What does Einstein’s General Relativity Tell Us about Black Holes?

Presentation Slides

We have today given a presentation at IU Science Slam with the same title as this post. The main themes are to present the black hole theory derived from Einstein’s General Relativity, and to point out the essential differences between the black hole theory and the viewpoint on black holes from Newton’s Law. The final message is

nothing gets out of the black hole, and nothing gets inside a black hole either.

This is based on a recent paper [Tian Ma & Shouhong Wang,  Astrophysical dynamics and cosmology, J. Math. Study, 47:4 (2014), 305-378]; see also the previous blog post: Singularity at the Black-Hole Horizon is Physical.

Singularity at the Black-Hole Horizon is Physical

1. Newtonian Viewpoint

Consider a massive body with mass inside a ball of radius . The Schwarzschild radius is defined by

Based on the Newtonian theory, a particle of mass will be trapped inside the ball and cannot escape from the ball, if its kinetic energy, , is smaller than gravitational energy:

which implies that

In other words, if the radius of the ball is less than or equal to , then all particles inside the ball are permanently trapped inside the ball .

It is clear that the main results of the Newton theory of black holes are as follows:

• the radius of the black hole may be smaller than the Schwarzschild radius ,
• all particles inside the ball are permanently trapped inside the ball , and
• particles outside of a black hole can be sucked into the black hole .

2. Einstein-Schwarzschild Theory

Black-Holes are closed

Now consider the case where . Based on the Einstein field equations, in the exterior of the body, the Schwarzschild solution is given by

and in the interior the Tolman-Oppenheimer-Volkoff (TOV) metric is

The both metrics have a singularity at , which is called the event horizon:

Both (3) and (4) show that time freezes at , and there is no motion crossing the event horizon:

Consequently the black hole enclosed by the event horizon is closed: Nothing gets inside a black hole, and nothing gets out of the black hole either.

Black holes are filled

We now demonstrate that black holes are filled. Suppose there is a body of matter field with mass trapped inside a ball of radius . Then on the vacuum region , the Schwarzschild solution would be valid, which leads to non-physical imaginary time and nonphysical imaginary distance:

Also, when , the TOV metric is given by

Then both time and radial distance would become imaginary near , and this is clearly non-physical.

This observation clearly demonstrates that the black is filled. In fact, we have proved the following black hole theorem:

Blackhole Theorem (Ma-Wang, 2014) Assume the validity of the Einstein theory of general relativity, then the following assertions hold true:

1. black holes are closed: matters can neither enter nor leave their interiors,
2.  black holes are innate: they are neither born to explosion of cosmic objects, nor born to gravitational collapsing, and
3.  black holes are filled and incompressible, and if the matter field is non-homogeneously distributed in a black hole, then there must be sub-blackholes in the interior of the black hole.

This theorem leads to drastically different view on the structure and geometry of black holes than the classical theory of black holes.

3. Singularity at is physical

A basic mathematical requirement for a partial differential equation system on a Riemannian manifold to generate correct mathematical results is that the local coordinate system that is used to express the system must have no singularity.

The Schwarzschild solution is derived from the Einstein equations under the spherical coordinate system, which has no singularity for . Consequently, the singularity of the Schwarzschild solution at must be intrinsic to the Einstein equations, and is not caused by the particular choice of the coordinate system. In other words, the singularity at is real and physical.

4. Mistakes of the classical view

Many writings on modern theory of black holes have taken a wrong viewpoint that the singularity at is the coordinate singularity, and is non-physical. This mistake can be viewed in the following two aspects:

A. Mathematically forbidden coordinate transformations are used. Classical transformations such as e.g. those by Eddington and Kruskal are singular, and therefore they are not valid for removing the singularity at the Schwarzschild radius. Consider for example, the Kruskal coordinates involving

This coordinate transformation is singular at , since becomes infinity when .

It is mathematically clear that by using singular coordinate transformations, any singularity can be either removed or created at will.

In fact, many people did not realize that what is hidden in the wrong transformations is that all the deduced new coordinate systems, such as the Kruskal coordinates, are themselves singular at :

all the coordinate systems, such as the Kruskal and Eddington-Finkelstein coordinates, that are derived by singular coordinate transformations, are singular and are mathematically forbidden.

B. Confirmation bias. Another likely reason for the perception that a black hole attracts everything nearby is the fixed thinking (confirmation bias) of Newtonian black hole picture. In their deep minds, people wanted to have the attraction, as produced by the Newtonian theory, and were trying to find the needed “proofs” for what they believe.

