Welcome to EverybodyWiki ! Sign in or create an account to improve, watchlist or create an article like a company page or a bio (yours ?)...

# Bimetric gravity

Bimetric models extend general relativity to explain precisely the nature of dark matter and dark energy.

Bimetric gravity or bigravity refers to two different sets of theories.[1] The first set of theories relies on modified mathematical theories of gravity (or gravitation) in which two metric tensors are used instead of one.[2] The second metric may be introduced at high density of energies, with the implication that the speed of light could be energy-dependent, enabling models with a variable speed of light.

If the two metrics are dynamical and interact, a first possibility implies two graviton modes, one massive and one massless; such bimetric theories are then closely related to massive gravity.[3] Several bimetric theories with massive gravitons exist, such as those attributed to Nathan Rosen (1909–1995)[4][5][6] or Mordehai Milgrom with Modified Newtonian Dynamics (MOND).[7] More recently, developments in massive gravity have also led to new consistent theories of bimetric gravity.[8] Though none has been shown to account for physical observations more accurately or more consistently than the theory of general relativity, Rosen's theory has been shown to be inconsistent with observations of the Hulse–Taylor binary pulsar. [5] Some of these theories lead to cosmic acceleration at late times and are therefore alternatives to dark energy.[9][10]

On the contrary, the second set of bimetric gravity theories does not rely on massive gravitons and does not modify Newton's law, but instead are extensions of general relativity, without exotic particles. These theories describes the universe as a manifold having two coupled Riemannian metrics, where matter populating the two sectors interact through gravitation (and antigravitation if the topology and the Newtonian approximation considered introduce negative mass and negative energy states in cosmology as an alternative to dark matter and dark energy). Some of these cosmological models also use a variable speed of light in the high energy density state of the radiation-dominated era of the universe, challenging the inflation hypothesis.[11][12][13][14] The Janus cosmological model belongs to this set.

## Janus cosmological model

Since 1977 Jean-Pierre Petit develops an atypical bimetric theory of gravity called the Janus cosmological model in reference to the two-faced god who "looks simultaneously to the future and to the past".[15]

The bimetric cosmological model Janus (JCM), a.k.a multi-sheeted model or theory of twins universes is a cosmological model representing the known universe as the mirror of a "shadow universe", forming two sides or sheets that interact only by gravitation. This enantiomorphic model was first introduced by Andrei Sakharov and then extended by others. It is currently more complete than the mainstream Lambda-CDM model, in terms of its ability to explain the observed phenomena, and without using free parameters.

JCM is not a theory of everything. The Janus Cosmological Model is built upon Albert Einstein's theory of general relativity, Andrei Sakharov's work in cosmology, and Jean-Marie Souriau's work in symplectic geometry.

The Janus model challenges many of the assumptions of the mainstream Lambda-CDM model, such as the absolute constancy of the speed of light or the existence of dark matter and dark energy in the observable universe for which it proposes alternatives. It is based on the Poincaré group, an extension of general relativity and quantum mechanics. Since it challenges many of the foundations of current theoretical physics (through the existence of negative masses and energies) and the current mainstream cosmological model, the Janus model is little studied by other cosmologists, although it has been the subject of articles in peer-reviewed scientific journals.[16][17]

The Janus Cosmological Model describes the universe as a Riemannian manifold with two different metrics that handle positive and negative masses in general relativity with no runaway paradox, in very good agreement with observational data.

Janus model's main author produces many science comic books and videos to popularize the various aspects of Janus cosmological model.[18][19][20][21]

The Janus theory originated in 1977, first as a non-relativistic model (Newtonian dynamics) of two enantiomorphic universes with opposite arrows of time: the twin universe theory. He learnt about Andrei Sakharov's work in this field in 1984 when his book 'Scientific Works' was published. In 1988 it was the first model to propose a variable speed of light (VSL) in cosmology, as a joint variation of all physical constants following a universal gauge relationship, letting the laws of physics invariant during the radiative era. This solution challenges the inflation hypothesis. As of 1994, both approaches were then merged into a VSL relativistic bimetric model of gravity.

The Janus model would explain various observational facts that the mainstream model cannot answer, the gravitational interaction of positive and negative masses being an alternative candidate for the explanation of dark matter, dark energy, cosmic inflation and the accelerating expansion of the universe.[22] Despite being peer reviewed, this non-standard cosmological model has not triggered much interest in the scientific community throughout the years, except with mathematicians and geometers who seem more interested than cosmologists in its topological subtleties.[23][24][25][26]

However, in particle physics, the theory shares similarities with the mirror matter of hidden sectors addressing CP violation.[27][28][29] In general relativity, later independent work about bimetric gravity with positive and negative masses lead to the same conclusions regarding the laws of gravitation.[11][12][13]

2D didactic image of Sakharov's twin universe model.

The Janus model uses only one mass: the relativistic mass m = E/c² (which is only a form of energy expressed in different units, valid even for so-called zero mass particles like the photon), simply affected by a plus or minus sign. All motions are then deduced from the geodesics secreted by the field equations. This is why the Janus model does not distinguish between inert mass, inertial mass, rest mass, gravitational mass, invariant, passive or active mass.

The Janus model has the same foundation as a model previously published by Andrei Sakharov ten years before.[30] In 1967, Andrei Sakharov addressed the baryon asymmetry of the universe considering for the first time events in CPT symmetry occurring before the Big Bang (i.e. with t < 0):

Andrei Sakharov was the first scientist to introduce twin universes he called "sheets". He achieved a complete CPT symmetry since the second sheet is populated by invisible "shadow matter" which is antimatter (C-symmetry) because of an opposite CP-violation there, and the two sheets are mirror of each other both in space (P-symmetry) and time (T-symmetry) through the same initial gravitational singularity. He continued this idea for twenty years.[32] · [33] · [34] · [35] · [36] · [37] · [38]

2D didactic image of Janus model.

Ignoring the prior existence of this work translated in a book only fifteen years after its Russian publication,[31] Jean-Pierre Petit published his first paper about two enantiomorphic universes with opposite arrows of time in 1977.[39] · [40] Unlike Sakharov, he makes the two parallel universes interacting through gravity straightforward. In this first non-relativistic Newtonian dynamics model, galaxies are imbedded in repellent invisible negative mass, so they can be modeled as an exact solution of two Vlasov equations, coupled by Poisson's equation.

In 1994, the model is developed as a bimetric description of the universe.[41]

The Janus model implies two sides of the same coin. The universe can be assimilated to a single 4D hypersurface, which would have a frontside and a backside.

The motion of positive mass particles and the motion of negative mass particles do not occur on the same "side", or sector. Each species evolves along its own family of geodesics, in its own metric. Negative mass is among us, but invisible to us, being located "on the other side of the sheet", in its own "negative sector".

In the Janus model, negative mass is therefore not an intrinsic feature of such "exotic matter", but stem from topology; it is purely relative.

However this bimetry is not similar to independent work done in the field of classical bimetric gravity where the second metric refers to gravitons with nonzero mass. In the janus model, the bigravity is an extension of general relativity describing the universe as a Riemannian manifold associated to two conjugated metrics generating their own geodesics, solutions of two coupled Einstein field equations:[42]

${\displaystyle R_{\mu \nu }^{(+)}-{\tfrac {1}{2}}\,R^{(+)}g_{\mu \nu }^{(+)}=\chi \left(T_{\mu \nu }^{(+)}+{\sqrt {\frac {g^{(-)}}{g^{(+)}}}}T_{\mu \nu }^{(-)}\right)}$
${\displaystyle R_{\mu \nu }^{(-)}-{\tfrac {1}{2}}\,R^{(-)}g_{\mu \nu }^{(-)}=-\chi \left({\sqrt {\frac {g^{(+)}}{g^{(-)}}}}T_{\mu \nu }^{(+)}+T_{\mu \nu }^{(-)}\right)}$

The Janus system of two coupled field equations reduces to Einstein's field equations in the case of a portion of spacetime where positive mass matter dominates and no negative mass is present, like in the Solar System. Similarly to this Einsteinian approximation, the Newtonian approximation allows to recover Newton's law of universal gravitation and formula for gravitational potentials from the field equations in the limit of weak fields and low velocities with respect to the speed of light. The Janus system of two coupled Einstein field equations do not need any cosmological constant Λ as a free parameter.

In yellow, the "preposterous" runaway motion of a positive and negative masses described by Bondi and Bonnor.
In green, gravitational movements in the Janus model which differ from those elaborated by Bondi and Bonnor, solving the runaway paradox.

The runaway paradox disappears: the Janus cosmological model finally enables the introduction of negative mass in cosmology. Moreover, accretion of negative mass particles in lumps of negative mass matter, through the Jean instability, is now possible as negative mass particles don't repulse themselves.

In the 5D hypersurface with two "sides", each type of mass belonging to its own metric. What is important to note however, is that the two field equations are coupled, i.e. a mass always creates a positive curvature in spacetime according to its own metric (where the mass appears visible), and it also always induces a negative curvature in the conjugate metric (where the mass appears invisible). "Negative mass" is not an intrinsic feature of some exotic matter. It is nothing but the (anti)gravitational effect produced by any mass, when it is felt from the opposite sector. According to an observer measuring such mass in the same sector where it lays in, that mass always appears positive.[43]

### The bimetric universe according to Janus model

The theory describes two sheets or parallel universes in CPT-symmetry interacting through gravity, both originating from the same initial singularity. In Janus model, four species of matter coexist:

• 1- positive mass matter (baryonic matter). Baryonic matter refers to all matter composed of elementary particles called baryons. In practice, this corresponds to protons, neutrons, their constituents (bosons, quarks), to which leptons are implicitly added (such as electrons and neutrinos) and which compose atoms and molecules and all directly visible structures in the observable universe (stars, galaxies, clusters of galaxies, etc.).
• 2- positive mass antimatter (C-symmetry with respect to the first specie). This is the antimatter according to Dirac definition which is not very abundant compared to the first type. C-symmetry reverses not only the sign of the electrical charge but also the other quantum charges qi including the baryonic number, but not the spin. ζ-symmetry is the translation into symplectic geometry of this C-symmetry between matter and Dirac's antimatter. ζ-symmetry, in the 5-dimensional evolution space used in the Janus model, causes symmetry C (called q-symmetry in Janus model) in the space of the moment.

Together with positive energy photons, these first two species are the components of the universe known until now: it is the first sheet of the universe (a.k.a. the positive sector).

The 4 species of matter in the positive and negative sectors according to the Janus model.
• 3- the negative mass material (CPT-symmetry with respect to the first specie, with an anti-linear and anti-unit operator T ), which is not very abundant with respect to the fourth specie. CPT-symmetry simultaneously reverses quantum charges, parity (the spatial image seen in a mirror) and time.
• 4- Negative mass antimatter (PT-symmetry with respect to the first specie, with a linear and unitary operator T ). Using symplectic geometry, the Janus model demonstrates that this PT-symmetry is also a ζ-symmetry and a q-symmetry which automatically go together, so the quantum charges are also reversed.

The fourth specie, the so-called « Feynman antimatter »,[44] is the primordial antimatter.

With negative energy photons, these last two types are the components of the second sheet of the universe (a.k.a. the negative sector).

This antimatter of the "shadow universe" is concentrated in huge conglomerates, radiating in the infrared and very dark red, structured like huge spheroidal proto-stars, but whose cooling time exceeds the age of the Universe.[45] The thermal agitation rate of these gigantic protostars creates a centrifugal force so strong that it prevents them from contracting further to induce a fusion reaction. Therefore, this universe sheet contains no stars, no planets, no life; only anti-hydrogen and anti-helium from negative masses that appeared after the primordial radiative era.

As positive mass matter emits positive energy photons travelling along null geodesics of the metric ${\displaystyle g_{\mu \nu }^{(+)}}$, and negative mass matter emits negative energy photons travelling along null geodesics of the metric ${\displaystyle g_{\mu \nu }^{(-)}}$, the exotic matter cannot be detected with optical instruments, besides its gravitational interaction with normal matter.

In blue, the attracting positive mass causes a gravitational well in the space-time. In red, a repulsing negative mass causes a gravitational hill. But this model implies negative masses are observable with others fundamental interactions : they have never been observed.
In blue, the attracting positive mass causes a gravitational well in the space-time. In red, the negative mass causes an attractive gravitational well on the other "side" according to the Janus model, hence a repulsing gravitational hill appears on our "side" of the space-time. This model implies negative masses are not observable except with gravitational effects, which are well observed.

