A pressure is defined by a force/surface. where the surface is twodimensional. So a pressure can only account for the inflation of a twodimensional surface (even curved).

I do not think that a pressure (force/r^2) can account for the inflation of the universe, it should be a force/r^3 that accounts for the inflation as the hypersurface of the universe is 3dimensional (3d and curved, it can be embedded in a 4d space).

The relation between the pressure (f/r^2) and f/r^3 where f is the force supposed to be constant is: d(-f/r^2)dr=2f/r^3.

So a negative pressure leads to a positive f/^3 with an entropy that increases.

The pressure is defined either as the negative of the derivative of the internal energy with respect to volume at constant entropy, or the derivative of the entropy with respect to volume at constant internal energy. In the former case, a negative pressure would mean that the energy increases with and an increase in volume, while the latter would mean that the volume must decrease if entropy is to increase. Since Einstein's equations are adiabatic, one must opt for the former definition. Then why should the internal energy increase volume?

No mention need be made about the curvature of space. The relation between curvature and energy density and pressure in the energy-stress tensor is supposedly taken care of by the Einstein equations which equates the latter to the Ricci tensor and scalar curvature.

How do you define a volume in the universe? The universe is 3dimensional and curved, so it may virtually be embedded in a 4dimensional euclidean space: a 'volume' in our universe has then to be written r^4. And straightforwardly, I question the demonstration of the second Friedmann equation (where the pressure is negative accounting for the acceleration of the expansion of the universe). Indeed, I wonder whether 'ordinary' thermodynamics apply in our universe...

About Einstein's equations, if you make the dimensional analysis of these, you find

length=time (see my corresponding article on my profile), so are they really adiabatic?

As Planck once said, the laws of thermodynamics and constants of nature must apply for all times and to universes for, otherwise, they would not be universally true. The only violations are are due to the misunderstanding of these laws, and have nothing to do with experimental refutations of these laws. We have nothing more general than the laws of thermodynamics, and if Friedmann predicts that a negative pressure causes expansion then Einstein's equations are to be questioned, not thermodynamics. The "coordinate" time does not enter into thermodynamics so I don't see how that would put the adiabatic constraint in jeopardy.

Negative temperatures simply do not exist, cf. Journal of Physics A 32 (1999) 4279-4297. Negative pressures cause implosion and not explosions whereby a liquid can spontaneously detach itself from its container to form cavities. Negative pressures are also shown to cause a negative mass defect when a gas condenses to form a star, That is, energy would be absorbed and not radiated away, cf. www.bernardhlavenda.com The pressure not only enters into Einstein's equations but also into the adiabatic condition (Bianchi's identity). The two are incompatible leading to the conclusion that Einstein's equations are wrong.

Thermodynamics was formulated much before discovery of

relativity. Charles Law (1787) Boyle's Law (1662), whereas

relativity (1905).

In modern theories a Lagrangian density is constructed

which is Lorentz invariant from the onset. Then there is no

dependence on length contraction/ time dilation etc when

shifting from one reference frame to another. Thus equations

of motions, obtained by Euler Lagrangian methods are

also Lorentz invariant.

Thermodynamic concepts of pressure and volume are

entirely non-relativistic. Changing a reference frame which

is moving with respect to the laboratory frame length

contraction will occur. Thus force/area=pressure as

well a volume=l^3 will change. Thus (a priori) I would not

think that Clausius Clapeyron equations are Lorentz

invariant. Instead of thinking that a negative pressure has driven

inflation, it safer to think in terms of energy released by a

scalar field in the early universe, and say that this energy

drives inflation.

Relativistic thermodynamics is almost as old as Planck's black body radiation (1900). In fact, it was developed by Planck's students as well as Planck himself in the period from 1903 to 1907, see Chapter 6 of my book "A New Perspective on Relativity: An Odyssey in Non-Euclidean Geometries (World Scientific, 2011). The pressure will not change under a change in the inertial reference frame, nor will the entropy. These results are due to Planck. And it was Hasenohrl who first pointed out that heat energy will increase with the mass of a body.

It is Einstein's equations that predict that negative pressure will cause expansion so it is not safer to think of a scalar field releasing energy that causes inflation. If Einstein's equations don't apply then you cannot use them to justify inflation, as in Guth's original publication. Since Einstein's equations are adiabatic there will be no generation of entropy.

Bernard: Thank you very much for the reference.

I can easily understand the entropy is a Lorentz invariant quantity.

Is there a easy way to understand that pressure is Lorentz invariant ?

This question is very relevant today as we think that there is

accelerated expansion of the universe.

Regarding entropy, I can understand that number of stable particles

have a relativistic invariance. Thus number of microstates given by

W=N ! / (n1 ! n2 ! n3 !...) has a relativistic invariance. Therefore

entropy= k_b ln W also has a relativistic invariance.

However, I will really appreciate if you kindly explain why

pressure is unchanged under Lorentz transformations.

