dear Hossein, a little bit apart from your question, I asked myself how could a natural constant take 3 different values: -1/+1/ or zero!?

Dear Paul

If I understood your question, these are relatives to Euclidean and Non-Euclidean Geometry of the universe.

zero for Euclidean Geometry (flat space), -1 for (open Universe/hyperbolic space) and +1 for (closed Universe/spherical space).

The value of the cosmological constant defines the vacuum energy.

Dear Stam

Surely, so we need to describe vacuum energy. What is your suggestion?

dear Hossein, that`s correct.

Hi dear Paul

Thank you so much.

This is an subject that has been deeply analysed. I recomend the study of Joan Sola , J.Phys.Conf.Ser. 453 (2013) 012015 , arXiv:1306.1527v3

Dear Mario

Thank you for interesting article.

In the six-dimensional cosmology, the rate of expansion of a three-dimensional universe does not depend on the density of matter in it. See

I. A. Urusovskii, Multidimensional Treatment of the Expanding Universe,

Physical Science International Journal, 4(8): 1110-1144, 2014.

I. A. Urusovskii, RADIATION SPECTRUM OF THE HYDROGEN ATOM WITH THE ACCOUNT OF MOTION OF ITS ELECTRON IN THE EXTRA SPACE, Journal of Modern Physics Vol. 8, No. 12, 2017.

Best Regards, I. A. Urusovskii

dear Igor, You are right. In my opinion evemn chandrasekhar border is superfluos.

Once more: the value of the vacuum energy of a gravitational system is given by the value of the cosmological constant. This is just a straightforward consequence of the definitions of what's vacuum energy and what's the cosmological constant.

they say that cosmological constant can have 3 different(!!) values! one constant: 3 values!? so it might be, it is postulated that the values -1,+1, or 0 do correspond to euclidian or non-euclidian geometries, but the euclidian space seems to be curved here and there (in black holes, singularities space could be hyperbolic, aso.) while space semms to be a mix of euclidian and non-euclidian geometries (so the big-bang theory). strange! very strange!! one constant has 3 values!?

Paul> they say that cosmological constant can have 3 different(!!) values! one constant: 3 values!? so it might be, it is postulated that the values -1,+1, or 0

No Paul, you are confusing two very different things. In the Friedmann-Lemaitre-Robertston-Walker Cosmological Model, https://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–Walker_metric, there is indeed a parameter k which can take values +1, 0, 1, describing a spatial geometry which is respectively spherical, flat, or hyperbolic; the three possibilities consistent with homogeneity and isotropy (position and orientation invariance). However, they combine with a scale factor which currently is so large that these possibilities cannot be very clearly distinguished observationally (like a small section of the Earth's surface looks pretty flat, in particular to flat-earthers).

The cosmological constant can in principle also be negative, zero or positive. Observations indicate that it is slightly positive, but of much smaller magnitude than one would naturally expect.

In the original formulation by Einstein the cosmological constant had to be truly constant, for consistency with the Einstein equations (without destroying the cherished principle of covariant conservation of the energy-momentum tensor). In other models, like models for the inflation phase of the universe expansion, there is another ("vacuum energy") contribution acting like a cosmological constant, but with the possibility of being time dependent. The latter is necessary to make the inflation phase end. Hence, it is usually not denoted cosmological constant, but quintessence instead https://en.wikipedia.org/wiki/Quintessence_(physics). I believe the term dark energy, which appears to make up about 70% of the mean energy density of the universe, is meant to cover both of the above possibilities, plus additional ones.

tldl: Cosmological constant and vacuum energy have similar effects on dynamical evolution of the universe, but they are not necessarily related.

Kare, ok, I agree! but, . I have the demonstration that ALL natural constants, are not constant indeed: they are time-dependent.

While history is fascinating, it isn't relevant for the technical issues. Once more: When coupling a classical matter system to gravity, the vacuum energy of the matter system is equal to the vacuum energy of the gravitational system, which is equal to the cosmological constant.

Dear Stam

It is true, the problem is energy distribution in vacuum and the mechanism of how gravitational energy is converted into electromagnetic energy.

.to G)?..and what about inflation, or expansion (which shouldn`t be equivalent

Inflation doesn't have anything to do with either Newton's constant or with the cosmological constant or with the vacuum energy; so it doesn't have anything to do with the question asked.

And the reason is that inflation isn't described by general relativity. The period after inflation is described by general relativity, when the only remnant of inflation is the inflaton field, a scalar field, with its potential.

Vacuum energy should partition into potential and kinetic parts that are almost equal in nearly flat space. The Lagrange Density is near zero and defines a small cosmological constant no matter how large the total energy is.

exactly that`s the point! And relativity has n

The separation of energy into kinetic and potential is a non-relativistic approximation.

In flat space the cosmological constant is zero-and that can be seen as a definition of flat space, in fact. The spacetime of our Universe isn't exactly flat, since the cosmological constant is non-zero.

Stam> Inflation doesn't have anything to do with either Newton's constant or with the cosmological constant or with the vacuum energy ... And the reason is that inflation isn't described by general relativity.

