his derivation. 

In retardation the crucial fact is that when the electron is moving at a constant velocity, its electric field is centered at the current position of the electron, not its position where the line of force "departed" the electron. There is no "retardation" at all if the electron moves at a constant speed. That is, the potential of the electron's electric field is fully up-to-date.

Connecting the lines of force in the diagram above requires very substantial deflection of the lines of force at the circle which marks the sudden acceleration of the electron.

A magnetic field is required to deflect the electric lines of force substantially. The wave thus contains both an electric field and a magnetic field.

The deflection of the lines of force makes them more dense at the circle, and increases the calculated energy density of the field, compared to a static field. The dense part of the field moves away at the speed of light, carrying the energy of the electromagnetic wave.

In the derivation we do not need to assume that a static electric field has any energy at all. It is enough to define that the extra density of the field lines in the circle carries energy.

To extract energy from the movement of the electron, the far field has to possess "inertia". We have suggested that the inertia is not from mass-energy of the far field, but a result of retardation: the lines of force of the far field cannot keep up, and simulate "inertia". Thus, retardation can replace inertia in the extraction of energy from a periodic movement of a charge.

Why the static electric field contains no energy, but the dynamic field does carry energy? The Fourier transform of the electric field

If the static electric field of an electron is just an abstract potential which does not contain energy, why does the retarded dynamic field then carry energy? Why cannot it be another abstract potential?

Let us determine the Fourier decomposition of the electric potential of the electron. Let us assume that the electron is eternally moving at a constant velocity.

If the Fourier transform is taken in an inertial frame comoving with the electron, then the decomposition only contains waves whose 4-momentum has zero energy. The waves are time-independent.

What if we take the Fourier transform in an inertial frame which is not comoving with the electron?

We have to Lorentz transform the 4-momenta of the time-independent waves to a moving frame. They are then time-dependent. Some waves acquire positive energy and some negative energy. One could claim that the total sum still has zero energy.

If we have an electromagnetic wave, then the Fourier decomposition of the electric field contains positive energy in the 4-momenta of the waves.

We gave a heuristic explanation why a static field contains no energy, but a dynamic field carries energy.

From where does the energy in the Coulomb interaction come? What about gravity?

                      Coulomb attractive force

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

         positron                                         electron

Let us have a positron and an electron. If we move them closer to each other, we can harvest energy from the Coulomb attractive force. From where does this energy come?

One way to explain the harvested energy is to say that the total energy of the electric field

                ∫ 1/2 ε₀ E² dV

       all space

grew smaller. The combined field grew weaker when we moved the particles closer to each other.

Another way is to say that we harvested energy from the Coulomb force between the particles.

If we claim that the energy of the static electric field is zero, then we have to explain the process with the Coulomb force. We cannot refer to the energy of the static field.

                     newtonian attractive force

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

        test mass                                         test mass

In the case of gravity, we apparently have to claim that the harvested energy comes from the newtonian attractive force between two test masses if we move the masses closer to each other.

In the case of gravity, the combined field of the masses grows stronger. If we define that the static field has zero energy, then the field is irrelevant for the process.

Note that in the rubber membrane model of our previous blog post, the energy of the gravity field is the deformation energy of rubber. It is positive.

In general relativity, there is no gravity force. From where does the harvested energy come? The ADM mass of the system does not change if we store the harvested energy close to the test masses. Apparently, general relativity has "implicit" potential energy associated with the system. In the "implicit" description, there exists a force of gravity.

Another interpretation is that the mass-energy of the test masses drops to a lower potential. A part of the released energy can be harvested, and the rest goes to the "deformation energy" of spacetime. This model would be analogous to the rubber membrane model.

There is positive energy in gravitational waves, but why it is 16-fold?

We can argue just like in the case of the electric field that a static gravity field contains no energy.

A dynamic gravitational wave does contain energy. Robert C. Hilborn calculated that the energy is 16-fold compared to a naive electromagnetic model of gravity.

We will try to find out the reason why it is 16-fold. We could take that as an axiom, but that would not be nice.

The simple way to harvest energy from a gravitational wave is to have a ring of test masses, and put rods as "shock absorbers" between them. The shock absorber harvests energy from (apparent) changes in the distance of the test masses.

                            shock absorber

                   ● =================== ●

            test mass                              test mass

The creation of a gravitational wave is the opposite process: the shock absorber stretches and contracts, moving the test masses.

We have to analyze the process in detail. It may explain the energy content of waves.

How much energy we can extract from a wave by destroying it?

If we have a spatial volume V with an electric field E, we can cancel the field E by putting suitable positive and negative charges in the volume. For example, consider a flat volume between capacitor plates.

When we cancel the field E, we extract

        ∫ 1/2 ε₀ E² dV

       V

of energy. If we cancel an electromagnetic wave, we actually get double the energy above because a half of the energy is in the magnetic field.

Canceling a wave with an equal amount of positive and negative test charges does not affect the creation of the wave since far away the field of the test charges is almost zero.