In your book have you proven that pressure is unchanged under

Lorentz transformations ? You mentioned in your comment that

pressure will not change under a change in the inertial reference

frame.

Biswajoy,

The Gibbs equations is TdS=dE+pdV+......

The volume undergoes Lorentz contraction along with the internal (not the total!) energy. Hence, p must be an invariant, and since S is an invariant, a body will cool when in motion. The total energy will increase with the velocity since it is the sum of the internal energy + the energy required to keep the system in a uniform state of motion.

The pressure enters with the wrong sign in the Einstein's equation for the acceleration of the scale factor. The pressure is defined as p/T=dS/dV (partials intended). The entropy will increase with volume at constant temperature and internal energy; hence, the pressure is positive. A negative pressure would mean a decrease in entropy with increasing volume thus causing implosion---not explosion! The requirement that pressure satisfy Einstein's equationwhile at the same time requiring it to satisfy the condition of adiabaticity, dE+pdV=0, has converted pressure into negative pressure in Einstein's equation for the acceleration of the scale factor..

Bernard,

generally momentum 4 vector is written as,

(p1,p2,p3,E/c)

it is re written in relativistic thermodynamics as

(p1,p2,p3,( E+ p V )/c )

is there an easy way to see why components of momentum

four vector will include pressure ?

 Sorry Biwajoy for not answering your question sooner but I just happened to see it now.

E+pV is the heat function, or enthalpy. Not only does it contain the internal energy, E, but also the mechanical work on the surface of the particle. According to Planck's contribution of January 1907,the equivalence of mass and energy should be mass and the heat function. You  can see this through the fact that it is not energy that transforms according to the Lorentz transform but the enthalphy.

Interesting question, which you’ve prompted me to examine (for my own benefit too).

First, as regards negative pressure in a cosmological sense (specifically for readers not familiar with the basic GR):

The clearest explanation I’ve found is in Scott Dodelson’s “Modern Cosmology” (2003), in which he explains, on p. 151, how the idea of negative pressure arises as a consequence of simply setting Einstein’s equations for the case of cosmic acceleration (inflation), [as I think Biswajoy has also stated, if in a slightly different way, above], as a result of which it is necessary that:

     Pressure < minus (Energy density/3)

…where energy density is always positive.

As Dodelson points out (which you can find in many other sources such as Wikipedia [6] and Weinberg's classic Cosmology):

For ordinary (non-relativistic) matter:

     Pressure = (essentially) zero  

For radiation or relativistic matter

     Pressure = + (Energy density/3)

However, as Dodelson also says:

“Negative pressure is not something with which we have any familiarity”.

I take this to mean that while accelerating inflation requires something with this property, we have no model for it, and hence I wonder whether we can be confident in treating it according to standard thermodynamics.

However, Dodelson does discuss entropy density (sic) on pp. 39-40, specifically: how, in an expanding universe, entropy density scales as the inverse of the cube of the scale factor – but it’s not clear to me whether this applies to an accelerating universe (involving negative pressure), rather than one that’s merely expanding (presumably, after some initial acceleration phase).

The only other source I can find on entropy in direct relation to negative pressure is from Patrick Greene (Fermilab Theoretical Astrophysics Group), but in this instance it’s not clear whether his argument must apply to negative pressure in the GR/cosmological sense.

“a system with negative pressure must be unstable, and thus, there are no thermodynamic states of a system with negative pressure. The reason for this is simple: The second law of thermodynamics tells us that all thermodynamic states of a system (aka your fluid with negative pressure) are such as to maximize the entropy of the system subject to any external constraints (fixed temperature, volume, etc.). However, a system with negative pressure can always increase its entropy by decreasing its volume. Mathematically this is:

dS = P dV = -|P| dV , for constant internal energy and particle number.

So, decreasing the volume ( dV < 0 ) increases the entropy ( dS > 0 ) if the pressure is negative ( P = -|P| ). Such a system will spontaneously collapse until the pressure again becomes positive.”

If you want to pursue this more technically, there are a couple of papers covering entropy increasing in inflationary universes [3rd and 4th refs below], while Steinhardt and Neil Turok discuss a cyclic model [5] with a slow accelerated expansion phase in which:

"the associated exponential expansion suppresses density perturbations and dilutes entropy, matter and black holes to negligible levels." [my bold].

As far as I can tell, it seems that cases can be made both for entropy increasing and for entropy decreasing, in inflationary universe models.