Surely You're Joking, Mr. Nicolis ;=D

Readers who wants reliable information about the inflation models may want to consult chapter 8 of the soon-to-be 30 year old textbook by Kolb and Turner (The Early Universe), or chapter 4 of the more recent (2008) text by Steven Weinberg (Cosmology), to pick two books of great merit. Or chapters 19+23 of the draft version, http://web.phys.ntnu.no/~mika/cpp.pdf, of the just 3 months old monograph by Michael Kachelriess (Quantum Fields). Or, more advisable, buy the printed text.

Or, if you just want a rapid confirmation that (the expansion part) of inflation in its most common version is described by general relativity, look up equations (2.1-2) of the article, https://arxiv.org/pdf/0705.0164.pdf, by one of the originators of inflation models, Andrej Linde.

For the record, in its simplest version, the inflation models are just special cases of the Friedman-Lemaitre-Robertson-Walker cosmological models, described by general relativity, with exponentially expanding (de Sitter like) solutions over a finite range of time. For outsiders this fact may not be entirely obvious from research texts or even popular presentations, just like it may be un-obvious for outsiders reading advanced texts on celestial mechanics that this is (mostly) a part of Newtonian mechanics.

Built on this very basic foundation, the main inflation topics of research interest are related to the right hand side of the Einstein equations, described by one or more scalar ("inflaton") fields, and their associated dynamics. An alternative approach were proposed by Alexei Starobinsky, who instead modified the left hand side of the Einstein equations, by adding an extra (R2) term to the Einstein-Hilbert Lagrangian. This has later been extended to general f(R) theories, and even further. There are also string related/inspired inflation models, involving extra space-time dimensions beyond our observed four.

Paul> I have the demonstration that ALL natural constants, are not constant indeed: they are time-dependent.

Yes, but talk is cheap! What is your predicted variation of the fine structure constant during the last 13.7 x 109 years? What about the ratio between electron and proton masses? Or the Weinberg angle?

Hossein> ...the problem is energy ...

Beware that in General relativity, or more specifically the FLRW models, it is not only the energy density ε that matters, but to an equally extent the pressure P. An important parameter characterising various forms of matter contributions is the ratio w = P/ε. For relativistic ("hot") matter w = 1/3, for ordinary ("cold") matter w ≈ 0. A cosmological constant has w = -1, for quintessence w ≈ -1 (a little bit larger, and slowly increasing). In such classification, spatial curvature can be described by w = -1/3. As the universe expands, the trend is that matter types with the smallest w increase in relative importance, everything else being equal.

Dear Kåre

"As the universe expands, the trend is that matter types with the smallest w increase in relative importance, everything else being equal."

Where does this matter come from?

You may start with something like some inflaton fields; some more matter may be created later in the form of fermionic matter plus antimatter, and radiation, when the temperature is high enough (it is unavoidable). Some of this matter/antimatter annihilate against each other again as the universe cools down, producing photons for the Cosmic Microwave Background (CMB). I don't think anyone knows the exact process very early on, but the behaviour from somewhat before nucleosynthesis is quite well understood and supported by observations. Because the behaviour only depends on general relativity, and known elementary particle physics.

There is a nice book by Steven Weinberg, The First Three Minutes, https://www.amazon.com/First-Three-Minutes-Modern-Universe/dp/0465024378, aimed at a general audience, which is well worth reading. Although quite old (almost 40 years), I don't think the general picture described there has changed very much.

So, this more or less pushes the question back to "Where do the inflaton fields come from?"

@ Javadi

Well here's a hint, the vacuum energy in space time is based upon Planck's constant and the speed of light, as is everything in the Universe (including matter). You can readily do the maths for the vacuum energy on the back of an envelope.

With Planck's constant and the speed of light alone it isn't possible to produce a quantity that has the dimensions of energy at all-Newton's constant is, also, necessary.

However this is just dimensional analysis: it must be shown that there do exist dynamical degrees of freedom, that are sensitive to that value. For the moment this isn't the case.

The cosmological constant is just a consequence of the properties of general relativity: Einstein's equations admit solutions for any value of Newton's constant and of the cosmological constant--and those solutions are valid when all effects, that can be attributed to Planck's constant, can be neglected. It's not known how to describe spacetime geometries that aren't solutions of Einstein's equations. The properties of the Universe that can be measured can be described by a solution of Einstein's equations, for definite values of Newton's constant and of the cosmological constant and the effects of Planck's constant can be neglected.

On the other hand, the known quantum properties of matter imply that it doesn't make sense to consider situations where one term of Einstein's equations, that contains the energy-momentum tensor of matter, is sensitive to quantum effects, while the other terms isn't.

While nothing special happens to the quantum properties of known matter at energies around 10^(19) GeV, if gravitational effects can"t be neglected, spacetime breaks up into causally disconnected patches, hidden behind event horizons.

Dear Andrew

Vacuum energy is the energy density of empty space.

Scientists are not in agreement about how much energy is contained in the vacuum. Quantum mechanics requires the energy to be large as Paul Dirac claimed it is, like a sea of energy.

https://en.wikipedia.org/wiki/Zero-point_energy

Problem is that how we can explain the vacuum energy without using the principle of uncertainty.

