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 slows 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.

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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?

Electric self-force: it is not private but visible to all

In the past few blog posts we have been claiming that the self-force on a gravity charge (= mass) must be private: other charges do not see it. But is it really so?

                           long jumping rope

      hand •--------------------------------------------● fixed point

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

Let us take children's jumping rope as an example. The rope is very long and tense, and we start to rotate it with our hand. We are sending a circularly polarized wave into the rope.

Our hand feels a force which resists the rotation. The force exists because the rope "lags behind" our hand and its tension pulls our hand back to its earlier position.

Self-force on an electric charge

If we replace the rope with an electron in our hand and its electric lines of force, could it be that we can explain the resisting force simply as

       F = E e ?

There ε₀ is vacuum permittivity, e is the charge of the electron, and E is the component of the electric field which resists the circular movement of our hand.

Let us rotate our hand at a velocity v along a circle whose radius is r. The acceleration is

       a = v² / r.

The Larmor formula says that the power of the outgoing radiation is

       P = 1 / (6 π ε₀) * e² a² / c³.

The energy density in the spatial volume whose radius is ~ r is then

       D = P r / c  *  1 / (4/3 π r³)

           = P / (4/3 π r² c)

           = 1 / (8 π² ε₀) * e² v⁴ / (r⁴ c⁴).

The force resisting our hand movement must be

       F_v = P / v

              = 1 / (6 π ε₀)  e² v³ / (r² c³).

The energy density of an electric field is (the factor 1/2 comes from the fact that a half of the energy of radiation is in its electric field):

       D / 2 = 1/2 ε₀ E²,

where E is the electric field strength. Within the volume whose radius is ~ r, the electric field strength of the radiation is

       E² = 1 / (8 π² ε₀²) * e² v⁴ / (r⁴ c⁴),

       E  = 1 / (2 sqrt(2) π ε₀) * e v² / (r² c²).

The force of the field E on the electron is

       F = 1 / (2 sqrt(2) π ε₀) * e² v² / (r² c²).

We have

      F_v / F ≅ 1/2 v / c.

We see that the electric field E can easily explain the force F_v. It is enough that a (small) component of E is opposite to v. Most of E has to be normal to v.

The jumping rope diagram explains this. Most of the tension force is along the x axis. A small component of the force pulls our hand toward the center of the rotating movement. A very small component is against the velocity vector v of our hand and we must do work against that component. The electric field in our calculation corresponds to the last two components.

Conjecture. The electric self-force on an electron is the electric force which its own electric field imposes on the electron. We can cut the electric field lines close to the electron and calculate how much the electric field differs from a spherically symmetric field. The difference is the electric field which pulls on the electron.

http://meta-phys-thoughts.blogspot.com/2020/12/electric-lines-of-force-are-elastic-new.html

On December 26, 2021 we wrote about this "tense field lines" model.

If the conjecture is true, then the electric self-force is not private. It is visible to all charges. An infinitesimal test charge, close to the electron which is generating the wave, will feel the same electric field pull it as the electron. The test charge, of course, feels a much larger force from the static field of the electron.

The steel wire model conveys a wrong impression. We visualize steel wires as attached to the electron. Then they would not affect the test charge.

Let us adopt the tense field line model from now on and forget about steel wires.

Lagging field lines versus an opposite charge nearby

                      \  lagging field line

                        \

                         e- -----> acceleration

                        /

                      /  lagging field line

                     

Above we have an electron being accelerated. The lagging electric field lines resist the acceleration.

                                   \

                                     \

             e+                     e-

                                     /

                                   /

We get similar bent field field lines if we place a positron behind a static electron. It may be that the electric force on an electron can be explained by the tense field lines model, and it does not matter if the field lines are bent because of another charge, or if they are bent because of acceleration.

This model would be a very simple description of the self-force, if true.

The centripetal self-force on a circling electron

On October 1, 2021 we calculated the self-force on an electron on a circular orbit, and got very roughly the correct result. We calculated the extra energy it requires to bend the field lines. From that we derived the centripetal force on the electron. Let us check if we get the same result by calculating the centripetal electric field which the bent field lines generate on the electron.

The deflection angle of the field lines is

       2 r / s,

where r is the radius of the circling motion, and s is the distance where we assume that the field lines are bent.

We calculated a centripetal force

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

where m_e is the mass of the electron and

       r_e = 1 / (4 π ε₀) * e² / (m_e c²).

We get

       F = 1 / (π ε₀) * e² r / s³.

The electric field strength at the distance s is

       E = 1 / (4 π ε₀) * e / s².

