Bimetric gravity

Bimetric gravity or bigravity refers to two different sets of theories. The first set of theories relies on modified mathematical theories of gravity (or gravitation) in which two metric tensors are used instead of one. 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. Several bimetric theories with massive gravitons exist, such as those attributed to Nathan Rosen (1909–1995) or Mordehai Milgrom with Modified Newtonian Dynamics (MOND). More recently, developments in massive gravity have also led to new consistent theories of bimetric gravity. 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. Some of these theories lead to cosmic acceleration at late times and are therefore alternatives to dark energy.

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. The Janus cosmological model introduced by Jean-Pierre Petit belongs to this set.

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

$1.~~~~ds^{2}=g_{ij}dx^{i}dx^{j}$

$2.~~~~d\sigma ^{2}=\gamma _{ij}dx^{i}dx^{j}$

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

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

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

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

A straightforward calculation yields the Riemann curvature tensor

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

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 $\Delta$ and ordinary differentiation by covariant $\gamma$-differentiation, ${\sqrt {-g}}$ by ${\sqrt {\tfrac {g}{\gamma }}},$ integration measure $d^{4}x$ by ${\sqrt {-\gamma }}\,d^{4}x,$ where $g=\det(g_{ij}),\gamma =det(\gamma _{ij})$ and $d^{4}x=dx^{1}dx^{2}dx^{3}dx^{4}$. Having once introduced $\gamma _{ij}$ 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

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

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

The quantity $\Delta ,$ 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

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

where

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

or

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

$N=g^{ij}N_{ij},\qquad \kappa ={\sqrt {\frac {g}{\gamma }}},$

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

The variational principle also leads to the relation

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

Hence from (3)

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

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

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.

Janus cosmological model

From 1977, Petit starts to build 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".

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 Jean-Pierre Petit. 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.

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.

Petit produces many science comic books and videos to popularize the various aspects of Janus cosmological model.

The theory originated in 1977 from French physicist Jean-Pierre Petit, 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 Petit was the first author to introduce 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. 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.

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

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. 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):

We can visualize that neutral spinless maximons (or photons) are produced at t < 0 from contracting matter having an excess of antiquarks, that they pass "one through the other" at the instant t = 0 when the density is infinite, and decay with an excess of quarks when t > 0, realizing total CPT symmetry of the universe. All the phenomena at t < 0 are assumed in this hypothesis to be CPT reflections of the phenomena at t > 0.

— Andrei Sakharov, in Collected Scientific Works (1982).

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. ·  ·  ·  ·  ·  ·

2D didactic image of Janus model.

Ignoring the prior existence of this work translated in a book only fifteen years after its Russian publication, Petit published his first paper about two enantiomorphic universes with opposite arrows of time in 1977. · 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.

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:

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

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

Petit's 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. Petit's 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 bimetric universe according to Janus model

The theory describes two parallel universes in CPT symmetry interacting through gravity, both originating from the same initial singularity. In Janus model, four types 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 type). 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 by Petit) in the space of the moment.

Together with positive energy photons, these first two types are the components of the universe known until now: it is the first sheet of the universe.

• 3- the negative mass material (CPT-symmetry with respect to the first type, with an anti-linear and anti-unit operator T ), which is not very abundant with respect to the fourth type. 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 type, with a linear and unitary operator T ). In symplectic geometry, Petit notes 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 type, the so-called « Feynman antimatter », is the primordial antimatter.

With negative energy photons, these last two types are the components of the second sheet of the universe.

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

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, and solve the runaway paradox, that usually makes scientists think negative mass can not physically exist:

I regard the runaway (or self-accelerating) motion […] so preposterous that I prefer to rule it out by supposing that inertial mass is all positive or all negative.

— William B. Bonnor, in Negative mass in general relativity.

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.

Cosmology with variation of constants

In 1988, Petit introduced the idea of the variation of the speed of light in cosmology 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.

Geometrization of Janus model

In 1995, Petit combines his bimetric model with his VSL theory into the first paper summarizing the twin universes cosmology.

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. 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, Petit thinks this problem should be reconsidered.

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.

In the 2000s, Petit integrated Souriau's mathematical physics and fully geometrized his model with this 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. ·  ·  ·

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. In a second paper this is extended to two metrics with their own speed of light, followed by the Lagrangian derivation of the model. A fourth paper is devoted to the cancellation of the central singularity in the Schwarzschild solution, questioning the classical black hole model.

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

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 $-m=-E/c^{2}$, so the Klein–Gordon probability density $E/m$ in relativistic quantum mechanics remains positive.

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.

Petit also explained using the Janus model the detailed phenomena that occur within a neutron star at the critical density threshold.

Jean-Pierre Petit provided a mathematical demonstration of the absence of 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 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)
• J.-P. Petit, Black holes do not exist, preprint, April and June 2014
• (fr) interview with J.-P. Petit by SputnikNews, June/July 2014, Les trous noirs n'existent pas - parts 1 and 2
• J.-P. Petit, Schwarzschild 1916 seminal paper revisited : A virtual singularity, preprint, July 2016

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

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

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.

