The zero-divergence property of the Einstein field equation gives the condition for an adiabatic expansion. Hence there can be no entropy increase due to reheating at the end of the inflationary era if Einstein's equation holds.
Yes, it is true that the entropy does not change in an exactly adiabatic Universe.
John A. Peacock points out in his book Cosmological Physics (p.277), that it is a good approximation to treat the expansion as adiabatic and reversible, because any effort to increase entropy on a local scale is swamped by the immense entropy residing in the microwave background. Thus Einstein's equation may be taken as true to a very good approximation.
In the old inflationary models, where the reheating was due to bubble wall collisions and/or dissipation of energy of the walls due to the coupling of inflaton to other fields, entropy indeed is generated. This is because only in a homogeneous and isotropic universe, Einstein eqs. imply no entropy generation, whereas when various bubbles are colliding with each other, the universe is not homogeneous and isotropic on the scales of bubble size.
I should have added the reminder, that Einstein's eqs. are expressed in space-time coordinates which cannot be exact in the Big Bang limit where quantum mechanics enters.
The entropy of the microwave background is "inside" the universe, and since there is no heat exchange with the "outside", entropy cannot be generated, i .e., the universe is isolated adiabatically according to general-relativistic cosmology. It is true that the entropy density can, and does, change but only in a way that the total entropy of the universe remains constant. This is a consequence of the Bianchi identity: the divergence of the stress-energy tensor vanishes. Therefore, the only conclusion that can be drawn is general-relativistic cosmology is incompatible with inflationary cosmology.
Matt, your remark would apply to any singularity--even a black hole.
Well, the expansion is adiabatic, even when accelerated, but there are a countless number of processes that are not, most particularly the interaction and decay of particles that yield relativistic particles in the final state. That is the case of the inflaton which releases the difference of energy before and after inflation through interaction terms with other fields, whose result reheats the universe.
If this is true there shouldn't there be an entropy source in Einstein's equation? Or, are you saying that Einstein's equation accounts for adiabatic expansion and nothing more? In either case it can't be the whole story. Or, if it is the whole story than it is incompatible with inflation.
There is no contradiction. The Einstein equations comply always with the Bianchi Identities, and so does the energy momentum tensor that is always conserved (technically, its covariant derivative is zero). When reheating happens a new term in the energy momentum tensor appears (due to particle physics properties) that explains the particle decay that Bertolami explained.
If it can't then Einstein's equation is meaningless for it predicts dE+pdV=TdS=0 for the universe.
Bernard: how do you define a volume V in the universe? V=r^3 or V=r^4? It is Friedmann's demonstration which fails not Einstein's equation.
During reheating TdS is not zero, and this is still consistent with the fact that the covariant derivative of the energy momentum tensor is zero, that is a demand of the theory.
All what theorizes greater speeds than the speed of light violates Einstein’s equations. What is strange and little understandable in the inflationary cosmological theory is that it supposes a violation of Einstein’s equations only in the immediately succeeding instants to the big-bang. In the Theory of Reference Frames greater speeds than the speed of light are exactly demonstrated whether in cosmology (relativistic theory of black holes) or in physics of high energy particles (leptonic and baryonic unstable particles). The debate on the nature of the space must take account of the difference between geometric space and physical space. I don’ t think solutions to the cosmological question can derive largely from thermodynamic considerations. From the universe we receive and measure electromagnetic radiations whose wavelength and frequency are basic physical characteristics.
Not if (17-5) and (17-6) on page 373 in Robertson and Noonan "Relativity and Cosmology" are correct, which follow directly from Bianchi's identity. The volume there is the physical volume.
The eq. for the covariant derivative of the energy momentum tensor is zero, as in the book, but it does not lead to eq. (17-5) during reheating.
The Einstein eqs. build in conservation of energy. In thermodynamics this is equivalent to the first law of thermodynamics. It is NOT equivalent to adiabatic expansion. In principle there can still be entropy generation, and non-adiabatic processes in the cosmological fluid. It depends on the form the energy-momentum tensor if these non-adiabatic processes are present or not, but the energy-momentum is always conserved in either case. Einstein's equations allow both possibilities and are no more inconsistent with entropy generation than the first law of thermodynamics is inconsistent with the second law of thermodynamics.
Jorge, right something IS definitely wrong.
Emil, there is no clear cut distinction between the first and second laws, even if Clausius would have us believe so [see Ch. 4 in "A New Perspective on Thermodynamics"]. What form of the stress-energy tensor would you have for non adiabatic processes and yet still satisfies Bianchi's identity? Such terms have been considered in Landau & Lfishitz, Fluid Mechanics, and appear as source terms in in the energy-momentum tensor (127-1) and in the particle flux density (127-2). They do not lead to conservation, they lead to an increase in entropy due to dissipative processes as they rightly should [see also Foundation of Physics Letters 5 (1992) 191-196].
Tmn;n = 0 is not a conservation law. A true conservation law must involve ordinary divergences integrated over a volume, otherwise Stokes' theorem may not be applied to convert the volume integral to one over the boundary. This is an elementary fact and it amazes me that no one seems to be aware of it. It was certainly known to Einstein, Weyl, Bergmann etc in the old days.
There is nothing equilibrium about an adiabatic expansion.
You forgot to mention Pauli, who wouldn't agree with you: The energy-momentum tensor whose divergence vanishes in equation (341) is a statement of conservation in his "The Theory of Relativity". Also Abraham and Laue wouldn't agree either. See also p. 333 in "A New Perspective on Relativity".
No it's not. It's only a conservation law under special circumstances. See Dirac, Weinberg, Weyl, Einstein, etc. etc. etc. There are hundreds of references. Google "pseudo-tensor of energy and momentum". The contracted Bianchi identities are *not* conservation laws in general, and not often specifically. Why would I make such a claim out of nowhere? Do you think I just pulled it out of my backside?