Its centripetal component E_c is E * 2 r / s, and the centripetal force

       F' = 1 / (2 π ε₀) * e² r / s³.

The order of magnitude is correct. If we adjust s to be a little smaller, then we get F' = F.

Our calculation suggests that the Conjecture might be true. The mysterious self-force on an electron is simply the force imposed on the electron by its own field, where the own field can be measured at some radius r from the electron.

Conclusions

It is time to forget the "rubber plate" model and the steel wire model of the static electric field of the electron, and move to the tense field line model.

We believe that the static, spherically symmetric electric field of an inertial electron has zero energy.

When the electric field lines bend, there is energy and momentum in the field of the electron. The force from bent lines can probably be calculated by looking at how much the field lines differ from spherically symmetric ones at some radius r from the electron. The difference gives us an electric field E which pulls on the electron. The self-force is

       F = E e.

If there is an infinitesimal test charge close to the electron, it feels the same field E as the electron. The self-force field is not private to the electron, but visible to all charges.

The force on the electron from another charge can probably be calculated in the same way: determine how much the lines of force differ from spherically symmetric ones.

Do objects follow geodesics in general relativity?

Our previous blog post suggests that a test mass does not obey the metric calculated from general relativity if another mass with pressure is present.

https://physics.stackexchange.com/questions/24359/why-do-objects-follow-geodesics-in-spacetime

Willie Wong (2012) in the link explains the current status of the problem. The claim that the test mass follows the metric is usually taken as an axiom of general relativity.

Several authors have tried to prove the axiom from the Einstein field equations.

The Ehlers and Geroch theorem (2003)

https://arxiv.org/abs/gr-qc/0309074

Jürgen Ehlers and Robert Geroch (2003) proved a theorem whose content is roughly the following:

Assume that we have a body of continuous matter (no pointlike particles). Let us have an orbit γ in spacetime. We assume that the body follows γ as we let the mass of the body tend to zero. We assume that we have a solution to the metric in an environment of the orbit γ, and that solution converges strongly everywhere toward the background metric. Then the orbit γ is a geodesic of the background metric.

"Converges strongly" means that the metric and its first derivatives only differ from the background metric by at most ε if we choose a suitable environment.

Let us analyze the theorem. For a pointlike particle, strong convergence cannot hold.

We cannot prove the existence of a metric which is a solution. Therefore, we do not know if the metric, if any, converges strongly toward the background metric. However, it is a reasonable conjecture to assume that such metrics exist.

Let us put a spherical mass with pressure close to the test body. Then the gravity field of the test body is coupled to the pressure. We believe that the gravity field of the test body imposes a self-force on the test body. We do not think that the self-force can be expressed with a metric which is visible to all observers. Ehlers and Geroch do not consider this mechanism. A self-force makes the body to deviate from a geodesic.

The Gralla and Wald model (2008)

https://arxiv.org/abs/0806.3293

Samuel E. Gralla and Robert M. Wald (2008) in their paper "A rigorous derivation of gravitational self-force" start from an orbit γ of a test body, just like Ehlers and Geroch. They study the near field and the far field of the test body. They allow the body to be even a mini black hole.

Gralla and Wald, too, assume a family of well-behaved metrics g(λ), where λ is the parameter, and that the family converges nicely toward a background metric as we reduce the mass of the test body.

We can object, just like in the case of Ehlers and Geroch, that we do not know if such well-behaved families exist. But we can make a conjecture that they do exist.

In the last half of the paper the authors seem to calculate some kind of a "vertex function" for gravity. When the test body is accelerated, its own field may impose a self-force on the test body.

We in this blog believe that gravity is almost totally analogous with electromagnetism. Therefore, a vertex function for gravity must exist.

The authors do not consider a self-force which would come from pressure or some complex interaction of the gravity field of the test body with matter fields.

The authors mention that there is no legitimate derivation of the self-force on a charged particle in electromagnetism, despite more than a century of work. We in this blog have tried to determine the electromagnetic self-force. Our October 1, 2021 blog post is a step to that direction.

Conclusions

In general relativity, the claim that an infinitesimal test mass follows a geodesic, is taken as an axiom.

There exist various attempted proofs of the axiom from the Einstein field equations, but the consensus is that the proofs are not general enough. The proofs have to assume the existence of well-behaved metrics.

Our blog posts from the past few days suggest that pressure, or other complex interaction, imposes a self-force on the test mass. By a self-force we mean that the field of the test mass is like steel wires attached to it, and these wires impose substantial forces on the test mass. It does not follow a geodesic, then.