The Newtonian approximation of the system of two coupled field equations provides the following gravitational interactions:

• particles of same energy attract each other according to Newton's law (positive mass attracts positive mass and negative mass attracts negative mass)
• particles of opposite energy repel each other according to "anti" Newton's law (positive mass and negative mass repel each other)

Those laws are different to the laws spelled out by Hermann Bondi and William Bonnor,[46][47] and solve the runaway paradox,[42] that usually makes scientists think negative mass can not physically exist: Template:Centered pull quote

Due to topological considerations, matter populating each fold appears to the other as having an opposite mass and an opposite arrow of time, although the proper time remains positive for both species.[18]

### Cosmology with variation of constants

In 1988, the Janus model's main author introduced the idea of the variation of the speed of light in cosmology[48][49][50][51] in conjunction with variations of all physical constants combined with changes in the scale factors of space and time, so that all the equations of physics and the relationships between these constants remain unchanged during the evolution of the universe. The Einstein field equations remain invariant thanks to adequate joint variations of c and G in the Einstein constant. The requirement of invariances of the Schrödinger and Maxwell equations provides the framework for the laws of joint variation of constants in gauge theory. The fine structure constant becomes an absolute constant. More recent work has restricted the variation of constants to the relativistic radiative era of the primordial universe, where space-time is identified with space-entropy with a flat metric.[52][53][54][55]

### Geometrization of Janus model

In 1995, the Janus model's main author combined his bimetric model with the VSL theory into the first paper summarizing the twin universes cosmology.[56]

The main hypotheses stating that negative energy particles exist and result from time reversal, that two particles of opposite mass repel each other, and that physical constants can vary, are in opposition with the standard models of particles physics and cosmology. In quantum field theory, the T operator acting on Hilbert spaces is complex, and can be either linear and unitary, or antilinear and antiunitary; but is arbitrarily chosen antilinear and antiunitary in order to prevent inversion of energy, as the vacuum state of the Zero-point energy must have the lowest possible ground state and can not have negative values.[57] But when this axiom was formulated, the accelerating expansion of the universe, which implies a negative pressure, was not known yet. As a pressure is a volumetric energy density, the Janus model's authors think this problem should be reconsidered.

Symplectic geometry is a branch of mathematics merging differential geometry and dynamical systems theory, based on the Hamiltonian formulation of classical mechanics. It is the study of manifolds with a symplectic form that allows to measure sizes. Indeed, in Riemannian geometry, the metric tensor gives lengths and angles, whereas the symplectic form measures areas.

Three mathematicians greatly developed symplectic geometry in the second half of the 20th century. Based on the orbit method from Russian mathematician Alexandre Kirillov in representation theory, American mathematician Bertram Kostant and French mathematician Jean-Marie Souriau developed the modern theory of geometric quantization, which allows to make fundamental quantities of physics (like energy and momentum) emerge as pure geometrical objects.

Souriau left a great legacy to mathematical physics in the fields of classical, relativistic and quantum mechanics. He notably introduced the important concepts of the moment space, the moment map (Hamiltonian action of a Lie group on a symplectic manifold, generalizing the notions of linear and angular momentum) and the coadjoint action, i.e. the action of a Lie group on the dual space to its Lie algebra. This allowed him, for instance, to give the first geometric and kinetic interpretations of spin.[58]

However, in group theory, the T operator is real and can reverse the energy. Dynamics of relativistic elementary particles is described by the Poincaré group. Currently physics uses the restricted Poincaré group, with only forward in time ("orthochronous") motions. As demonstrated by Jean-Marie Souriau using the complete Poincaré group, including backward in time ("antichronous") motions, arrow of time reversal equals mass inversion of a particle.[59]

In the 2000s, the Janus model's authors integrated Souriau's mathematical physics and fully geometrized the cosmological model with his dynamical group theory. The addition of dynamical group theory explains why time reversal goes with energy and mass inversion, a fundamental property of particles secreted by the complete Poincaré group.[60] · [61] · [24] · [62]

In 2014 and 2015 he publishes a set of four papers detailing the most recent developments of the Janus model. The first paper produces an exact solution to the coupled field equations referring to the matter-dominated era which resolves the runaway paradox of negative mass and challenges dark energy to account for the accelerating expansion of the universe.[42] In a second paper this is extended to two metrics with their own speed of light,[22] followed by the Lagrangian derivation of the model.[63] A fourth paper is devoted to the cancellation of the central singularity in the Schwarzschild solution, questioning the classical black hole model.[64]

A comparison of the Janus model in agreement with latest observational data has been published in 2018.[65]

The same year, the model also shows how negative energy states are compatible with quantum mechanics. A paper discussed the arbitrary decision of preventing negative energy states in quantum field theory, as such negative energy is compatible with the Dirac equation when considering a unitary time-reversal operator, provided that one considers that negative energy goes with negative mass according to ${\displaystyle -m=-E/c^{2}}$, so the Klein–Gordon probability density ${\displaystyle E/m}$ in relativistic quantum mechanics remains positive.[66][67]

### The bridges between the sheets of the universe (Black Holes reconsidered)

In the classic model, when a neutron star exceeds the stability limit, neutrons that compose it "break" and the neutron star collapses because of the excessive gravity due to its mass. The gravitational collapse leads to the black hole, whose centre is described in the classic black hole model as a mathematical and physical singularity where the temperature, density and curvature of space-time are infinite.

In the Janus bimetric model, when a neutron star exceeds the stability limit, neutrons that compose it "break", and then the star imaginatively "pierces" space-time forming a bridge connecting the two sheets of the universe. Part of the positive mass at the heart of the neutron star becomes negative during this process and is therefore found in the "shadow universe" where it is repelled by the gravitation of the positive mass remaining on the other sheet. This process intrinsically lasts for a very short time (in the proper time of the neutron star), and it allows the neutron star to pass below the critical threshold of positive mass density. There is no longer any singularity. It can then continue to attract positive mass matter by gravitation until the next time the critical density threshold is reached, the cycle repeating itself. This gives the impression of a black hole that swallows matter all the time.[68]

The Janus model also explained the detailed phenomena that occur within a neutron star at the critical density threshold.[69][70]

The Janus model's main author provided a mathematical demonstration of the cancellation of the central singularity, which according to the author invalidates the black hole theory as it had been understood since 1916: there was an error in the interpretation of a demonstration of Schwarzschild by Hilbert, which remained unnoticed until its discovery in 1989 by Abrams[71][72][73][74][75][76] and which has not yet been taken into account in the classic black hole model.

• J.-P. Petit, P. Midy, Questionable black hole, Geometrical Physics A, 11 (1998) and 12 (2000) [77]
• J.-P. Petit, Black holes do not exist, preprint, April and June 2014[78]
• (fr) interview with J.-P. Petit by SputnikNews, June/July 2014, Les trous noirs n'existent pas - parts 1[79] and 2[80]
• J.-P. Petit, Schwarzschild 1916 seminal paper revisited : A virtual singularity, preprint, July 2016[81]

These studies have been explained in detail in the following videos:

• JANUS 22 - 1 : Black hole VS mass inversion. Conferences 2017. Negative energy states[82] (September 2017)
• JANUS 22 - 2: (fr) Géométrie, la part du réel et de l'imaginaire[83]
• JANUS 22 - 3 : Schwarzschild's forgotten solution[84]
• JANUS 22 - 4 : Contestation in good standing of the black hole model[85]
• JANUS 22 - 5 : Gravitational waves. Fusion of neutron stars. Mass with inverted sign[86][87]
• JANUS 24 : The Black Hole in the JANUS COSMOLOGICAL MODEL[88] (January 2018)

Shortly before his death, Stephen Hawking made hypotheses consistent with the absence of singularity within the black hole, which is thus seen as a passage to another universe.[89]

On December 10, 2018, Abhay Ashtekar, Javier Olmedo, and Parampreet Singh published a scientific paper in the field of the theory of loop gravity demonstrating the absence of central singularity within the black hole, without geometrically specifying the future of matter at this point while the Janus model proposes an explanation.[90][91][92]

### Formation of large structures in the universe according to the Janus model

#### Density fluctuations in the universe

The peer-review committee of the journal Progress on Physics[93] validated a study that rely on the Janus model to explain the origin of fluctuations in the cosmological background (CMB). It was published in October 2018.[94]

#### Formation and stability of galaxies and structures on a large scale

The Janus model would explain the shape and stability of spiral galaxies by using negative mass in interaction with the galaxy that would confine it and be repellent for the latter.

In the first non-relativist Newtonian model developed by the Janus model's authors, galaxies were buried in a halo of invisible negative masses that repel them, and they can be modeled as the exact solution of two Vlasov equations, coupled by the Poisson equation.[95][96] In the current Janus model, the modeling of these negative masses is geometrized; moreover, the nature and origin of these negative masses are defined.[97][98]

The very large-scale structures of the observable universe (filament structures separated by large apparent voids, Great Repeller) are explained by the gravitational repulsion between positive and negative masses.[99][97][100]

### Past and future of the bi-sheeted universe according to the Janus model

The cosmological scenario of the past and future of the bi-sheeted universe was developed in an article published in May 2008 and at several conferences.[101] · [102] · [103]

## Viability and relevance of the Janus model

You can refer to the relevance criteria of a valid cosmological model to check that the Janus model meets all these criteria.

All the elements gathered below show that the Janus cosmological model is in many ways the simplest solution (Ockham's razor).

### The differences between the Janus model and the mainstream cosmological model ΛCDM

This section summarizes the observations that support the Janus model.

The Janus model explains the accelerating expansion of the universe.[104] The ΛCDM Model does not explain it : it introduces dark energy (without explaining what it is) to take it into account in the ad-hoc model.

In a scientific publication published in the journal Astrophysics and Space Science, July 2018, the Janus model's authors set out the differences in the explanatory scope between the ΛCDM Model and the Janus Model.[105] Quote :

• "JCM explains the absence of observation of the so-called primeval antimatter, opposite to the mainstream ΛCDM model.
• JCM describes precisely the nature of the invisible components of the universe, opposite to the  mainstream ΛCDM model.
• In addition, JCM predicts that the antimatter produced in laboratory[106] will react as the matter with respect to the gravitational field of the Earth (it will fall).
• Because positive and negative matter are repelling each other, the negative matter content in the solar system is almost zero. So, JCM fits the classical relativistic observations, as presented in former papers.[107][108][109] JCM suggests a clear scheme for VLS formation[110] when the mainstream ΛCDM model seems to struggle more to give one.
• JCM  explains  the  observed  repellent  effect  due  to  “the Great Repeller”. The measured escape velocities of galaxies are due to the presence of an invisible repellent cluster made of negative mass in  the centre of a big void. The mainstream model’s supporters suggest that such a repellent effect could be due to some kind of hole in the dark matter field of the universe (positive masses). But, if the gravitational instability leads to the setting up of massive clusters, it does not provide any scheme for such void formations. So that the mainstream ΛCDM model does not provide any explanation for this observation.
• JCM explains the confinement of galaxies and the shape of their rotation curves. As we showed,[111] if one introduces a surrounding repellent negative matter environment, it gives larger rotation velocities at distance. Mysterious dark matter is no longer required, while the mainstream ΛCDM model does.
• After JCM, the intensity of the observed gravitational lensing effect is mainly due to the negative matter that surrounds galaxies and clusters of galaxies. Mysterious dark matter is no longer required, while the ΛCDM model does.
• JCM suggests an explanation of the low magnitude of very young galaxies: this would be due to negative lensing weakening, when their light are crossing the negative mass clusters located at the center of the big void. Mysterious dark matter is no longer required, while the ΛCDM model does.
• JCM explains the spiral structure of galaxies,[111] due to dynamical friction with the surrounding negative mass. The ΛCDM model don’t give any model explaining the spiral structure. As a conclusion JCM is definitively not a simple or pure speculative product of theoretical mathematics. It had been compared with many observations and happened to fit with them. Opposite to the today’s mainstream ΛCDM model, JCM does not carry unknown and mystery like dark matter or dark energy."
The positive and negative lensing effect imply different curvature of space according to the Janus model.
The positive curvature in the positive sector caused by a positive mass imply a conjugated, induced negative curvature in the negative sector, according to the Janus model. Respectively, the positive curvature in the negative sector caused by a negative mass imply a conjugated, induced negative curvature and an observable negative lensing effect in the positive sector.

The Janus model does not involve any hypothetical exotic particles or mirror matter (P-symmetry only). He explains why the only interaction between gemellar matter (whose components are all described) and observable matter is gravitation.

The Janus model explains the origin of CMB fluctuations.[94] The ΛCDM Model provides no detailed explanation for this origin. In the same October 2018 publication, the authors add that Janus model explains the homogeneity of the primordial universe, without resorting to the hypothesis of cosmic inflation, which so far remains a problem without a complete theoretical solution in the mainstream ΛCDM model.

The Janus model takes into account the absence of central singularity within the black hole, and it is the first model that geometrically specified the fate of matter at this point. The candidate theory of loop quantum gravity followed it,[112][113][92] but still not the ΛCDM Model.

In summary, the Janus model makes it possible to explain both, as any realistic cosmological model :

1) What is part of the primordial cosmology

• how during the Big Bang the universe could have been in the very homogeneous state observed by the cosmological scattered background: because of the joint variation of the physical constants;
• why at that time small irregularities already existed: because of quantum effects and symmetry violations;
• how the different forms of matter could have come from the Big Bang, in our sheet of the universe (i.e. as presented in the standard model), and in the other sheet according to the quantum properties and density of the negative masses found there;
• what is the shape of the universe (local and global curvature geometries).[102][114][115]

2) What is more of an observational cosmology

• the current distribution of galaxies, clusters and superclusters of galaxies revealed by galaxy catalogues (i.e. as explained by simulations according to mainstream cosmology), and by explaining in addition the origin of the structure of the universe on a very large scale;
• the physical properties of these (size, mass, temperature, stability, etc.): Janus explains the formation and stability of galaxies, but also the observed effects of negative gravitational lens[116] due to negative mass conglomerates;
• the evolution of their distribution that can be observed by comparing the current distribution of these objects with the one they had in the past by observing the older periods in the history of the universe (i.e. as explained by simulations according to mainstream cosmology), and by further explaining the origin of the Great Repeller.