Bernard, the Einstein eqs. or covaraint conservation of the stress tensor do NOT imply adiabatic expansion or perfect fluid behavior. That is an additional assumption about the form of the stress tensor. There are numerous examples of dissipative fluid stress tensors in the literature. If one insists on a spatially homogeneous, isotropic Robertson-Walker geometry, then Zimdahl and others have shown a bulk viscosity term can be included in 2-fluid models. These generate entropy consistent with the second law but are perfectly consistent with Einstein's eqs. If one relaxes the condition of perfect homogeneity and isotropy, then additional shear and conduction terms are also allowed in the stress tensor as sources of entropy generation. Go to the arxiv and look up Zimdahl and/or "dissipation" or "viscosity" with "cosmology." You will find dozens of papers and examples. These are not the usual cosmological models and it is an open question if they have anything to do with real cosmology, but certainly dissipation and entropy generation (second law) are not in any fundamental conflict with Einstein's eqs. which only build in energy conservation--the first law.
There is no general law of conservation of energy in GR, so all arguments proceeding from energy conservation, including this one, are cooked from the beginning.
It simply amazes me that this elementary fact is routinely ignored, and even when it is explicitly pointed out and the reason given, it is still ignored. It makes me wonder how many people actually understand GR. In particular, it is no wonder that GR cannot be quantized.
D. R., the covariant conservation of matter or radiation energy-momentum is
D_a T^{ab} = 0
where D_a is the covariant derivative.
D. R. , you seem to arguing about words. GR builds in the COVARIANT conservation law I and others have mentioned. This word "covariant" is important because it takes into account the interaction with the gravitational field, if there is one. Of course it is not an "ordinary" or "true" conservation law in the sense that energy can be exchanged between the matter or radiation sources and the gravitational field, so the energy of the matter alone is not conserved. This is probably what you are referring to in the references to Dirac, Einstein, etc. and is well-known.
There is nothing surprising or controversial about this.
The upshot is - any question about cosmology and what its models mean for GR is utterly pointless, because GR does not even meet the demands of a local field theory featuring a conserved energy tensor. In all likelihood, GR is a weak-to-very-weak-field approximation that is only applicable to either test particles in the vicinity of mass points (e.g. the planets around the Sun), or thinly smeared out matter distributions such as single galaxies or dust clouds. And in the latter case, the original blunder of ignoring the lack of energy conservation in GR is compounded by the perhaps even worse blunder of ignoring its essential non-linearity (see work of Cooperstock), leading to further absurdities (dark matter and energy). Totally amazed that these elementary errors are repeated again and again.
There is no such thing as a "covariant conservation law" - this is just an astonishingly elementary error - a local conservation law in continuum physics, as pointed out, amounts to conversion of a volume integral into one over the boundary by Stokes' theorem. The only way to have a conservation law involving a covariant divergence is under the condition that by some manipulation involving components of the metric, the covariance divergence may be converted into an ordinary one. And the circumstances are precisely known - the tensor must be totally antisymmetric. So for example, the covariant divergence of a vector may be converted into an ordinary divergence,
Jm;m = 1/sqrt(g) d/dx_m ( sqrt(g) Jm)
But only under extremely special conditions can the covariant divergence of any symmetric tensor be converted into an ordinary divergence.
-drl
Sorry, D. R. there is no blunder.
GR passes all experimental tests so far.
If you choose to disbelieve that GR works in cosmology, that is your prerogative, but so far it's the best we have.
Questioning GR is also not germane to the original question Bernard asked, and which I and others answered.
In further support of GR and against inflationary cosmology, from Wikipedia on Emmy Noether:
"Noether was brought to Göttingen in 1915 by David Hilbert and Felix Klein, who wanted her expertise in invariant theory to help them in understanding general relativity, a geometrical theory of gravitation developed mainly by Albert Einstein. Hilbert had observed that the conservation of energy seemed to be violated in general relativity, due to the fact that gravitational energy could itself gravitate. Noether provided the resolution of this paradox, and a fundamental tool of modern theoretical physics, with Noether's first theorem, which she proved in 1915, but did not publish until 1918.[102] She solved the problem not only for general relativity, but determined the conserved quantities for every system of physical laws that possesses some continuous symmetry.
"Upon receiving her work, Einstein wrote to Hilbert: 'Yesterday I received from Miss Noether a very interesting paper on invariants. I'm impressed that such things can be understood in such a general way. The old guard at Göttingen should take some lessons from Miss Noether! She seems to know her stuff.'[103]
"For illustration, if a physical system behaves the same, regardless of how it is oriented in space, the physical laws that govern it are rotationally symmetric; from this symmetry, Noether's theorem shows the angular momentum of the system must be conserved.[104] The physical system itself need not be symmetric; a jagged asteroid tumbling in space conserves angular momentum despite its asymmetry. Rather, the symmetry of the physical laws governing the system is responsible for the conservation law. As another example, if a physical experiment has the same outcome at any place and at any time, then its laws are symmetric under continuous translations in space and time; by Noether's theorem, these symmetries account for the conservation laws of linear momentum and energy within this system, respectively.
"Noether's theorem has become a fundamental tool of modern theoretical physics, both because of the insight it gives into conservation laws, and also, as a practical calculation tool.[4] Her theorem allows researchers to determine the conserved quantities from the observed symmetries of a physical system. Conversely, it facilitates the description of a physical system based on classes of hypothetical physical laws. For illustration, suppose that a new physical phenomenon is discovered. Noether's theorem provides a test for theoretical models of the phenomenon: if the theory has a continuous symmetry, then Noether's theorem guarantees that the theory has a conserved quantity, and for the theory to be correct, this conservation must be observable in experiments."