The paper by Gralla and Wald raised the possibility that there exists a vertex function, just like in electromagnetism, where the field of a particle interacts with the particle itself and steers it away from the geodesic. We believe that a vertex function exists for gravity.

The vertex function, too, can be understood with the steel wire model of the field of the particle. If the particle is accelerated, the wires resist the change in the velocity vector. Our "rubber plate" model of electromagnetism is equivalent to the steel wire model.

The geodesic equation in general relativity

The geodesic equation of general relativity shows how an infinitesimal test mass "free from all external, non-gravitational forces" moves under a metric.

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

The metric is supposed to determine the orbit of a freely falling particle of infinitesimal mass.

Extra inertia which is caused by pressure in a spherically symmetric mass: does the mass have time to adjust?

               spherically symmetric mass

               _____

            /   -  -  -   \     negative energy

            \ +  +  +  /     positive energy

               --------

                    •  ---->

                    infinitesimal test mass dm

In the diagram we have an infinitesimal test mass flying fast past a spherically symmetric mass. The test mass probably stretches the radial metric on the far side of the sphere and contracts it on the near side. Pressure does work on the far side but gains energy on the near side. We have marked the areas of energy loss and gain in the diagram.

When test mass moves, it causes energy (or mass) to flow inside the spherical mass. That means that the inertia of the test mass has become bigger.

Let us calculate if the spherical mass has time to adjust to the new metric.

Let us imagine a 1 kg = 10¹⁷ J mass flying past a neutron star whose mass is M ~ 10³⁰ kg and the radius is r = 10 km. The pressure inside the neutron star is p ~ 10³⁴ Pa.

The 1 kg mass stretches the radial metric by a factor

       f ~ r_kg / r

         = 10⁻³¹,

where r_kg is the Schwarzschild radius 10⁻²⁷ m of a mass of 1 kg.

The pressure drop on the far side is something like

       Δp ~ f p

             = 1,000 Pa.

The total force pushing the mass M to new positions is

       F ~ Δp r²

           = 10¹¹ N.

Let the test mass be mildly relativistic and pass the neutron star in a time

       t = 10⁻⁴ s.

The mass M of the neutron star can move during that time a distance

       d ~ 1/2 F / M * t²

          = 5 * 10⁻²⁸ m.

That is about the same as f r = r_kg. We conclude that the neutron star does have time to adjust significantly, even if the test mass is relativistic. If the test mass is slow, then the adjustment is essentially complete.

Let us calculate the figure for an arbitrary star with the same mass M. If we keep M constant

       f ~ 1 / r,

       p ~ 1 / r⁴,

       F ~ 1 / r³,

       t ~ r,

       d ~ 1 / r.

We see that d is much less than f r = r_kg if r is much larger than 10 km. We conclude that ordinary stars do not have time to adjust to a relativistic test mass. The Sun, or any star, does have time to adjust significantly to a test mass whose velocity is the escape velocity of the star.

Is the metric of general relativity aware of the increase in the inertia?

Is the metric of general relativity aware of this effect? We doubt that.

Suppose that we have a Sun-like star and a relativistic test mass.

When a test mass is flying by, we temporarily, significantly, increase the pressure inside the star. Then the test mass starts to drag with it a larger package of pressure energy inside the star: the inertia of the test mass probably grows because of the extra pressure. The relative change in the inertia of the test mass may be significant, around 10⁻¹⁰ if we double the pressure inside the Sun.

But Birkhoff's theorem says that the metric around a spherically symmetric system cannot change if we change pressure. The inertia cannot change.

Let us analyze the energy flow in the gravity field of the star. The energy of a non-zero gravity field is negative. The test mass makes the field stronger on the far side of the spherical mass. Thus, there is negative energy there.

If we increase the pressure, the gravity field becomes stronger, and there is even more negative energy on the far side. Thus, the energy flow in the gravity field adds more inertia to the test mass. Doubling the pressure of the Sun may add another fraction 10⁻¹⁰ to the inertia of the test mass through the changes in the gravity field.

There may be a general rule that if the interaction between a test charge and a macroscopic system is made stronger, then the test charge acquires more inertia. Increasing the pressure inside the spherical mass makes it more sensitive to the perturbation which the test mass causes. It is no surprise that more pressure means more inertia.

The reaction of the metric to the test mass cannot explain the increased inertia effect

What if we calculate the metric of the spherical mass taking into account its reaction, the pressure changes, caused by the infinitesimal test mass?

Since the equations are nonlinear, we are not able to prove mathematically anything about the reaction.