How would we cancel a gravitational wave? We cannot use positive and negative charges because negative mass-energy does not exist. What is the correct analogy in electromagnetism? Consider canceling a wave created by a negative electric charge with a large number of positive test charges. The test charges create an electric field which may affect the creation of the wave.

We need to analyze this. It may be that a wave of a purely attractive force must hold 16-fold energy compared to an electromagnetic wave.

The "true" gravity force is 4 times the newtonian force?

Our previous blog post uncovered the surprising fact that the energy drain for a mass doing periodic motion is 16-fold relative to an "equivalent" electric charge.

We define the mass equivalent m of an electric charge q with the following formula:

       1 / (4 π ε₀)  * q² = G m²,

where ε₀ is vacuum permittivity and G is the gravity constant. That is, the electric Coulomb force between the two charges q should be equal in absolute numbers to the gravity force between the two masses m.

1 coulomb is equivalent to 1.16 * 10¹⁰ kilograms.

Since gravity is attractive, it is not equivalent to the Coulomb force

The "equivalence" which we defined in above is dubious. If we move two opposite charges closer to each other, then their combined electric field grows weaker.

We can harvest the entire energy which is freed from the weakening electric field, where we define the field energy density with the familiar formula

       1/2 ε₀ E²,

where E is the strength of the electric field.

However, if we move two masses closer to each other, their gravity field grows stronger. This is a big difference from the electric case. Should we define that the energy of the gravity field is negative, and it becomes even more negative when we move the masses close to each other?

Gravitational waves are kind of a gravity field "broken free" from their original source, a massive object. If the energy of the field is negative, how can the waves carry positive energy?

Robert C. Hilborn in his paper mentions that James Clerk Maxwell considered building an analogy between newtonian gravity and electromagnetics, but Maxwell thought that the apparent negative energy of the gravity field prevents us from forming of an analogy.

https://royalsocietypublishing.org/doi/pdf/10.1098/rstl.1865.0008

The famous 1865 paper by James Clerk Maxwell is freely readable on the internet. He writes that energy is "essentially positive". He suggests that the negative energy density of the gravity field should be added to some, ad hoc, constant positive energy density, to make the total energy density positive. However, he does not like the idea and does not explore gravity further.

The model of metal spheres on a rubber membrane explains things?

Let us have a tense rubber membrane stretched horizontally. If we put two heavy metal spheres on it, the spheres tend to roll together. Potential energy is freed when the spheres can sink deeper on the membrane.

There is an apparent "attractive force" between the spheres. We can harvest a part of the released potential energy by letting the spheres pull on a spring when the spheres roll together. But another part goes to the deformation of the rubber membrane.

If we identify the deformation of the rubber membrane with the gravity field of a sphere, or a set of spheres, then the gravity field has positive energy. Now we no longer have the problem of gravitational waves carrying negative energy.

Furthermore, the energy of the gravity field is not dictated by the attractive "force" between the spheres.

The energy of the gravity field is positive, and 16-fold compared to the naive analogue?

Robert C. Hilborn observed that the if the field in his electromagnetic model of gravity would be 4 times as strong, then the model would describe well gravitational waves. The field energy would be 16-fold.

In the previous section we argued that the gravity field can have arbitrary energy which is not determined by the strength of the newtonian attractive gravity force.

Conjecture. The energy of the gravity field is positive and 16-fold compared to the naive electromagnetic model.

In general relativity, there has been debate about what is meant by the energy of gravitational waves. The energy is measured with a pseudotensor, and the energy does not appear in the stress-energy tensor. Thus, is it "real" energy?

In the rubber membrane model, it certainly is real, positive energy.

Our Minkowski & newtonian model needs a major overhaul

Gravitational waves revealed a major flaw in our Minkowski & newtonian model. The structure of the gravity field is more complex than we thought.

It may be that one cannot build a model of an attractive force from the simple model of electromagnetism. Negative energy of the field cannot work.

How does the new, more complex, model affect the inertia of a test mass or a photon in the Schwarzschild solution? Does it affect the rate of clocks?

How do gravitational waves in the new model affect the apparent metric of time and space?

Hilborn's electromagnetic model of gravitational waves

Robert C. Hilborn (2017) has written a very interesting paper where he treats the newtonian gravity force exactly like the electric force in electromagnetics is treated.

https://arxiv.org/abs/1710.04635

Hilborn calculates the radiation for an orbiting binary system. Such a system is a quadrupole.

Why is the energy density in a gravitational wave 16-fold compared to its exact electromagnetic analogue?

The analogy between gravitational waves in linearized general relativity and electromagnetics looks very good, except for one thing: the energy drain from the binary system is 16-fold in general relativity! We need to find out why that happens.

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

For an oscillating (not rotating) electric quadrupole, the energy drain seems to be even less than for a rotating electric quadrupole.

Question 1. We know that the electromagnetic dipole wave is "detached" from the "quasi-static" electric field of the charge at the distance of roughly 1 radian. Where is the gravitational wave detached from the Schwarzschild metrics of the orbiting masses? (We use the word quasi-static because the "static" electric field of the charge is actually moving, and even accelerating.)

