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Lorentz Ether Interpretation of the Einstein Equations of GR
There exists a simple extension of the Lorentz ether to gravity - a Lorentz ether interpretation of the the Einstein equations of General Relativity (GR).  I give an introduction on my website.

The basic formulas of this interpretation are:
  • There exists a global absolute space \(\mathbb{R}^3\) and absolute time \(\mathbb{R}\), which are defined by preferred coordinates \(X^\mu\) with \(X^0=T\).
  • The preferred coordinates are defined by the harmonic coordinate condition \[\square X^\mu = \partial_\nu (g^{\mu\nu}\sqrt{-g}) = 0,\]which, together with the Einstein equations, and the usual general-relativistic equations for matter fields, define the physical equations of the theory.  
  • The gravitational field defines: 
    • the density \(\rho(X,T)\) of the ether by \(\rho=g^{00}\sqrt{-g}\);
    • the velocity \(v^i(X,T)\) of the ether by \(v^i=g^{0i}/g^{00}\),
    • the stress tensor \(\sigma^{ij}(X,T)\) of the ether by \(\rho v^i v^j +\sigma^{ij} = g^{ij}\sqrt{-g}\).
  • This identification transforms the harmonic conditions into continuity and Euler equations for the ether:
    \[\partial_0 \rho + \partial_i (\rho v^i) = 0\]\[\partial_0 (\rho v^j) + \partial_i(\rho v^i v^j + \sigma^{ij}) = 0\]
The ether density \(\rho=g^{00}\sqrt{-g}>0\) has to be positive. This is equivalent to the requirement that the preferred time coordinate T is time-like.  

Let's note that this generalization of the Lorentz ether to gravity removes some important arguments against the ether.  
  • First of all, the argument which, it seems, has killed ether theory:  That there was no viable ether theory of gravity.  There was a theory which has developed the Minkowski spacetime to a curved spacetime - general relativity - but no ether-theoretical alternative. That the harmonic coordinates, which are used in this ether interpretaion, essentially simplify the equations of GR, was known, and they have always been popular. But that they could have been easily used to define an ether interpretation remained unknown.  But if there was no ether theory of gravity, but a successful spacetime theory of gravity, was a very strong, a decisive argument against the ether.  
  • A known argument against the Lorentz ether was that he was static and incompressible.  The ether of this interpretation is no longer static and incompressible.  Instead, we have classical equations, known from classical condensed matter theory for compressible media, for this ether.
  • Given that the ether fulfills continuity and Euler equations, which are classical conservation laws for mass and momentum, there is no interaction of the ether with other matter - else, an exchange of momentum between ether and other matter would lead to additional terms in the Euler equation.  This leads to an interpretation of all matter fields as fields describing other properties of the ether itself.  But this removes yet another argument against the ether - that, on the one hand, he should be quite rigid, but, on the other hand, we do not feel any resistance if we move.  But waves of the ether should not feel any resistance by the ether.

It should be noted that this ether interpretation of the Einstein equations is not completely equivalent, as a physical theory, to general relativity. Here are some differences:
  • GR allows solutions with nontrivial topology. The most well-known example are wormholes.  Another important one a spherical closed universe. These solutions do not allow an ether interpretation - the ether interpretation requires a flat background with trivial topology \(\mathbb{R}^3\times \mathbb{R}\).
  • GR allows solutions with closed causal loops.  The most well-known example is the Gödel solution for a rotating universe. But in some variants of the Kerr solution for rotating black holes there are also parts with closed causal loops.  These solutions do not allow an ether interpretation: The preferred time T is time-like everywhere.  Such a global time-like function does not exist for solutions with closed causal loops.  An ether interpretation would be possible only for parts of such a solution. For other parts, the ether interpretation would be meaningless because the ether density would have to be negative there.  
  • Another unexpected difference is that the solution of some part of the complete GR solution may already complete as a solution of the ether interpretation.  The point is that they have different notions of completeness.  A solution of ether theory is complete if it is defined for the whole spacetime \(-\infty < X^i, T <  \infty\).  A GR solution is complete if it is geodesically complete.  These are different notion.  And there are also important physical examples:  The first is the gravitational collapse, where the part before horizon formation is already complete.   The other one the Big Bang. It is incomplete, singular from point of view of GR, but harmonic time goes for usual matter to \(-\infty\).  Above cases illustrate the conceptual difference between GR proper time and the absolute time of the ether.  In the ether interpretation, "proper time" is interpreted as clock time, shown by clocks which are distorted by the ether.  If such a clock stops completely, forever, it does not mean that something will happen to this clock after infinity, which is defined by absolute time. 
From point of view of Popper's criterion of empirical content, these differences give additional possibilities to falsify the ether interpretation in comparison with the spacetime interpretation. (So, in a strong sense, these two interpretations could be considered as two different theories. But, given that these differences are only minimal, I will name them, nonetheless, interpretations.)  And, once these are additional possibilities to falsify the ether interpretation, this gives additional empirical content to the ether interpretation. Thus, from point of view of Popper's criterion, the ether interpretation is preferable, because it has higher empirical content.  

