The newtonian expanding ball mimics the FLRW universe perfectly

In newtonian gravity, the gravitational field does not hold any special privilege to determine the "curved geometry of spacetime". The true metric of spacetime is the Minkowski metric. Newton's gravity force is almost like the Coulomb force.

The Coulomb electric field of a charge q is

       E = k q / r²

where k is the Coulomb constant and the energy density of the field is

       D = 1/2 * 1 / (4 π k) * E².

The gravity field of a mass m is

       g = G m / r²,

and we claim that the energy density of a weak gravity field is

       D = -1/2 * 1 / (4 π G) * g².

For a strong field, the formula is probably more complex.

At what radius R does the negative energy of a weak gravity field of a mass m match its positive energy? We can integrate from R to infinity to obtain the energy:

                    ∞

       m c² = ∫ 1 / (8 π) G m² / r⁴ * 4 π r² dr

                  R

                = 1/2 G m² / R.

We get R = 1/2 G m / c², which is 1/4 of the Schwarzschild radius of m.

The energy of the gravity field has to be negative because when masses come together, energy is freed, and at the same time the field grows stronger.

The negative energy of the gravity field is the viewpoint of a Minkowski observer who believes that the "true geometry" is flat. General relativity interprets that the gravity field has no energy which should appear in the stress-energy tensor. In the Einstein-Hilbert action we can interpret that the geometric part calculates the energy of the gravity field.

The newtonian expanding ball mimics the FLRW universe perfectly

https://arxiv.org/abs/2006.07418

Valerio Faraoni and Farah Atieh in their 2020 paper review various solutions of general relativity for a newtonian expanding ball of dust.

Mashhoon and Partovi (1980) among others have calculated that the FLRW metric inside, and the Schwarzschild metric outside is the general relativity solution of the newtonian expanding ball.

The cosmic microwave background inside the ball is just like in the FLRW universe as long as our line of sight only sees the hot gas of the 380,000 years old universe. Eventually, the CMB will disappear when we start to see the emptiness of the Minkowski space.

Dark energy in a newtonian expanding ball

In an asymptotically Minkowski space, Birkhoff's theorem and possibly the ADM formalism enforce energy conservation. A positive energy field, which grows with a newtonian expanding ball, is prohibited.

We need to find out if the accelerated expansion of the universe might be caused by emission and absorption of light inside the ball.

In the FLRW metric, a photon in the expanding universe seems to lose its energy through a redshift. In a newtonian ball the energy loss is only an illusion which is caused by the high velocities of the observers. Nevertheless, photons exert some pressure on matter. We have to figure out a way to calculate this effect.

In the interior Schwarzschild metric, pressure is able stop the contraction of the uniform, incompressible, fluid ball if its radius is at least 9/8 of the Schwarzschild radius.

Suppose that our newtonian expanding ball is not inside 9/8 of the Schwarzschild radius. Then positive pressure probably would accelerate the expansion.

A low gravitational potential slows down light to a crawl in the Minkowski view of things. An analogy is a block of glass which slows down photons which are emitted from the lamp inside glass. In a sense, the photons cause an outward pressure on the glass.

If we put two observable universes side by side, then their newtonian gravity force is ~ 10⁴⁴ N, while the force from pressure is only ~ 10³⁸ N. We conclude that the radiation pressure cannot explain dark energy.

Dark matter which turned into radiation might explain the accelerated expansion

Let us assume an asymptotically Minkowski geometry and a newtonian expanding ball.

Let us assume that most of the matter in the universe is of a new type of dark matter. Initially it was particles which had mass. They decayed into radiation some 4 billion years ago.

The dark radiation has a similar mass-energy as the hypothetical dark energy field. The dark radiation has a similar pressure as the dark energy field, except that the pressure is positive.

If we are not inside 9/8 of the Schwarzschild radius, then the positive pressure will speed up acceleration.

Our model has the advantage that it conserves energy. The dark energy field of standard cosmology creates energy from nothing - that is not allowed in quantum mechanics or classical mechanics.

Albert Einstein believed in the geometrical interpretation of general relativity

The collected papers of Albert Einstein are freely readable at the Princeton University Press web page.

https://einsteinpapers.press.princeton.edu/

There we can read his papers about general relativity, both in German and in an English translation.