However, we conjecture that the reaction on the different sides of the spherical mass is infinitesimal. We further conjecture that the metric which we derive from the spherical mass is then only infinitesimally changed. An infinitesimal change in the metric cannot explain a significant relative change in the inertia of the test mass.

What if we replace the spherical mass with a pressurized spherical vessel?

https://arxiv.org/abs/gr-qc/0510041

Ehlers et al. (2005) in their paper showed that the calculated static external metric remains the same if we change the pressure in incompressible liquid by tightening a membrane around it. That is, Birkhoff's theorem is satisfied for a static configuration.

Let us have an infinitesimal test mass flying by the vessel.

The pressure inside the liquid does work just like in the case of the spherical mass.

               

              spherical vessel

                ______

            /    -  -  -     \         negative energy

        + |                   | +

            \   +  +  +   /         positive energy

                ----------

                    •  ---->

                    infinitesimal test mass dm  

The negative pressure in the membrane is tangential. Our test mass makes a ring of positive energy to move there as it passes by. In the picture, the + at the equator denotes positive energy in the membrane.

If we tighten the membrane, our test mass drags more pressure energy along.

It probably also drags more negative energy of the gravity field along.

The inertia of the test mass probably increases with pressure.

The variation of a hamiltonian

Suppose that we have an ordinary lagrangian, with no metric. We have a test particle with an infinitesimal charge.

Close to the test charge is a system of macroscopic charges.

Let us assume that we can convert the lagrangian to a hamiltonian.

If we vary the location of the test charge and calculate the difference in the total potential energy, we get the force on the test charge.

The field of the test charge moves along with the test charge. The variation of the position of the test charge by dx involves the variation of its field throughout the space by a small amount.

How do we calculate the inertia of the test charge? We need to find out how much displacement of field energy happens if we move the test charge a distance dx.

This all is complicated.

In the special case of the Coulomb law and electric charges, we can calculate the electric field of the macroscopic charges through a variational principle, and that tells us the force on the test charge. But there is no simple expression for the inertia of the test charge.

For weak electric fields we may assume that the inertia of the test charge is constant, and we may calculate its orbit from the electric field. But if the inertia changes during its orbit, the calculation does not yield the correct orbit.

The self-force of gravity on a test mass

The analysis of the previous sections leaves open the possibility that the metric which the test mass creates around itself could somehow "understand" that the inertia has grown. The question is about the self-force which the gravitational field of the test mass imposes on the test mass itself, and if that force can be expressed with a metric.

The self-force is the obvious mechanism with which the test mass interacts with pressure far away. It is like steel wires attached to the test mass. If the steel wires start to drag along mass-energy from pressure, they slow down the progress of the test mass.

The self-force is private to the test mass. We do not think that one can describe it with a metric which is visible to all observers.

Conclusions

A basic idea of the metric is similar to the electric field: derive a field which we can calculate from the configuration of macroscopic charges, and which determines the orbit of an infinitesimal test charge.

However, taking into account changes in the inertia of a test charge is difficult. An electric field does not claim to know the inertia of a charge at each position of the orbit.

General relativity, on the other hand, claims to know the orbit, which means that it must know the inertia. Can it know it? If it does, that is quite a feat.

We need to check the literature. Has anyone proved that test mass orbits obey the metric?

Few exact solutions of general relativity are known. It is hard to prove mathematically that general relativity does not calculate correct orbits, since we do not know what it calculates, if anything. Demetrios Christodoulou and Sergiu Klainerman (1994) have proved that a small perturbation of the Minkowski metric has a solution in general relativity, but in a more general case we do not know if general relativity has any solutions at all.

The metric of general relativity is only approximate?

Let us call the "true geometry" of spacetime something which is determined by orbits of an infinitesimal test mass. The test mass only interacts through gravity.

Our "antigravity device" example on November 5, 2021 shows that if we calculate the metric for a rigid iron grid, without including an infinitesimal test mass, then the metric does not describe correctly the orbits of the test mass.

The metric of general relativity is approximately the true geometry in the case of a spherically symmetric mass. However, it does not include tidal effects, which close to the mass can be substantial.

The stress-energy tensor in the Einstein field equations includes essentially the mass-energy density, movement of energy, and pressure, if we ignore shear stresses.

Thus, some effects of pressure do affect the calculated metric. Let us try to analyze what is included.

Adding extra pressure: the Ehlers et al. metric for a spherical mass which is squeezed by a membrane on the surface

https://arxiv.org/abs/gr-qc/0510041

Jürgen Ehlers et al. (2005) calculated the metric. How does pressure from the external membrane affect the metric inside the mass?