I think the crux of the matter is this: Einstein's equation demands adiabaticity. Temperature falls when the volume increases. But the inflationary scenario requires a negative pressure so that temperature can now increase with volume under the same adiabatic conditions! [This is impossible because negative pressures lead to implosion not explosion.] Phase transitions do not "create entropy that we observe in the thermal cosmic background radiation". The latent heat in the phase transition cannot be "released" to reheat the system. According to Maxwell "Latent heat is the quantity of heat which must be communicated to a body in a given state in order to convert it into another state WITHOUT changing its temperature." Hence, latent heat cannot be used to reheat the system to some higher temperature and create entropy. It can cause changes in the system, like a change in volume so that liquid can condense or evaporate so as to leave the vapor pressure constant, but without increasing its temperature. The fact that the system underwent supercooling prior to the phase transition is extraneous since the latent heat refers to the temperature at which the phase transition takes place.
The crux of the matter is this: Einstein's equations do not demand adiabaticity.
Did no-one listen? You cannot invoke ideas from thermodynamics when there is not even a conservation law for energy. The energy content depends on the coordinate system. Therefore, so does the entropy content. What does any of this have to do with GR as such??
-drl
The energy content does not depend upon the coordinate system. If it did you could throw thermodynamics out! Recall what Einstein said about the universality of the laws of thermodynamics.
My question is what does GR have to do with reality---not even cosmology? When it predicts that a negative pressure will cause expansion something is definitely wrong!
How can latent heat cause a rise in temperature when it is "latent"? All the scenarios of the "standard model" of cosmology use thermodynamical concepts--- wrongly! de Sitter got expansion from a parameter--- the cosmological constant! Guth got it by changing the sign of the Gruneisen constant.
By no means am I defending GR. Many people (including myself) have shown that all the tests of GR can be obtained through physical processes beginning with Ritz in 1909. GR is full of circular reasoning. I have listed all this in my latest book, "A New Perspective on Relativity". It asks the question of why "invent" new non-Euclidean geometries when you already know that three exist with constant curvature? And the metrics of GR do not reduce to these when curvature is constant. There is no consensus of whether gravity propagates with the velocity of light or instantaneously.
About conservation and the divergence of the pseudo-stress-energy tensor. In Landau & Lifshitz "Fluid Mechanics" their equation (126.4) supposedly expresses conservation, a 4-vector with ALL zero components. They then say "We multiply this equation by u^i, i.e. project it on the direction of the 4-velocity. Then they get a conservation equation involving the enthalpy density. How can you have two equations expressing conservation? The latter equation is thermodynamically correct.
1. All shapes of energy which depend on the speed, as the kinetic energy, also depend on the reference frame (system).
2. If the universe is finite and closed, also if in expansion, it necessarily must be adiabatic.
Conservation of energy depends on coordinates and frames as much as Carnot's principle depends on the unit of temperature.
!!! If the energy content did NOT depend on the coordinate system, then there would be a tensor expression for it. That is why TENSORS exist!! But there is NOT a tensor expression for the conservation of energy in GR. NONE. There is NO conservation law for energy in GR!!!
-drl
Dear colleagues,
this discussion is very important for modern gravity physics and cosmology, please see discussions in my page of ReseachGate
This is worth reading, if needlessly complicated..
http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html
It is important to understand that there is no escape from this problem. GR as it stands is lop-sided - energy-momentum, or equivalently any distribution of matter, generates curvature, but curvature does not correspond uniquely to matter, even modulo some gauge group. This is precisely opposite of say the Maxwell-Lorentz electrodynamics, which has two complementary aspects, the field equations with charge distribution as source, and the Lorentz force showing how the fields act back on the sources.
My point is therefore that the topic of this thread is backward - the question is not to make something that is compatible with GR, but to find out how to extend, repair, and complete GR in light of what is observed. All evidence points to GR being a weak-to-very-weak field approximation. All accepted observational evidence amounts to tests in the very weak field regime. Cooperstock's work amounts to an accurate test in the weak field regime. There is no other test as of now. The full equations in the strong regime are completely untested.
-drl
Maxwell's equations do not discriminate between advanced and retarded potentials, and thus cannot explain radiation phenomena (Ritz). If advanced potentials are admitted, charges would not be the source of the field. Therefore, Maxwell's equations admit unphysical solutions. The two components in the Lorentz force are incompatible: how can an electron be both stationary and moving at the same time? (O'Rahilly) [see "A New Perspective on Relativity" Ch. 4]
And what does this have to do with anything? The mass in the Schwarzschild solution is chosen to correspond to reality. The retarded potentials and picked to correspond to reality. Both things come from equations second order in time.
-drl
Hello Bernard, I am glad that you started the discussion about the Einstein equation and cosmology. As you indicate it is difficult with this approach to quantitatively model the expansion of the Universe using this methodology.
However, there is now another, different model of the Universe available. This model is solely based on classical science and predicts and explains the expansion and its acceleration from first principles.
The novel model is based on the process of the formation of the Universe (electron and protons) from the pre-Big Bang energy singularity. The reverse process is the present massive conversion of Mass to Energy in the form of Photons (e.g., by solar radiation). Thus, this model of the life-cycle of the Universe is referred to as the Big-Bang-> Fizzle -> Big (BFB) Bang model.
For the solar system, the model predicts that, because of the solar mass-loss, the solar system is expanding, predicting the experimental results that Mars changed from liquid to frozen water about 3.7 Byr ago.