Question 2. What is the energy density of the wave if we use the electromagnetic model? The energy density does not need to be

        ε₀ E²,

like for an electromagnetic wave. (A half of the energy resides in the magnetic field: that is why we do not have the coefficient 1/2 in front of the formula above.) What matters is how much work the wave can do, or equivalently, how much work is required to create the wave.

Harvesting energy from a wave

When an electric charge is accelerated, the charge has do work against the self-force which is imposed on it by its own quasi-static electric field. For a reason unknown to us, the formula ε₀ E² tells us the energy density of the wave. The formula could be different, since we do not completely understand the self-force.

We cannot pinpoint the exact location of the energy in an electromagnetic wave. Harvesting energy from a wave is a complex process, and it does not reveal us where the energy "exactly" was. The same holds for a gravitational wave: the pseudotensor tells the energy contained in a spatial volume whose size has to be several wavelengths.

(The diagram by MOBIe.

https://commons.wikimedia.org/wiki/File:GravitationalWave_PlusPolarization.gif#mw-jump-to-license)

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

Energy is fed to a gravitational wave through a quadrupole, and people usually think that the energy has to be harvested in the same way, using a quadrupole.

The wave squeezes and stretches a ring of test masses. In the diagram we have a ring of test masses which has been squeezed horizontally by a + polarization gravitational wave. In principle, we could extract energy by putting rods between the test masses.

We can extract energy from an electromagnetic quadrupole wave with a dipole antenna where opposite charges travel to opposite directions. For gravity, we do not have that option.

Is the gravity field 4 times as strong in the wave, or is its energy for some other reason 16-fold?

If an electric field is 4 times as strong, then its energy density is 16-fold. If the gravity field somehow is detached from the quasi-static field at a distance of 1/4 radians, then the field would be 4 times as strong as in the electromagnetic analogue.

The bending of light close to the Sun comes 1/2 from the newtonian attractive force, and 1/2 from the stretching of the radial metric in the Schwarzschild solution. This suggests that the gravitational wave might have 2 times the amplitude of the electromagnetic analogue. But how do we get a factor of 4?

A wave as an image of the near-field behavior

If we have a ring of test masses very close to a binary star, we can harvest energy both from the movement of the masses and from the stretching of the radial metric in the Schwarzschild solution. We have to calculate how much do we get from each effect. Could that explain the 16-fold energy flow?

We believe that waves, in some sense, "transfer" the environment which is close to the source, close to the receiver. Can we find a reason why in the case of gravity, the receiver gets even closer to the source than in the case of a rotating electric quadrupole?

Conclusions

Our Minkowski & newtonian model must be able to explain the structure and the energy flux of gravitational waves. We were hoping that the simple analogue with electromagnetism would suffice. That is not the case. We need to investigate this.

The Minkowski & newtonian model: what is the inertia of a test mass inside a gravitational wave?

A basic principle of our Minkowski & newtonian model is that gravity is an ordinary force in the Minkowski geometry of spacetime. Gravity can in some cases imitate a "geometry" of spacetime, but the true geometry is always the flat Minkowski geometry.

This basic principle has as the consequence that superluminal communication in the Minkowski background metric cannot happen.

Gravitational waves in the Minkowski & newtonian model are analogous to electromagnetic waves. Gravitational waves impose forces on charges of gravity, that is, on photons and other elementary particles. Those forces imitate a change in the spatial metric, but the true metric does not change.

Let us study gravitational wave phenomena in more detail in our Minkowski & newtonian model.

Why clocks slow down in the Minkowski & newtonian model?

The slowing down of mechanical clocks close to a mass is due to two effects:

1. Any packet of energy, when lowered down in the gravitational potential, does work when it is lowered. There is less energy available in a low potential. Forces are weaker.

2. The inertia of any particle increases when lowered down in a gravitational potential. Besides the mass-energy of the particle, we also have to move some (negative) energy in the combined gravity field.

Time itself does not slow down close to a mass. It is clocks which slow down. Light slows down because a photon has to carry besides its own inertia, also some inertia of the gravity field. These effects create the illusion that time itself would have slowed down.

Why the radial Schwarzschild metric is stretched in the Minkowski & newtonian model?

The stretching of the radial metric in the Schwarzschild solution is in our model explained by the fact that the inertia in a radial movement is larger than in a horizontal movement. If we move a test mass deeper in the potential, it receives energy from distant parts of the gravity field. Moving this energy around causes extra inertia. Light propagates slower to the radial direction than to a horizontal direction.

We have to assume that the local geometry of all physical phenomena between elementary particles is controlled by the light speed. Then all things and phenomena are squeezed in the radial direction according to the ratio

       radial light speed / horizontal light speed.

The fact that inertia is, say, 1% larger to the radial direction, cannot alone explain the squeezing by 1%. Consider a particle which is moving horizontally and collides elastically with a wall at a 45 degree angle. The particle after the collision moves radially, 1% slower. The absolute momentum |p| of the particle is preserved, but its kinetic energy is now lower:

       E = p² / (2 m),

where m is the inertia of the particle. Where did some energy go? It had to go to a deformation of the force fields. The deformation energy is freed if the particle starts to move horizontally again.