The Janus model approach is:

• The universe would not have experienced cosmic inflation;
• Dark matter and dark energy do not exist as components of our positive sheet of the universe;
• The structure of the universe is composed around lacuna (as seen from the positive sheet);
• The universe is isotropic in each sheet of the universe;
• The universe is spatially homogeneous in each metric (sheets of universe);
• The universe is not a single space-time continuum if one considers (wrongly) the two sheets as strictly separated; however, by antigravity these two sheets interact continuously. In addition, the natural (or artificial) mass inversion allows to move to the other sheet of universe;
• Physical constants would not be absolute constants. They would have varied jointly during the radiative era.

### Other criteria for the falsifiability of Janus model

#### Evidence by direct observation

In the future, a conclusive experiment of sudden "de-materialization" in the laboratory of a small quantity of matter by artificial inversion of the sign of its mass (and without equivalent energy release according to E=m*c²) would be a direct proof of the existence of the negative sheet of the universe.

#### Prediction on the acceleration of cosmic expansion

The ΛCDM model predicts an exponential acceleration of cosmic expansion.

The Janus model predicts an attenuation of this acceleration with the passing of time, tending towards a linear function of time.

### Are there any observations that ΛCDM model explains and that Janus model does not explain to date?

The ΛCDM model has had several predictive successes:

• the existence of the phenomenon of baryon acoustic oscillations, discovered in 2005
• the statistical calculation of weak gravitational lenses, which was first observed in 2000
• in the data collection carried out in 2015 by the Planck satellite, 7 peaks are observed in the power spectrum of the CMB temperature as a function of the angular scale (TT spectrum), 6 peaks in the crossed temperature-polarization (TE) spectrum, and 5 peaks in the polarization (EE) spectrum. All six free and related parameters of the ΛCDM model can be deduced from the TT spectrum alone. The TE and EE spectra can then be deduced with an accuracy of a few percent.

The Janus model has not yet been the subject of such calculations and numerical simulations.

## Perspectives on the application of the Janus model

### In astronautics: the non-impossibility of faster-than-light (FTL) interstellar travel

The first theoretical study of quantum mechanics on this subject dates back to 2007.[117]

The model finally considers the possibility of apparent faster-than-light interstellar travel with limited energy. The mechanism would involve an artificial version of the black hole natural inversion mass process.[64] The transferred vehicle would cruise along geodesics of the metric ${\displaystyle g_{\mu \nu }^{(-)}}$ where the speed of light is greater, and the distances shorter. The inverted particles of the ship and its passengers would have to appear at a relativistic speed in the new frame of reference through Lorentz contraction, in order for the energy to be conserved, with no acceleration. After mass inversion, a craft would go so fast that it could not slow down, but arriving at its destination, a new mass inversion would give back its former kinetic parameters, with no deceleration.[22]

A study onto this subject was presented in a scientific conference "Negative Energy states and Interstellar travel", October 2018;[118] the video of this intervention is available.[119]

The same principles and the same spacecraft could be used to move from one point on Earth to any other in a fraction of a second. It would require the same energy for the ship to travel 20 m or 20,000 km. The Janus model's main author explained that such a vessel could also remain motionless in the air and silent, levitating, as if it were apparently no longer subjected to Earth's gravity.

The Janus model's main author estimates that such a technology could be operational in a century or two.

### Other applications

If we limit ourselves to the one-way transfer without considering a return to our sheet of the universe, the Janus model's main author explained that the transfert principles could be used to instantly dispose of all types of waste, including radioactive elements with very long half-life periods generated in nuclear power plants, in a clean and cost-effective way.

## Scientific critiques of the Janus model

The number of scientific critiques expressed is low. The Janus model's main author has publicly testified on many occasions since the 1980s to the great lack of interest shown by most cosmology specialists who do not read his publications and who do not respond to his requests for seminars, whether in France or abroad. Nevertheless, the following can be cited as criticisms, in chronological order:

1) In August 2006, the French astrophysicist Alain Riazuelo published a short public commentary on the IAP website[120] criticising only the article "Twin Universes Cosmology" published in Astrophysics and Space Science in 1995. The tone used is very discourteous, and the author does not detail all his arguments. We also note the concluding sentence that takes on particular meaning if we look at the quantitative predictions (successes and failures) of the ΛCDM model over the past 30 years:

"No quantitative prediction, and therefore no comparison to current data. It was forgivable in 1995, but certainly not anymore. In cosmology, as elsewhere in astrophysics, we cannot claim to be serious if we do not make quantitative predictions."

Following Riazuelo's comment republished by a third party on the article Jean-Pierre Petit on french Wikipedia and Petit's own contributions that followed between August 14 and October 9, 2006 over 21 days, Jean-Pierre Petit was banned for life from the french Wikipedia project[121] by 16 french-speaking administrators.[122] Petit expressed his deep disagreement[123] and responded on his personal site and on Wikibuster,[124] whose URLs have since been blacklisted by the french Wikipedia.[125] On the scientific and sociological background, the Janus model's main author has published an open journal for more than 10 years[126] in which he details the answers he provided to Riazuelo and the management of his institution, in writing and then in video, concerning in particular his requests for scientific debate in public seminars,[127] which have remained systematically ignored by IAP, contrary to any rule of integrity and scientific ethics.[128][129][130][131]

2) Ellis' 2007 general critique of the variable speed of light in cosmology[132] does not contradict the Janus model.

Indeed, the latter meets all the conditions:[133]

(1) The Janus model defines the distance measurements that result from the coupled field equations, each of which is structured in a similar way to Einstein's field equation (without cosmological variables)

(2) the Janus model provides an alternative expression for the metric tensor in general relativity with the consideration of the second universe-sheet

(3) the Janus model does not contradict the Lorentz invariance during the radiative period when the speed of light varied in conjunction with the other fundamental constants of physics

(4) the Janus model does not modify the Maxwell equations during the radiative period when the speed of light has varied in conjunction with the other fundamental constants of physics

(5) the variation of the speed of light in conjunction with the other fundamental constants of physics is carried out in a manner consistent with all other established physical theories.

3) In the April 2017 issue 522 (p. 74-75) of La Recherche journal,[134] Luc Blanchet briefly mentions the principles of the Janus bimetric model, but without naming it: "Question: Is there not another way to obtain negative mass particles? L.B.: Absolutely, and this is possible in the strict case of general relativity, using an extension of general relativity in which the graviton - the mediating particle of gravitational interaction - has a mass. However, it happens that in a formulation of this theory, everything happens as if there were two different ways of measuring distances - two space-time or two "metrics". In each space-time, we can have particles, and since the two metrics behave differently (with a single coupling term between the two), particles from one space-time can appear to have a nagative mass when measured relative to the other space-time. So we have an antigravity effect." The Janus model's main author then commented on this article.[135]

4) "Versus?" channel on YouTube broadcast a 4-part series devoted to an attempt to evaluate the Janus model by a panel of specialists who wanted to remain anonymous even after the end of the series, and who were talking with the Janus model's main author through the channel's host. The diffusion took place over a period of 8 months between June 2017 and February 1, 2018 (i.e. before the publication of the publications from June to November 2018 on observational results, on the quantum mechanics of negative states, on the origin of CMB fluctuations), with totally divergent conclusions within the panel.[136]

François Cauneau, a physicist, professor at a french major engineering school and director of a research site, participated in the comments after the broadcast of the third episode, and published a favourable public opinion on the Janus model's main author's approach.[137]

5) In J.S. Farnes' article accepted on October 20, 2018, it reads:

"The introduction of bimetric models has allowed for extensions of general relativity with two different metrics (e.g. Hossenfelder 2008). One application of these models has been to explore cosmological theories with negative masses as a form of dark energy (Petit & D'Agostini 2014), however such theories have remained incompatible with observations."[138]

The Janus model papers of November 2014[139] and June 2018[140] on the observations explained by the Janus model are not cited and have remained ignored by the reading committee of Farnes' article, too.

6) On December 8, 2018, Marc Lachièze-Rey, a specialist in general relativity at the CNRS, answered a question on the Janus model during a public conference he led at Marseille.[141] He says he studied Petit's published work 20 years ago, and that he cannot comment on Petit's more recent work. He says he found calculation errors at that time but does not specify which ones. He says "that this idea of negative mass should be included in the rest of physics, that it should be compatible with quantum physics. Petit's work has been read by CNRS committees, so far nothing interesting seems to come out of it". He is unaware that these results have been published.[142] He says that the scientific community does not refuse to debate with Petit. In this regard, it is useful to mention two complementary points:

• Jean-Pierre Luminet's written public comment in response to a reminder from Petit: "I have never "categorically" refused to meet you, I did not answer you because I was far too overwhelmed (which is not the same thing), it is also true that I did not see the importance of the issue as you did." [143]
• the electronic petition launched in August 2018: "Right to debate for a new cosmology"[144]

### Ongoing debate between specialists

a) The article published in 2015 [145] presents the derivation of a Lagrangian. The article of 2018,[146] dealing with Janus' constraints, presents in reference an article by Sabine Hossenfelder to obtain a Lagrangian, without referring to the derivation published in 2015.

In the comments section of Janus video Episode 25, a user with the login "Frédéric Henry-Couannier" [147] writes that the 2015 demonstration would include some errors.

b) Another but linked objection was argued by F. Henry-Couannier to the antigravity theory of the Janus model in the case of gravitational wave emission in the article published on his own website : "Consistency of JP. P and S.H Janus anti-gravity theories".[148]

## Communication on the Janus model

The contribution of Jean-Marie Souriau's geometry was decisive for the development of the Janus model. Janus' diffusion is therefore also a diffusion of Souriau's works. He thanked Jean-Pierre Petit who convinced him to make his work available on the Internet.[149]

An official letter from the Ministry of Foreign Affairs of the Russian Federation dated 9-19-2018 and reproduced in the book by Jean-Claude Bourret and Jean-Pierre Petit, "Contacts Cosmiques : Jusqu'où peut-on penser trop loin", Guy Trédaniel ed., October 2018 (p. 398), states that "Mr Jean-Pierre Petit's research in astrophysics, theoretical physics and fluid mechanics is well known in the Russian scientific community. I am pleased to confirm our full support for his entire initiative to establish mutually beneficial cooperation with interested partners in Russia."

### Conferences on the Janus model

The papers given by Jean-Pierre Petit at scientific conferences specializing in cosmology are available in the Bibliography section.

On March 8, 2006, Petit gave a lecture on the geometry of the universe at the Académie lorraine des sciences.

Vincent Borrelli led a conference on 21 October 2013, organised by the mathematics department of INSA Lyon, entitled: "Quelle est la forme de l'univers?".[150] Jean-Pierre Petit's science comic book Le Géométricon is cited as a reference.

The Janus model's main author has given lectures to the general public on the Janus model:

• conference in Paris on the barge "Le Vaisseau Fantôme" on Wednesday, January 15, 2003 for the release of the book "OVNIS et armes secrètes américaines", part of which deals with the twin universes theory
• on Friday, October 1, 2016 at the premises of the Andromeda Astronomy Association in Marseille[151][152]
• communication at the UFO601 Congress on Saturday, October 8, 2016 in Montreal (via Skype)[153]
• on March 24, 2018 in Hyères, lecture entitled "Les voyages interstellaires : C'est pour demain !!!! ». Former American colonel Robert Salas made a second lecture.[154]
• on 27 October 2018 in Hyères, lecture entitled "OVNIS : une science venue d'ailleurs".[155]

### Teaching the Janus model through videos

The Janus model's main author has been creating, editing, subtitling and publishing on YouTube since January 2017 a series of educational videos for teaching purposes explaining the development of the Janus cosmological model. This is a complete open and massive online course (or MOOC).