The same model, applies to the Universe, however, using a closed-system approach. This model predicts the linear expansion from the center of the Universe, and the acceleration of its expansion. Further, it leads to the prediction that the Hubble-constant is H0= ku/2, where ku is the radiative mass loss constant of the Universe. Evaluation of 366 measured Hubble-data between 1996 and 2010 shows that the Hubble constant varies systematically between 50 and about 120. This is a strong indication that the Hubble constant is not a constant when measured from Earth.
There are two apparent mechanisms for this variability.
(1) The variability may indicate that the various Hubble values reflect different radiative mass loss rates in different parts of the Universe. From the Hubble values, the mass-loss rate in these areas may be determined. Another implication is that Hubble values of equal magnitude might arise at given volumes of the Universe.
(2) Since Earth is not located at the Universe Center, determination of the Hubble value from Earth is not a constant, but a variable. This variability depends on the relative location and Hubble expansion of Earth and the measured galaxy. This approach may be modeled by standard physics and may allow determining the relative position of the Earth vs. the Universe Center.
Hello Ingo,
I am not familiar with the model you describe. However, Hubble's relation has come under increasing scrutiny because there appears to be many exceptions in every category of celestial objects. Some astronomers have proposed a non-velocity dependent red-shift. I don't know what a non-velocity dependent red-shift would be caused by because the only source known to cause a shift is uniform motion.
Regarding DRL, I only pointed out that no theory is perfect. The short-comings of GR are well-known and documented ["A New Perspective on Relativity" (World Scientific, 2011)]. There are many ways to "skin the cat" [Sec. 7.6.1, 7.6.2, and 7.6.3 on the tests of GR]. To the list, regarding the Schwarzschild solution, you may add that there is a change in curvature on going from the outer solution to the inner one. Why? Regarding the arbitrary constant of integration--mass-- [see Sec. 9.10.3].
From Einstein's equation for acceleration, it is seen that a negative pressure produces acceleration leading to an increase in the cosmological scale parameter, instead of a decrease as one would expect from a negative pressure. This has already been noted by Robertson and Noonan [Relativity and Cosmology, p. 373]: "Thus regardless of how much the cosmological fluid may be compressed due to a decrease in S(t), any pressure which is built up by the compression only tends to aid in the compression." For a positive pressure, a decrease in volume [decrease in S(t)] would require an increase in the internal energy at constant entropy, or a decrease in the entropy at constant internal energy. These are not spontaneous processes whereby entropy should increase with volume or energy should decrease with volume. And this is precisely the reason why the inflationary proposal gets expansion with negative pressure. It comes from Einstein's equation for acceleration! [de Sitter got acceleration through a positive cosmological constant introduced into that equation for an empty universe!]
Hello Bernard,
Thank you for your reply and comments. I am appreciating that you are clearly pointing out where the presently accepted models fall short. Personally, I am of the opinion that this is not surprising. Einstein’s ideas and mathematical models have now been studied and expanded for a hundred years. It is time to bring new aspects to his brilliant work to more fully understand the Universe and the processes within it.
The BFB model, by inclusion of the universal ‘fizzle’, is contributing a previously disregarded aspect, which from observational and rational points of view cannot be rejected. I have presented the BFB model and its consequences at a number of national and international meetings and conferences. I also submitted manuscripts for publication, which regularly were returned without evaluation and comments. I am in the process of re-reviewing the manuscripts, and I shall be delighted to share them with those who are interested.
Using Einstein modeling and classical physics models, modeling the process of the Big Bang from a prior energy singularity has thus far been futile. As you show, arbitrary variation of model parameters has neither predicted experimental observations in the Universe, nor provided novel insights or quantitatively predicted novel suggestions for experiments and observations.
Within the BFB model, viewing the process from a crystallization point of view has solved a number of enigmas, like the preservation of energy during the Big Bang process, the unforeseen appearance of gravity and identification as anti-energy, the present mass –loss of the Universe and many others. In conversations with physicists, the idea of applying the radiative mass-low concept to the solar system met with vehement opposition. The general argument was that the mass-loss rate of the Sun was too insignificant to affect planetary orbits. This justified the serious disregard of Newton’s action = reaction law. The fundamental error in this argument is that the cohesion of the planets and other solar objects had never been modeled. Thus, I set out and succeeded in the modeling of the stability of planetary orbits. The result of this modeling reveals that solar mass loss due to radiative mass loss and solar wind significantly weakens the stability of planetary orbits. One result is the transition of Mars from liquid to frozen water state.
Einstein’s models and those of all his followers ignored the radiative mass to energy (photon) conversion of the Universe and the associated loss of gravitational cohesion of its parts. The BFB model and its application to classical physics require the expansion of the Universe, including the acceleration of expansion, and the existence of a Hubble constant. BFG predicts that the latter is a constant when determined from the center of the Universe, but not if measured from outside the center. Since Earth is outside the Universe center, Hubble value determinations naturally lead to dispersity of the results, dependent on the place of Earth and the observed galaxies relative to and their separation rate from the Universe center.
Note: I will present my results on the life-cycle of the Solar system at the AGU 2013 Meeting of the Americas, May 14-17, 2013, at Cancun, Mexico. My presentation will be on Tuesday, May 14, as a poster session. This will allow time for detailed scientific exchanges. Please, anyone who will be attending the meeting is invited to contact me for a meeting during this time.
P.S.: The development of the BFB model was supported my prior scientific activities. One of these activities included the development of practical and science supported models of nucleation and growth of materials. The work is summarized in ‘Precision Crystallization, theory and practice of controlling crystal size (Ingo H. Leubner, CRC Press, Boca Raton, Fl. USA, 2009). Since 2009, significant contributions have been made modeling the processes and results for standard laboratory crystallizations.
If we return to the initial question on the Violation of the Einstein equations by Inflation,
We should start from discussion of the mathematical properties and physical interpretation of the Einstein equations of the General Relativity.