Of course, this holds only until this additional empirical prediction does not actually falsify the ether interpretation.  But, up to now, this has not yet happened:  We have not observed no nontrivial topologies like wormholes, nor closed causal loops.  

Beyond the experiment, there are also other, theoretical arguments in favor of interpretations.  Namely the compatibility with other, independent physical principles.  Here, the following differences exist:
  • With absolute space as a background and absolute time, the ether interpretation does not have a "problem of time" of quantization.
  • Once the ether theory has a preferred frame, causality is connected with this preferred frame.  So, the ether allows faster than light hidden causal influences.  This makes the ether interpretation compatible with the violation of Bell's inequality.   
I am a layman with no background in science or maths.
Is there anything you can offer by way of a simple description of the ether.
How does the MM experiment fail to detect an ether?
I will try. The simplest answer is that the Lorentz ether has the same mathematics, the same equations, as special relativity. And, similarly, this ether interpretation of the Einstein equations of GR also uses the same equations as general relativity - the Einstein equations. From the same math it follows that the physical predictions are the same too. What is predicted by relativity, is a consequence of the equations used by relativity, thus, once the same equations are used, the predictions are the same too. But, once the predictions are the same, there is simply no possibility that relativity survives the MM experiment but the ether interpretation not.

But this argument holds only as long as the math is really the same. It does not hold for other ether theories, which use equations which are different from the Einstein equations. Whenever an ether theory would have different equations than relativity, one would have to check if it survives the MM experiment.

And, in fact, it is not even completely correct: The math is completely the same only for special relativity. For general relativity, the equations are the same - but there are some additional assumptions, namely that there is some global preferred frame with some special properties. This excludes some solutions of GR, namely solutions with a nontrivial topology, that means, with wormholes and so on. And it also excludes solutions like Gödel's rotating universe solution, which have causal loops, because it has no preferred global time coordinate. So, if one would observe wormholes, or causal loops, the ether interpretation would be falsified, while GR itself would not have a problem.

But MM does not fall into this category. It is simply some local measurement, in a situation where to define a global time coordinate is unproblematic, so that there is no difference to SR or GR in the equations.
Thank you for taking the time to help me.
I seem to detect that supporters of GR reject the proposition of an ether with some vigor.
I think I understand that it is not needed but some reject the very existence of a physical ether.
Do you have a description of the ether in a physical sence.
What is its composition, for example is it a mix of photons and neutrinos and other known particles or a composition of "ether" particles hypothesised for the purpose?
Is there any reason why mainstream dislikes the concept and even the use of the word ether.
It is a composition of some completely different objects.

Quantum condensed matter theory gives an example: A solid, say a crystal, consists of some atoms, a lattice of atoms. There are sound waves - oscillations of these atoms. The sound waves distribute because the atoms are connected with each other by forces.

But if we consider the large picture, we can describe the sound waves by continuous fields - density, average velocity, stress tensor and so on. And we have macroscopic wave equations. And then we have quantum effects of these wave equations. Mathematically this is similar to atomic energy levels, where we have discrete energy levels. If we restrict ourselves to waves of a particular frequency, we also obtain discrete energy levels. Moreover, these different energy levels follow a very simple rule, namely the n-th energy level has the energy \(E = E_0 + n E_1\). So, we have the same energy formula which we would have for n independent particles with energy \(E_1\). Of course, sound waves are not particles, so these sound waves with discrete energies have been named "quasi-particles": Roughly, one can use the same mathematical formulas for them as those used in particle theory, even if one recognizes that these are not really particles.