Each episode of his videos, although lasting about an hour and dealing with very complex subjects, have an audience of several tens of thousands of views (figures recorded in November 2018). The episodes of the JANUS series are:

• JANUS 1 (eng. subtitles):[156] Aristotle, Ptolemy, Poliedric world[157]
• (fr) JANUS 2: Tycho Brahé, Kepler
• JANUS 3 (eng. subtitles) : Galileo the heretic
• (fr) JANUS 4: Newton and Laplace
• (fr) JANUS 5: The bankruptcy of common sense
• (fr) JANUS 6: The EPR paradox
• (fr) JANUS 7: The non-existence of the void
• JANUS 8 (eng. subtitles) : Special Relativity 1st part
• (fr) JANUS 9: Special Relativity, Part 2
• (fr) JANUS 10: General Relativity
• (fr) JANUS 11: The contemporary crisis of cosmology
• (fr) JANUS 12: The problem of primordial antimatter
• JANUS 13 (eng. subtitles) : Time reversal. Groups of people
• JANUS 14 (eng. subtitles): Measurement of the curvature of negachips
• JANUS 15 (eng. subtitles): Two field equations instead of one
• JANUS 16 (eng. subtitles): The Janus model explains cosmic acceleration
• JANUS 17 (eng. subtitles): The only coherent explanation of the Great Repeller
• JANUS 18 (eng. subtitles): We explain why the primitive universe is so homogeneous
• JANUS 19 (eng. subtitles): the speed of light must be infinite at the time of the Big Bang
• JANUS 20 (eng. subtitles): An observational test for this model
• JANUS 21 (eng. subtitles): Theoretical physicists no longer believe in dark matter
• JANUS 22 - 1 (eng. subtitles) : Black hole VS mass inversion. Negative energy states[158]
• (fr) JANUS 22 - 2: Geometry, the part of the real and the imaginary
• JANUS 22 - 3 (eng. subtitles): Schwarzschild's forgotten solution
• JANUS 22 - 4 (eng. subtitles): Contestation in good standing of the black hole model
• JANUS 22 - 5 (eng. subtitles): Gravitational waves. Fusion of neutron stars. Mass reversal.
• JANUS 23 (eng. subtitles): The alternative to the cosmic inflation model[159]
• (fr) JANUS 24 : How the system of coupled field equations is constructed. The geometric context of Janus[160]
• (en) JANUS 24 : The Black Hole in the JANUS COSMOLOGICAL MODEL[161]
• JANUS 25 (eng. subtitles): A model validated by two publications in high-level scientific journals
• (fr) JANUS 26 : quantum physics and negative energy states. Refutation of Farnes critique.

### Teaching bimetric cosmology through science comic books

The Janus model's main author created the script, drew and had published a series of science comic books for teaching purposes (The adventures of Archibald Higgins) explaining in particular astrophysics, relativity and cosmology with its geometric subtleties that are used in the Janus model. These comic books of a new kind created by the Janus model's main author (the science comic book) has been published since 1980. All the books are now distributed free of charge by the association Savoirs sans Frontieres.[162] They are translated by volunteers into 39 languages (figure recorded in December 2018).

The books on astrophysics and cosmology topics are: Geometricon, Everything is relative, Topo the world, Chronologicon, The black hole, The twin universe,[163][164] Faster than light, A thousand billion suns, Big Bang, Cosmic Story.

### Teaching astrophysics and the bimetric model using software

On April 11, 2004, the Janus model's main author launched the collaborative project EPISTEMOTRON, attempting to perform distributed computing simulations involving an increasing number of personal computers, in the same way as Seti@home, to simulate the behavior of an N-body system. This project focused entirely on the simulation of cosmological and astrophysical phenomena of all kinds within a "twin dynamics" and corresponding to work by the Janus model's main author, whose first computer simulation work dates back to 1992. Several tutorials were published online[165] and then this project was frozen at the end of May following a split of the group.

In 2014 the Janus model's main author published two preprints on computing simulations results :

• Bimetric theory : the only model which explains the nature of spiral structure, as the result of dynamical friction between galaxy and surrounding negative matter, providing good looking barred spiral, stable over 20 turns, (Jul. 2014) [166]
• Very Large Structure numerical simulation in a compact computational space, (Jul. 2014) [167]

As a result of the increase in PC power, this type of simulation begins to be implemented in 2018 on a single computer, with 100,000 to 3 million mass points,[168] based on Newtonian models modified with negative masses (with nVIDIA's CUDA[169] toy model software) that are not as sophisticated as Janus model equations.

### Popularization of the Janus model through books for the general public

The Janus model's main author has written several books that aim in particular to popularize the main principles of the Janus model during its construction, by mixing them with anecdotes and other research or personal subjects. Rich technical appendices with reproductions of some of his scientific articles published in peer-reviewed journals are systematically available.

• (fr) Enquête sur les OVNI: voyage aux frontières de la science - preface by Jacques Benveniste, ed. Albin Michel, 1990 (ISBN 9782402507875)
• (fr) Enquête sur des extra-terrestres qui sont déjà parmi nous : le mystère des Ummites, ed. Albin Michel, 1991 and 1993 (ISBN 2226055150)
• (fr) Le mystère des Ummites : une science venue d'une autre planète, ed. Albin Michel, 1995 ( ISBN 2226078452 )
• (fr) (foreword by Jean-Claude Pecker), On a perdu la moitié de l'Univers, Paris, Albin Michel, 1997 (ISBN 2226093931) and 2001 (ISBN 978-2012789357)
• (fr) a digital edition dated September 1999 entitled Le versant obscur de l'Univers[170] was distributed by the association Savoirs sans Frontieres.
• (en) It is translated into English as The Dark side of the universe: Outposts and perspectives of astrophysics and contemporary cosmology and dated June 1998.[171]
• (fr) OVNI, le message, published by the author, 2008, ( ISBN 978-2918564003 ) (Annex 1 with 17 p.)
• (fr) OVNI et armes secrètes américaines : l'extraordinaire témoignage d'un scientifique, ed. Albin Michel, 2003 and 2014
• (fr) with Jean-Claude Bourret, OVNI : L'extraordinaire découverte, Guy Trédaniel ed., February 2017, ( ISBN 978-2813213907 )[172]
• (fr) with Jean-Claude Bourret, Contacts Cosmiques : Jusqu'où peut-on penser trop loin, Guy Trédaniel ed., October 2018, ( ISBN 978-2-8132-1811-7)

### The Janus model in politics and philosophy

In his paper at the third Euro-BRICS seminar (Cannes, Sept. 27-28, 2012)[173] organized by LEAP and MGIMO University in Moscow, Bruno Paul argued in favour of the interest in building new scientific networks between Europe and Russia by developing the case of the Janus model's main author work in cosmology.[174]

In his philosophical essay Qu'est-ce que le temps ? published in 2015, Michele Angelo Murgia presents a kind of inventory of knowledge about the fragile and evasive concept of time, drawing on recent contributions from physicists such as Stephen Hawking, Leonard Susskind, Julian Barbour, Carlo Rovelli, Étienne Klein, Lee Smolin and Jean-Pierre Petit.[175]

In his essay on metaphysics and formal logic, La signature du quaternaire, published in 2018, Bruno Paul mentions the Janus cosmological model (p. 100).[176]

### The Janus model in the media

The Janus model's main author has been interviewed several times by french media about his work in cosmology, the link with Sakharov's work and interstellar travel:

• intervention in 1981 by the Janus model's main author in Temps X show on TF1 to present the release of the science comic The Black Hole (The adventures of Archibald Higgins) by Belin Editions[177]
• report on French television (date unknown, perhaps in the late 1980s)[178]
• interview by Quebec television in Montreal in 1991 [179]
• the public television channel TF1 broadcast on Monday 13 January 2003 a show hosted by Bernard Tapie, with the following guests: Jean-Pierre Petit (for the publication of the book "OVNI et armes secrètes américaines"), François Louange, Edmond Campagnac, Pierre Beake, Michel Chevalet, René Pellat, Couturier, and Gildas Bourdais.[180]
• The Janus model's main author is Laurent Ruquier's guest on his show "On a tout essayé" on France 2 on Friday 21 March 2003, for the release of the book "OVNI et armes secrètes américaines".
• The Janus model's main author is Damien Hammouchi's guest on his show "La grande soirée - spéciale OVNI" on Direct8 in May 2009, for the release of the book "OVNI, le message"[181]
• interview by RFI on August 28, 2008 in the "Têtes Chercheuses" program of RFI's Sciences division: "Jean-Pierre Petit, chercheur au CNRS, à la recherche des ovnis" [182]
• a series of 9 DVDs entitled "Interviews with Jean-Pierre Petit" was published and marketed by Enquête et Débat in 2011. DVDs #8 and 9 entitled "CNES, Cosmology and Astrophysics" are Petit's place of explanations on the theory of twin universes and interstellar travel. Each DVD lasts from 1h30 to 2h approximately.
• intervention by Jean-Claude Bourret in Thierry Ardisson's "Salut les Terriens" program on May 4, 2017 for the publication of the book "OVNI: l'extraordinaire découverte", a large part of which is devoted to the Janus model
• Sciences et Conscience n°21 - Broadcast on Thursday, October 5, 2017 on Radio Agora - Cote d'azur[183]
• The "Versus?" channel on YouTube broadcast a 4-part series devoted to an attempt of a scientific critique of the Janus model by a panel of anonymous specialists who were asking questions to the Janus model's main author through the host. The series was broadcast between June 2017 and February 1, 2018[184]
• The "ThinkerView" channel on YouTube broadcast a 2-hour interview devoted to the Janus model in June 2018[185] (431,000 views recorded in December 2018)
• The "Nuréa TV" channel on YouTube broadcast a 3-hour interview (in 2 episodes) devoted to the Janus model in June 2018 [186]
• The private channel "Bob dit toute la vérité" (BTLV) broadcast an interview with the Janus model's main author on 5 November 2018, about the publishing of the book "Contacts cosmiques", which partly deals with the Janus model.

Articles in french journals related to the Janus model's main author's work on cosmology are as follows:

• L'Est Républicain published an article on 8 March 2006, entitled "Je suis incombustible", on the occasion of the Janus model's main author's conference on the geometry of the universe given at the Académie lorraine des sciences.[187]
• interview with the newspaper La Voix de la Russie / SputnikNews in 2013:
• Part 1 "A new vision of the universe inspired by Sakharov", 10/23/2013[188]
• Part 2 "Interstellar travel and extraterrestrial contacts", 11/1/2013[189]
• interview with the newspaper La Voix de la Russie / SputnikNews in 2014, entitled "Black holes do not exist":
• Part 1 on 6/30/2014 [190]
• Part 2 on 7/1/2014 [191]
• Nexus published the article "Ovni : changement de figure avec Janus"[192] on the occasion of an interview in their magazine (No. 109, March–April 2017, p. 94) with the authors of the book "OVNI: L'extraordinaire découverte" published by Guy Trédaniel in February 2017 which half of its content deals with Janus model.[193] The newspaper Le Soir,[194] the websites Aeromorning.com,[195] notre-siècle.com,[196] as well as L'Union (4/9/2017)[197] did the same.

## Bibliography of studies on the Janus model since its inception

Complete bibliography of A. D. Sakharov's work in cosmology and astrophysics, including :

• (en) Andreï Sakharov, CP violation and baryonic asymmetry of the Universe. ZhETF Pis'ma 5, 32-35, 1967, Transl. JETP Lett. 5 : 24-27 (1967)
• (en) Andreï Sakharov, A multisheet Cosmological Model, Preprint of the Institute for Applied Mathematics of the USSR Academy of Sciences, 7, (1970)
• (en) Andreï Sakharov, Topological structure of elementary particles and CPT asymmetry, "Problems in Theoretical Physics", in memory of I.E.Tamm, Nauka, Moscow, (1972), p. 243-247
• (en) A.D.Sakharov , The baryonic asymmetry of the Universe, ZhETF Pis’ma 76 : 1172 (1979) ; Transl. JETP 49 : 594 (1979)
• (en) Andreï Sakharov, Cosmological Model of the Universe with a time-vector inversion. ZhETF 79, 689-693 (1980), Transl. Sov. Phys. JETP 52 : 349-351, (1980) ;
• (en) A.D.Sakharov, Collected scientific works, (1982) ; transl. in 1984 in french: Œuvres Scientifiques - foreword by Professor Louis Michel de l'Académie des Sciences, éditions Anthropos.

Complete bibliography of J.-P. Petit's work in cosmology and astrophysics:[198] publications in scientific journals, pre-prints, books, videos, DVDs, science comics, conferences, including :

• (foreword by Jean-Claude Pecker), On a perdu la moitié de l'Univers, Paris, Albin Michel, 1997 (ISBN 2226093931)
• reprint: On a perdu la moitié de l'Univers, Hachette Littérature, "Pluriel" series, January 2, 2001 (ISBN 978-2012789357)
• a digital edition dated September 1999 entitled Le versant obscur de l'Univers[170] was distributed by the association Savoirs sans Frontieres.
• It is translated into English as The Dark side of the universe: Outposts and perspectives of astrophysics and contemporary cosmology and dated June 1998.[171]
• conference "Quelle géometrie pour l'univers?", Académie lorraine des sciences, Nancy, March 8, 2006[102]
• (en) J.-P. Petit, G. D'Agostini, Negative mass hypothesis and the nature of dark energy. Astrophysics and Space Science (2014) 354 : 611-615 ; DOI 10.1007/s10509-014-2106-5
• (en) J.-P. Petit, G. D'Agostini, Cosmological bimetric model with interacting positive and negative masses and two different speeds of light in agreement with the observed acceleration of the Universe. Modern Physics Letters A. Vol. 29, No. 34, 1450182, Nov. 2014; DOI: 10.1142/S021773231450182X
• (en) J.-P. Petit, G. D'Agostini, Constraints on Janus Cosmological model from recent observations of supernovae type Ia, Astrophysics And Space Science, Vol. 363, Issue 7, article id. 139, 7 pp., accepted 3 June 2018, DOI: 10.1007 / s10509-018-3365-3
• (en) J.-P. Petit, N. Debergh, G. D'Agostini, Negative Energy states and Interstellar travel, Oct. 2018, conference paper presented at the 2018 Estes Park, Advanced Propulsion Workshop, Colorado, September 14, 2018
• (en) J.-P. Petit, The JANUS Cosmological Model and the Fluctuations of the CMB, Progress in Physics, vol.14, issue 4, October 2018, p. 226-229
• (en) N. Debergh, J.-P. Petit, G. D'Agostini, On evidence for negative energies and masses in the Dirac equation through a unitary time-reversal operator; accepted Nov. 2018, Journal of Physics Communications, vol.2, 11

## Theory of massive bigravity

Since 2010 there has been renewed interest in bigravity after the development by Claudia de Rham, Gregory Gabadadze, and Andrew Tolley (dRGT) of a healthy theory of massive gravity.[199] Massive gravity is a bimetric theory in the sense that nontrivial interaction terms for the metric ${\displaystyle g_{\mu \nu }}$ can only be written down with the help of a second metric, as the only nonderivative term that can be written using one metric is a cosmological constant. In the dRGT theory, a nondynamical "reference metric" ${\displaystyle f_{\mu \nu }}$ is introduced, and the interaction terms are built out of the matrix square root of ${\displaystyle g^{-1}f}$.