Then we will see that the Friedman’s equations is exactly identical to the non-relativistic ordinary Newtonian acceleration d^2 R / dt^2 = - GM(R) / R^2 , with
M(R)=c^2 V(R) (rho c^2 + 3p), where V(R) = (4 pi/3) R^3 is the volume of the gravitating sphere. This form of the Friedman equation explains why the galaxies recede from us with the velocities more than the velocity of light and why there is no relativistic energy conservations in the standard cosmological model.
What is more, the Bianchi identity applied to Einstein equations leads to the continuity equation in the form dE = -pdV and adiabaticity of the Space Expansion.
The details one may find in
http://arxiv.org/abs/0810.0153
http://arxiv.org/abs/0809.2323
and in recent monograph:
“Fundamental Questions of Practical Cosmology. Exploring the Realm of Galaxies
Series: Astrophysics and Space Science Library, Vol. 383
Baryshev Yurij & Teerikorpi Pekka
2012, XVI, 332 p. 46 illus.
The condition for inflation is precisely that your mass is negative,
rho c^2+3p
Dear Bernard,
the equations which I mentioned are exact mathematical derivations and well known
(Peebles just used them in his book, see his note on p.139 on the violation of energy conservation).
The identity of non-relativistic Newtonian equation of motion to the relativistic Friedman equation is the most strange feature of the relativistic standard cosmological model.
By the way the constant Lambda means that p = -e, and this is why it is called vacuum.
The equation de/dt=-3(e+p)dS/dt/S (S – is the scale factor) is not a Conservation of Energy
(see Landau & Lifshitz “The classical theory of fields”, 1971, paragraph 101 “The energy-momentum peudotensor”, p.304, eq.101.1), because it does not contain the energy of gravity field itself. This equation is the strict consequence of the Bianchi identity and can be called as continuity equation (but not as “conservation of energy”).
Dear Yurij,
There is a great deal of confusion so let's follow Peebles. First, the radial coordinate in the Robertson-Walker metric is NOT r=sinh chi, but, rather, r=tanh chi, hyperbolic distance. Otherwise the Robertson-Walker metric does not coincide the stereographic inner product metric, which is a known metric of negative constant curvature [see p. 487 of "A New Perspective on Relativity" The same error appears in Landau and Lifshitz "Classical Fields" (111.12)] The confusion about "local" as opposed to "non-global" energy conservation is a way of trying to get out of mess. (6.19) is NOT the Stefan-Boltzmann law because there is no reason for assuming (6.3) IF the black-body spectrum is exact. It's only when you assume the wavelength lambda is proportional to the scale factor a(t) and require that the black-body spectrum holds exactly that makes (6.3) necessary. In fact Peebles is mixing an adiabatic condition (6.18) with black-body radiation S=(aT)^3, which is non-adiabatic. This is impossible, the two don't mix!
Why the similarity with a Newtonian force law? Because you start off with the virial, then transform from mass to mass density, and then confuse mass density with the internal energy density. A factor of 2 is also thrown in. These are all illogical steps. The fundamental error is (4.31) "if the pressure is high the source of gravity changes from the mass density rho to rho+3p where p is the pressure." This has nothing to do with matter but radiation, for matter you might want to try rho+(3/2)p because the Gruneisen parameter for a material non-relativistic gas is 2/3, not 1/3 as for a photon gas.
Finally, the paragraph following(6.19) is hopelessly confused. If the radiation energy in a closed universe decreases as 1/a(t) how does the black-body radiation maintain itself? Why should the universe be "closed"? If the "pressure does not act to accelerate the expansion of the universe", it must decelerate it. Then you come back to the unpalatable conclusion that pressure built up by compression [i.e, a decrease in a(t)] aids compression. This is a runaway effect, and unphysical. It is the seat of all the woes, and the reason why inflation assumes that negative pressure causes expansion.
Dear Yurij,
I forgot to mention the error in Landau and Lifshitz "Classical Fields" "The general property (34.2) of the energy-momentum tensor of an arbitrary system now shows that the following inequality is always valid for the pressure and density of a macroscopic body
p
The equation of state of the vacuum is e=-p--both being constants independent of the temperature and particle number. From Einstein's second equation, the acceleration is constant, and so we are back to a de Sitter model of expansion without having to introduce, ad hoc, a positive cosmological constant into that equation, and consider an empty universe.
Dear Bernard,
the mentioned by you formula (34.2) and (35.6) from Landau & Lifshitz is not a Mistake but is the Truth for the systems of real particles which they consider – system of charged particles.
This formula is used in modern theoretical physics (and confirmed all physical experiments) and state that the trace of the EMT:
T^i_i > 0 for all real particles and T^i_i = 0 for massless particles (e.g. photones),
So it is not a mistake, that p < e/3 but reality which is experimentally tested for all real particles.
Your mistake is that you take equation of state p > e/3 and consider it as an experimentally
tested. It is well known that in modern cosmology physicists use any fantastic equation of state which has no experimental verification. This is a weak point of the standard cosmology and this is why many alternatives suggested for alternative interpretation of observational data.
Dear Bernard,
What is your claim? The inflation is error or not?
I cannot understand your discussion because you did not define your initial suggestions and mathematical formulation for that words which you use in the text.
To continue our discussion I suggest to use available in the free internet my paper
http://arxiv.org/abs/0810.0153 where mathematical and physical bases of the Friedman equations is formulated. So let us use it as the starting point.
Dear Yurij,
pV=NkT and E=(3/2)NkT for an ideal gas. Therefore p=(2/3)e where e=E/V.
My claim is that inflation is definitely wrong for two main reasons: (1) negative pressures cause implosion--not explosion, and (2) latent heat cannot heat.