And my ether theory interprets now the "particles" of the standard model of particle physics as such quasi-particles - thus, not really particles, but quantum excitations of sound waves of the ether.
I read in one of your other posts that your ether has a velocity. Is that correct?
GLET makes sense except for one thing: the ether. (I see no problem at all with the preferred frame, although many of today's physicists probably would focus on that.) I understand not all details of ether are worked out and it's a large, ongoing effort, but would like to ask a question. First let me summarize the situation, tell me if this is correct.

The ether in GLET is essentially all there is in the universe. Everything else is modelled as aspects of this ether.

Fermions are essentially phonons in the ether.
Bosons are something similar but gauge fields are not yet entirely understood.
Electromagnetic waves travel in the ether medium; this is the original role of ether, back in 1900.
Gravity waves also travel in the ether.
Gravitational field g is represented in a fairly standard fashion as density, velocity and stress tensor of the ether.

I'm not sure about electroweak and strong forces, but suppose they must be expressible as an aspect of ether as well? But that's not an important issue.

Note that all these things are found in conventional physics but they avoid a lot of problems by ignoring any substrate, or medium, for EM waves, fermionic fields etc. From the ether point of view this is "cheating" - how do these waves travel with no medium? Gravity and gravitational waves are considered curvature of "spacetime" and "ripples in spacetime". From ether point of view, this "spacetime" is nothing but the ether, poorly understood.

But there's a reason conventional physics just ignores the issue: it seems almost impossible to solve.

One small point: grav waves are quadrupolar, EM waves dipolar. How can the same medium support both types? I see this can be handled, but the solution seems ad hoc.

Anyway here is one key question I'd like to understand. First, note ether must be condensed matter (of some sort) if only to support transverse waves.

Now GLET includes expansion of the universe. Evidently, originally (about 13.4 billion years ago) there was a very condensed ball of ether less than about 13 million light years across. (It must have been small enough to allow for a surface of last scattering, to produce CMBR.)

Since then this ball of ether has expanded more than 1000 times - smoothly, and more-or-less linearly. Note this expansion is only in 3-d Euclidean space not the usual 4-d Minkowski. The time axis didn't expand, rather it is (has always been) absolute and fixed.

Now here's my question. Such a smooth expansion can happen only with a gas, not with condensed matter! But ether must be condensed matter as noted above. So how to reconcile this discrepancy?

I'm aware of a couple possible solutions; also some points I may be misunderstanding that would resolve the issue (but bring up other questions). But let me just leave it at this. If you will please consider my statements, (briefly) point out misunderstandings, and try to answer the question, I would appreciate it, and can take it from there.
(05-09-2016, 02:36 AM)secur Wrote: GLET makes sense except for one thing: the ether. (I see no problem at all with the preferred frame, although many of today's physicists probably would focus on that.)

The ether in GLET is essentially all there is in the universe. Everything else is modelled as aspects of this ether.
Fermions are essentially phonons in the ether. Bosons are something similar but gauge fields are not yet entirely understood.
I'm not sure about electroweak and strong forces, but suppose they must be expressible as an aspect of ether as well?
GLET itself is only the theory of gravity, but it indeed prescribes that all the other matter fields have to be some sort of sound waves of the ether. It is this property which allows to prove the Einstein Equivalence Principle.

What properties of the ether are described by the other fields and forces is what I have tried to identify in my ether model for the standard model. I think, the gauge fields are already understood in a sufficiently good way, even if there are weak places - but, last but not least, I have been able to derive, essentially, the gauge group of the SM as being one of the greatest possible one.