In dRGT massive gravity, the reference metric must be specified by hand. One can give the reference metric an Einstein-Hilbert term, in which case ${\displaystyle f_{\mu \nu }}$ is not chosen but instead evolves dynamically in response to ${\displaystyle g_{\mu \nu }}$ and possibly matter. This massive bigravity was introduced by Fawad Hassan and Rachel Rosen[3] as an extension of dRGT massive gravity.

The dRGT theory is crucial to developing a theory with two dynamical metrics because general bimetric theories are plagued by the Boulware-Deser ghost, a possible sixth polarization for a massive graviton.[200] The dRGT potential is constructed specifically to render this ghost nondynamical, and as long as the kinetic term for the second metric is of the Einstein-Hilbert form, the resulting theory remains ghost-free.[3]

The action for the ghost-free massive bigravity is given by[201]

${\displaystyle S=-{\frac {M_{g}^{2}}{2}}\int d^{4}x{\sqrt {-g}}R(g)-{\frac {M_{f}^{2}}{2}}\int d^{4}x{\sqrt {-f}}R(f)+m^{2}M_{g}^{2}\int d^{4}x{\sqrt {-g}}\displaystyle \sum _{n=0}^{4}\beta _{n}e_{n}(\mathbb {X} )+\int d^{4}x{\sqrt {-g}}{\mathcal {L}}_{\mathrm {m} }(g,\Phi _{i}).}$

As in standard general relativity, the metric ${\displaystyle g_{\mu \nu }}$ has an Einstein-Hilbert kinetic term proportional to the Ricci scalar ${\displaystyle R(g)}$ and a minimal coupling to the matter Lagrangian ${\displaystyle {\mathcal {L}}_{\mathrm {m} }}$, with ${\displaystyle \Phi _{i}}$ representing all of the matter fields, such as those of the Standard Model. An Einstein-Hilbert term is also given for ${\displaystyle f_{\mu \nu }}$. Each metric has its own Planck mass, ${\displaystyle M_{g}}$ and ${\displaystyle M_{f}}$. The interaction potential is the same as in dRGT massive gravity. The ${\displaystyle \beta _{i}}$ are dimensionless coupling constants and ${\displaystyle m}$ (or specifically ${\displaystyle \beta _{i}^{1/2}m}$) is related to the mass of the massive graviton. This theory propagates seven degrees of freedom, corresponding to a massless graviton and a massive graviton (although the massive and massless states do not align with either of the metrics).

The interaction potential is built out of the elementary symmetric polynomials ${\displaystyle e_{n}}$ of the eigenvalues of the matrices ${\displaystyle \mathbb {K} =\mathbb {I} -{\sqrt {g^{-1}f}}}$ or ${\displaystyle \mathbb {X} ={\sqrt {g^{-1}f}}}$, parametrized by dimensionless coupling constants ${\displaystyle \alpha _{i}}$ or ${\displaystyle \beta _{i}}$, respectively. Here ${\displaystyle {\sqrt {g^{-1}f}}}$ is the matrix square root of the matrix ${\displaystyle g^{-1}f}$. Written in index notation, ${\displaystyle \mathbb {X} }$ is defined by the relation

${\displaystyle X^{\mu }{}_{\alpha }X^{\alpha }{}_{\nu }=g^{\mu \alpha }f_{\nu \alpha }.}$

The ${\displaystyle e_{n}}$ can be written directly in terms of ${\displaystyle \mathbb {X} }$ as

{\displaystyle {\begin{aligned}e_{0}(\mathbb {X} )&=1,\\e_{1}(\mathbb {X} )&=[\mathbb {X} ],\\e_{2}(\mathbb {X} )&={\frac {1}{2}}\left([\mathbb {X} ]^{2}-[\mathbb {X} ^{2}]\right),\\e_{3}(\mathbb {X} )&={\frac {1}{6}}\left([\mathbb {X} ]^{3}-3[\mathbb {X} ][\mathbb {X} ^{2}]+2[\mathbb {X} ^{3}]\right),\\e_{4}(\mathbb {X} )&=\operatorname {det} \mathbb {X} ,\end{aligned}}}

where brackets indicate a trace, ${\displaystyle [\mathbb {X} ]\equiv X^{\mu }{}_{\mu }}$. It is the particular antisymmetric combination of terms in each of the ${\displaystyle e_{n}}$ which is responsible for rendering the Boulware-Deser ghost nondynamical.

## Rosen's bigravity (1940)

In general relativity (GR), it is assumed that the distance between two points in spacetime is given by the metric tensor. Einstein's field equation is then used to calculate the form of the metric based on the distribution of energy and momentum.

Rosen (1940)[202][203] has proposed that at each point of space-time, there is a Euclidean metric tensor ${\displaystyle \gamma _{ij}}$ in addition to the Riemannian metric tensor ${\displaystyle g_{ij}}$ . Thus at each point of space-time there are two metrics:

${\displaystyle 1.[[User:WikiMasterBot2|WikiMasterBot2]]([[Usertalk:WikiMasterBot2|talk]])00:19,21January2019(CET)ds^{2}=g_{ij}dx^{i}dx^{j}}$
${\displaystyle 2.[[User:WikiMasterBot2|WikiMasterBot2]]([[Usertalk:WikiMasterBot2|talk]])00:19,21January2019(CET)d\sigma ^{2}=\gamma _{ij}dx^{i}dx^{j}}$

The first metric tensor, ${\displaystyle g_{ij},}$ describes the geometry of space-time and thus the gravitational field. The second metric tensor, ${\displaystyle \gamma _{ij},}$refers to the flat space-time and describes the inertial forces. The Christoffel symbols formed from ${\displaystyle g_{ij}}$ and ${\displaystyle \gamma _{ij}}$ are denoted by ${\displaystyle \{_{jk}^{i}\}}$ and ${\displaystyle \Gamma _{jk}^{i}}$ respectively.

Since the difference of two connections is a tensor, one can define the tensor field ${\displaystyle \Delta _{jk}^{i}}$ given by:

${\displaystyle \Delta _{jk}^{i}=\{_{jk}^{i}\}-\Gamma _{jk}^{i}\qquad (1)}$

Two kinds of covariant differentiation then arise: ${\displaystyle g}$-differentiation based on ${\displaystyle g_{ij}}$ (denoted by a semicolon), and covariant differentiation based on ${\displaystyle \gamma _{ij}}$ (denoted by a slash). Ordinary partial derivatives are represented by a comma. Let ${\displaystyle R_{ijk}^{h}}$ and ${\displaystyle P_{ijk}^{h}}$ be the Riemann curvature tensors calculated from ${\displaystyle g_{ij}}$ and ${\displaystyle \gamma _{ij},}$ respectively. In the above approach the curvature tensor ${\displaystyle P_{ijk}^{h}}$ is zero, since ${\displaystyle \gamma _{ij}}$ is the flat space-time metric.

A straightforward calculation yields the Riemann curvature tensor

${\displaystyle R_{ijk}^{h}=P_{ijk}^{h}-\Delta _{ij/k}^{h}+\Delta _{ik/j}^{h}+\Delta _{mj}^{h}\Delta _{ik}^{m}-\Delta _{mk}^{h}\Delta _{ij}^{m}=-\Delta _{ij/k}^{h}+\Delta _{ik/j}^{h}+\Delta _{mj}^{h}\Delta _{ik}^{m}-\Delta _{mk}^{h}\Delta _{ij}^{m}}$

Each term on right hand side is a tensor. It is seen that from GR one can go to the new formulation just by replacing {:} by ${\displaystyle \Delta }$ and ordinary differentiation by covariant ${\displaystyle \gamma }$-differentiation, ${\displaystyle {\sqrt {-g}}}$ by ${\displaystyle {\sqrt {\tfrac {g}{\gamma }}},}$ integration measure ${\displaystyle d^{4}x}$ by ${\displaystyle {\sqrt {-\gamma }}\,d^{4}x,}$ where ${\displaystyle g=\det(g_{ij}),\gamma =det(\gamma _{ij})}$ and ${\displaystyle d^{4}x=dx^{1}dx^{2}dx^{3}dx^{4}}$. Having once introduced ${\displaystyle \gamma _{ij}}$ into the theory, one has a great number of new tensors and scalars at one's disposal. One can set up other field equations other than Einstein's. It is possible that some of these will be more satisfactory for the description of nature.

The geodesic equation in bimetric relativity (BR) takes the form

${\displaystyle {\frac {d^{2}x^{i}}{ds^{2}}}+\Gamma _{jk}^{i}{\frac {dx^{j}}{ds}}{\frac {dx^{k}}{ds}}+\Delta _{jk}^{i}{\frac {dx^{j}}{ds}}{\frac {dx^{k}}{ds}}=0\qquad (2)}$

It is seen from equations (1) and (2) that ${\displaystyle \Gamma }$ can be regarded as describing the inertial field because it vanishes by a suitable coordinate transformation.

The quantity ${\displaystyle \Delta ,}$ being a tensor, is independent of any coordinate system and hence may be regarded as describing the permanent gravitational field.

Rosen (1973) has found BR satisfying the covariance and equivalence principle. In 1966, Rosen showed that the introduction of the space metric into the framework of general relativity not only enables one to get the energy momentum density tensor of the gravitational field, but also enables one to obtain this tensor from a variational principle. The field equations of BR derived from the variational principle are

${\displaystyle K_{j}^{i}=N_{j}^{i}-{\frac {1}{2}}\delta _{j}^{i}N=-8\pi \kappa T_{j}^{i}\qquad (3)}$

where

${\displaystyle N_{j}^{i}={\frac {1}{2}}\gamma ^{\alpha \beta }(g^{hi}g_{hj/\alpha })_{/\beta }}$

or

{\displaystyle {\begin{aligned}N_{j}^{i}&={\frac {1}{2}}\gamma ^{\alpha \beta }\left\{\left(g^{hi}g_{hj,\alpha }\right)_{,\beta }-\left(g^{hi}g_{mj}\Gamma _{h\alpha }^{m}\right)_{,\beta }-\gamma ^{\alpha \beta }\left(\Gamma _{j\alpha }^{i}\right)_{,\beta }+\Gamma _{\lambda \beta }^{i}\left[g^{h\lambda }g_{hj,\alpha }-g^{h\lambda }g_{mj}\Gamma _{h\alpha }^{m}-\Gamma _{j\alpha }^{\lambda }\right]-\right.\\&\qquad \Gamma _{j\beta }^{\lambda }\left[g^{hi}g_{h\lambda ,\alpha }-g^{hi}g_{m\lambda }\Gamma _{h\alpha }^{m}-\Gamma _{\lambda \alpha }^{i}\right]+\Gamma _{\alpha \beta }^{\lambda }\left.\left[g^{hi}g_{hj,\lambda }-g^{hi}g_{mj}\Gamma _{h\lambda }^{m}-\Gamma _{j\lambda }^{i}\right]\right\}\end{aligned}}}
${\displaystyle N=g^{ij}N_{ij},\qquad \kappa ={\sqrt {\frac {g}{\gamma }}},}$

and ${\displaystyle T_{j}^{i}}$ is the energy-momentum tensor.

The variational principle also leads to the relation

${\displaystyle T_{j;i}^{i}=0.}$

Hence from (3)

${\displaystyle K_{j;i}^{i}=0,}$

which implies that in a BR, a test particle in a gravitational field moves on a geodesic with respect to ${\displaystyle g_{ij}.}$

It is found that the BR and GR theories differ in the following cases:

• propagation of electromagnetic waves
• the external field of a high density star
• the behaviour of intense gravitational waves propagating through a strong static gravitational field.