Bernard, you say that inflation is adiabatic. So you are able to define the volume => the boundaries of the universe?
Dear Yurij,
I forgot to mention that the negative pressure results from the Einstein equations
e=3(R'/R)^2 and p=-(2RR''+R'^2)/R^2 in flat space with no cosmological constant. The prime denotes differentiation with respect to time of the scale factor, R.
From the thermal equation of state s e=p where s is the Gruneisen constant (that varies between 1/3 for an ultrarelativistic gas to 2/3 for a nonrelativistic material gas), s must satisfy:
3s+1=-2R R''/R'^2
In order for there to be acceleration the right-hand side must be negative which requires that the left-hand side be negative, s
No Nathalie, I am saying that inflation doesn't (or didn't) exist. If it existed it would contradict Einstein's equations which are adiabatic. Moreover, Einstein's equations allow a negative pressure to cause expansion, which is wrong. I am not proposing anything new about the volume of the universe, or its boundaries.
Bernard, we already discussed about that:
https://www.researchgate.net/post/Why_is_the_pressure_P_in_the_Friedmann_cosmological_model_negative_to_account_for_the_accelerated_expansion_of_the_universe
Dear Bernard,
It seems that you miss the fact that the energy density in relativistic thermodynamics must include the rest mass energy of the particles
Dear Yurij,
According to Landau and Lifshitz, the density of energy is the average of the total relativistic energy divided by the volume. When expanded there will be a constant factor the rest energy cf. (35.9) theory of fields. It would be then impossible to satisfy the equation of state (35.8) in "The Classical Theory of Fields". Rather, if the ratio of the pressure to the rest energy is considered, it will be equal to the ratio of the Schwarzschild radius to the radius of a star, which for normal stars relativistic effects are of order 10^{-6}. This would make the pressure 10^{-6} times the rest energy density and it certainly would not satisfy a thermodynamic equation of state. When the pressure is set equal to zero together with the curvature and the cosmological constant, the Einstein equations derived from the Robertson-Walker metric give the relativistic virial theorem and Newton's equation. Taking the pressure into account gives a correction to the latter precisely the ratio of the pressure to the rest energy density, which, as I mentioned, does not even come close to being a thermal equation of state for anything but neutron stars where the radius is of the order of the Schwarzschild radius.
In the expression for the energy-momentum tensor of a macroscopic body
T_ik = diag (e, p, p, p) the energy density is the total quantity which includes rest mass
energy of the particles. Then the definition of equation of state is p = w e, so for ordinary classical conditions in cosmology (cold dark matter) usually used so called
“dust” universe with equation of state p = 0 (as you correctly noted).
Dear Yurij
If you assume the equation of state p=w e, w is the ratio of the Schwarzschild radius to the radius of a star. It is of the order 10^{-6} for normal relativistic stars, 10^{-4} for white dwarfs, and 1 for black holes, and 0 for dust. For inflation even worse, w
I have come to the conclusion that the original question can be answered:
Einstein's equations are inadequate to predict and explain the inflation of the Universe .
The reason is that the equations do not include significant processes in the Universe and these processes have been neglected ever since. See my earlier contribution.
Dear Professors excuse me; I think the inflation is related to the mechanism of universe expansion which Einstein's equations did not take into account. The expansion is happened like a ripple of a stone in water this manner may be continued even in nowadays??? so there are heterogeneity in expansion these are not taken into account because these equations assume only homogenous expansion radially outward
Sadeem, Then how can you use the same equations to explain the inhomogeneities which they supposedly don't take into account?
Sadeem and Joseph,
Einstein realized that his equations could predict expansion or contraction, but since the universe known at his time comprised only the Milky Way which appeared stable, he introduced the cosmological constant. This he called his greatest blunder when Hubble proved that the universe was indeed expanding.
When the expansion of the universe turned out to be accelerating, which is also permitted by Einstein's equations, the cosmological constant was reintroduced as one possible explanation. The true reason is not yet known.
The primordial inflation and the accelerated expanion of today could be related, but that is just one hypothesis among a thousand. The simplified mechanical pictures offered by you are without any value.
Dear Professor Matts
I think you miss understand me. This simplified mechanical picture as you said is really complicated if you try to make it conjugated in Einstein's equations. This can be added as a disturbance in the expansion. But this is not easy because we are talking about phenomenon that happened in microscopic dimensions while Einstein's equations are describing classical large dimensions. Until a new theory of quantum gravity is appeared this will be under research.
I have given much thought and have done much modeling of the expansion of the Solar System and of the Universe. I discussed this in my previous contributions in more detail.
I am not kidding:
‘Einstein's equations are inadequate to predict and explain the inflation of the Universe’
I followed the discussion above, and my statement is supported by the many approaches to force Einstein’s hundred year old science into the present observations.
I will present the model on the stability of the solar system at the Planetary Session of the ‘Meeting of the Americas’, organized by the American Geophysical Union. I will be showing the increase of the planetary orbits from the beginning of the solar system to 4.5 billion years in the future. The model correctly predicts the transition of Mars from water to ice 3.7 billion years ago. It also predicts the orbital expansion of Earth, and that all planets were significantly closer to the Sun at the planetary formation. Attempts to explain these events have not been successful using Einstein’s science. The same applies to the expansion of the Universe and the existence of the Hubble expansion. The universe modeling uses the same fundamental science as that for the solar system. Thus, my statement above.
http://physics.ucsc.edu/~joel/10Phys224/10_Wk7-Before%20and%20After%20Inflation.pdf
Read this link. It makes the case of how inflation can be commensurate with the Einstein equations. In particular, note how Primack models the Friedman equations taking into account inflation. His example is the chaotic inflationary potential which is proportional to field phi ^ 2
Inflation cannot be commensurate with Einstein's equations because it demands S/= constant while Einstein's equation require S=const. It is well-known in thermodynamics that irreversible processes cannot be derived from the stationary value of the time integral of a Lagrangian. Pressure behaves incorrectly in Einstein's equation: Positive pressure is defined as the increase in entropy with volume at constant internal energy. Inflation claims just the opposite and so is entirely wrong.
http://physics.ucsc.edu/~joel/10Phys224/10_Wk7-Before%20and%20After%20Inflation.pdf
I disagree with you. If inflation can be made commensurate with the Friedman equations, it is comensurate with the Einstein equations. read what Primack derived and learn.