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

The metric in the bulk of the sphere is the interior Schwarzschild metric. The spatial metric of the equatorial plane inside the sphere is the surface of a polar region of a 3-dimensional ball whose radius is R.

(Picture by Djexplo https://commons.wikimedia.org/wiki/File:Latitude_and_Longitude_of_the_Earth.svg )

Let

       A₁ = the latitude of the "effective" border of the polar region, including the pressure of the membrane;

 

       A₀ = the latitude of the border of the bulk of the sphere.

We can increase the pressure inside the bulk of the sphere either by

1. adding more mass of the density ρ around it, down to the latitude A₁, or

2. increasing the pressure by tightening the membrane on the surface.

Above, a₁ = sin(A₁). Let us define a₀ = sin(A₀).

We are interested in how much slower does time run at the center of the sphere relative to its border, if we increase pressure by squeezing the sphere with the membrane.

The metric shows that the closer we are to the pole, the slower time runs. Furthermore the rate of time depends on the sine of the latitude.

According to Wikipedia,

       R = sqr(r₀³ / r_s),

where r₀ is the coordinate radius of the sphere (not proper), and r_s is the Schwarzschild radius of it.

The rate of time at the center is

       1/2 (3 a₁ - 1)

while at the border it is

       1/2 (3 a₁ - a₀).

Let us calculate a case where the latitudes A₀ and A₁ are almost 90 degrees. The pressure at the center is not very big and we can calculate most things using newtonian gravity.

Let us have a sphere of a radius r₀ and a density ρ. We add more matter, so that the new radius is r₁.

We have

       R = sqrt(r₀³ / r_s)

           = sqrt( r₀³ c² / (2 G ρ * 4/3 π r₀³) )

           = sqrt( 3 c² / (8 π G ρ) ),

       a₀ = sin(A₀)

            = sin(r₀ / R)

            = 1 - b₀

            ≅ 1 - 1/2 r₀² / R²,

        a₁ = 1 - b₁

            ≅ 1 - 1/2 r₁² / R².

We introduced b₀ and b₁ which are small positive numbers.

The rate of time at r₀ divided by the rate at the center is

       T = (1 - 3/2 b₁ + 1/2 b₀) / (1 - 3/2 b₁)

          ≅ 1 + 1/2 b₀ + 3/4 b₀ b₁,

where we used the fact that b₁ is small.

       T ≅ 1 + 1/4 r₀² / R² + 3/16 r₀² r₁² / R⁴.

We assume that r₀ >> 1. We let

       r₁ = r₀ + 1.

The increase in T, relative to the setting r₁ = r₀, is approximately

       3/8 r₀³ / R⁴

       = 8/3 π² r₀³ G² ρ² / c⁴.

The energy of an extra 1 kg test mass placed at the center of the sphere, as seen by an  observer at r₀, is reduced by

       E = 8/3 π² r₀³ G² ρ² / c²,

because of the slowdown of time. That is, if an observer at the center would send that 1 kg of mass-energy as radiation to the r₀ observer, the redshift would cut off that much energy.

The extra pressure from 1 meter of matter is

       p = G ρ * ρ * 4/3 π r₀³ / r₀²

          = 4/3 π r₀ G ρ².

Let us put 1 kg of extra matter to the center of the sphere and calculate how much it increases the volume of the sphere. We assume that the 1 kg mass stretches the radial metric according to the Schwarzschild metric of that 1 kg mass. We denote by r_kg the Schwarzschild radius of 1 kg mass:

               r₀

       V = ∫ 1/2 r_kg / r * 4 π r² dr

            0

           = 2 π r₀² G / c².

The extra pressure p does the work

       E' = p V

           = 8/3 π² r₀³ G² ρ² / c².

We see that E' = E. If we increase the pressure by p, we can explain the lower potential E of the 1 kg mass at the center solely by the fact that when the 1 kg is lowered, the volume of the sphere increases by V, and the extra pressure p does the extra work E.

Conjecture. General relativity calculates correctly the direct effect on the metric of time caused by the volume increase around a test mass combined with pressure. The rate of time slows down according to the work done by pressure.

However, it may be that general relativity miscalculates the effect of pressure on the radial metric. General relativity claims that the spatial metric remains the same, even if we increase the pressure. In the interior Schwarzschild metric, R only depends on ρ, not on the pressure.