What is not understood is the Higgs sector and dark matter. There are some candidates for scalar fields in the model, so there are candidates for Higgs fields as well as dark matter, but this situation is far from being clear.
(05-09-2016, 02:36 AM)secur Wrote: Note that all these things are found in conventional physics but they avoid a lot of problems by ignoring any substrate, or medium, for EM waves, fermionic fields etc. From the ether point of view this is "cheating" - how do these waves travel with no medium? Gravity and gravitational waves are considered curvature of "spacetime" and "ripples in spacetime". From ether point of view, this "spacetime" is nothing but the ether, poorly understood.
I would not say cheating. I would say that one loses important possibilities to understand nature if one does not try to interpret the fields we observe in terms of what they describe on the fundamental level. One could have developed thermodynamics as some "field theory", with a "temperature field", a "pressure field" and so on, without searching for a fundamental understanding of what these fields describe - properties of a medium consisting of atoms. That we have gone beyond this "field theory" version of thermodynamics is the greatest success of fundamental science which has been reached up to now, the atomic theory.
(05-09-2016, 02:36 AM)secur Wrote: But there's a reason conventional physics just ignores the issue: it seems almost impossible to solve.
I disagree. The problem why ether theories have not been studied is a strong prejudice against such theories. I have found them alone, without even the support of a small group, a larger group with some specialists in field theory and renormalization techniques could have reached much better results in much shorter time. So I can tell you this was easy enough.
(05-09-2016, 02:36 AM)secur Wrote: First, note ether must be condensed matter (of some sort) if only to support transverse waves.
Now GLET includes expansion of the universe. Evidently, originally (about 13.4 billion years ago) there was a very condensed ball of ether less than about 13 million light years across. (It must have been small enough to allow for a surface of last scattering, to produce CMBR.)
For the expansion of the universe you have to distinguish between distances measured by rulers and absolute distances of the background. (Same for clocks vs. absolute time, that they are different is already part of the old Lorentz ether.) The expanding universe is described by a variant of the FLRW ansatz, one where time is harmonic (this is what the equations require from absolute time):

\[ ds^2 = d\tau^2 -a^2(\tau) (dx^2+dy^2+dz^2) = a^6(t) dt^2 - a^2(t) (dx^2+dy^2+dz^2) \]

Then, the ether density is defined by \(\rho = g^{00}\sqrt{-g} = 1\), so that the density is constant, and the ether velocity is \(v^i = g^{0i}/g^{00}=0\). So the ether does not expand, but the stress tensor changes, and, as a result, the rulers are shrinking.

The classical condensed matter example for something similar is the size of vapor bubbles near the boiling point - they becomes greater and greater near the boiling temperature. Similarly one can think about us as the analogs of such vapor bubbles which are shrinking once we move away from the boiling point of the big bang.
Thanks Schmelzer, that makes sense; I suspected it might be something like that.

Volovik [1] and similar papers are very interesting; superfluid properties are exactly what's needed for an ether theory. I thought ether "impossible" because I was looking at it from Lorentz's view. At that time it was clear ether couldn't be any known state of matter, so it took a real leap of faith to believe in it. But now it can be modelled as something that really exists, and can be played with in the lab. Suddenly it's much more believable.

Here is an overview of ether theory as I understand it. Note it goes beyond only GR, the topic of this subforum; but I don't know where else to put it.


There is a permanent, fixed 3-d Euclidean space of unknown dimensions; we can imagine it's "infinite". There's a 1-d time axis, increasing linearly. So the 3-d framework evolves regularly in time. At any instant there is a state of the universe.

This 3-d space is filled by ether, some sort of condensed matter. Not sure if it should be conceived of as a liquid; I think of it as gelatinous. Its properties are similar to He3 superfluid (bosonic Cooper pairs of supercooled He3). He3's properties happen to be closest to what ether theory requires; I guess any superfluid, like He4, is similar. He3 properties may need some modification to match our universe exactly; no reason ether must be exactly like He3.

Ether is a bit like "Dark Matter", and indeed could fill that role. No one has yet worked out whether it can match galactic rotation curves, and supercluster "excess gravity", but it might. It could make "Dark Energy" superfluous also. That's not important at the moment; the reason I mention DM: it's supposed to be non-baryonic matter with no electromagnetic or strong force coupling, just gravity and weak. That's why it fills our space, but we can't sense it. So it's natural to model ether as similar non-baryonic matter. This is another advance over Lorentz's day. Back then they couldn't explain how ether could be unnoticeable; today we have this clear understanding of how forces work, and the vital fact that matter must couple to a force to be noticed.

So, ether is non-baryonic matter with superfluid properties, in Euclidean 3-d space evolving along a linear time axis.

Modern physics asks questions like "what is time?", "what is space?", "why the arrow of time?", "Where did it all come from?", "Why does anything exist?", "Why does the world behave mathematically?", "What is consciousness?" An endless parade of philosophy masquerading as science. In ether theory all these (except, indeed, consciousness) are dealt with in this initial framework. This doesn't answer - in the deep, philosophical sense - what time is, why it goes from past to future, or any of that. What it does do is postulate, at the very beginning, a clear picture that encompasses all such issues. The picture is simple, complete, comprehensible, effective and useful without further cogitation. Metaphysics can be left for philosophy, where it belongs. BTW I don't mean to imply these questions aren't important - they are very important - but they're not, as far as I know, science.