Hope this helps - PGE

https://www.amazon.co.uk/Modern-Cosmology-Anisotropies-Inhomogeneities-Universe/dp/0122191412/ref=sr_1_3?s=books&ie=UTF8&qid=1497031666&sr=1-3&keywords=scott+dodelson

http://www.fnal.gov/pub/science/inquiring/questions/negative_pressure.html

https://arxiv.org/pdf/1404.6444

https://arxiv.org/pdf/1605.02241

https://en.wikipedia.org/wiki/Equation_of_state_(cosmology)

It amuses me how cosmologists twist thermodynamics. Even if there were such a possibility of a negative pressure governing inflation, you couldn't use Einstein's equations to show it. There it is clear that dE+pdV=0, which is the condition of adiabaticity. This is why Guth's 1980 was completely wrong when he spoke about increasing the entropy of the universe. Sure, the entropy density can change, but the total entropy remains constant, at least if Einstein's equations apply. It is also amusing to see more recent papers on inflation skirting the problem of an increase in entropy of the universe completely. The expansion of the universe is supposedly occurring in a void, i.e., there is nothing "outside" with which to exchange heat.

If you consult Landau and Lifshitz's "Statistical Physics", a 'negative' pressure is not ruled out entirely. However, a "negative" pressure will always cause implosion---not explosion---where material attached to the walls of the container will spontaneously "detach" itself  leading to "the formation of a new surface" which would be called a "metastable state". This is a far cry from a negative pressure driving an accelerating universe!

One other point which irks me is that if Einstein's equations give geodesic equations of motion, where do you get radiation and acceleration?

Bernard,

I can sympathise with anyone, like myself, trying to make sense of the current state of physics.

Your latest post is, I believe, a response to my own sincere (if comparatively inexpert) attempt, using available sources, to try to answer your question, posed originally in 2013, prior to publication of your book “Where Physics went Wrong”.

I am not a cosmologist, nor am I an expert such as yourself, in thermodynamics.

In making my own judgement, I am inclined to accept Einstein’s GR as a good working hypothesis, especially as publications reporting observations of what appear to be gravitational waves (whose existence your book had earlier questioned) have now been made.

From your R-G answer above, on March 20, 2013 to another’s replies to your original question, you obviously understand the relationship of energy density and pressure in the stress-energy tensor in the EFEs (and indeed EFEs as a whole, better than I do). My point was only to quote Dodelson and others, that if  inflation occurs, then EFEs require negative pressure – and they say so without attempting to explain what negative pressure might be.

The point of you question, it seems to me, was not really

“How can a negative pressure drive inflation,..”

(as I’m sure you understand EFEs well enough to see how it arises, but for other interested readers, see [1]) but rather whether the idea of negative pressure is consistent with TD. You saw that I added, in my post, a rider wondering

“whether we can be confident in treating it [inflation] according to standard thermodynamics.”

Might such an assumptions allay your discomfort?

On your last point, as I understand it, Einstein’s original geodesic equations of motion were not initially intended to model the ‘evolution’ of an assumed-static universe. Radiation and matter etc. were (as I’m sure you’re also aware) introduced into the stress-energy tensor via an equation of state; and acceleration can only occur after the radius of curvature, R, was allowed to vary with time, by Friedman in his 1922 model [2].

I’m now digging into the history of the cosmological constant in an effort to understand the issues better, and perhaps move forward my own understanding, if not this conversation (so please don’t hold your breath awaiting my further reply).

But if you happen still to be interested in whether inflation is compatible with thermodynamics, you might try posting a question on Physics Stack Exchange [3], where snippy postgrads and post-docs (mainly, but also the occasional Nobellist) seem to enjoy tackling questions such as yours.  

Regards - PGE

http://www.preposterousuniverse.com/blog/2013/11/16/why-does-dark-energy-make-the-universe-accelerate/

https://arxiv.org/abs/1311.2763

https://physics.stackexchange.com/

Paul,

I appreciate your answer, I will answer in more detail.

But, I just wanted to ask how can we be sure that gravitational waves were detected? From a very naive point of view, gravitational waves would affect the whole interferometer, not just the mirrors. How do we know that it is (just) an electromagnetic phenomenon due to a cataclysmic event? I believe--and maybe I'm wrong---that gravitational phenomena manifest themselves through the propagation of light rays as if the latter were propagating through media of different indices of refraction. Gravity itself propagating at the speed of light would raise havoc with planetary orbits, etc. as Eddington pointed out long ago.

Feynman, in his Lectures on Gravitation, said it quite poignantly: "We don't know how to write a [energy-stress tensor] to represent a rotating rod, so that we cannot calculate exactly [!] its radiation of gravity waves. We cannot calculate the [energy-stress tensor] for a system consisting of the earth and moon because the tidal forces and elasticity of the earth change the gravity fields significantly."

So, the problem, I believe, does not lie with an appropriate equation of state, but, rather, with the Einstein equation itself and its inability to encompass phenomena related to radiation and dissipation. The key limitation is its covariant divergentless property. What can we ask more of a poor energy-stress tensor? Also remember that his equation was formulated upon Einstein belief of a static universe--one that does not dissipate or radiate. One cannot, therefore, add (cosmological) constants, etc,  to patch of its inadequacies.

Regards,

Bernard