In summary, the classical theory of black holes is essentially the Newton theory of black holes. The correct theory, following the Einstein theory of relativity, is given in the black hole theorem above.

Tian Ma & Shouhong Wang

PID Weak Interaction Theory

This post presents a brief introduction to the field theory of weak interactions, developed recently by the authors, based only on a few fundamental principles:

• the action is the classical Yang-Mills action dictated uniquely by the gauge invariance and the Lorentz invariance; and
• the field equations and the Higgs fields are then derived using the principle of interaction dynamics (PID) and the principle of representation invariance (PRI).

The essence of the new field theory is that the Higgs fields are the natural outcome of the PID, which takes variation under energy-momentum conservation constraints. We call this new field theory the PID weak interaction theory.

This new theory leads to layered weak interaction potential formulas, provides duality between intermediate vector bosons , , and their dual neutral Higgs and two charged Higgs , and offers first principle approach for Higgs mechanism.

1. PID field equations of the weak interaction

The new weak interaction theory based on PID and PRI was first discovered by the authors in [1]; see also the new book by the authors.

First the weak interaction obeys the gauge symmetry, which, together with the Lorentz invariance and PRI, dictates the standard Yang-Mills action, as we have explained in our previous post.

Then the field equations of the weak interaction and the Higgs fields are determined by by PID and PRI, and are given by:

where represents the mass of the Higgs particle, are the Pauli matrices, () are the three gauge potentials, are the three Higgs fields, and

2. Prediction of Charged Higgs

The right-hand side of (1) is due to PID, leading naturally to the introduction of three scalar dual fields. The left-hand side of (1) represents the intermediate vector bosons and , and the dual fields represent two charged Higgs (to be discovered) and the neutral Higgs , with the later being discovered by LHC in 2012.

It is worth mentioning that the right-hand side of (1), involving the Higgs fields, are non-variational, and can not be generated by directly adding certain terms in the Lagrangian action. This might be the very reason why for a long time one has to use logically-inconsistent electroweak theory, as we explained in the previous post here.

3. First principle approach to spontaneous gauge symmetry-breaking and mass generation

PID induces naturally spontaneous symmetry breaking mechanism. By construction, the action obeys the gauge symmetry, the PRI and the Lorentz invariance. Both the Lorentz invariance and PRI are universal principles, and, consequently, the field equations (1) and (2) are covariant under these symmetries.

The gauge symmetry is spontaneously breaking in the field equations (1), due to the presence of the term, , in the right-hand side, derived by PID. This term generates the mass for the vector bosons.

4. Weak charge and weak potential

As we mentioned in the previous posts here, elements in are expressed as , where is a basis of the set of traceless Hermitian matrices, and plays the role of a coordinate system in this representation. Consequently, an gauge theory should be independent of choices of the representation basis. This leads to the principle of representation invariance (PRI), and is simply a logic requirement for any gauge theory. This was first discovered by the authors in 2012.

With PRI applied to gauge theory for the weak interaction, two important physical consequences are clear.

First, by PRI, the gauge coupling constant plays the role of weak charge, responsible for the weak interaction.

In fact, the weak charge concept can only be properly introduced by using PRI, and it is clear now that the weak charge is the source of the weak interaction.

Second, PRI induces an important constant vector . The components of this vector represent the portions distributed to the gauge potentials by the weak charge . Hence the (total) weak interaction potential is given by the following PRI representation invariant

and the weak charge potential is the temporal component of this total weak interaction potential , and weak force is

where is the weak charge of a reference particle with radius .

5. Layered formulas for the weak interaction potential

The weak interaction is also layered, and we derive from the field equations (1) and (2) the following

where is the weak force potential of a particle with radius and carrying weak charges , taken as the unit of weak charge for each weakton, is the weakton radius, is a parameter depending on the particles, and represents the force-range of weak interactions.

The layered weak interaction potential formula (5) shows clearly that the weak interaction is short-ranged. Also, it is clear that the weak interaction is repulsive, asymptotically free, and attractive when the distance of two particles increases.

Tian Ma & Shouhong Wang

Transition Theory and Phase Transition Dynamics

[1] Tian Ma and Shouhong Wang, Phase Transition Dynamics, Springer, pp 555, 2013

This book synthesizes the mathematical and physical theories for a phase transition phenomena, established by the authors. This blog post intends to give a brief introduction to the theories developed in this book, and to point out some main differences between the transition theory and the classical bifurcation theory.