Ether theory papers I've read address fermionic matter and bosons; and the forces gravity and EM. I don't see ether theorists explicitly addressing weak or strong force, mass, charge, Planck's constant, wave function, quantum fields, Thermodynamics, and many more topics. But if we thoroughly understand the particles and the forces, we probably understand all the rest of physics as well. Some of these topics will probably require special attention, however.

So how does ether theory explain, and how does ether support, the particles and forces? I'm not on top of it, but get the basic idea. All particles (or fields - equivalent concept) are "sound waves" (phonons) in the ether. Nothing exists but ether! Schmelzer's paper "A CONDENSED MATTER INTERPRETATION OF SM FERMIONS AND GAUGE FIELDS" [2] is a detailed look at how fermions work, with a sketchier idea of bosons. It draws on Volikov et al's work on superfluids. At this point I have to take on faith (subject to later correction) that phonons in He3 really do behave rather like our familiar particles. Schmelzer's paper "A GENERALIZATION OF THE LORENTZ ETHER TO GRAVITY WITH GENERAL-RELATIVISTIC LIMIT" [3] gives details on how gravity works in ether. Basically the phonons (particles) move along geodesics determined by the stresses and strains of the ether. These equations, for He3, happen to be identical to Einstein's Field Equations from GR!

EM and the other forces can be modelled similarly but there's still a lot of work to do. It's very promising but ether theory could run into a show-stopper as these details are worked out. That's another wonderful thing about it, as compared to string theory, quantum loop gravity, supersymmetry etc. Ether theory is falsifiable. It presents a quite specific mechanical model for the universe; if it doesn't work it can't be easily fixed with a few ad-hoc parameters. If it needs a little of that to get by, that's alright; something can be left for future researchers. But if there are more than just a handful of problems, the theory will be rejected. A layman probably sees that as a bad thing, but in fact it's very good: Ether theory can be proven false! Of course, let's hope it won't be.

I'll give one example of the superiority of this type of model over today's physics. If you concentrate a lot of energy in a small space a "zoo" of particles will appear from the vacuum. This "E=Mc2" trick takes place constantly in all the matter around you, but is most noticeable at very high energies (like LHC at CERN). In the standard picture it "just happens". Why do electrons (for instance) appear, with 511 KeV rest mass, but not some other rest mass, or some other properties like 1/2 charge? Because the vacuum is capable of producing just those particles. Why? No answer. Since we know experimentally what particles actually exist, physicists put them into the vacuum (with suitable creation operators) "by hand". OTOH in ether theory, when energy appears in the medium, it causes well-understood, mathematically precise, excitations - waves - which travel away from the disturbed region according to specific He3-like mechanics. Admittedly there's room for some adjustments ("fudge factors"), to match experimental data, but not much. We can hope the ether model will predict exactly the particles we see created in an energy event such as protons colliding at the LHC. It can provide a mechanical, intuitively understandable model explaining why we see those particles and no others. This is vastly superior to the current approach - assuming it works.


First some basic definitions. Time "t" is the absolute time of the underlying evolution of the 3-d frame. I'll call our perceived time "tau". t is unknowable to us: it's a hidden variable. tau is exact only when we're at rest in the absolute space frame. That frame is the one where CMBR, and the universe's distribution of matter, are isotropic. I don't know if this is part of standard ether theory but it should be. Earth is not far from that frame (600 kps) so Earth's time is (almost) tau. The underlying spatial metric of the 3-d frame is also a hidden variable (3-vector). We can determine how our x, y, z differentials change as time goes on but can't relate them to the underlying metric, except proportionally. While the underlying x, y, z, and t are perfectly linear, the rulers and clocks we experience are not.

The early history of the universe ("Big Bang") is based, minimally, on three types of observations: galactic redshifts ("expanding universe"), CMBR, and elemental abundances. Here is ether theory's explanation.