Consider an almost light-speed particle which starts to interact with other particles. For example, we may have a photon propagating in air: it moves at almost the light speed. Then the photon enters a pane of glass. The interaction is much stronger inside glass. The photon gains more inertia and moves considerably slower. But again we have the problem: where did the extra kinetic energy go? Also in this case the extra energy had to go to a deformation of the system. The extra energy is given back to the photon when it exits the glass pane.

Any interaction increases the inertia of a light-speed particle and slows it down?

We can slow down light easily: let it go through a glass pane, or fly past Earth at a close distance.

We are not aware of any process which could speed up the travel of light.

Conjecture. If a light-speed particle interacts with some other particle or field, the particle always slows down. The speed of the particle is the largest in otherwise empty space.

Gravitational waves in general relativity seem to break the conjecture because they allow time to run faster in some areas.

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

The weak energy condition is a hypothesis which bans negative mass in general relativity. Negative mass would make time to run faster in its vicinity. The purpose of the weak energy condition is to prevent paradoxes which would come from too fast a speed of light.

What is the inertia of a test charge or a test mass inside an electromagnetic or gravitational wave?

In the Minkowski & newtonian model, a gravitational wave is similar to an electromagnetic wave. There is a force field which is analogous to the electric field, and another field which is analogous to the magnetic field.

The movement of test masses (= test charges) is due to the "electric field". If we could have dipole gravitational waves, they would make masses to oscillate up and down in the plane of polarization.

Quadrupole waves squeeze a ring of test particles. The squeezing has two components: + and ×. The component + squeezes horizontally, and the component × at a 45 degree angle to horizontal.

Let us analyze a dipole wave.

We have the field of a test mass, and the "electric" field of the wave. We want to find out what is the inertia of the test mass. This is just like finding the inertia of an electric test charge inside an electromagnetic dipole wave.

   

                  ^  E electric field of the wave

                  |

                  |       ● 

                           q test charge    

Suppose that the wave has an almost zero frequency. Then E is almost constant. How much does the ambient field E increase the inertia of the charge q?

If the charge q is positive, it makes the total field stronger than E above q and weaker than E below q. The field of q is

        E' ~ 1 / r²,

where r is the distance. The increase in energy density is

        ~ E E'

and the volume element is

       ~ r² dr

This is strange. The inertia could increase without bounds inside a large plane wave. The energy of such a plane wave is infinite, which is not realistic, though.

The quadrupole wave from a black hole merger carries huge energy. If our test mass would move significant energy around the field of the wave, then the inertia of a small test mass could be astronomical? Let us calculate an example.

The mass-energy of a wave from a merger of black holes can be 10³⁰ kg. If our test mass is 1 kg, it might be able to increase the field by a factor 10⁻³⁰. The inertia of the test mass might double. Moreover, the wave does not need to be close to us. A distance of a few billion light years is no problem because the gravity field of our 1 kg test mass extends that far. Everything would happen much slower because of gravitational waves which exist in the visible universe. That does not seem plausible.

With an electric charge we face a similar problem in its coupling to all electromagnetic radiation in the universe. If moving the charge alters the energy distribution of every electromagnetic wave in the whole universe, the inertia of the charge might be huge.

Estimating the inertia of an electric charge inside an electromagnetic wave

We believe that the inertia of a test charge q increases close to macroscopic charge Q, because by moving q we can transfer energy from one place to another. Maybe inertia is not about the energy density of fields, but about the ability to transfer concrete packets of energy from one place to another?

For a static field of a charge Q, moving a test charge q for a distance s does allow us to transfer the associated potential energy. But for a light-speed wave we only have a certain time to store potential energy to q and harvest the potential energy from it.

              ^       ^       ^       ^

              |        |        |        |       E

                ● q

                 |                   |  s

                 |                   |

                  ------------------

If the cycle time of the wave is t, then we, in principle, might be able to store and harvest energy during a trip whose vertical displacement in the diagram

       s = 1/4 c t.

The stored energy:

       W = q E * 1/4 c t.

That gives us an estimate of the increase of inertia of the charge:

       W / c².

That is, only the electric field E relatively close to the test charge q can increase the inertia of q. Electromagnetic fields light-years away do not contribute.

Estimating the increase in the inertia of a test mass, caused by a gravitational wave

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

The first observation of a gravitational wave happened on September 14, 2015. The strain was 2 * 10⁻²¹, the frequency 35 ... 250 Hz, the distance 1.3 billion light-years, and the radiated energy 3 solar masses. The masses of the merging objects were 29 and 36 solar masses.

Let us calculate the effect on the inertia of a test mass. Both black holes had a radius of 100 km. The radius of the merging system was ~ 300 km at the final stage.

Close to the system, the extra inertia of a test mass comes from the energy deficit in the gravity field of the system, caused by the infinitesimal field of the test mass.

In the final round of the spiral, the black holes have relativistic speeds. We can guess (or calculate) that a signigicant part of their field g is "detached" and escapes as gravitational waves.