1. Mathematical theory

The mathematical theory is also called dynamic transition theory, and we summarize hereafter the main ingredients of the theory.

First, we have proved a general transition classification theorem, which states that all dynamic transitions are classified into three categories: the Type-I, Type-II and Type-III.

Second, one important tool of the dynamic transition theory is the central manifold reduction, and we have derived for the first time approximate central manifold reduction formulas, which are crucial for many applications of the theory to physical problems. Also, these formulas has been generalized to random dynamical systems by Chekroun et al.

Third, we have systematically developed theorems and criteria for different transition types. For example, for the first time, we introduced the concept of attractor bifurcation, and proved a general attractor bifurcation theorem, which can be used to handle most Type-I transitions. A sequence of theorems are established also for the Type-II, and Type-III transitions. These theorems are easy to use in applications and are crucial in deriving the needed physical theory for the related physical problems.

2. Physical theory

Our physical theory of transition dynamics involves a wide range of scientific fields, including statistical physics, fluid dynamics, atmospheric and ocean physics, and biology and chemistry. Hereafter we present a few physical theory we have derived, and we refer interesting readers to the book [1] for more details and for other physical applications.

Principle of phase transition dynamics: We discovered a general principle of phase transitions for dissipative physical systems, which we call principle of phase transition dynamics. Namely, phase transitions of all dissipative physical systems are classified into three categories: continuous, catastrophic, and random.

General dynamic model for equilibrium phase transitions: We introduced a unified dynamical model for equilibrium phase transitions, based on the Le Chatelier principle and the Ginzburg-Landau mean field theory.

Discovery of third-order phase transition: It is well-known that the gas-liquid coexistence curve terminates at a critical point, also called the Andrews critical point, and gas-liquid transition is of first order before the critical point and of the second-order at the critical point. Going beyond the critical point, physical phenomena indicates that a high-order phase should occur. However, it is a longstanding open question why the Andrews critical point exists and what is the order of transition going beyond this critical point. For the first time, 1) we derived the gas-liquid co-existence curve beyond the Andrews critical point, and 2) we show that the transition is first order before the critical point, second-order at the critical point, and third order beyond the Andrews critical point. This gives rise to the mechanism of the Andrews critical point, and the reason why it is hard to observe the gas-liquid phase transition beyond the Andrews critical point.

Prediction of a new superfluid phase in liquid helium-3: We have derived new dynamical models for liquid helium-3, helium-4 and their mixture, leading to various physical predictions, such as the existence of a new phase for helium-3. Although these predictions need yet to be verified experimentally, they certainly offer new insights to both theoretical and experimental studies for a better understanding of the underlying physical problems.

New mechanism of El Nino Southern Oscillation (ENSO): We discovered a new mechanism of the ENSO, as a self-organizing and self-excitation system, with two highly coupled oscillatory processes: 1) the oscillation between the two metastable warm (El Nino phase) and cold events (La Nina phase), and 2) the spatiotemporal oscillation of the sea surface temperature (SST) field. The interplay between these two processes gives rises the climate variability associated with the ENSO, leads to both the random and deterministic features of the ENSO, and defines a new natural feedback mechanism, which drives the sporadic oscillation of the ENSO.

3. Differences between the dynamic transition theory and classical bifurcation theory

It is important to emphasize the main differences between the dynamical transition theory and the classical bifurcation theory.

First the key difference is that the transition states derived in our dynamic transition theory are physical, and the bifurcation states derived from the classical bifurcation theory may not be physical.

In fact, the classical bifurcation theory first seeks bifurcation solutions and then decides the stability of the bifurcated solutions. The main drawback for this approach is that there is no way to know if the bifurcated solutions represent all transition physical states. In addition, it is always technically difficult to derive the stability of the bifurcated solution.

Instead, our dynamic transition theory finds all physical phase transition states.

Second, our theory indicates that transition always happen at the critical point, but bifurcation may not occur.

Third, it is clear that the general physical principle for phase transitions can only be discovered by using the dynamic transition theory.

4. Summary

The dynamic transition theory can be viewed as a true mathematical representation of a physical theory. The general principle of phase transition dynamics clearly offers guidance to the understanding of dissipative physical systems.

Tian Ma & Shouhong Wang

Problems in Classical Electroweak Theory

The classical electroweak theory forms the core of the standard model of particle physics. The great success of both the electroweak theory and the standard model include e.g. the prediction of the intermediate vector bosons and the Higgs boson. In spite of its success, there are a number of issues and difficulties for the classical electroweak theory, which we will address in this blog post.

