***I'm new. Been lurking for a bit. I was pleasantly surprised at the civility/intelligence of this forum, even in ATM threads, so I decided to give it a shot. The model presented here isn’t mine, I simply discovered it through an odd series of clicks (much like cosmoquest actually). There is a good bit published on the topic, in peer reviewed journals (http://www.jp-petit.org/papers/cosmo), two educational comics (http://www.savoir-sans-frontieres.co...than_light.pdf http://www.savoir-sans-frontieres.co...n_universe.pdf), a website http://www.januscosmologicalmodel.com and a 20-something part series of youtube lectures in french with English subtitles (https://www.youtube.com/watch?v=kYIu...LaK0-2_A_TQrKB). Still, from what I’ve seen the model hasn’t gotten any traction in the mainstream scientific community due to the creator (Jean-Pierre Petit) having advocated quite strongly for some less-than-scientific ideas regarding extraterrestrials (specifically Ummites). As I am not able to find major problems with the model itself, I have been seeking out individuals more intelligent (or at least more astronomically literate) who might find and falsify what I couldn’t. After all, I wanna make Popper proud.Preface***

JCM is a bimetric conception of gravity based on two coupled field equations which take into account both positive and negative masses. It’s the only sensible way I’ve heard negative masses and negative energies talked about, but including those ideas at all puts you firmly against the mainstream. The negative mass it claims to deal with is already present and we can observe its effects, we just haven't explained it as negative mass yet. This shift in perspective, according to Petit, solves or makes irrelevant at least a dozen major cosmological issues including but not limited to: the nature of dark matter/energy, the isotropy of the CMB, the missing baryonic matter, the behavior of the early universe (an alternative to inflation), an explanation for what happens to matter that enters a black hole, etc.

Obviously I’m skeptical.

Though, to his credit, all of these appear to follow directly from Petite’s bimetric conception of the universe (and his idea of a variable speed of light as a joint variation of all physical constants following a universal gauge relationship). He doesn’t need to appeal to anything except observational data, his model, and a bunch of crazy maths I can barely follow. But then again... I can barely follow it.

***This is synthesized from a bunch of places, but it's a lot of copypasta- I hope that's okay.Foundation***

The precedent for negative mass begins with the Poincarė group and David Hilbert. In quantum field theory, the T operator acting on Hilbert spaces is complex, and can be either linear and unitary, or anti-linear and anti-unitary; but is arbitrarily chosen anti-linear and anti-unitary 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. However, when this axiom was formulated the accelerating expansion of the universe, which implies a negative pressure (pressure being understood as a volumetric energy density), had not been recorded, therefore it seems sensible to reconsider how we view negative energy states.

According to Jean-Marie Soureau’s dynamical group theory, the action of elements of this group on movements in space-time result in some movements which follow a time reversal; they appear to be going "backwards through time". Modern physicists use the restricted Poincarė group, limited to only orthochronous movements, and therefore ignore this issue. In group theory, however, the T operator is real and we can explore retrochronic movements. Smarter people than I have shown, (http://ayuba.fr/souriau/Souriau-time...-inversion.pdf) using maths I can only begin to understand, that when using the full Poincare group time reversal goes with mass and energy inversion. So at least the precedent for negative mass is there, determined mathematically by an independent researcher.

I’ve developed an illustrative way to explain this concept of negative mass in a bimetric universe using the classic rubber sheet example that Sagan and others have adopted over the years, but given the character constrains I'll leave it. You're all smart, you'll get it.

So if we want to consider the universe this way we need two metrics, g(+)µν and g(−)µν, from which two different families of geodesics are calculated, referring to positive mass particles and negative mass particles respectively. From these metrics, we calculate Ricci tensors R (+) µν and R (−) µν as well as Ricci scalars R(+) and R(−) .

Personally, I prefer to refer to "negative mass" as "inverted mass" simply because, the way that this model requires you to look at it, negative mass is not an intrinsic feature of some exotic matter, but instead stems from topological consequences; it is purely relative. The "bigravity" associated with inverted masses 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:

where χ is Einstein's constant

Notice that if there is very little negative mass, the negative energy-tensor, T(-)µν, approaches zero and the upper equation for space-time according to the positive metric gets closer to Einstein's original field equation with a cosmological constant of zero. Thus we see that the two coupled field equations reduce to the Einstein field equations of general relativity for regions of space-time where positive mass largely dominates. Therefore this model automatically fits with local relativistic observations and measurements without any ad hoc changes in the same way GR reduces to Newtonian mechanics with small gravitational potential and at low velocities relative to c.