The energy density of the gravity field is

       ~ - (g + g')²,

where g' is the field of the test mass. The change in the total energy of the field due to g' is

       ~ - g g'.

The inertia of a test mass at a distance 500 km might have been elevated 10% because of its interaction with the detaching field g of the system.

The field in a gravitational wave decreases as

       ~ 1 / r

with the distance.

We assume that far away, the extra inertia of the test mass m still comes from its interaction with the gravitational wave field in a volume whose diameter is only

       ~ 500 km.

We can then apply the 1 / r rule to calculate the inertia at a great distance.

The increase in the inertia at the distance of 500 km = 5 * 10⁵ m was ~ 0.1. The distance 1.3 billion light years is 1.3 * 10²⁵ m. We get

       ~ 0.1 * 5 * 10⁵ / 1.3 * 10²⁵

       = 4 * 10⁻²¹

as the increase of the inertia at the distance of Earth. The figure is of the same order of magnitude as the measured strain. Because of the extra inertia, a mechanical clock on Earth may tick 2 * 10⁻²¹ slower when inside the gravitational wave.

What causes the undulation of the spatial metric in a gravitational wave?

In the Schwarzschild metric, the slowing down of time has the same ratio as the stretching of the radial metric. Close to the source, the changes in the rate of a clock and measured distances have similar magnitudes. If this carries over to large distances, the strain and the slowing down of time should have similar magnitudes on Earth. Our very crude calculation agrees with this.

However, we are not sure if the apparent change in the spatial metric in a gravitational wave is due to the force of the "electric" field, or if it is caused by differences of inertia, like in the Schwarzschild metric. We need to investigate this.

It is difficult to visualize quadrupole waves. What does the electric field look like in them?

Question. If we have a very rigid rod which connects very large masses, then the strain of a gravitational wave can put very large energy to the deformation of the rod. The energy obviously cannot exceed the total energy of the wave. How do we model this? What is the maximum force that the wave can impose on the very large masses?

What is the rate of a clock inside a gravitational wave?

If the inertia of a photon increases 1%, it will fly 1% slower, and the speed of light is 1% slower.

If we have a clock which measures time by letting a photon to bounce between two walls, the rate of the clock depends on the inertia as well as the distance of the walls. The rate of the clock depends both on the inertia and the stretching of the spatial metric.

If we have a mechanical clock, how should we apply the rule 1 of the first section? Do forces grow weaker inside a gravitational wave?

In the Schwarzschild metric, the light clock and the mechanical clock tick at the same rate. However, it may be that the rates are different inside a gravitational wave. This might offer a possibility to test experimentally if general relativity is correct versus if the Minkowski & newtonian model is correct.

Another way to test is to try to detect if the speed of light is > c inside a gravitational wave. General relativity may predict superluminal speeds.

Conclusions

The Minkowski & newtonian model predicts gravitational waves where the metric of time and spatial distances undulate, and the amplitude of undulation has the same order of magnitude as in general relativity.

However, in the Minkowski model, the rate of clocks and the speed of light can only decrease when measured in the global Minkowski coordinates. A gravitational wave slows down clocks and the speed of light. In general relativity, these might grow above the values of the asymptotic Minkowski metric. This offers us an opportunity to show experimentally that general relativity is incorrect.

We do not understand well enough how electromagnetic or gravitational waves change the inertia of objects, or if they affect the strength of other forces. We do not know what causes the spatial metric to change. We need to do more research.

The metric around a wave packet of light or gravitational waves

UPDATE November 20, 2021: We added a link to the Physics stack exchange question where an author claims that the metric of time cannot change in a gravitational wave. We added a link to a paper by A. Loeb and D. Maoz where practical measurements of the oscillating metric of time are discussed.

----

Our previous blog post brought up the question what is the metric inside and around a wave packet which moves at the speed of light.

The limiting case of a long rod of ordinary matter moving at almost the speed of light

                          long rod

            =======================  

                              ----> v ≅ c

Our first guess, naturally, is that the metric is similar to ever longer rods of ordinary matter which we make to move ever faster.

The length of the rod is kept constant, say, 1 meter in the frame of the observer. The total energy of the rod is kept constant, say,

       E = m c².

The electric field of a fast-moving charge is squeezed in the direction of its movement. The newtonian gravity field of the rod might be cylindrical?

Let us put an initially static test mass close to the orbit of the rod. The rod feels the gravity field of the test mass long before the test mass knows anything of the approaching rod and the orbit of the rod starts to bend.

Momentum has to be conserved. When the rod has passed, the test mass must have received the momentum which the rod exchanged with the gravity field of the test mass.

A cylindrical field might be the right solution.

Our own syrup model of gravity suggests that the rod makes test pulses of light to travel along it. The speed of a test pulse in the global Minkowski coordinates is almost c to the direction of the rod movement. The speed of light may be considerably less than c to any other direction.

What kind of a metric could describe this behavior?

The rod pulls on the test mass. The flow of time must be slower than the Minkowski time close to the rod, to implement the pulling force. For slow speeds of the rod, this is certainly true.