In the next blog post, we shall introduce the PID electroweak theory, resolving all these difficulties. In particular, the PID approach provides a first principle approach for introducing the Higgs field.

1. Classical electroweak theory

In essence, the electroweak theory is the generalization of the Fermi theory, and provides a useful computational tool for transition probability and amplitudes. It is a gauge theory incorporating the Higgs field, and its main ingredients include

• It involves three gauge potentials, , and and one potential ;
• The Higgs scalar doublet was introduced into the Yang-Mills Lagrangian action in order to derive proper mass generation mechanism for the intermediate bosons.
• With the gauge potentials, the following combinations are introduced to represent the intermediate vector bosons , and the electromagnetic potential , respectively:

  

  

  

2. Lack of weak force formulas

This problem is that all weak interaction theories have to face, and it is also that all existing theories cannot solve.

In fact, the classical electroweak theory, there are four gauge field potentials:

and we don’t know which of these potentials plays the role of weak interaction potential.

3. Violation of Principle of Representation Invariance (PRI)

We have discovered a basic principle, called the principle of representation invariance (PRI), for the gauge theory, which describes an {interacting} particle system; see the previous post for details about PRI.

Elements in are expressed as , where is a basis of the set of traceless Hermitian matrices, and plays the role of a coordinate system in this representation. Consequently, an gauge theory should be invariant under the following global transformation of the representation bases:

where is a a nondegenerate complex matrix. We call such invariance of the gauge theory the principle of representation invariance (PRI).

PRI is a logic requirement for any gauge theory, and has profound physical consequences. In particular, by PRI, any linear combination of gauge potentials from two different gauge groups are prohibited.

In the classical electroweak theory, a key ingredient is the linear combinations of and . By PRI,

Hence, for the combinations of two different types of tensors:

used in the classical electroweak theory and the standard model of particle physics, violate PRI.

4. Decoupling obstacle

The classical electroweak theory has a difficulty for decoupling the electromagnetic and the weak interactions. In reality, electromagnetism and weak interaction often are independent to each other. Hence, as a unified theory for both interactions, one should be able to decouple the model to study individual interactions. However, the classical electroweak theory manifests a radical decoupling obstacle.

For example, if there is no weak interaction involved, then

hold true. In this case, the theory should return to the gauge invariant Maxwell equations. But we see that

where is a gauge field, and is a component of gauge field. Therefore, is not independent of gauge transformation. In particular, the condition (5) means

Now we take the transformation (4) for the generators of , becomes

It implies that under a transformation (4), a nonzero weak interaction can be generated from a zero weak interaction field of (5)-(6):

and the nonzero electromagnetic field will become zero:

Obviously, it is not reality.

5. Artificial Higgs mechanism

In the classical electroweak action, the Higgs sector is not based on a first principle, and is artificially added into the action.

6. Presence of a massless and charged boson

In the WS theory, the Higgs scalar doublet contains a massless boson with positive electric charge. Obviously there are no such particles in reality. In particular, the particle is formally suppressed in the classical electroweak theory by transforming it to zero. However, from a field theoretical point of view, this particle field still represents a particle. This is one of major flaws for the electroweak theory and for the standard model.

Angular Momentum Rule and Scalar Photons

[1] Tian Ma and Shouhong Wang, Quantum Rule of Angular Momentum, AIMS Mathematics, 1:2(2016), 137-143.

[2] Tian Ma and Shouhong Wang, Mathematical Principles of Theoretical Physics, Science Press, 2015

1. Angular Momentum Rule of Quantum Systems

Quantum physics is the study of the behavior of matter and energy at molecular, atomic, nuclear, and sub-atomic levels. Two most distinct features of quantum mechanics, drastically different from classical mechanics, are the Heisenberg uncertainty relation and the Pauli exclusion principle.

We present a new feature, the angular momentum rule, discovered recently by the authors [1, 2], This new angular momentum rule can be considered as an addition to the Heisenberg uncertainty relation and the Pauli exclusion principle in quantum mechanics.

Quantum Rule of Angular Momentum [1, 2]. Only fermions with spin and bosons with can rotate around a center with zero moment of force, and particles with will move on a straight line unless there is a nonzero moment of force present.

This quantum mechanical rule is important for the structure of atomic and sub-atomic particles. In fact, the rule gives the very reason why the basic constituents of atomic and sub-atomic particles are all spin- fermions.

The angular momentum rule provides the theoretical evidence and support of scalar photons, a recent prediction from our unified field theory and the weakton model of elementary particles.