Dear Kåre

All our theories today seem to imply that the universe should contain a tremendous concentration of energy, even in the emptiest regions of space. The gravitational effects of this so-called vacuum energy would have either quickly curled up the universe long ago or expanded it too much greater size. The Standard Model cannot help us understand this puzzle, called the cosmological constant problem.

Kane, G., 2003, the dawn of physics beyond the standard model, Scientific American, vol. 288(6), p.68-75.

As I wrote in my last post, the problem is that how we can explain the vacuum energy without using the principle of uncertainty.

Once more, the statement about a tremendous energy density of the Universe is wrong, since it isn't computed consistently: it's identified with the zero point energy, in flat spacetime, of the free particles of the Standard Model. How such a quantum effect affects spacetime geometry can't be described using Einstein's equations.

This isn't a puzzle, since, as posed, the question doesn't make sense. The energy density of the Universe can be described by the cosmological constant consistently-its value can be found by a consistent measurement and at the scales where all this makes sense the microscopic properties of matter aren't relevant.

So the only ``puzzle'' is the wrong identification of a classical quantity, the cosmological constant, with a quantum quantity, the zero point energy of a theory that isn't consistently coupled to gravity, in order for the identification to be possible at all.

The ``puzzle'' was relevant, when it was thought that the cosmological constant for our Universe, in fact, was equal to zero-so a mechanism that would lead to such a value was sought. Now that it's known that it isn't, in fact, zero, there isn't any ``puzzle''. The question of what can affect its value can only be resolved within a quantum theory of gravity, not within a classical one.

(0) In the language of distribution for set functions [5], the mass of an “entity” can be described as m(A)=Integral(rdqA where qA is an adequate measure and the density function. (i) E = mc² in which E is the energy, m is the mass and c is the speed of light. Let now consider the following constant functional f (c², m) = (mc²)‘ in the sense of [5]. Then, the integral as well as the derivative are operators depending on time and there are two cases to analyse: 1. If the energy of the entire universe is zero then the derivative of the functional must be zero, what means a symmetrical behavior for the function of genesis of mass (considered as set function [5] outside of the (a,t) time interval ). It is notable that if and only if the mass genesis function has the properties of a test function, then the energy of the universe is zero, which means that the universe contents an equal quantity of negative and positive energy at each moment. Also remarkable is that a test function allows asym-metry within the (a,t) time interval and that in this case the universe is closed, whereas the existence of a critical mass is not required and - attention- where-by a symmetrical behavior of the variation of the function (i.e. the derivative, that means the density) is also allowed according to the cosmological theories the temperature of the universe is inversly proportional to its radius and pro-portional to its contained mass. Thus, at symmetrical moments (outside of (a,t)-interval) not only the equal quantity of mass of the universe exists, but also its temperature, its volume and its mass density (i.e. the derivative of the mass genesis function).

Hossein> The gravitational effects of this so-called vacuum energy would have either quickly curled up the universe long ago or expanded it too much greater size.

In which cases there would have been no physicists around to worry about any tremendous concentration of energy :=D

Kepler once had a theory that the (then observed) six planets were distributed around the Sun at relative distances determined by the five platonic solids, https://en.wikipedia.org/wiki/Platonic_solid. This beautiful theory is described in his 1596 work Mysterium Cosmographicum, https://en.wikipedia.org/wiki/Mysterium_Cosmographicum. Fortunately, he did careful observations and data analysis for verification of his theory, to come up with a much better and real scientific discovery. Today, nobody would think of developing a theory for the sizes of the planetary orbits; they are determined by unpredictable events. There are many other solar systems looking somewhat different, and only a fraction of these will provide an habitable environment for life and physicists.

In a multiverse world, only a few of all existing universes will expand in a way to provide suitable conditions for the development of solar systems like ours. It may be highly unlikely that this should happen. But it has happened; otherwise we wouldn't be here.

So, like the details of our planetary system is mostly determined by random events, it may be that the details of our universe is mostly determined by equally random events, one of which could be an almost complete cancellation of the various contributions to a normally tremendous concentration of (positive or negative) vacuum energy.

It is not an attractive scenario, but it is a possible scenario. And the about 10500 possible vacuum solutions of string theory provide some support for this idea.

Dear Kåre

When the number of variables is high or variables are low but the conditions are constantly changing, we have no choice but to use random events.

But I am trying to describe the photon structure from Photon behavior in the gravitational field.

Generalization of the Dirac's Equation and Sea

https://www.researchgate.net/publication/303988070_Generalization_of_the_Dirac%27s_Equation_and_Sea

Hossein> I am trying to describe the photon structure from Photon behavior in the gravitational field.

That seems like the most difficult^* way to investigate photons.

^*) Maybe in competition with photon-neutrino interactions.

Dear Kåre

You are right, it is most difficult.

This is the reason that I have started with photon-graviton interaction. You can see it in all my articles.

This is a nice papper discussed new and old idea for cosmological equation and cosmological constant by joan sola try to check it out :Article Cosmological constant and vacuum energy: Old and new ideas