Birkhoff's theorem and non-conserved mass or changing pressure

UPDATE January 19, 2022: Birkhoff's theorem requires strict spherical symmetry. The test mass has to be a spherical shell of matter. A shell does not affect the spatial metric inside it. Thus, pressure does not attract the test mass (= shell) if the test mass is outside the spherical mass. Birkhoff's theorem is saved.

----

General relativity is derived from the Einstein-Hilbert action. The path of a system should be a local extreme value of the action.

Birkhoff's theorem is derived from the Einstein field equations, and is extremely strict: the theorem claims that the metric in the vacuum area of a spherically symmetric system has to be the static Schwarzschild solution. It enforces conservation of energy.

If we allow energy non-conservation in the system, then obviously general relativity does not have a solution.

Let us analyze how having no solution is expressed in the action. The first thought is that since the action only looks for a local extreme value, there would be a solution for all systems.

However, if a singularity develops, then there is no solution.

Suppose that we want to remove energy from a spherically symmetric mass, while obeying conservation of energy. The mass may radiate, for example.

There exists a smooth solution since the outgoing radiation takes away the extra curvature of spacetime.

If we would just let the radiation disappear, then a discontinuity would appear in the attempted solution. A discontinuity is a kind of a singularity.

The problem of pressure changes - a rubber membrane model

One can temporarily change the pressure of a spherically symmetric system. For example, we may have springs which we temporarily release.

Pressure acts as a source of gravity in general relativity. The metric inside the spherical mass will change.

How can we match the change in the internal metric to the static external metric?

In a rubber membrane model, Birkhoff's theorem would mean that the membrane is frozen outside the spherical mass. If we increase the pressure inside the mass, we can make the depression in the membrane deeper. Is it possible to match the membrane outside the mass and inside?

Increased pressure starts to eject matter from the borders of the spherical mass. Could this help in matching?

Pressure probably increases the positive curvature inside the mass. If we want to keep the curvature outside static, we probably have to introduce negative curvature to the borders of the mass. What could produce such negative curvature? In general relativity, negative curvature is prohibited by a weak energy condition.

If we do not enforce Birkhoff's theorem to the rubber membrane, negative curvature does not appear, or does it? Yes it does. Faraway parts of the membrane do not have time to react when pressure pushes the depression deeper. A wave of negative curvature starts spreading from the mass.

Does general relativity prohibit negative curvature?

In a rubber membrane model there is nothing which prohibits negative curvature. What about general relativity? Can negative curvature exist in the vacuum?

No. If the stress-energy tensor is zero, then Ricci curvature has to be zero.

Could it be that the ejected matter at the border of the mass generates negative curvature?

Probably not. It is a common belief that moving matter only creates positive curvature.

We may imagine that the mass is enclosed into a rigid shell. When pressure tries to push matter out, it bumps into the shell. Negative pressure starts to build up in the shell. Could this negative pressure cancel the effect of a sudden pressure increase in the mass?

Probably not. It takes time for the negative pressure to build up, while the positive pressure increase is immediate.

Does general relativity have a solution for a change in pressure?

In the interior Schwarzschild solution pressure contributes to positive curvature along with mass-energy.

If we increase the mass-energy temporarily through energy non-conservation, then general relativity probably does not have a solution.

We can increase the pressure temporarily. Why would general relativity have a solution in that case?

If there is no solution, then the Einstein-Hilbert action is broken. The probable culprit is using the Ricci scalar R as the lagrangian of spacetime curvature. In our previous blog post we showed that the derived metric does not describe the orbit of an infinitesimal test mass correctly. The status of the metric and R as the "curvature of spacetime" is not clear.

If we increase the pressure symmetrically in a spherically symmetric mass, then the attraction from pressure probably does not grow outside the mass but the inertia of the test mass grows

In our Minkowski & newtonian model, stretching of the spatial metric is due to a potential gradient.

If we put an infinitesimal test mass dm close to the spherical mass, but not inside it, then the test mass makes the potential gradient steeper on one side of the spherical mass and less steep on the other side. The summed effect probably is zero.

The radial metric stretches on the far side, and pressure does work there. On the near side, pressure receives work when the radial metric contracts.

If we want to create an extra pulling force on the test mass, we should increase the pressure on the far side of the spherical mass, or we should lower the test mass inside the spherical mass.

                  ____

               / -  -  -  \       negative energy

               \ + + + /       positive energy

                  ------ 

                      ^

                      |

                      •  test mass dm

Pressure does not increase the force on the test mass when it comes outside the spherical mass. But pressure does cause energy to move inside the spherical mass. This, in turn, increases the inertia of the test mass and changes the metric.