Einstein writes very well and explains clearly the motivation for his claims. The language is somewhat convoluted. It probably was customary to write long sentences in the German language during that time.

It turns out that Albert Einstein after 1915 immediately suggested that gravitational waves exist, presented a plane wave solution, and calculated the power output of a binary star.

He also described the familiar cosmological model where the spatial slice is the 3D surface of a 4D sphere, and matter is uniformly spread over space. In 1917, Einstein introduced the cosmological constant Λ to make the model static.

The cosmological model shows that Einstein believed in the geometric interpretation of general relativity. In the Minkowski space, a spatial slice cannot have that topology.

That is, Einstein believed that spacetime truly is curved. He did not think that gravity only simulates a modified geometry of spacetime.

What about a cosmological model where we have an island of matter in an asymptotically Minkowski space? Einstein writes that then the average density of matter in the infinite space has to be zero. He did not like the idea that there is an infinite almost empty extent of space, and a small island of matter. He writes that light would escape from that island, and would never come back.

Motivation of general relativity

Albert Einstein several times writes that he wants Mach's principle to be true.

https://en.wikipedia.org/wiki/Mach%27s_principle

However, the principle is not true in general relativity.

A rotating massive body creates a "magnetic" field which deflects masses moving close to it, just like a magnetic field deflects electric charges. But that is really not a realization of Mach's principle, which requires that masses even very far away would determine which frame is inertial and which is rotating.

Einstein refers to the Galilei symmetry of inertial frames in newtonian physics.

He also writes about the equivalence principle of gravity and acceleration. If a scientist in a laboratory feels acceleration, there is no way for him to detect inside the laboratory if it really is gravity or genuine acceleration in space.

In 1917 Einstein published a popular science book, in which he gives a very detailed account of the motivation behind special and general relativity. The book is an excellent piece of science.

The rotating disk

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

One of the examples in Einstein's popular science book is the rotating disk. People living on the disk would measure that the circumference of the disk is more than 2 π times the radius of the disk. This is because a measuring rod on the circumference is length-contracted. A measuring rod on the radius is not length-contracted because its velocity is normal to the rod.

People living on the disk might think that their spacetime is curved.

The rotating disk is actually a good example of the illusion of curved space. People who are inertial, and not on the rotating disk, see that the spacetime around the disk is not curved. It is the ordinary Minkowski space. The illusion of curved space is due to the acceleration of the observers on the disk.

Observers on the rotating disk have centrifugal acceleration and measure the circumference to be greater than 2 π r.

If the observers would form a ring around the Schwarzschild metric, they would have centripetal acceleration, and measure the circumference to be less than 2 π r.

We in our blog claim that in both cases, the curved metric of spacetime is just an illusion.

The problem of complex interactions

Albert Einstein apparently did not write anything about gravity interacting with complex matter fields, though he did spend a lot of time later in his life trying to find a "geometric" interpretation for electromagnetism.

On October 10, 2021 in our blog post we wrote about the complex system S, where the stress energy tensor and the Einstein field equations say that the electric charge q of the particle affects the metric of spacetime through changing the pressure around it.

In the system S there is no meaningful way to divide the potential energy between gravity and electromagnetism. Why would one of the fields deform the metric of spacetime, and the other not?

Quantum gravity of weak gravity fields is almost totally equivalent to QED?

Yesterday we wrote about the perfect analogy of "magnetism" for Newton's gravity force and the Coulomb force.

Newtonian gravity is an extremely precise approximation for weak gravitational fields. In quantum field theory we study scattering between particles whose mass is very small. Their gravity is very weak. Consequently, newtonian gravity should be an excellent approximation for scattering processes of particles.

A graviton is not completely analogous to a photon. A real graviton carries a mass-energy, that is, a gravity charge. A photon is electrically neutral.

The polarization of the graviton in general relativity is a 4 × 4 matrix and its spin is assumed to be 2. In newtonian gravity, the spin of the graviton probably is 1 because it is analogous to a photon.