Look at any book on relativity, e.g. Robertson-Noonan, "Relativity and Cosmology" Eqn (17.5) and Eqn (17.6). The original error of Guth's was to take dS/dt=0 as separate and distinct from dE/dt+pdV/dt=0. They're not two separate statements! The error then is to write S=sR^3, where s is the entropy density. Anything derived from the Einstein equations is by definition adiabatic!
Benard
Once again, you are WRONG
If inflation fits within the Friedman equations, and it does as explained by Primack, then it is commensurate with GR
Andrew,
http://milesmathis.com/dunning.pdf
Then tell me where I'm wrong!
I will refer to what you wrote about entropy in your document. I.e. the problem with entropy is serious, and is well known. Your equations 7,8, 9, 10, and 11 miss the point. I.e. the paper is a good one, but you overlook several details . I.e. if one has a cosmic singularity then your critique of the equations 7,8,9, 10, and 11 are bullet proof.
If one does not have a singularity, but a quantum bounce, as in LQG, or a situation as given by Penrose in his CCC cosmology, i.e .where one does not even have a collapse but a recycling of space time according to a conformal mapping, then one can have an initial entropy at the start which is non zero. I.e. inflation by its own choices will have to have zero entropy at its start.
Your statement is that Eq. 7 is a combination of two thermodynamic laws, and not just one. This is your counter pose to inflation.
If there is NOT an initial singularity then what you said about Eq. 7 being a combination of two, not one entropy laws, is FALSE. I will PROVE IT and put it in a reply to your paper.
I will write a full reply to your paper in the next 10 hours and make my own paper in response with regards to YOUR paper. Thanks for bringing your paper to my attention.
Oops, amend my reply as follows: I typed too fast>
I will refer to what you wrote about entropy in your document. I.e. the problem with entropy is serious, and is well known. Your equations 7,8, 9, 10, and 11 miss the point. I.e. the paper is a good one, but you overlook several details . I.e. if one has a cosmic singularity then your critique of the equations 7,8,9, 10, and 11 are bullet proof.
If one does not have a singularity, but a quantum bounce, as in LQG, or a situation as given by Penrose in his CCC cosmology, i.e .where one does not even have a collapse but a recycling of space time according to a conformal mapping, then one can have an initial entropy at the start which is non zero. I.e. inflation by its own choices will have to have zero entropy at its start ONLY if there is a cosmic singularity at the beginning of spacetime.
This leads to my MAIN POINT about your paper...
Your statement is that Eq. 7 is a combination of two thermodynamic laws, and not just one. This is your counter pose to inflation.
If there is NOT an initial singularity then what you said about Eq. 7 being a combination of two, not one entropy laws, is FALSE. I will PROVE IT and put it in a reply to your paper.
I will write a full reply to your paper in the next 10 hours and make my own paper in response with regards to YOUR paper. Thanks for bringing your paper to my attention.
Andrew,
For greater detail, see Sec. V.2.3 of "Thermodynamics of Extremes" (Albion, 1995)
and Fads and fallacies in cosmology, Physics Essays 7 (1994) 442-449.
http://vixra.org/pdf/1305.0070v1.pdf'
I am fixing the second level of typos in it but I will go over it with you.
It doesn't open.
Once you write down the Friedmann equation and the acceleration equation which defines the pressure, the game is over. You get the adiabatic condition. There is nothing that prevents me from adding what I want to the energy-stress tensor. It is only when I use the Einstein equation to find the energy density and pressure in terms of the non-vanishing components of the Ricci tensor--using the energy-stress tensor of a perfect fluid---do I find the condition of adiabaticity. The condition of adabiaticity makes the inflation scenario illusory. This has nothing whatsoever to do with there being or not being an initial singularity.
bernard, I am uploading another document to t hat server and I will post my answer to you Just wait to you read it
Andrew,
It's a fortunate coincidence that the two Einstein equations are compatible with the vanishing of the divergence of the energy-stress tensor. Or is it? The Friedmann equation is OK; its the virial theorem if the energy density reduces to the rest energy density. What is wrong is the acceleration equation which defines the pressure. It says regardless of how much the perfect fluid may be compressed the pressure caused by the compression only helps to compress the fluid more. This is not what pressure does. And this is why negative pressure causes the opposite effect (which id doesn't) and why Guth and de Sitter used it. (de Sitter used the cosmological constant which has the same effect as a negative pressure.)
And due to the adiabatic condition, I can tell you the form of the entropy. It will not be the isothermal entropy of black body radiation, S=sR^3,which is incompatible with it [cf. Sec. 6.2 "A New Perspective on Thermodynamics"] No entropy increase due to expansion!
Equation (16) of my paper has the right hand side vanishing--not of the divergence of the energy-stress tensor but the inner product of the 4-velocity and the divergence. The first condition is the Gibbs equation, the second condition is the entropy balance (17). The real test would be to put the energy-stress tensor (10) into the Einstein equations and see what you come out with!