This is evidence against Birkhoff's theorem. By increasing pressure inside the spherical mass we can increase the inertia of the test mass outside it. Increased inertia means a slower speed of light, which can be explained by slower time or increased distances.

The effect on the rate of a mechanical clock seems to be ~ 1 / r², while for gravity it is ~ 1 / r. The effect is a tidal effect. This makes sense: Birkhoff's theorem is not aware of tidal effects on an infinitesimal test mass.

A slower speed of light means that rays of light are bent more close to the spherical mass. That implies that gravity feels stronger with increased pressure inside the spherical mass.

A higher pressure increases the interaction between masses. A strong interaction increases inertia, and slows down the speed of light. In our syrup model of gravity, pressure increases the viscosity of the syrup.

We must analyze the process in more detail. The effects of pressure can be surprising.

Variation of the Einstein-Hilbert action does not understand complex mechanisms: the metric is not the "true" geometry of spacetime

UPDATE November 6, 2021: The answer to Question 1 is incorrect if we keep the test mass outside the body of the spherical mass. The increase dV in the volume only happens if we lower the test mass inside the spherical mass. The whole process caused by pressure is very complicated. See the last section of our blog post today.

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Let us cast from iron a rigid grid in otherwise empty space. The grid is in its minimum energy state after the casting.

           ----------------------

             |        |        |               rigid grid

           ----------------------

             |        |        |

           ----------------------

                       ^

                       |

                      ●    infinitesimal test mass m

There is some positive pressure in the grid because it has to resist contraction under its own gravity.

Let us then calculate the metric around the grid from the Einstein field equations.

Does the calculated metric describe right the orbit of an infinitesimal test mass? Probably not. When the test mass approaches the grid, the Schwarzschild metric around the test mass distorts the spatial metric in the grid. The grid would become deformed, and that requires energy. The test mass feels a repulsive force from the rigidity. On May 28, 2019 we wrote about this "antigravity device".

In the idealized case where the mass-energy of the grid is zero, there is no mass-energy nor any pressure in the grid. The Einstein field equations for the grid, ignoring the test mass, calculate that the metric around the grid is the flat Minkowski metric. That metric does not describe the real behavior of the test mass. The test mass feels a repulsive force.

The orbit of an infinitesimal test mass defines the "true" geometry of spacetime. If there is a complex interaction, and we ignore the test mass when solving the Einstein field equations, we obtain a different metric which is not the "true" geometry of spacetime.

The Einstein field equations are derived from the Einstein-Hilbert action using a simple variation of the system. That simple variation ignores complex interactions.

Thus, in the case of a rigid grid, the metric derived from the Einstein field equations does not describe right the orbit of an infinitesimal test mass.

The infinitesimal test mass causes a deformation of the flat metric. That, in turn, introduces negative and positive pressure in various parts of the grid.

Question 1. If we calculate the corrected metric for the grid, including this "backreaction" from the test mass, does the corrected metric describe the orbit of the test mass correctly?

Question 2. In which cases does the metric derived from the Einstein field equations describe right the orbit of an infinitesimal test mass?

We believe that the external Schwarzschild metric is approximately the true geometry of spacetime around a spherical mass, though we have not seen a proof.

In popular science books it is claimed that a ray of light obeys the metric which we calculate from the Einstein equations. The ray moves along a "straight" path. But in our grid example, a test mass does not move along a straight path of the calculated metric.

A thought experiment: a spherical mass supported by springs, and an infinitesimal test mass lowered close to it

                 ●   large mass supported with springs

                 ^

                 |

                 •   infinitesimal test mass dm

We assume that the infinitesimal test mass is surrounded by a Schwarzschild metric. We assume that the deviations from the flat metric are small, so that we can, in some way, sum the deviations.

Initially the large mass is static in its lowest energy state.

We lower the test mass very slowly to the large mass. The spatial metric inside the large mass gets slightly stretched in the direction of the test mass. The pressure in the springs does some work as the spatial metric stretches. The person doing the lowering can harvest the work done by pressure. He interprets that pressure contributes to the gravitational attraction of the large mass.

The person can then use the energy which he harvested and pull the mass very slowly back up.

However, if the person pulls the test mass quickly up, then the large mass ends up being slightly displaced from its lowest energy state. The springs become compressed on the side where the test mass was.

The end result of a swift pull is that energy was transferred from the person into vibration of the central mass. 

During the whole process, all the interaction of the test mass was through its gravity.

We may say that the gravity field around the central mass for swift movements of the infinitesimal test mass is not a conservative force field.