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 validated a study by Petit that uses the Janus model to explain the origin of fluctuations in the cosmological background (CMB). It was published in October 2018.

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 Petit, 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. In the current Janus model, the modeling of these negative masses is geometrized; moreover, the nature and origin of these negative masses are defined.

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.

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. ·  ·

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. 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, Petit and D'Agostini set out the differences in the explanatory scope between the ΛCDM Model and the Janus Model. 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 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. JCM suggests a clear scheme for VLS formation 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, 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, 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 Janus model does not involve any hypothetical exotic particles or mirror matter. 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. The ΛCDM Model provides no detailed explanation for this origin. In the same October 2018 publication, Petit adds that Janus 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, 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).

 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 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.

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. The transferred vehicle would cruise along geodesics of the metric $g_{\mu \nu }^{(-)}$ 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.

A study onto this subject was presented in a scientific conference:

• J.-P. Petit, N. Debergh, G. D'Agostini, Negative Energy states and Interstellar travel, October 2018; the video of this intervention is available.

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 20000 km. Petit 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.

Petit 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, Petit 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. Petit 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 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 Wikipedia France 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 Wikipedia France project by 16 french-speaking administrators. Petit expressed his deep disagreement and responded on his personal site and on Wikibuster, whose URLs have since been blacklisted by Wikipedia France. On the scientific and sociological background, Jean-Pierre Petit has published an open journal for more than 10 years 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, which have remained systematically ignored by IAP, contrary to any rule of integrity and scientific ethics.

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

Indeed, the latter meets all the conditions:

   (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 magazine, 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." Jean-Pierre Petit then commented on this article.

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 Petit 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.

François Cauneau, a physicist, professor at a french major league 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 Petit's approach.

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)172, however such theories have remained incompatible with observations."

Petit and D'Agostini's articles of November 2014 and June 2018 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. 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. 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."
• the electronic petition launched in August 2018: "Right to debate for a new cosmology"

Looking for a Lagrangian formulation of the Janus model

a) Petit's article published in 2015 presents the derivation of a Lagrangian. The article of 2018, 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 J.P. Petit's video Janus 25, physicist Frédéric Henry-Couannier writes that the 2015 demonstration would include an error. A hypothesis is made on the variations of the two metrics and this hypothesis is no longer used in the rest of the article. Quote from the article: "A link between the two variations is required. Let's write:

(14) $\delta g^{(+)\mu \nu }=-\delta g^{(-)\mu \nu }$ "

According to F. Henry-Couannier, we should obtain not two field equations, but only one. The absence of a Lagrangian formulation, still according to F. Henry-Couannier, would give little chance of obtaining mathematically coherent equations. Petit responded in the following comments precisely to the objection about the identities of Bianchi who initiated the question on the Lagrangian derivation (see Chapter 11 of the book "Introduction to general relativity" by Ronald Adler, Maurice Bazin and Menahem Schiffer, ed. McGraw-Hill, 1965). Petit indicated that he would give more detailed explanations in a forthcoming Janus video about the transition step from strict mathematical reasoning to theoretical physics reasoning with its inevitable approximations that make it possible to produce a workable result for calculating predictions: equations without infinite members, without divergence, without instabilities or so-called "ghosts".

b) A 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 "Consistency of JP. P and S.H Janus anti-gravity theories".

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.

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: "What is the shape of the universe?". Jean-Pierre Petit's album Le Géométricon is cited as a reference.

Jean-Pierre Petit 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 free "UFOs and American secret weapons" part of which deals with the twin universes
• on Friday, October 1, 2016 at the premises of the Andromeda Astronomy Association in Marseille
• communication at the UFO601 Congress on Saturday, October 8, 2016 in Montreal (via Skype)
• on March 24, 2018 in Hyères, lecture entitled "Les voyages interstellaires : C'est pour demain !!!! ». Former American colonel Robert Salas made a second lecture.
• on 27 October 2018 in Hyères, lecture entitled "OVNIS : a science from elsewhere".

Teaching the Janus model through videos

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

Teaching bimetric cosmology through science comic books

Jean-Pierre Petit 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 Jean-Pierre Petit (the science comic book) has been published since 1980. All the books are now distributed free of charge by the association Savoirs sans Frontieres. 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, Topologicon, Chronologicon, The black hole, The twin universe, Faster than light, A thousand billion suns, Big Bang, Cosmic Story.

Teaching astrophysics and the bimetric model using software

On April 11, 2004, Petit 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 Petit's work, whose first computer simulation work dates back to 1992. Several tutorials were published online and then this project was frozen at the end of May following a split of the group.

In 2014 Petit 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)
• Very Large Structure numerical simulation in a compact computational space, (Jul. 2014)

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, based on Newtonian models modified with negative masses (with nVIDIA's CUDA toy model software) that are not as sophisticated as Janus model equations.