Andrew,
Do you believe what you wrote? If so please submit it to Foundations of Physics so that I can give a rebuttal. You don't even define your terms. Equation (13) is wrong. Ng cannot go from the thermal wavelength (which you don't define) of a nonrelativistic gas to a relativistic one. The logarithmic form of entropy, which is valid for a nonrelativistic gas, is incorrect for a ultrarelativistic gas. You have only one component in your energy-stress tensor (mass density) and you want to show that the tensor doesn't include terms of heat flux and other dissipative terms that contain the second law? Take a look at Sec 127 of Landau and Lifshitz, Fluid Mechanics, especially eqn (127.1).
I am not presenting it to be a foundations of physics piece. You already gave me your rebuttal and so I will reviews it.
Bernard, cool off. I presented something in VIXRA, so you could have your say. Now what are you so outraged by ? This is an informal place to give opinions, so cool off. As it is, I put something together so you could have your say. You had your say, so once again< COOL IT >.
I will continue to do what I want, and you are free to shoot back. You just did. And I will get a copy of Landau fluid mechanics and continue to work with it. the problem.
Incidently, Bernard, I DID review the problem again, and an even more general stress energy tensor which has some of the factors you mentioned from Landau STILL gives the same answer.
I will put it up in Vixra soon and then submit it to a journal. You will be told what Journal it is, and I will invite you to attack what I wrote, if the article is accepted for publication.
Bernard, prepare for the Deluge.
Go to " A primer on the physics of the cosmic microwave background" by Massimo Giovannini, go to pages 437-439 for its treatment of the Einstein Stress-energy tensors and their fluctuations.
I will use them on the problem, and guess what ? YOU are going to have to evolve. Expect an answer in about a day, put here for you to have fun with.
Toots.
Andrew,
You mean you derived the Einstein equations for the tensor (127.1) with the dissipative part given by (127.7) in Landau and Lifshitz? You can clearly see that the entropy flux density 4-vector in (127.6) appears in the inner product of the velocity 4-vector and the divergence of the energy-stress tensor in the equation just before (127.5).
Since you are attacking a paper which appeared in Foundations of Physics Letters, which doesn't exist any longer, it is only correct that you submit it to its parent publication, Foundations of Physics, in which I have the right to a rebuttal.
I would also remove Ng's equation (13) for the relativistic entropy because the thermal wavelength must be the nonrelativistic wavelength, not the ultrarelativistic one. Consulting his paper, he took it for granted that any thermal wavelength could be put in the argument of the logarithm. Not so.
Just wait. This is going to get interesting.
TOOTS
Gotta eat Bernard. I will be writing out my equations during breakfast and then putting them in for the afternoon for you to have fun with tomorrow :-)
Yeah, I will remove Ng, for the reason you mentioned, but it does not change the situation
Andrew,
Before claiming I'm wrong it would be helpful to read what is written. What you say:
"Where Lavenda and Davies [2] develop their theme is in claiming that for general relativity, that the covariant derivative of the stress energy tensor being zero,
if one has inflation, is a combination of the first and second laws of thermodynamics."
is totally unfounded and untruthful. According to Einstein the vanishing of the covariant derivative of the stress-energy tensor is a condition of conservation. So what you are saying is that all conservative systems have inflation. Nice.
The second statement on the combination of the two laws can be found in almost any text on relativity. So what you are saying is that the tensor excludes dissipative processes. I cite Landau and Lifshitz, Fluid Mechanics not because it is new, but because the mathematics hasn't changed from the time of Levi-Civita and Bianchi. Nothing has evolved, only the errors have undergone inflation.
Bernard,
You are free to fight me in Foundations of physics. Ground rules, and if you show my derivation is wrong, I will withdraw it.
To help you do that, before I send the article in to Foundations of physics, I will
send it to you, and I will delay sending it to Foundations of physics , to there is
absolutely no chance of any commonality of agreement.
IMO, I will give you first shot at knocking it down.
Yes, you are very likely WRONG. I will give this dispute another four days to go and if by the end of it there is a stand off, still, I will put it in the article and it will go to Foundations of physics and it depends upon the fate of a term I will analyze which is
h(a,b) = g(a,b) + ( four vector (a)) * (four vector (b)).
g(a,b)= flat space contribution + GW contribution.
Expanding this out, and putting in the Visser values will lead to
T(a,b) = T(a,b) [ non gravitational] + T(a,b) [ Massive gravitons]
I spent all day on the problem, and I am sorry but you blew it. I will explain what will show up in a paper, HERE, and you can await the executive summary in the paper itself
The proof starts in the new book by George Ellis, Maartens, and Mas Callum
Relativistic cosmology
Go to look at page 91 of the above text, go to formula 5.9
The stress momentum tensor has the most general expression available.
Now note the term, with :P ( pressure) times h (a,b)
h(a,b) = g(a,b) + ( four vector (a)) * (four vector (b))
g(a,b) = flat space term + gravitational metric term
If one takes this , then T(a,b) = T(a,b) [ non gravitational] + T(a,b) [ Massive gravitons]
At the very start of inflation, and I will explain this in my paper, the pressure P is what to watch. If after the beginning of inflation, if P is anytime not equal to zero, then one can put in the Visser T(a,b) [ massive graviton] term and perform the analysis and evaluation I put in,.
If the pressure is canceled out at the START of inflation, your " qualms .... " paper holds. If there is any chance that is not the case, and this is a model dependent phenomenon, then your " qualms.." paper results are flat out wrong.
I was surprised at how important P ( pressure) times h (a,b) really is for the
T(a,b) term, but it really is, and so I will write the full analysis up in the next two days submit it in a link here and you can do your very best to wreck it,.
If after several back and fourth goes at it, we do not reach agreement, the article will be sent to Foundations of physics
If on the other hand you conclusively show what I put is wrong, then the paper will not go to foundations of physics,.
Andrew Beckwith
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