The predictions of gravitational radiation in Rosen's theory have been shown to be in conflict with observations of the Hulse–Taylor binary pulsar.[5]

## References

1. Clifton, Timothy; Pedro G. Ferreira; Antonio Padilla; Constantinos Skordis (2012). "Modified Gravity and Cosmology". Physics Reports. 513 num.3 (1): 1–189. arXiv:1106.2476. Bibcode:2012PhR...513....1C. doi:10.1016/j.physrep.2012.01.001.
2. Rosen, Nathan (1940), "General Relativity and Flat Space. I", Phys. Rev., 57 (2): 147–150, Bibcode:1940PhRv...57..147R, doi:10.1103/PhysRev.57.147
3. Hassan, S.F.; Rosen, Rachel A. (2012). "Bimetric Gravity from Ghost-free Massive Gravity". JHEP. 1202 (2): 126. arXiv:1109.3515. Bibcode:2012JHEP...02..126H. doi:10.1007/JHEP02(2012)126.
4. Rosen, Nathan (1973), "A bi-metric Theory of Gravitation", Gen. Rel. Grav., 4 (6): 435–447, Bibcode:1973GReGr...4..435R, doi:10.1007/BF01215403
5. The New Physics, Paul Davies, 1992, 526 pages, web: Books-Google-ak.
6. "Nathan Rosen — The Man and His Life-Work", Technion.ac.il, 2011, web: Technion-rosen.
7. M. Milgrom, Matter and twin matter in bimetric MOND, Monthly Notices Roy. Astronomical Soc., 405, 2, (2010), 1129–1139 https://academic.oup.com/mnras/article-abstract/405/2/1129/1182555
8. Zyga, Lisa (21 September 2017). "Gravitational waves may oscillate, just like neutrinos". Phys.org. Omicron Technology Limited.
9. Akrami, Yashar; Koivisto, Tomi S.; Sandstad, Marit (2013). "Accelerated expansion from ghost-free bigravity: a statistical analysis with improved generality". JHEP. 1303 (3): 099. arXiv:1209.0457. Bibcode:2013JHEP...03..099A. doi:10.1007/JHEP03(2013)099.
10. Akrami, Yashar; Hassan, S.F.; Könnig, Frank; Schmidt-May, Angnis; Solomon, Adam R. (2015). "Bimetric gravity is cosmologically viable". Physics Letters B. 748: 37–44. arXiv:1503.07521. Bibcode:2015PhLB..748...37A. doi:10.1016/j.physletb.2015.06.062.
11. Henry-Couannier, F. (30 April 2005). "Discrete symmetries and general relativity, the dark side of gravity". International Journal of Modern Physics A. 20 (11): 2341–2345. arXiv:gr-qc/0410055. Bibcode:2005IJMPA..20.2341H. doi:10.1142/S0217751X05024602.
12. Hossenfelder, S. (15 August 2008). "A Bi-Metric Theory with Exchange Symmetry". Physical Review D. 78 (4): 044015. arXiv:0807.2838. Bibcode:2008PhRvD..78d4015H. doi:10.1103/PhysRevD.78.044015.
13. Hossenfelder, Sabine (June 2009). Antigravitation. 17th International Conference on Supersymmetry and the Unification of Fundamental Interactions. Boston: American Institute of Physics. arXiv:0909.3456. doi:10.1063/1.3327545.
14. Petit, J.-P.; d'Agostini, G. (10 November 2014). "Cosmological bimetric model with interacting positive and negative masses and two different speeds of light, in agreement with the observed acceleration of the Universe" (PDF). Modern Physics Letters A. 29 (34): 1450182. Bibcode:2014MPLA...2950182P. doi:10.1142/S021773231450182X.
15. Petit, J.-P. (12 November 2016). "Presentation for the public at large of the Janus Cosmological Model" (PDF). Savoir Sans Frontières.
16. Petit, J.-P. (2018-12-12). "Articles by J.P. Petit in INSPIRE". inspirehep.net.
18. Petit, J.-P. (2008). The Twin Universe (PDF). Savoir Sans Frontières.
19. Petit, J.-P. (2008). The Twin Universe: Scientific Appendix (PDF). Savoir Sans Frontières.
20. Petit, J.-P. (2008). Faster Than Light (PDF). Savoir Sans Frontières.
21. Petit, J.-P. (2017). The Janus Cosmological Model (English subtitles). YouTube.
22. Petit, J.-P.; d'Agostini, G. (10 November 2014). "Cosmological bimetric model with interacting positive and negative masses and two different speeds of light, in agreement with the observed acceleration of the Universe" (PDF). Modern Physics Letters A. 29 (34): 1450182. Bibcode:2014MPLA...2950182P. doi:10.1142/S021773231450182X.
23. In 2007, Janus model's main author enlisted a restricted group of mathematicians and geometers working in the field of mathematical physics through functional analysis and focusing on the falsifiability of theories with respect to observations. The members share their work at the International Meeting on Variational Techniques, a workshop originally created by Jean-Marie Souriau in 1950.
24. Petit, J.-P.; d'Agostini, G. (August 2007). Bigravity as an interpretation of the cosmic acceleration. International Meeting on Variational Techniques. Le Mont-Dore (France). arXiv:0712.0067. Bibcode:2007arXiv0712.0067P.
25. Petit, J.-P.; d'Agostini, G. (August 2007). Bigravity : A bimetric model of the Universe. Positive and negative gravitational lensings. International Meeting on Variational Techniques. Le Mont-Dore (France). arXiv:0801.1477. Bibcode:2008arXiv0801.1477P.
26. Petit, J.-P.; d'Agostini, G. (August 2007). Bigravity: a bimetric model of the Universe with variable constants, including VSL (variable speed of light). International Meeting on Variational Techniques. Le Mont-Dore (France). arXiv:0803.1362. Bibcode:2008arXiv0803.1362P.
27. Foot, R.; Volkas, R. R. (1 December 1995). "Neutrino physics and the mirror world: How exact parity symmetry explains the solar neutrino deficit, the atmospheric neutrino anomaly, and the LSND experiment" (PDF). Physical Review D. 52 (11): 6595–6606. arXiv:hep-ph/9505359. Bibcode:1995PhRvD..52.6595F. doi:10.1103/PhysRevD.52.6595.
28. Berezhiani, Zurab G.; Mohapatra, Rabindra N. (1 December 1995). "Reconciling present neutrino puzzles: Sterile neutrinos as mirror neutrinos". Physical Review D. 52 (11): 6607–6611. arXiv:hep-ph/9505385. Bibcode:2001PhLB..503..355F. doi:10.1016/S0370-2693(01)00228-3.
29. Okun, Lev B. (2007). "Mirror particles and mirror matter: 50 years of speculation and search". Physics-Uspekhi. 50 (4): 380–389. arXiv:hep-ph/0606202. Bibcode:2007PhyU...50..380O. doi:10.1070/PU2007v050n04ABEH006227.
30. Sakharov, A. D. (January 1967). "Нарушение СР–инвариантности, С–асимметрия и барионная асимметрия Вселенной". Pi'sma ZhÉTF (in Russian). 5 (1): 32–35.CS1 maint: Unrecognized language (link) Translated as: Sakharov, A. D. (January 1967). "Violation of CP invariance, C asymmetry, and baryon asymmetry of the universe" (PDF). JETP Letters. 5 (1): 24–26. Republished as Sakharov, A. D. (May 1991). "Violation of CP invariance, C asymmetry, and baryon asymmetry of the universe" (PDF). Soviet Physics Uspekhi. 34 (5): 392–393. Bibcode:1991SvPhU..34..392S. doi:10.1070/PU1991v034n05ABEH002497.
31. Sakharov, A. D. (7 December 1982). Collected Scientific Works. Marcel Dekker. ISBN 978-0824717148.
32. Sakharov, A. D. (January 1967). "Кварк–мюонные токи и нарушение СР–инвариантности". Pi'sma ZhÉTF (in Russian). 5 (1): 36–39.CS1 maint: Unrecognized language (link) Translated as: Sakharov, A. D. (January 1967). "Quark-Muonic Currents and Violation of CP Invariance" (PDF). JETP Letters. 5 (1): 27–30.
33. Sakharov, A. D. (1969). "Антикварки во Вселенной" [Antiquarks in the Universe]. Problems in Theoretical Physics (in Russian): 35–44.CS1 maint: Unrecognized language (link) Dedicated to the 30th anniversary of N. N. Bogolyubov.
34. Sakharov, A. D. (1972). "Топологическая структура элементарных зарядов и СРТ–симметрия" [The topological structure of elementary charges and CPT symmetry]. Problems in Theoretical Physics (in Russian): 243–247.CS1 maint: Unrecognized language (link) Dedicated to the memory of I. E. Tamm.
35. Sakharov, A. D. (April 1979). "Барионная асимметрия Вселенной". Pi'sma ZhÉTF (in Russian). 76 (4): 1172–1181.CS1 maint: Unrecognized language (link)Translated as: Sakharov, A. D. (April 1979). "The baryonic asymmetry of the Universe" (PDF). JETP Letters. 49 (4): 594–599.
36. Sakharov, A. D. (September 1980). "Космологические модели Вселенной с поворотом стрелы времени". Pi'sma ZhÉTF (in Russian). 79 (3): 689–693.CS1 maint: Unrecognized language (link)Translated as: Sakharov, A. D. (September 1980). "Cosmological models of the Universe with reversal of time's arrow" (PDF). JETP Letters. 52 (3): 349–351.
37. Sakharov, A. D. (October 1982). "Многолистные модели Вселенной". Pi'sma ZhÉTF (in Russian). 82 (3): 1233–1240.CS1 maint: Unrecognized language (link)Translated as: Sakharov, A. D. (October 1982). "Many-sheeted models of the Universe" (PDF). JETP. 56 (4): 705–709.
38. Sakharov, A. D. (September 1986). "Испарение черных мини–дыр и физика высоких энергий". Pi'sma ZhÉTF (in Russian). 44 (6): 295–298.CS1 maint: Unrecognized language (link)Translated as: Sakharov, A. D. (September 1986). "Evaporation of black mini-holes and high-energy physics" (PDF). JETP Letters. 44 (6): 379–383.
39. Petit, J.-P. (23 May 1977). "Univers jumeaux, énantiomorphes, à temps propre opposées" [Enantiomorphic twin universes with opposite proper times] (PDF). Comptes Rendus de l'Académie des Sciences (in French). 263: 1315–1318.CS1 maint: Unrecognized language (link)
40. Petit, J.-P. (6 June 1977). "Univers en interaction avec leurs images dans le miroir du temps" [Universes interacting with their opposite time-arrow fold] (PDF). Comptes Rendus de l'Académie des Sciences (in French). 284: 1413–1416.CS1 maint: Unrecognized language (link)
41. in Petit, J.-P. (July 1994). "The missing-mass problem" (PDF). Il Nuovo Cimento B. 109 (7): 697–709. Bibcode:1994NCimB.109..697P. doi:10.1007/BF02722527.
42. Petit, J.-P.; d'Agostini, G. (December 2014). "Negative mass hypothesis in cosmology and the nature of dark energy" (PDF). Astrophysics and Space Science. 354 (2): 611. Bibcode:2014Ap&SS.354..611P. doi:10.1007/s10509-014-2106-5.
43. Petit, Jean-Pierre. "Janus cosmology: what is negative mass?". januscosmologicalmodel.com. Retrieved 2019-01-07.
44. Richard Feynman suggested that an electron, cruising backwards in time and observed through a mirror (P-symmetry) would behave like a positron.
45. Petit, J.-P.; d'Agostini, G. (10 November 2014). "Cosmological bimetric model with interacting positive and negative masses and two different speeds of light, in agreement with the observed acceleration of the Universe" (PDF). Modern Physics Letters A. 29 (34): 1450182. Bibcode:2014MPLA...2950182P. doi:10.1142/S021773231450182X.
46. Bondi, H. (July 1957). "Negative Mass in General Relativity" (PDF). Reviews of Modern Physics. 29 (3): 423–428. Bibcode:1957RvMP...29..423B. doi:10.1103/RevModPhys.29.423.
47. Bonnor, W. B. (November 1989). "Negative mass in general relativity" (PDF). General Relativity and Gravitation. 21 (11): 1143–1157. Bibcode:1989GReGr..21.1143B. doi:10.1007/BF00763458.
48. Petit, J.-P. (16 November 1988). "An interpretation of cosmological model with variable light velocity" (PDF). Modern Physics Letters A. 3 (16): 1527–1532. Bibcode:1988MPLA....3.1527P. CiteSeerX 10.1.1.692.9603. doi:10.1142/S0217732388001823.
49. Petit, J.-P. (18 December 1988). "Cosmological model with variable light velocity: the interpretation of red shifts" (PDF). Modern Physics Letters A. 3 (18): 1733–1744. Bibcode:1988MPLA....3.1733P. CiteSeerX 10.1.1.692.9067. doi:10.1142/S0217732388002099.