Let us study interactions between photons and gravitons in newtonian gravity.

         photon ~~~~~~~~~~~~~~

                                   |

                                   |  graviton

                                   |

         photon ~~~~~~~~~~~~~~

Tree-level "Coulomb scattering" in newtonian gravity is a very simple process, just like in QED.

Vacuum polarization

What happens in vacuum polarization?

          photon ~~~~~~~~~~~~~

                                    | graviton q

                                 __|__

                              /           \   virtual photon

                        k    \______/   q - k

                                    |

                                    | graviton q

          photon ~~~~~~~~~~~~~

In QED, the virtual pair pulls both charges together. In the diagram above, one of the virtual photons might have negative mass-energy. Is this allowed? There exist no "positrons" for gravity. No known particle has a negative charge in gravity.

If a virtual photon has zero mass-energy, then its interaction is zero and it cannot participate in the diagram.

Let us assume that negative mass-energy is allowed. If the virtual pair is located between the photons, it pulls one photon and pushes the other. That does not work.

                             ● photon

                             +  positive energy virtual

                                 photon

                              ● photon

The configuration where the positive energy photon is between the real photons, increases the pull of gravity. But where do we put the negative energy virtual photon? It is hard to find a place where momenta and angular momenta gained by the virtual pair would cancel.

The effect of vacuum polarization in QED is quite small relative to Coulomb attraction. It might be that the effect in newtonian gravity is negligible.

What happens at a black hole horizon?

In general relativity, close to the event horizon, the pull of gravity is immense for a static observer. This has made some people to think that there might be powerful quantum effects there.

In newtonian gravity, for an external observer, all non-gravity forces have grown extremely weak there, while the inertia is larger than in faraway space.

In astronomical black holes we observe huge radiation from the accretion disk, but there is no evidence of similar radiation from the event horizon. If we hold the view that the frame of the external observer is the "true physics" of the events, then the horizon seems to be quiet.

In practice, observers close to the horizon are in a free fall. General relativity says that they will not see anything special. In the frame of the external observer, the falling matter slows down and never reaches the horizon. This all suggests that there are no powerful quantum phenomena at the horizon.

The Lorentz ether theory

Hendrik Lorentz, Oliver Heaviside, Joseph Larmor, and Henri Poincare introduced almost all the ideas of special relativity in the late 19th century in various ether theories.

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

Light is transverse waves in the ether. Light is described by the ordinary wave equation (= the massless Klein-Gordon equation). Massive particles might be described with the massive Klein-Gordon equation.

Let us consider the special theory of relativity. We define the inertial frame of our laboratory as the frame where the "ether" is static. It might be possible to reproduce all phenomena of special relativity in that ether, with suitable waves.

                                            _________ mirror

                                                   / \

                                                 /     \   bouncing light

                                               /         \

                                            _________ mirror

                 ●                               v ------>

        our laboratory          moving laboratory

For example, time slows down for a moving observer because if he measures time by looking at light bouncing between two mirrors, normal to his direction of movement, the light will travel a longer path in the ether.

Should we claim that special relativity is just an illusion and that newtonian wave mechanics in the ether, which is static in our frame, is the "true physics"?

The claim is somewhat awkward because it requires us to declare that all matter is waves, and requires us to pick one inertial frame as the ether frame. In newtonian mechanics, all inertial frames are created equal. Why pick some specific one as the ether frame?

What about our claim that the Minkowski metric is the "true physics" and curved spacetime is just an illusion?

The ether theory on top of newtonian mechanics requires us to break the Galilei symmetry by picking one inertial frame over others. Also, the wave theory of matter is a serious complication.

Special relativity retains the galilean symmetry of inertial frames. If we have any frames which are moving at a constant velocity v relative to each other, the frames have the same status.

General relativity is an attempt to establish equality among all freely falling laboratories. Does that make sense?

A laboratory falling freely under the gravitation of any celestial body will observe tidal effects. The laboratories cannot be strictly equal.

In special relativity, inertial frames extend over the whole universe. In general relativity we cannot extend the frame of the laboratory over the whole universe. The symmetry between freely falling laboratories is much weaker than between inertial frames in special relativity.

Moreover, general relativity breaks the symmetry of forces by picking one force, gravity, as the paramount force which determines the "spacetime geometry". Yesterday we showed that for a complex system S, we cannot split apart the effect of various forces. Lifting gravity above other forces cannot succeed cleanly.