Comoving clocks versus static clocks close to the rod

Let us have an observer comoving with the rod at almost the speed of light. He sees the rod as very long, and its mass is very small. He thinks that the metric is very close to the flat Minkowski metric. He sees that the speed of light is slightly below c close to the rod, to every direction. The radial metric is slightly stretched.

The comoving observer sees the mass of the rod as

       m / γ

where

       γ = 1 / sqrt(1 - v² / c²).

If the comoving observer shoots a ray of light to the left in the diagram, past the rod, the ray is only deflected by an angle

       α ~ m / γ

by the gravity of the rod. However, because of length contraction, a static observer sees the angle as

       α' ~ m.

This makes sense: the static observer sees the whole mass-energy of the fast moving rod to deflect the light.

What about the flow of time? If there is a clock attached to the rod, the comoving observer sees it tick only slightly slower than his own clock, because the mass is only m / γ.

If we have a static observer normal to the rod very far away observing time signals from both clocks, he sees both clocks ticking very slowly, and the clock attached to the rod ticking just slightly slower than the comoving clock.

What about a static observer close to the rod? How much has his time slowed down?

If a static clock very close to the rod send a signal, there is a considerable redshift relative to a static clock far away from the rod.

The relative redshift in the comoving clocks is much less than in the static clocks. How can we explain this?

It is probably frame dragging. The "effective" velocity of the clock attached to the rod is less than one would expect because the rod drags the frame along with it. If we have a rotating large mass, we can make a clock close to the mass to tick faster by letting it comove with the mass.

The mass-energy of a gravitational wave packet

Time for a static observer is slowed down considerably near the rod. We want to find out if this slowdown is enough to cancel the speeding up of time in the "crests" of the wave. That is, if clocks close to the wave never tick faster than clocks far away in the Minkowski space.

https://core.ac.uk/download/pdf/25257063.pdf

S. V. Babak and L. P. Grishchuk calculate in the link the stress-energy pseudotensor for a perturbation h of the Minkowski space metric. It is the formula (27) in the paper, and we see that every term is proportional to the product of two partial derivatives of components of h.

The speedup of time is proportional to h₀₀. If we divide h by a large number N, then the energy density is only 1 / N². We immediately see that the potential generated by the energy density cannot cancel a possible speedup of time in a gravitational wave.

What about frame dragging? Could it cause the signal to take a longer path, so that communication cannot be too fast? If frame dragging is proportional to the mass-energy of the wave, then it cannot slow it enough. Also, frame dragging could be used to move the signal to the right direction. Then it would not slow down communication.

What about a gravitational wave which contracts spatial metric in the Minkowski space?

Let us image that we have an orthogonal spatial coordinate grid drawn into the Minkowski space.

Let a gravitational wave packet pass by and contract the spatial distance between two observers A and B. If A sends a light signal to B during that time, then the signal appears to have moved faster than light. Does this bring us all the paradoxes of superluminal communication?

We may imagine that the wave just temporarily moves A and B, and the true metric remains exactly the flat Minkowski metric. Then there is no paradox.

However, if A and B are not inside the wave packet, and the distance anyway gets contracted, then we have true superluminal communication which brings the paradoxes.

https://physics.stackexchange.com/questions/553487/do-gravitational-waves-cause-time-dilation-or-not

In the answers to the Physics stack exchange question (2020) above, an author Paul T. claims that a gravitational wave changes the spatial metric but not the metric of time. That is a strange claim. If we in the Minkowski space in the frame 1 have a spatial distance, then in a moving frame 2 the distance is both spatial and temporal.

Paul T. writes that by "gauge fixing" we can show that the metric of time is constant. But the choice of the gauge cannot affect observed physical phenomena. If someone observes that clocks tick at different rates, that cannot be altered in any gauge.

Several authors have a consensus that the component h₀₀ and other components of the perturbation h obey the standard wave equation in linearized Einstein equations. Thus, there are waves in the metric of time. 

https://arxiv.org/abs/1501.00996

Abraham Loeb and Dan Maoz (2015) write about observing mHz gravitational waves by distributing atomic clocks to the Solar system.

https://www.smithsonianmag.com/science-nature/send-atomic-clocks-space-find-gravitational-waves-180954359/

The consensus seems to be that gravitational waves do affect the metric of time.

Conclusions

If a gravitational wave allows superluminal communication in an asymptotic Minkowski space, we get all the causality paradoxes. Breaking causality is not accepted in a robust physical theory. We must correct general relativity in a way which prevents superluminal communication.

The existence of timelike loops in, e.g., the Gödel rotating universe, is evidence against general relativity. If timelike loops exist with gravitational waves, that is strong evidence against general relativity.

Our own Minkowski & newtonian model of gravity probably does not allow superluminal communication in a gravitational wave. We will analyze it in the next blog post.