50. Petit, J.-P.; Viton, M. (10 November 1989). "Gauge cosmological model with variable light velocity: Comparizon with QSO observational data" (PDF). Modern Physics Letters A. 4 (23): 2201–2210. Bibcode:1989MPLA....4.2201P. doi:10.1142/S0217732389002471.
51. Midy, P.; Petit, J.-P. (June 1999). "Scale invariant cosmology" (PDF). International Journal of Modern Physics D. 8 (3): 271–289. arXiv:gr-qc/9909086. Bibcode:1999IJMPD...8..271M. doi:10.1142/S0218271899000213.
52. Petit, J.-P. (2008). Savoir Sans Frontières, ed. Faster Than Light (PDF).
53. Petit, J.-P.; Midy, P.; Landsheat, F. (June 2001). "Twin matter against dark matter, International Meeting on Atrophysics and Cosmology "Where is the matter?"" (PDF).
54. Petit, J.-P.; d'Agostini, G. (August 2007). "Bigravity: a bimetric model of the Universe with variable constants, including VSL (variable speed of light), International Meeting on Variational Techniques, Le Mont-Dore (France)". arXiv:0803.1362. Bibcode:2008arXiv0803.1362P.
55. Petit, J.-P.; d'Agostini, G. (10 November 2014). "Cosmological bimetric model with interacting positive and negative masses and two different speeds of light, in agreement with the observed acceleration of the Universe" (PDF). Modern Physics Letters A. 29 (34): 1450182. Bibcode:2014MPLA...2950182P. doi:10.1142/S021773231450182X.
56. Petit, J.-P. (April 1995). "Twin universes cosmology" (PDF). Astrophysics and Space Science. 227 (2): 273–307. Bibcode:1995Ap&SS.226..273P. CiteSeerX 10.1.1.692.7762. doi:10.1007/BF00627375.
57. Weinberg, Steven (2005). "Relativistic Quantum Mechanics: Space Inversion and Time-Reversal" (PDF). The Quantum Theory of Field. 1: Foundations. Cambridge University Press. ISBN 9780521670531.
58. Petit, Jean-Pierre. "Janus cosmology: CPT symmetry, the primeval antimatter and energy inversion". januscosmologicalmodel.com. Retrieved 2019-01-07.
59. Souriau, J.-M. (1997). "14. A mechanistic description of elementary particles: Inversions of space and time" (PDF). Structure of Dynamical Systems. Progress in Mathematics. Boston: Birkhäuser. pp. 189–193. doi:10.1007/978-1-4612-0281-3_14. ISBN 978-1-4612-6692-1.
60. Petit, J.-P.; Midy, P.; Landsheat, F. (June 2001). "Twin matter against dark matter" (PDF). "Where is the matter?". International Meeting on Atrophysics and Cosmology.
61. Henry-Couannier, F.; d'Agostini, G.; Petit, J.-P. (2005). "I- Matter, antimatter and geometry. II- The twin universe model: a solution to the problem of negative energy particles. III- The twin universe model plus electric charges and matter-antimatter symmetry". arXiv:math-ph/0502042.
62. Petit, J.-P. (January 2018). "A Symplectic Cosmological Model" (PDF). Progress in Physics. 14 (1): 38–40.
63. Petit, J.-P.; d'Agostini, G. (May 2015). "Lagrangian derivation of the two coupled field equations in the Janus cosmological model" (PDF). Astrophysics and Space Science. 357 (67): 67. Bibcode:2015Ap&SS.357...67P. doi:10.1007/s10509-015-2250-6.
64. Petit, J.-P.; d'Agostini, G. (21 March 2015). "Cancellation of the central singularity of the Schwarzschild solution with natural mass inversion process" (PDF). Modern Physics Letters A. 30 (9): 1550051. Bibcode:2015MPLA...3050051P. doi:10.1142/S0217732315500510.
65. D'Agostini, G.; Petit, J.-P. (June 2018). "Constraints on Janus Cosmological model from recent observations of supernovae type Ia" (PDF). Astrophysics and Space Science. 363 (7): 139. Bibcode:2018Ap&SS.363..139D. doi:10.1007/s10509-018-3365-3.
66. Debergh, N.; Petit, J.-P.; D'Agostini, G. (November 2018). "On evidence for negative energies and masses in the Dirac equation through a unitary time-reversal operator". Journal of Physics Communications. 2 (11): 115012. arXiv:1809.05046. Bibcode:2018JPhCo...2k5012D. doi:10.1088/2399-6528/aaedcc.
67. Debergh, N.; Petit, J-P; d'Agostini, G. (2018). "On evidence for negative energies and masses in the Dirac equation through a unitary time-reversal operator". Journal of Physics Communications. 2 (11): 115012. arXiv:1809.05046. Bibcode:2018JPhCo...2k5012D. doi:10.1088/2399-6528/aaedcc.
68. (en) J.-P. Petit, The Dark Side of the universe, p. 273-282, june 1998
69. (en) J.-P. Petit, G. D'Agostini, Cancellation of the central singularity of the Schwarzschild solution with natural mass inversion process, Modern Physics Letters A, 30 (09):1550051, March 2015
70. (en) J.-P. Petit, Mass inversion in a critical neutron star: An alternative to the black hole model, COSMO-17, Paris, Aug. 2017
71. Abrams, L. S. (1979-11-15). "Alternative space-time for the point mass". Physical Review D. 20 (10): 2474–2479. arXiv:gr-qc/0201044. Bibcode:1979PhRvD..20.2474A. doi:10.1103/PhysRevD.20.2474.
72. Abrams, L. S. (1989). "Black Holes: The Legacy of Hilbert's Error". Canadian Journal of Physics 67 (9): 919–926. doi:10.1139/p89-158. arXiv:gr-qc/0102055.
73. Antoci, S.; Liebscher, D.-E. (July 2001). "Reconsidering Schwarzschild's original solution". Astronomische Nachrichten, Issn2=1521-3994. 322 (3): 137–142. arXiv:gr-qc/0102084. Bibcode:2001AN....322..137A. doi:10.1002/1521-3994(200107)322:3<137::AID-ASNA137>3.0.CO;2-1. ISSN 0004-6337.
74. Antoci, Salvatore (2003-10-21). David Hilbert and the origin of the "Schwarzschild solution". Meteorological and Geophysical Fluid Dynamics. p. 343. arXiv:physics/0310104. Bibcode:2004mgfd.book..343A.
75. Fromholz, Pierre; Poisson, Eric; Will, Clifford M. (April 2014). "The Schwarzschild metric: It's the coordinates, stupid!". American Journal of Physics, Issn2=1943-2909. 82 (4): 295–300. arXiv:1308.0394. Bibcode:2014AmJPh..82..295F. doi:10.1119/1.4850396. ISSN 0002-9505.
76. Mol, Igor (2014-03-10). "Revisiting the Schwarzschild and the Hilbert-Droste Solutions of Einstein Equation and the Maximal Extension of the Latter". arXiv:1403.2371 [math-ph].
77. Petit, Jean-Pierre (2000). "Questionable black hole" (PDF). www.researchgate.net. Retrieved 2019-01-08.
78. Petit, Jean-Pierre (April 2014). "Black holes do not exist". Researchgate.
79. "Les trous noirs n'existent pas - Partie 1". La Voie de la Russie / Sputnik News (in français). 30 June 2014.
80. "Les trous noirs n'existent pas - Partie 2". La Voie de la Russie - SputnikNews (in français). 1 July 2014.
81. Petit, Jean-Pierre (2016-07-04). "Schwarzschild 1916 seminal paper revisited : A virtual singularity". ResearchGate.
82. "Janus 22 - partie 1". youtube (in français and English).
83. "Janus 22-partie 2". youtube (in français).
84. "Janus 22 - partie 3" (in français and English).
85. "Janus 22 - partie 4" (in français and English).
86. "Janus 22 - partie 5" (in français and English).
87. Petit, Jean-Pierre (2018). "Annex paper to JANUS 22 - 5 : Gravitational waves. Fusion of neutron stars. Mass with inverted sign" (PDF). www.researchgate.net. Retrieved 2019-01-08.
88. "Stephen Hawking suggested that black holes could be a route to another universe". The Independent. 2018-03-14. Retrieved 2018-12-30.
89. Ashtekar, Abhay; Olmedo, Javier; Singh, Parampreet (2018-12-10). "Quantum Transfiguration of Kruskal Black Holes". Physical Review Letters. 121 (24): 241301. arXiv:1806.00648. Bibcode:2018PhRvL.121x1301A. doi:10.1103/PhysRevLett.121.241301. PMID 30608746.
90. Rovelli, Carlo (2018-12-10). "Viewpoint: Black Hole Evolution Traced Out with Loop Quantum Gravity". Physics. 11.
91. Boisson, Thomas (2018-12-21). "La gravité quantique à boucles fait disparaître la singularité centrale des trous noirs". Trust My Science (in français). Retrieved 2018-12-22.
92. "Progress in Physics homepage". Progress in Physics.
93. (en) J.-P. Petit, The JANUS Cosmological Model and the Fluctuations of the CMB, Progress in Physics, octobre 2018, vol. 14, issue 4, p. 226–229 [1]
94. (fr) J.-P. Petit, Une méthode de résolution de l'équation de Vlasov. Application à une théorie globale, non-linéaire, de la rotation galactique et de la structure spirale galactique, CRAS B.275..755P (1972)
95. (en) J.-P. Petit, The Dark Side of the universe, p. 238-243, juin 1998
96. Petit, Jean-Pierre; d'Agostini, Gilles (2014). "An alternative model, explaining the VLS, created by the gravitational interaction of two matter populations, one composed of positive mass and the other of negative mass". Journal of Physics: Conference Series. 490 (1): 012096. Bibcode:2014JPhCS.490a2096P. doi:10.1088/1742-6596/490/1/012096. ISSN 1742-6596.
97. D’Agostini, G.; Petit, Jean-Pierre (2015-05-01). "Lagrangian derivation of the two coupled field equations in the Janus cosmological model". Astrophysics and Space Science, Issn2=0004-640X. 357 (1): 67. Bibcode:2015Ap&SS.357...67P. doi:10.1007/s10509-015-2250-6. ISSN 1572-946X.
98. d'Agostini, Gilles; Petit, Jean Pierre (2008-05-09). "Five Dimensional bigravity. New topological description of the Universe". arXiv:0805.1423 [physics.gen-ph].
100. Petit, J. P.; D'Agostini, G. (2014-10-27). "Cosmological bimetric model with interacting positive and negative masses and two different speeds of light, in agreement with the observed acceleration of the Universe". Modern Physics Letters A. 29 (34): 1450182. Bibcode:2014MPLA...2950182P. doi:10.1142/S021773231450182X. ISSN 0217-7323.
101. Petit, J. P.; D’Agostini, G. (2018-07-01). "Constraints on Janus Cosmological model from recent observations of supernovae type Ia". Astrophysics and Space Science. 363 (7): 139. Bibcode:2018Ap&SS.363..139D. doi:10.1007/s10509-018-3365-3. ISSN 1572-946X.
102. Petit, J.P.: Twin universe cosmology. Astrophys. Space Sci. 226, 273 (1995)
103. Petit, J.P., D’Agostini, G.: Negative mass hypothesis in cosmology and the nature of dark energy. Astrophys. Space Sci. 353 (2) (2014)
104. Petit, J.P., D’Agostini, G.: Cosmological bimetric model with interacting positive and negative masses and two different speeds of light, in agreement with the observed acceleration of the Universe. Mod. Phys. Lett. A 29 (34) (2014)
105. El-Ad, H., Piran, T., da Costa, L.N.: Automated detection of voids in redshift surveys. Astrophys. J. 462, L13–L16 (1996)
106. Petit, J.P., Midy, P., Landsheat, F.: Twin matter against dark matter. In: International Meeting on Astrophysical and Cosmology “Where is the matter?”, Marseille, pp. 25–29 (2001)
107. Ashtekar, Abhay; Olmedo, Javier; Singh, Parampreet (2018-12-10). "Quantum Transfiguration of Kruskal Black Holes". Physical Review Letters. 121 (24): 241301. arXiv:1806.00648. Bibcode:2018PhRvL.121x1301A. doi:10.1103/PhysRevLett.121.241301. PMID 30608746.
108. Rovelli, Carlo (2018-12-10). "Viewpoint: Black Hole Evolution Traced Out with Loop Quantum Gravity". Physics. 11. Retrieved 2018-12-22.
109. d'Agostini, Gilles; Petit, Jean Pierre (2008-05-09). "Five Dimensional bigravity. New topological description of the Universe". arXiv:0805.1423 [physics.gen-ph].
110. Izumi, Koji; Hagiwara, Chisaki; Nakajima, Koki (2013-07-29). "Gravitational lensing shear by an exotic lens object with negative convergence or negative mass". Physical Review D. 88 (2): 024049. arXiv:1305.5037. Bibcode:2013PhRvD..88b4049I. doi:10.1103/PhysRevD.88.024049.
111. (en) Fabrice Petit, Michel Sarrazin, Plausible FTL displacements in a two-sheeted spacetime, Phys. Rev. D 76, 2007, p. 085005