2. Prediction of Scalar Photons

First, we recall that the photon, denoted by , is the mediator of the electromagnetic force. The photon is a massless spin-1 particle, described by a vector field defined on the space-time manifold, which obeys the Maxwell equations:

Second, the scalar photon, denoted by , was first introduced as a natural byproduct of our unified field theory based on the principle interaction dynamics (PID), which we have discussed in the previous posts. The scalar photon is a massless, spin-0 particle, described by a scalar field , satisfying the following Klein-Gordon equation:

Third, the puzzling decay and reaction behavior of subatomic particles suggest that there must be interior structure of charged leptons, quarks and mediators. Careful examinations of subatomic decays/reactions lead us to propose six elementary particles, which we call weaktons, and their anti-particles:

where are the three generation neutrinos, and are three new particles, which we call -weaktons.

Remarkably, the weakton model offers a perfect explanation for all sub-atomic decays. In particular, all decays are achieved by 1) exchanging weaktons and consequently exchanging newly formed quarks, producing new composite particles, and 2) separating the new composite particles by weak and/or strong forces.

In the weakton model, the constituents of the photon is given as follows:

and different spin arrangements of the weaktons give rise naturally to the scalar photon with the following constituents:

3. Bremsstrahlung as an Experimental Evidence for Scalar Photons

It is known that an electron emits photons as its velocity changes, which is called the bremsstrahlung. The reasons why bremsstrahlung can occur is unknown in classical theories.

In fact, our viewpoint is that the bremsstrahlung suggest that a mediator cloud is present near a naked electron, and the mediator cloud contains photons. The angular momentum rule demonstrates that the photons circling the naked electron must be scalar photons, as free vector photons can only take straight line motion. We refer the interested readers to Section 5.4 of [2] for more detailed discussions.

In summary, bremsstrahlung, together with the angular momentum rule, offers an experimental evidence for scalar photons. Of course, further direct experimental verification and discovery of scalar photons are certainly important feasible.

Tian Ma & Shouhong Wang

Contradictions between the Black Hole and the Big-Bang Theories, and the Structure of the Universe

Based on the authors’ recent work [1, 2, 3], the main objectives of this post are as follows:

• First, assuming the Einstein theory of general relativity, we demonstrate that the theory of black holes and the theory of Big-Bang are contradicting to each other.
• Second, we introduce a theorem, proved recently by the authors, on the structure and geometry of our Universe, assuming the Einstein theory of general relativity and the principle of cosmological principle. This theorem gives rise to a new theory of the Universe, which is consistent with redshift and CMB observations.

1. Conditions for the existence of black holes

Consider a massive body with mass and radius . The Schwarzschild radius is defined by . Based on the Einstein field equations, in the exterior of the body, the Schwarzschild solution is given by

and in the interior the Tolman-Oppenheimer-Volkoff (TOV) metric is

The theory of black holes then shows that the massive body is a black hole if and only if . Here we note that if , the massive body is not a black hole. Also, it is impossible to have . Otherwise, the Schwarzschild solution would be valid in the vacuum region , where the factor changes sign and the metric becomes nonphysical. In other words, a black hole must be filled with matter in the entire interior.

The above necessary and sufficient condition implies that a massive body with radius and density is a black hole whenever

2. Contradictions between Big-Bang and Black-Hole Theories

First, if the Universe began with the big-bang, the energy density of the early universe would be so big that the universe would be filled with many black holes of infinitesimal radius defined by (3).

Second, if the Universe were born to a Big-Bang and expanded continuously, then in the expansion process it would generate successively a large number of black holes, whose radii vary as follows:

where is the total mass in the universe, is the initial radius, is the expanding radius, is the radius of sub-black holes, and is the radius of the Universe viewed as a black hole.

To see this, we consider a homogeneous universe with radius . Then the mass density is given by

which implies that the mass of a ball of radius with this mass density is

Recall that the condition for the ball to form a black hole is

The combination of (5) and (6) implies immediately (4).

In summary, (4) demonstrates clearly that if the Universe were born to a Big-Bang and continuously expands, then it would contain many black holes with smaller ones being embedded in the larger ones. This is not what we observed in our Universe.

3. Theorem on Structure of the Universe

The aforementioned contradictions force us to examine the black hole and Big-Bang theories from both the fundamental level and the observational evidences.

It is clear that the large scale structure of our Universe is essentially dictated by the law of gravity, which is based on Einstein’s two principles: the principle of general relativity and the principle of equivalence. Also, strong cosmological observational evidence suggests that the large scale Universe obey the cosmological principle that the Universe is homogeneous and isotropic.