Gravitational waves: the metric of time and superluminal communication

C. Denson Hill and Pawel Nurowski (2017) have written an excellent historical account of wave solutions to the Einstein equations.

https://www.ams.org/publications/journals/notices/201707/rnoti-p684.pdf

Linearized Einstein equations and a perturbation of the metric of the Minkowski space: superluminal communication

Albert Einstein in 1916 linearized his equations for a small perturbation h to the flat Minkowski space metric η:

If the stress-energy tensor T = 0, then we have:

where the box is the d'Alembert operator

       □   =  -1 / c² * d²/dt² + d²/dx² + d²/dy + d²/dz².

That is, each component of the perturbation h satisfies the familiar wave equation for light-speed waves. We use the East coast signature (- + + +) in the metric.

Now we see an immediate problem: if the metric of time is allowed to oscillate around  the Minkowski metric, where

       η₀₀ = -1,

then in some zones of spacetime, time flows faster than in the asymptotic Minkowski space. If the spatial metric is not stretched accordingly, then the speed of light defined in the global Minkowski coordinates would exceed c in those zones. That brings all the paradoxes of superluminal communication.

For electromagnetism, the analogous oscillation is not a problem. There are no paradoxes if the electric potential swings around the baseline potential of faraway space.

Could it be that the spatial metric is stretched enough to prevent superluminal messages?

The stretching should be in-sync with the undulation in the metric of time. If a gravitational wave is born from a binary star, we do not see why the spatial metric to every direction would be stretched in that way.

In the Schwarzschild metric, time has slowed down and the radial metric has been stretched. If the distorted metric sends a wave, we expect the metric of time to undulate, and there is also stretching of spatial metric to a certain direction at each point.

We conclude that linearized Einstein equations would produce metrics which are not satisfactory.

We do not know if Albert Einstein recognized this problem. Linearized equations allow the gravitational potential to swing above the potential of faraway space. That has similar consequences as matter of negative mass. Paradoxes abound.

How do you implement waves in a drum skin which is not allowed to swing above the horizontal level?

Should we change to comoving coordinates?

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

Wikipedia mentions the synchronous gauge which "requires that the metric does not distort measurements of time." Can that work?

Let us try to define comoving coordinates by putting a clock at each Minkowski spatial coordinate position

       (n, m, l),

where n, m, l are integers.

        y

        ^

        |           O          O          O     clocks

        |

        |           O          O          O

        |

         ----------------------------------------> x

The clocks initially show the global Minkowski time. The positions of the clocks and the time which they show define comoving coordinates.

Let then a gravitational wave pass through the system of clocks.

What could go wrong? If the wave puts the clocks in a spatial disorder, then our spatial coordinates are of no use afterwards.

Also, if the wave leaves the clocks showing different times, then our coordinates are awkward. Traveling back in coordinate time would become possible.

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

Demetrios Christodoulou has shown that a wave can leave permanent changes in the relative positions of the clocks. It is called the gravitational wave memory effect.

Why would we define new coordinates? All physical phenomena stay exactly the same with the new coordinates. Also, if there are permanent changes in the relative positions or times shown by the clocks, the new coordinates become misleading.

We conclude that defining new coordinates is not a good idea.

The background metric around a wave packet

Could it be that nonlinearity somehow prevents superluminal communication inside gravitational waves?

The waves themselves carry mass-energy. If they bend the background metric in a suitable way, then the speed of light inside the waves might be slow enough to prevent superluminal communication.

However, if the wave propagates at the speed of light in the global Minkowski coordinates, it cannot slow down the speed of light toward the direction of its propagation.

This question is connected to the metric around a pulse of light. People believe that the background metric around a packet of gravitational waves is similar to the metric around a wave packet of light. Let us write another blog post about this question.

Gravitational waves and the self-force: can we describe the force with a metric?

UPDATE November 17, 2021: We removed the mention of gauge freedom. We added a section about frame dragging.

----

We believe that gravity is almost exactly analogous to electromagnetism. Our analysis about the electric self-force in the previous blog post is relevant for gravitational waves.

Since gravitational waves are quadrupole, only little energy will escape, compared to dipole waves. There is almost complete destructive interference of the outgoing dipole waves. This complicates the detailed analysis of the process. Let us forget those problems for now.

Does a test mass follow a geodesic?

Let us assume that the metric around the Sun is the Schwarzschild metric. Is there a proof that geodesic orbits in the Schwarzschild metric conserve energy?

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

Yes. The total energy and the specific angular momentum are constants of motion.

Earth loses its total energy in gravitational waves at a rate of 200 W. Thus, the precise orbit of Earth is not a geodesic of the Schwarzschild metric.

Could it be that the orbit is a geodesic in the metric which includes the bent gravity field of Earth?

Is there an analogue of a metric in electromagnetism? Probably not

In an earlier blog post we had the thought experiment in which all particles have the same ratio

       q / m,

where q is the (negative) charge of the particle, and m is its mass. Electrons satisfy this condition.

Can we define a "metric" in electromagnetism which would describe the orbits of such particles?

The negative potential of gravity in the metric of general relativity mainly shows up as slowing down of time, or a redshift.

It sounds strange if we have to manipulate the flow of time in an electromagnetic metric.

However, the speed of light does slow down in a polarizable material.

Also, when a negative charge is close to another charge, it has acquired more inertial mass in the interaction. It moves slower than we would expect. This sounds like slowing down of time.