Thus, the case for general relativity is much weaker than for special relativity.

The equivalence principle of general relativity is a local, approximate, symmetry while the equivalence of inertial frames in special relativity is a global symmetry.

In newtonian mechanics, an accelerating frame is not equal to an inertial frame. Locally, we can fool people in a freely falling laboratory to think that their accelerating frame is equal to an inertial frame. That does not mean that the "true physics" in that laboratory is the physics of an inertial frame.

There is a perfect analogy of "magnetism" for Newton's gravity force and the Coulomb force

Imagine a system where we have a set of objects moving in space. Each object has a mass m and a negative electric charge q. The ratio

      C = q / m

is the same for all the objects. The Coulomb force and Newton's gravitational force are completely analogous between any two objects, except that the forces have opposite directions.

         <---- ●           

                                ● -->

                     ●

                      |

                      v

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

https://en.wikipedia.org/wiki/Biot%E2%80%93Savart_law

The electromagnetic force between any two objects we calculate from the the electric and the magnetic fields produced by them, using the Hendrik Lorentz force (1895) formula

      F = q E + q v × B

and the Oliver Heaviside (1888) formula for the magnetic field of a point charge:

      B = μ₀ / (4π) q v × r₀ / r²,

where r₀ = r / |r| is the unit vector to the direction of the vector r.

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

Oliver Heaviside in 1893 suggested that Newton's gravity should be extended by a magnetic gravity field which is perfectly analogous to electromagnetism.

Let us assume that the magnetic field B is nothing more than the Lorentz transform of the Coulomb force. Then there must be an analogous gravitomagnetic field B_g for Newton's gravity force, and we must be able to calculate the effects of gravitomagnetism using equations which are completely analogous to electromagnetism.

This is indeed the case. According to Wikipedia, general relativity implies the gravitomagnetic laws for weak fields and slowly moving objects.

In this blog we are claiming that general relativity can be derived from special relativity and Newton's gravity. Now we got more evidence for our claim. Frame dragging close to a rotating black hole is explained by gravitomagnetism, which is a consequence of special relativity and Newton's gravity force.

Cosmology in a flat Minkowski space

UPDATE October 11, 2021: our model predicts anisotropy in redshifts on different sides of the sky, because we probably are not at the center of the expanding explosion cloud in the Minkowski space.

https://phys.org/news/2019-11-evidence-anisotropy-cosmic.html

Something like that has been observed:

"The supernova data indicate, with a statistical significance of 3.9, a dipole anisotropy in the inferred acceleration (see figure) in the same direction as we are moving locally, which is indicated by a similar, well-known, dipole in the CMB."

----

Under less-than-extreme conditions it may be impossible to distinguish our new special relativity & newtonian model from the geometric interpretation of general relativity. It is just a mapping of the Minkowski geometry, under the effect of all fields, to a geometry of general relativity under the effect of all fields except gravity.

Extreme conditions happen in black holes and in cosmology. The predictions of the different models may be distinguishable there.

If we believe that the Minkowski metric is the true metric of spacetime, then the FLRW model of the universe probably is not possible because its spatial topology is different from the Minkowski space.

https://en.wikipedia.org/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric

The FLRW metric is very simple. A spatial slice of the universe is simply the 3-dimensional surface of a 4D sphere. There is a uniform distribution of matter in that spatial slice.

Can we embed something like a part of the FLRW metric into an asymptotically Minkowski space?

We certainly can simulate the expansion of the visible universe with an ordinary newtonian explosion. An observer at the center of an expanding cloud sees a redshift which depends approximately linearly on the distance of the object which is receding.

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

Dark energy is actually evidence against the FLRW universe. The model has been patched together through speculating a cosmological constant Λ which accelerates the expansion.

In an ordinary newtonian explosion, the far edges of the expanding cloud start to slow down because of gravitation. We should observe a surprisingly small redshift in faraway stars.

That is what we actually are seeing. There is a surprisingly high redshift in near objects closer than some 5 billion light years from us. Or the redshift in faraway objects is surprisingly small.

Could it be that dark energy is evidence for Minkowski cosmology?