About 13.7 billion years ago (in our current clock time) the ether was in a state of very high energy. This means that the disturbances representing matter were very large, and that clocks ran slower - I don't actually know why. Presumably it makes sense in the light of superfluid properties. Schmelzer writes a metric evolution equation identical to normal FLRW except for one thing: time differential goes by the cube of a(t). In terms of our current metric, then, space metric has been shrinking linearly (proportional to t) while tau has been shrinking by the cube of t. That's why it appears the universe is expanding: it's not; rather, we're shrinking. The most important effect is that speed of light has been slowing since the "Big Bang" by a factor of ... I'm not sure, I think it goes by the 4th power. (There are many details I haven't gotten straight; this overview is the best I can do at the moment.) The effect of this metric evolution matches observational evidence, such as Hubble's constant. (At least I hope it does.) This metric is not ad hoc; it's inevitable (I hope) from ether properties, modelled on He3 superfluid.

In the beginning of this scenario, the perceived universe radius was less than 13 million or so light-years - in terms of today's metric. But in terms of light-years at that time, it was much smaller. That takes care of the so-called "horizon problem": evidently there's no need for inflation. The universe could not have been a whole lot smaller: there was no singularity. Presumably this is a natural consequence of He3-like properties. BTW this is yet another benefit of the model: no singularities happen: no wormholes, time-travel, or any of those uncomfortable consequences of GR. However the universe was small enough to account for our observation of the surface of last scattering, the CMBR. One problem: I don't think this model gets close enough to Big Bang thermal conditions to account for elemental abundances of H, He, Li etc. No doubt there's some way to handle that.

Since then, the ether has cooled. Where did the energy go? Perhaps, into the outer ether which is beyond our horizon; or, into internal fermionic degrees of freedom; or, ? From our point of view this looks like the expanding universe we see, with galactic redshifts. Schmelzer mentions that he favors a type of "Big Bounce" scenario. When expansion reaches some point it turns around and contracts, until it's again about 13 million light years; then bounces back. I have no idea why it would turn around, which corresponds to the ether heating up again after cooling for a long time. How about the problem of galactic formation? As with galactic rotation curves and perhaps a million other details, I don't know.


That completes the rough outline of ether theory as I understand (or, not) it. Many more details have been worked out, including relativistic behavior. Many more haven't been touched, such as Planck's constant (I imagine it's related to the size of ether atoms). Ultimately every aspect of known physics must be explained in the ether model: a large effort is needed. The surface has, in my estimation, been more than scratched, but a lot more digging remains.

A final question, how do we get from here (a good beginning) to there (a complete, validated, accepted theory)? The key: participation of more physicists. I want them to read Volovik et al, and Schmelzer, and tell us what's wrong with them. Then let Schmelzer et al answer. Let competent people debate both sides, the truth will become apparent. How to make that happen? This web site, "Hidden Variables", is a beginning. Further steps are a beyond the scope of this post.

[1] G.E. Volovik, Induced gravity in superfluid 3He, J. Low. Temp. Phys. 113, 667-680 (1998), arXiv:cond-mat/9806010
[2] I. Schmelzer, A Condensed Matter Interpretation of SM Fermions and Gauge Fields, Found. Phys. 39, 73-107 (2009), arXiv:0908.0591
[3] Schmelzer, I.: A generalization of the Lorentz ether to gravity with general-relativistic limit. arXiv:gr-qc/0205035 (2002)
Thanks, a nice text. Some notes:

The interesting point of Volovik's \(^3He\) research is that it gives some qualitative insights. It is very good and useful research, but I have not used it. So there is nothing in the evolution of the cosmos "modelled on \(^3He\)".

In the ether interpretation, the ether has a stress tensor or pressure tensor, instead of simply some scalar pressure. This is what one has to use in condensed matter theory to define solids.

About dark matter: There appear some fields in the ether which may play the role of dark matter. For every electroweak pair of fermions there has to be a massive scalar field with much greater mass and similar color. Those corresponding to leptons would not have any interaction at all, thus, would be good candidates. But the ether as a whole differs from dark matter - it is everywhere. There can be places without dark matter, but not without the ether.

The metric is not a hidden variable. What is hidden is what makes the coordinates I use preferred coordinates. But they are valid coordinates and the can be used in standard cosmology. And, in fact, the spatial coordinates are the ones which are used. No GR theoretician has anything to object if I use these coordinates - except that he can ask the question "why these? why not others?" And the spatial part of the metric - as visible as all other parts of the metric \(g_{\mu\nu}(x,t)\) - is not a 3-vector, but a 3-metric, a symmetric tensor, with two indices \(g_{ij}(x)\) instead of one \(a^i(x)\) of a vector field.

With a small enough parameter \(\Upsilon>0\) the solution gets as close to the Big Bang solution as one wants.

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