The black theory is a direct consequence of the Einstein theory of general relativity, and is a more trustable theory. Consequently, a careful examination of modern cosmological theories on the structure of the Universe is inevitable. In fact, we have recently shown a theorem on the geometry and structure of our Universe under the assumption of general relativity and the cosmological principle:

Theorem on Structure of our Universe [2, 3]. Assume the Einstein theory of general relativity, and the principle of cosmological principle, then the following assertions hold true:

1.  our Universe is not originated from a Big-Bang, and is static;
2. the topological structure of our Universe is the 3D sphere such that to each observer, the corresponding equator with the observer at the center of the hemisphere can be viewed as the black hole horizon;
3. the total mass in the Universe includes both the cosmic observable mass and the non-observable mass, regarded as dark matter, due to the space curvature energy; and
4. a negative pressure is present in our Universe to balance the gravitational attracting force, and is due to the gravitational repelling force, also called dark energy.It is clear that this theorem changes drastically our view on the geometry and the origin of the Universe.

Inevitably, a number of important questions need to be addressed for this scenario of our Universe in the following sections.

 4. Redshift problem

The natural and important question that one has to answer is the consistency with astronomical observations, including the cosmic edge, the flatness, the horizon, the redshift, and the cosmic microwave background (CMB) problems. These problems can now be easily understood based on the structure of the Universe and the blackhole theorem we derived. Hereafter we focus only on the redshift and the CMB problems.

The most fundamental problem is the redshift problem. Observations clearly show that light coming from a remote galaxy is redshifted, and the farther away the galaxy is, the larger the redshift. In modern astronomy and cosmology, it is customary to characterize the redshift by a dimensionless quantity in the formula

where and represent the observed and emitting wavelenths.

There are three sources of redshifts: the Doppler effect, the cosmological redshift, and the gravitational redshift. If the Universe is not considered as a black hole, then the gravitational redshift and the cosmological redshift are both too small to be significant. Hence, modern astronomers have to think that the large port of the redshift is due to the Doppler effect.

However, due to black hole properties of our Universe, the black hole and cosmological redshifts cannot be ignored. Due to the horizon of the sphere, for an arbitrary point in the spherical Universe, its opposite hemisphere relative to the point is regarded as a black hole. Hence, can be approximatively taken as the Schwarzschild solution for distant objects as follows

where for is the distance from the light source to the opposite radial point, and is the distance from the light source to the point. Then by the gravitational redshift formula:

we derive the following redshift formula, which is consistent with the observed redshifts:

5. CMB problem

In 1965, two physicists A. Penzias and R. Wilson discovered the low-temperature cosmic microwave background (CMB) radiation, which fills the Universe, and it has been regarded as the smoking gun for the Big-Bang theory. However, based on the unique scenario of our Universe we derived, it is the most natural thing that there exists a CMB, because the Universe has always been there as a black-body, and CMB is a result of blackbody equilibrium radiation.

6. PID-cosmological model

We have demonstrated that the right cosmological model should be derived from the new gravitational field equations [1], taking into consideration the presence of dark matter and dark energy:

In this case, based on the cosmological principle, the the metric of a homogeneous spherical universe is of the form

where is the cosmic radius. The new gravitational field equations provide then the following PID-cosmological model:

supplemented with the equation of state:

Note that only two equations in (10)-(12) are independent.

Also, the model describing the static Universe is the equation of state (14) together with the stationary equations of (10)-(12), which are equivalent to the form

Now we consider a perturbation of the steady state solution, which gives

which implies , and consequently, . By (13), , which implies that . Then by (12), , which contradicts with (16). In other words, this simple argument tells us that the dual field plays a role of balancing, preventing any perturbation of the combined quantity:

In a nutshell, we have shown that the steady state solution of (10)-(14) is physically stable.

References

[1] Tian Ma and Shouhong Wang, Gravitational field equations and theory of dark matter and dark energy, Discrete and Continuous Dynamical Systems, Ser. A, 34:2(2014), 335-366; see also arXiv:1206.5078v2.

[2] Tian Ma and Shouhong Wang, Astrophysical dynamics and cosmology, Journal of Mathematical Study, 47:4(2014), 305–378.

[3] Tian Ma and Shouhong Wang,  Mathematical Principles of Theoretical Physics, Science Press, Beijing, 524pp., August, 2015.

Tian Ma and Shouhong Wang