If we lower a positron so close to an electron that we could harvest the entire mass-energy of the pair, they annihilate. This sounds like a black hole.

In the case of pair: a single positive charge and a single negative charge, a metric of general relativity has these effects qualitatively right. Inertia increases when the charges are close to each other.

When the charges have the same sign, it is hard to make a metric which is qualitatively right. We cannot speed up time when the two charges are close to each other, since that would lead to paradoxes. Can we get the effect by contracting the radial metric around a charge? Then a static test charge would not feel a force. That does not work.

We in this blog hold the opinion that curvature of spacetime is just an illusion which happens with an attractive force of gravity. There is no need to describe the electric force with a metric - and we neither can see how it could succeed. The electric force simply does not create the illusion of a metric.

The self-force which a gravitational wave imposes on the mass producing it

In our previous blog post we conjectured that the electric self-force on an electron can be calculated in a very simple manner: just measure how much the electric field differs from a spherically symmetric field at some distance r. That is the "self-field" E which explains the self-force with the formula

       F = E e.

In gravity we probably can do the same trick.

But can we describe the self-field with a metric in gravity?

            self-field E

              <-------

                  • --->

            test mass doing circular motion

Let us assume that the mass is a point particle. We want to impose a force on it. The self-field E would resist its movement in a circular orbit. How do we implement the force in a metric? By slowing down time in the direction of the force?

But if we slow down time at some position P, that position will attract mass from every direction. Is that right? Why would the self-field E attract mass from other directions?

                        \     bent electric field line

                          \

     P   •                e-  ---> acceleration

                          /

                        /

The bending of electric lines of force is a one-sided phenomenon. The self-force F tries to pull the electron back to its old position. Let us assume that the electron started the acceleration from the position P. Could it be that the effect of the self-field would be to pull a negative test charge toward P, even if the charge is to the left of P?

We are interested in the force on a negative test charge relative to the state where the field around the electron would be spherically symmetric around its current position.

The far field of the electron still "lags behind" around the position P. It looks like the repulsion on the test charge close to P is larger than in the case where the electron would be static in its current position.

Thus, the self-field E seems to be just to one direction. In the diagram it pulls to the left. Can we describe such a one-way force with a metric?

Slowing down time is a good way to implement a force. Objects are "moving" in time and they steer to that direction where time flows slower. Can we implement the same effect by manipulating the spatial metric? That is hard since the effect of a force depends on the velocity of an object.

                       ---------

                   /                \

                 /                   ●  grenade

              

            / /

           O   cannon

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

Imagine a cannon shooting grenades at different speeds at different directions. We can implement their parabolic orbits by making time to flow slower at lower locations. But could we implement the orbits just by tampering with the spatial metric? The orbits cross each other. How could we remap the distances so that the final locations would be correct? That looks very hard.

It may be that one cannot describe dynamic, changing fields with a metric? One must fall back to a description as a force?

Frame dragging

Or maybe frame dragging is the way to implement the self-force? Around a rotating mass inertial frames are dragged along to the rotating movement.

Frame-dragging around a large rotating mass can be explained by a model where the (negative) energy of the gravity field rotates along the mass. If an infinitesimal test mass is close to it, the test mass has the minimum inertia relative to the large rotating mass if it moves along the large mass. A mechanical clock ticks the fastest if it comoves with the large mass.

Gravitational waves and the flow of time

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

Our Minkowski & newtonian model suggests that any interacting object gains inertia. A gravitational wave is interacting with the clocks, which can be seen from the fact that the distances which we measure with a laser change as the wave passes.

If mechanical clock parts gain inertia, then the clock will run slower. Also, light will propagate slower.

Linearization of gravity is somewhat suspicious, because in electromagnetism we have charges of both signs, but in gravity we only have positive charges.

Imagine a drum skin. Time runs slower in depressions of the skin. In gravity, depressions of the skin are allowed but hills not. How do we make waves in such a skin?

In the Minkowski & newtonian model gravitational waves are like electromagnetic waves: they make charges to move. Slowing down of time, and apparent changes in distances, is a side effect of the interaction. The metric is just an illusion.

Conclusions

The big question is if we can describe dynamic phenomena, like gravitational waves, with a metric at all.

A metric works when we describe the Schwarzschild field around a spherical mass. It is a static configuration.

Even though Albert Einstein derived gravitational waves from linearized equations in 1916, he continued being unsure if the phenomenon is possible with the full, nonlinear equations.

https://www.ams.org/publications/journals/notices/201707/rnoti-p684.pdf

C. Denson Hill and Pawel Nurowski (2017) write about exact solutions of waves in the full nonlinear theory of general relativity. Andrzej Trautman was able to find solutions. Let us check what they are like.

https://en.wikipedia.org/wiki/Pp-wave_spacetime

Plane-fronted waves with parallel propagation (pp-waves) are a description of spacetimes where a plane wave moves. We need to check if these have problems with the metric of time. There is an obvious problem with pp-waves: what kind of a physical process could produce infinite planar waves?