This is in contrast to Birkhoff's theorem which states that the gravity field around a spherical mass is the static Schwarzschild metric. Even an infinitesimal test mass perturbs the system in such a way that it no longer is spherically symmetric.

Our thought experiment suggests that if we temporarily increase the pressure inside the central mass, then the test mass will feel a stronger pull from the central mass.  Increasing the pressure can temporarily increase the gravity of a spherically symmetric object. Birkhoff's theorem does not prevent this.

Tolman's paradox is that changing slow mass into radiation increases its gravity. Our thought experiment suggests that this indeed is the case. There is no paradox if one realizes that the Birkhoff theorem is not valid for an infinitesimal test mass.

Based on our new thought experiment, we can now answer Question 1 negatively:

Answer to Question 1. Correcting the metric with the backreaction to the test mass dm does not yield the right value for the gravitational pull. The following reasoning proves that:

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UPDATE November 7, 2021: See our blog post on November 6, 2021, the last section. The process is much more complicated, and the analysis below is incorrect.

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When we very slowly lower the infinitesimal test mass dm close to the spherical mass, it stretches the metric inside the spherical mass and increases the volume by some

       dV ~ V dm,

where V is the volume of the spherical mass.

We can harvest the energy which is roughly

       dE = p dV

             ~ p V dm,

where p is the pressure of the spherical mass.

When we swiftly pull the test mass out, a significant portion of dE becomes vibrational energy of the spherical mass.

The extra pressure that is created inside the spherical mass during the swift pull is

       dp ~ p dV / V.

The gravitational pulling force F on the test mass from the pressure p inside the spherical body is

       F ~ p V dm.

The added pressure adds something like

       dF ~ dp V dm

             ~ p dV dm

             ~ p V dm²

to the gravitational pulling force of the pressure. The energy that we must spend to win the extra pull is

       ~ s p V dm²,

where s is the distance that we lift the test mass. If dm is small, this energy is much less than dE.

Birkhoff's theorem

Let us, instead of an infinitesimal test mass, use a spherical shell of infinitesimal mass which we lower close to the central mass. Then there is no increase dV in the volume of the central mass? How does pressure in that case create attraction?

The answer may be that the spherical shell lowers the potential of the central mass so much that it compensates the missing contribution of pressure.

Question 3. What about changing the pressure of the central mass temporarily? Can we find a solution where the external metric stays constant as the static Schwarzschild metric?

In a static configuration, the attraction of the pressure is visible in the Schwarzschild external metric. It may be so that in a dynamic configuration the contribution of the pressure change is not visible in the external metric. If this is true, then we avoid a contradiction with Birkhoff's theorem. If this is the right explanation, then the static external Schwarzschild metric is the correct solution for the Einstein-Hilbert action, but the orbit of an infinitesimal test mass does not obey that metric. Rather, the test mass feels the pull from the changing pressure.

Is the Ricci scalar R the right term in the Einstein-Hilbert action?

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

The metric which we calculate from the Einstein-Hilbert action does not predict correctly the orbit of an infinitesimal test mass. It only predicts the orbit approximately right under most circumstances.

The question arises if the R term in the action is correct. Why calculate the Ricci scalar for a metric which is not the "true" geometry of spacetime?

We do not yet know the answer to that question. Our own approach is to treat the newtonian gravity force as an ordinary force, and not to claim that there is a curved metric of spacetime at all. That is, in our own approach the metric, or its Ricci scalar, does not appear.

Our own approach seems to reproduce effectively the external Schwarzschild metric. Since the Einstein-Hilbert action gives the same metric, the term R must be approximately right.

The slowing down of time in the Schwarzschild metric has been tested empirically. There probably is no empirical data on how well general relativity fares with gravity of pressure.

Conclusions

The grid experiment shows that solving the Einstein field equations for a system alone does not necessarily produce a metric which would describe correctly the orbit of an infinitesimal test mass. The "backreaction" of the system to the test mass is important.

Our second thought experiment showed that even in the basic case of a spherical central mass supported by pressure, the static Schwarzschild external metric does not predict correctly the orbit for a swift movement of an infinitesimal test mass.

Also, the interpretation of the metric as the "geometry" of spacetime is not correct under a complex interaction. Even if we calculate a new metric from the backreaction of the system, it does not predict the orbit of an infinitesimal test mass correctly.

In our October 10, 2021 blog post we claimed that one cannot meaningfully define the "geometry of spacetime" if we have a complex lagrangian. We got more evidence which supports our claim.

The Schwarzschild interior solution assumes that the Einstein field equations calculate the correct metric for incompressible fluid. We need to check if that really is the case.