One of the weaknesses of general relativity is the singularity at the birth of the universe. In Minkowski cosmology we might be able to avoid such singularity.

Note that the Big Bang is not exactly like a black hole collapse run backwards. Entropy decreases with time if we reverse time. Also, the radiation goes in the wrong direction. The baryonic matter content may be the same, whether we run a collapse or an explosion, but the radiation is different.

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

The Vaidya metric describes a star which is emitting or absorbing radiation.

Energy non-conservation in the FLRW model

As the universe expands in the FLRW model, photons get redshifted and lose energy. Where does this energy go?

In the Minkowski model, photons lose energy when they climb up from the side of potential well of the exploded matter. There is also an illusion of lost energy, when an observer absorbs a photon which was sent from a faraway galaxy which has a large redshift. But there is no loss of energy if measured by an inertial observer in the asymptotic Minkowski space.

We see that Minkowski cosmology solves the problem of energy non-conservation in general relativity.

A supernova is a mini Big Bang

A large explosion requires a lot of energy which is in a low-entropy state, and can suddenly escape to a higher entropy state.

A supernova is such an object. It can also be regarded as a mini Big Bang, because new stars and planets will form from its explosion cloud of gas and dust.

Scientists living inside the explosion cloud might think that they are inside an expanding FLRW universe. When gravity slows down the outer fringes of the cloud, the scientists might be led to believe that dark energy has created a cosmological constant Λ. Later, very accurate measurements would reveal that the redshift is not the same on different sides of the cloud. That is the result of the scientists not being at the center of the cloud.

Dark energy is a symptom that something is happening to the uniform expansion of the cloud. In the FLRW model, people interpret it as a sign of an accelerated expansion.

If Minkowski cosmology is correct, we are starting to see deceleration at the edges of the explosion cloud. The diameter of the explosion cloud might be only 10 times the diameter of the observable universe.

Expanding balls in an asymptotically Minkowski space, and the cosmic microwave background

An expanding ball of dust in newtonian gravity is an analogue which is used to teach the FLRW universe to students.

https://arxiv.org/abs/2006.07418

Valerio Faraoni and Farah Atieh have written a review article of expanding balls in general relativity. It turns out that some solutions of an expanding ball do have the FLRW metric inside the ball, and the Schwarzschild metric outside the ball.

https://journals.aps.org/pr/abstract/10.1103/PhysRev.56.455

The Oppenheimer-Snyder collapsing star, time-reversed, is a well-known model of an expanding ball.

What about the cosmic microwave background? We are seeing radiation whose intensity is uniform with a precision 1 / 100,000 in every direction of the sky.

Let us reverse time and treat the expanding ball as a collapse. Let an observer inside the ball point a flashlight at some direction in the sky.

If the ray of light ends up in the final hot and dense state of the collapse, then the observer in the expanding ball at the same point of spacetime would see hot gas in that direction. He would interpret it as the cosmic microwave background.

If the flashlight is inside the Schwarzschild radius in the collapse, then the light will end up in the singularity. However, if the observer is standing on the surface of the collapsing ball, the light might never meet the hot gas state. On what condition the light will meet hot gas?

The FLRW model probably solves this question. In FLRW, the spatial metric of the universe has grown 1,100-fold since the hot gas phase. In the CMB we are seeing a hot gas shell whose radius was around 60 million light years at the time when the light was emitted. If an expanding ball mimics the FLRW metric well enough, then our position at the epoch of the hot gas must have been at least 60 million light years inside the expanding ball.

In the FLRW model, the size of the universe can be arbitrarily large. In the expanding ball model, the ball probably can be arbitrarily large.

Question. Can we make the expanding ball such that the metric explains the redshift measurements that people interpret as "dark energy"?

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

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

----

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

              _______________

             |                            |

             |    ● M, q             |

             |______________|

              pressurized vessel

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

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

The particle interacts with the atoms in the liquid.

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

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

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

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

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

A complex system under gravity and electromagnetism

         •       •       •          system S

         •       •       •          of many particles

         •       •       •

                    ● test particle M, q

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

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

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

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

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

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

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

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

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

Could this work?

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

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

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

Conclusions

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

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

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