Popularization of the Janus model through books for the general public

Jean-Pierre Petit 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) Jean-Pierre Petit, Enquête sur les OVNI: voyage aux frontières de la science - preface by Jacques Benveniste, ed. Albin Michel, 1990 (ISBN 9782402507875)
• (fr) Jean-Pierre Petit, Enquête sur des extra-terrestres qui sont déjà parmi nous : le mystère des Ummites, ed. Albin Michel, 1991 and 1993 (ISBN 2226055150)
• (fr) Jean-Pierre Petit, Le mystère des Ummites : une science venue d'une autre planète, ed. Albin Michel, 1995 ( ISBN 2226078452 )
• (fr) Jean-Pierre Petit, OVNI, le message, published by the author, 2008, ( ISBN 978-2918564003 ) (Annex 1 with 17 p.)
• (fr) Jean-Pierre Petit, OVNI et armes secrètes américaines : l'extraordinaire témoignage d'un scientifique, ed. Albin Michel, 2003 and 2014
• (fr) Jean-Claude Bourret and Jean-Pierre Petit, OVNI : L'extraordinaire découverte, Guy Trédaniel ed., February 2017, ( ISBN 978-2813213907 )
• (fr) Jean-Claude Bourret and Jean-Pierre Petit, 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) 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 Jean-Pierre Petit's work in cosmology.

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.

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

The Janus model in the media

Petit 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 Petit 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
• report on French television (date unknown, perhaps in the late 1980s)
• interview by Quebec television in Montreal in 1991
• 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.
• Jean-Pierre Petit 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".
• Jean-Pierre Petit 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"
• 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"
• 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
• 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 Petit through the host. The series was broadcast between June 2017 and February 1, 2018
• The "ThinkerView" channel on YouTube broadcast a 2-hour interview of Petit devoted to the Janus model in June 2018 (431,000 views recorded in December 2018)
• The "Nuréa TV" channel on YouTube broadcast a 3-hour interview (in 2 episodes) of Petit devoted to the Janus model in June 2018
• The private channel "Bob dit toute la vérité" (BTLV) broadcast an interview with Petit 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 Petit'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 Petit conference on the geometry of the universe given at the Académie lorraine des sciences.
• 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
  • Part 2 "Interstellar travel and extraterrestrial contacts", 11/1/2013
• interview with the newspaper La Voix de la Russie / SputnikNews in 2014, entitled "Black holes do not exist":
  • Part 1 on 6/30/2014
  • Part 2 on 7/1/2014
• Nexus published the article "Ovni : changement de figure avec Janus" 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 within which half of its content is devoted to Janus model. The newspaper Le Soir, the websites Aeromorning.com, notre-siècle.com, as well as L'Union (4/9/2017) 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: publications in scientific journals, pre-prints, books, videos, DVDs, science comics, conferences, including :

• Jean-Pierre Petit (foreword by Jean-Claude Pecker), On a perdu la moitié de l'Univers, Paris, Albin Michel, 1997 (ISBN 2226093931)
• reprint: Jean-Pierre Petit, 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 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.
• conference "Quelle géometrie pour l'univers?", Académie lorraine des sciences, Nancy, March 8, 2006
• (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. Massive gravity is a bimetric theory in the sense that nontrivial interaction terms for the metric $g_{\mu \nu }$ 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" $f_{\mu \nu }$ is introduced, and the interaction terms are built out of the matrix square root of $g^{-1}f$.

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 $f_{\mu \nu }$ is not chosen but instead evolves dynamically in response to $g_{\mu \nu }$ and possibly matter. This massive bigravity was introduced by Fawad Hassan and Rachel Rosen 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. 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.

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

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

As in standard general relativity, the metric $g_{\mu \nu }$ has an Einstein-Hilbert kinetic term proportional to the Ricci scalar $R(g)$ and a minimal coupling to the matter Lagrangian ${\mathcal {L}}_{\mathrm {m} }$, with $\Phi _{i}$ representing all of the matter fields, such as those of the Standard Model. An Einstein-Hilbert term is also given for $f_{\mu \nu }$. Each metric has its own Planck mass, $M_{g}$ and $M_{f}$. The interaction potential is the same as in dRGT massive gravity. The $\beta _{i}$ are dimensionless coupling constants and $m$ (or specifically $\beta _{i}^{1/2}m$) 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 $e_{n}$ of the eigenvalues of the matrices $\mathbb {K} =\mathbb {I} -{\sqrt {g^{-1}f}}$ or $\mathbb {X} ={\sqrt {g^{-1}f}}$, parametrized by dimensionless coupling constants $\alpha _{i}$ or $\beta _{i}$, respectively. Here ${\sqrt {g^{-1}f}}$ is the matrix square root of the matrix $g^{-1}f$. Written in index notation, $\mathbb {X}$ is defined by the relation

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

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

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

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

See also

• Accelerating universe
• Alternatives to general relativity
• DGP model
• Scalar–tensor theory
• Gravitational interaction of antimatter
• Symplectic geometry
• general relativity
• Dirac equation
• Quantum field theory
• gravity
• Schwarzschild metric
• Alternatives to general relativity
• Accelerating expansion of the universe

External links

• Janus cosmological model : summary and main papers
• short synthetic sheet about bimetric gravity theories

Notes

Some of the elements of this article have been translated from the french Wikipedia article.