112. "Negative Energy states and Interstellar travel". 14 September 2018.
113. "mon sentiment sur la pertinence des travaux effectués en cosmologie par Jean-Pierre Petit". Institut d'Astrophysique de Paris (in français). 18 December 2006. Retrieved 7 December 2018.
115. (fr) Archive of the article 'Jean-Pierre Petit' on Wikipédia France|version=16186826| 24 April 2007 https://fr.wikipedia.org/w/index.php?title=Discussion:Jean-Pierre_Petit/archive2&oldid=10418947
116. "(fr) Ma réponse aux attaques d'Alain Riazuelo de l'Institut d'Astrophysique de Paris" (PDF). researchGate (in français).
117. Wikibuster (2013-01-12). "Le témoignage de Jean-Pierre PETIT : " Repenser à Wikipédia me donne la nausée "" (PDF). Wikibuster : Les dessous de Wikipédia (in français). Retrieved 2018-12-20.
118. Petit's website is the oldest website permanently blacklisted by the french Wikipedia : https://fr.wikipedia.org/wiki/MediaWiki:Spam-blacklist
119. "Le temps de la lâcheté" (PDF) (in français) (published 2 May 2017). 8 December 2006. Retrieved 7 December 2018.
120. "Jean-Pierre PETIT : "Les Z-machines permettent d'envisager une fusion nucléaire pratiquement sans déchets"". www.20minutes.fr (in français). 14 January 2013. Retrieved 17 December 2018.
121. Vergès, Etienne (13 March 2018). "Ethique et déontologie de la recherche scientifique : Un système normatif communautaire, p.131-149" (PDF). (fr) Qu'en est-il du droit de la recherche ?. Presses de l’Université Toulouse 1 Capitole, collection: Travaux de l’IFR. pp. 131–149. ISBN 9782379280214.
122. Jean-Pierre Petit (2011). "la liberté d'expression (interview)" (in français). Retrieved 2018-12-14.
123. La-Croix.com (2018-06-05). "(fr) Doit-on légiférer pour assurer l'intégrité scientifique ?". La Croix (in français). Retrieved 2018-12-14.
124. conférence "Faut-il être hérétique pour innover en science ?", XVIIème festival "Science frontières", Cavaillon (January 2000). "Faut-il être hérétique pour innover en science ?" (in français). Retrieved 2018-12-14.CS1 maint: Multiple names: authors list (link)
125. Ellis, George F. R. (April 2007). "Note on varying speed of light cosmologies". General Relativity and Gravitation, Issn2=0001-7701. 39 (4): 511–520. arXiv:astro-ph/0703751. Bibcode:2007GReGr..39..511E. doi:10.1007/s10714-007-0396-4. ISSN 1572-9532.
126. Petit, J. P.; D'Agostini, G. (27 October 2014). "Cosmological bimetric model with interacting positive and negative masses and two different speeds of light, in agreement with the observed acceleration of the Universe". Modern Physics Letters A. 29 (34): 1450182. Bibcode:2014MPLA...2950182P. doi:10.1142/S021773231450182X. ISSN 0217-7323.
127. "3. Gabriel Chardin et Luc Blanchet : " Si nous déterminions qu'un antiatome antigravite, ce serait une révolution "". www.larecherche.fr (in français).
129. Mr. Sam - Point d'interrogation (1 February 2018). "JEAN-PIERRE PETIT: conclusion - Versus? Ep1.4" (in français).
130. François Cauneau. "Commentaire sur JEAN-PIERRE PETIT: confrontations - Versus? Ep.1.3" (in français).
131. J.S. Farnes (December 2018). "A unifying theory of dark energy and dark matter: Negative masses and matter creation within a modified ΛCDM framework". Astronomy & Astrophysics, Vol. 620. 620: A92. arXiv:1712.07962. Bibcode:2018A&A...620A..92F. doi:10.1051/0004-6361/201832898.
132. Petit, J. P.; D'Agostini, G. (27 October 2014). "Cosmological bimetric model with interacting positive and negative masses and two different speeds of light, in agreement with the observed acceleration of the Universe". Modern Physics Letters A. 29 (34): 1450182. Bibcode:2014MPLA...2950182P. doi:10.1142/S021773231450182X. ISSN 0217-7323.
133. D'Agostini, G.; Petit, J.-P. (June 2018). "Constraints on Janus Cosmological model from recent observations of supernovae type Ia" (PDF). Astrophysics and Space Science. 363 (7): 139. Bibcode:2018Ap&SS.363..139D. doi:10.1007/s10509-018-3365-3.
134. Debergh, N.; Petit, J.-P.; D’Agostini, G. (2018). "On evidence for negative energies and masses in the Dirac equation through a unitary time-reversal operator". Journal of Physics Communications. 2 (11): 115012. arXiv:1809.05046. Bibcode:2018JPhCo...2k5012D. doi:10.1088/2399-6528/aaedcc. ISSN 2399-6528.
135. Jean-Pierre Luminet (January 2018). "commentaire de Jean-Pierre Luminet suite à sa conférence " L'univers, le big bang et au-delà "" (in français).
136. MesOpinions.com. "Pétition : Droit au débat pour une nouvelle cosmologie". mesopinions.com (in français).
137. Petit, J.-P.; d'Agostini, G. (May 2015). "Lagrangian derivation of the two coupled field equations in the Janus cosmological model" (PDF). Astrophysics and Space Science. 357 (67): 67. Bibcode:2015Ap&SS.357...67P. doi:10.1007/s10509-015-2250-6.
138. D'Agostini, G.; Petit, J.-P. (June 2018). "Constraints on Janus Cosmological model from recent observations of supernovae type Ia" (PDF). Astrophysics and Space Science. 363 (7): 139. Bibcode:2018Ap&SS.363..139D. doi:10.1007/s10509-018-3365-3.
139. "Fil de commentaires sur "JANUS 25 : Du souffle épique sinon rien !"" (in français). December 2018.
140. Henry-Couannier, Frederic (17 June 2017). "Consistency of JP. P and S.H Janus anti-gravity theories" (PDF). darksideofgravity.com.
141. "(fr) Jean Marie Souriau - le site officiel". www.jmsouriau.com. Retrieved 2018-12-20.
142. "Conférence "Quelle est la forme de l'univers ?" | Portail documentaire SCD Doc'INSA | INSA de Lyon (in French)". scd.docinsa.insa-lyon.fr. Retrieved 2018-12-20.
143. Petit, Jean-Pierre. "(fr) 2017-inversion-masse-etoile_neutrons part b" (PDF). researchgate.net. Retrieved 2019-01-08.
144. Petit, Jean-Pierre. "(en) 2017-mass-inversion-neutron-star part a" (PDF). www.jp-petit.org. Retrieved 2019-01-08.
145. OrandiaTV. "(fr) VOR161015 B - Retour sur le Congrès UFO601 "La Forme"". Retrieved 2018-12-20.
146. "Conférence Robert Salas – Jean Pierre Petit : une réussite totale ! – MUFON FRANCE" (in français). Retrieved 2018-12-20.
147. CROPS. "(fr) Jean Pierre PETIT " OVNIS : Une science venue d'ailleurs " - 27 October 2018". Evensi. Retrieved 2018-12-20.
148. Petit, J.-P. (2017). YouTube, ed. The Janus Cosmological Model (English subtitles).
149. Jean-Pierre PETIT (2017-01-07). "JANUS 1 : Naissance de la cosmologie avec Platon, Aristote, Ptolémée, Copernic (in French with EN subtitles)".
150. Jean-Pierre PETIT, JANUS 22 – 1/5 : Trou noir VS inversion de la masse. Colloques 2017. États d'énergie négative, retrieved 2019-01-08
151. Petit, Jean-Pierre. "(fr) Annexe à Janus 23" (PDF). www.jp-petit.org. Retrieved 2019-01-08.
152. Petit, Jean-Pierre. "(en) Annex to Janus 24" (PDF). www.jp-petit.org. Retrieved 2019-01-08.
153. Jean-Pierre PETIT (2018-01-21). "The Black Hole in the JANUS COSMOLOGICAL MODEL (episode 24a eng)".
154. "(fr) Site officiel de l'Association Savoir sans Frontières". savoir-sans-frontieres.com.
155. Petit, J.-P. (2008). The Twin Universe (PDF). Savoir Sans Frontières.
156. Petit, J.-P. (2008). The Twin Universe: Scientific Appendix (PDF). Savoir Sans Frontières.
157. "(fr) Le projet EPISTEMOTRON (archive)". ayuba.fr.
158. "(PDF) Very Large Structure numerical simulation in a compact computational space". ResearchGate. Retrieved 2019-01-04.
159. franck Malers. "Janus 3 million points N-Body".
160. "CUDA Zone". NVIDIA Developer. 2017-07-18. Retrieved 2019-01-08.
161. Petit, Jean-Pierre. "(fr) LE VERSANT OBSCUR DE L'UNIVERS - PDF". docplayer.fr. Retrieved 2018-12-14.
162. "NROER - Science Comic Books;Jean-Pierre Petit". nroer.gov.in. Retrieved 2018-12-14.
163. Alain Moreau (2018-05-31). "(fr) OVNI. L'extraordinaire découverte – le livre de Jean-Pierre Petit et Jean-Claude Bourret". mondenouveau.fr (in français).
164. Page archivée : https://www.europe2020.org/spip.php?article739&lang=en
165. Paul, Bruno (2012-12-22). "(fr) Les réseaux Euro-BRICS comme réponse à la crise scientifique". Conscience Sociale.
166. Michele Angelo Murgia (21 September 2015). Qu'est-ce que le temps ?, p.8, 125, 155, 325, 348, 458, 481, 485, 531, 583, 606 (in français). Atramenta. p. 676. ISBN 978-9522737014.
167. Bruno Paul (2018-05-25). La Signature Du Quaternaire - Logique, sémantique et Tradition.
168. (fr), JEAN-PIERRE PETIT - LA MHD ET LES UNIVERS JUMEAUX, retrieved 2019-01-08
170. "(2003) A tort ou à raison ovni l'émission de TF1 du lundi 13 janvier ( vers 22h30 )". www.forum-ovni-ufologie.com (in français). Retrieved 2019-01-03.
171. Prise De Conscience, (fr). "La réalité du phénomène OVNI - Emission spéciale".
172. (fr). "Sciences et Conscience n°21 – Émission du Jeudi 05 Octobre 2017 – Agora Côte d'Azur". agoracotedazur.fr.
173. Mr. Sam - Point d'interrogation, (fr). "JEAN-PIERRE PETIT: Janus, une vraie solution ? - Versus? 1.1".
174. Thinkerview, (fr). "Jean-Pierre Petit : Modèle Janus et Armes russes [EN DIRECT]".
175. NURÉA TV. "" LE MODÈLE JANUS " (Partie 1) avec Jean-Pierre Petit - NURÉA TV" (in français). YouTube.
176. Sputnik. "Voyage interstellaire et contacts extraterrestres, par Jean-Pierre Petit (Partie 2)". fr.sputniknews.com (in français). Retrieved 2019-01-08.
177. Sputnik. "Les trous noirs n'existent pas (Partie 1)". fr.sputniknews.com (in français).
178. Sputnik. "Les trous noirs n'existent pas (Partie 2)". fr.sputniknews.com (in français).
179. Nexus. "Ovni : changement de figure avec Janus". Nexus (in français).
180. Petit, Jean-Pierre; Bourret, Jean-Claude (2017-02-24). OVNI: L'extraordinaire découverte (in français). Guy Trédaniel. ISBN 9782813215147.
181. "Prétendre que nous sommes seuls dans l'univers est une absurdité totale". lesoir.be (in français). 11 April 2017.
182. aeromorning (2017-02-27). "OVNI L'EXTRAORDINAIRE DECOUVERTE Edition Guy Trédaniel". AeroMorning.com (in français).
183. Kauf, +Olivier (2017-03-22). "OVNI : L'extraordinaire découverte de JC Bourret et JP Petit chez GuyTrédaniel". Notre-Siècle (votre e-mag. du XXIème siècle) (in français).
184. http://www.lunion.fr/archive/recup/25081/article/2017-04-09/jean-claude-bourret-il-existe-un-autre-univers-cote-du-notre
185. Paul, Bruno (2012-05-18). "(fr) Une théorie et un homme en crise : les cordes et Alain Riazuelo". Conscience Sociale. Retrieved 2019-01-04.
186. de Rham, Claudia; Gabadadze, Gregory; Tolley, Andrew J. (2011). "Resummation of Massive Gravity". Phys. Rev. Lett. 106 (23): 231101. arXiv:1011.1232. Bibcode:2011PhRvL.106w1101D. doi:10.1103/PhysRevLett.106.231101. PMID 21770493.
187. Boulware, David G.; Deser, Stanley (1972). "Can gravitation have a finite range?" (PDF). Phys. Rev. D6 (12): 3368–3382. Bibcode:1972PhRvD...6.3368B. doi:10.1103/PhysRevD.6.3368.
188. Hassan, S.F.; Rosen, Rachel A. (2011). "On Non-Linear Actions for Massive Gravity". JHEP. 1107 (7): 009. arXiv:1103.6055. Bibcode:2011JHEP...07..009H. doi:10.1007/JHEP07(2011)009.
189. Rosen, Nathan (1940), "General Relativity and Flat Space. I", Phys. Rev., 57 (2): 147, Bibcode:1940PhRv...57..147R, doi:10.1103/PhysRev.57.147
190. Rosen, Nathan (1940), "General Relativity and Flat Space. II", Phys. Rev., 57 (2): 150, Bibcode:1940PhRv...57..150R, doi:10.1103/PhysRev.57.150