I can't go through the proofs for the field equations here, but there are Lagrangian Derivations (https://www.researchgate.net/publica...ological_model) as well as the original formulation (http://www.jp-petit.org/papers/cosmo...voCimentoB.pdf) for the extremely invested.

You may be asking about the bottom equation for space-time according to the negative metric. Wouldn't this mean the same material can be forced to follow two different equations of motion? No, the second equation gives the metric and spatial curvature for the "negative sector" of the manifold. This is the equivalent to being on the balloon side of the sheet. Positive mass (marbles), viewed from the underside of the sheet, will produce a negative curvature and an anti-gravitational effect. The system of two coupled field equations involves a 4D 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 space-time 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). To reiterate, "negative mass" is not an intrinsic feature of some exotic matter. According to an observer measuring any mass in the same sector where it lays, that mass always appears positive. Fundamentally, a mass is not "positive" or "negative"; it is both. Its sign depends on from "which side" we are measuring that mass. It is therefore a purely relative concept.

There’s a lot to unpack, and it makes sense to talk about the observational evidence for negative masses, but there's a character limit so I may not do it justice. The 1995 paper goes far deeper into this.

***Evidence***

Consider the three boxes above. Current cosmological models insist there's far more dark matter than regular matter. If visible matter takes the pattern of the far left box, we can assume there is more inverted mass in the area. If visible matter takes the pattern of the middle box, we can assume there is more regular mass in the area. For reference, and to illustrate the reality, the far right box has been included. This would be the full distribution of all matter on both sides of the manifold, assuming there is a large difference in how much mass/energy is present in either sector.

Standard models of the universe describe large galaxy clusters linked by hundred-lightyear long threads of galaxies surrounding massive empty voids. This can be pictured like bubbles from a pot full of dish soap. Planes of soap would be fairly rarified areas of matter, lines would be the galactic threads, and vertices of the bubbles would be galactic clusters. The ΛCDM model proposes "galactic filaments" which accrete around threads of dark matter that I have yet to see anyone convincingly locate.

When we look at the universe on a grand scale, it seems to imply there is a greater distribution of inverted masses due to the filamentous shape (like the left box). But if we deign to zoom in on a section of the universe, say our solar system or galaxy, we find something like the middle box. What this implies is that the third box describes reality at both the grandest and more local cosmic scales- but at some point they switch. So on the universal scale the dark clusters are inverted mass matter pushing regular matter into filaments, but on more local scales we consider the dark clusters galaxies held in rotation by the inverted matter surrounding them. The fuss made by Dr. Farnes’s negative-mass fluid (which I’m sure took this place by storm) does a similar, but less ontologically justified, thing by nestling galaxies in a field of repulsive material.

"Ah, Okay," I can almost hear you mutter, "but what about gravitational lensing?”

This illustration shows the effects of negative and positive lensing. You can mimic it at home with two paper circles, a pair of scissors and some tape if you have them.

This negative curving would, in the presence of a large hole empty of any matter, give a focalization effect. This focalization would be reinforced by a positive mass in the center of that empty space, accounting for the massive discrepancy between apparent matter and matter required for strong [positive] gravitational lensing. An illustration follows:

In the JCM, there is no cosmological constant because it’s unnecessary. The math is well beyond me (found https://januscosmologicalmodel.com/p...ysSpaceSci.pdf in section 6,non-steady state solution for dust universe). It seems the identity of matter/energy in the negative sector is sufficient to calculate the deceleration parameter, which Petit gives as −0.087 ± 0.015. The JCM can calculate this due to precise knowledge of what comprises dark matter, which is determined using group theory. A recently published peer-reviewed article (http://www.jp-petit.org/papers/cosmo...ysSpaceSci.pdf) shows how the Janus Cosmological Model is compatible with modern observational data.

There's a lot more, but let's start here.