joigus

Here's my analysis of the situation. I did this many years ago to help myself understand the details of this --very well known but quite academic, and at the same time somewhat puzzling-- example. Diagramatics: Obviously, there must be an initial pressure in the gas (P1) that is in excess to the environment's pressure (P1>Pex.) Otherwise the gas wouldn't expand --let's assume is an expansion. One problem is this pressure is not mentioned in the statement. That's one reason why it's confusing. There is, of course, a fiducial isotherm wich goes through this point (P1,V1), and we're not going to use either. Isotherms 1 and 2 are there just as a reference, two coordinate curves on a P,V diagram if you will, but they play no role in the problem. They would look as straight lines on a P,T diagram, and they look like hyperbolae on a P,V diagram. That's perhaps another reason why it's confusing. Now, very important: The straight line in red does not represent the actual trajectory on the P-V plane of the system. The system is jumping outside of the P,V,T surface of possible states of the system, the reason being that it's going through a series of states that are not equilibrium states. It would look something like the orange line that joins (P1,V1) and (P2,V2). So @sethoflagos has a point, if I understood them correctly. Something's going on of which we're given no account. What happens in between? That's the final and most important reason why discussing the problem of the "in between non-thermodynamic states" is so confusing. During this time, the system is no longer described by equilibrium thermodynamics, and sure enough time starts playing a role. We must appeal to a mix between thermodynamical and mechanical arguments, if only to qualitatively understand what's going on. Here's what's going on: If the initial and final internal energies of the gas are resp. U1 and U2, we have an energy tradeoff that looks like: \[ U\left(t\right)=U_{1}+\delta U\left(t\right)+\delta K_{\textrm{piston}}\left(t\right) \] But it stands to reason that the energy balance can be expressed as a function of only initial and final states, because the "mechanical boundary conditions" if you will, are constrained by the thermodynamics of the problem. The air from the environment acts as an inexhaustible reservoir of pressure, for lack of a better word. Both external air and internal pressure act on the piston, increasing its kinetic energy as they go. They counteract each other, but don't exactly cancel: \[ \delta W_{\textrm{gas -> piston}}=-\int P_{\textrm{gas}}\left(x,t\right)dV_{\textrm{gas}}\left(t\right) \] \[ \delta W_{\textrm{ex ->piston}}=-\int P_{\textrm{ex}}\left(x,t\right)dV_{\textrm{ex}}\left(t\right)=+\int P_{\textrm{ex}}\left(x,t\right)dV_{\textrm{gas}}\left(t\right) \] \[ \delta U_{\textrm{gas}}\left(t\right)=+\int P_{\textrm{ex}}\left(x,t\right)dV\left(t\right)-\int P_{\textrm{gas}}\left(x,t\right)dV_{\textrm{gas}}\left(t\right)=-\int\left[P_{\textrm{gas}}\left(x,t\right)-P_{\textrm{ex}}\left(x,t\right)\right]dV_{\textrm{gas}}\left(t\right) \] Finally, the piston stops. How does it do that? The gas, no doubt, overshoots a little bit --see diagram--, so even though the reservoir to a good approximation stays where it is thermodynamically, locally it exerts an excess pressure to take the piston to its final position. So, initially, the piston works against the air with its local field P(x,t) averaged on the piston's surface; but finally, the air works against the piston to compensate for the overshooting and leaving it at its final rest position. We must assume also that the walls leave heat go in and out. So, \[ K_{\textrm{piston}}\left(2\right)=0 \] So no matter how complicated the details are, the overall effect of the air is to compensate for the unbalance in the work terms so as to deliver the unbalance back to the gas, so to speak, in such a way that, energetically, it's as if all the time the pressure Pex had been the only pressure that's been doing all the work. Final note: The proposed curve is, of course, a simplification. Really, the gas doesn't have a thermodynamic pressure, as @studiot and myself have pointed out. But even so we can talk, I think, about an average pressure that the gas exerts on the piston near the surface. That's the pressure I'm talking about when I wrote things like Pgas(t)

No. Curvature is defined by going from one point to another by applying equal changes in 2 coordinates in different order, and thereby learning if there's a difference in going through different paths across your space. If there is, your space is curved. You can also have curvature as solutions to Einstein's vacuum field equations. It so happens that there are 20 numbers that give you the curvatures in dimension 4 or more. Such is the nature of Riemannian curvature. That's the curvature we talk about when we do GR. Mind you, you can also define a special curvature for a curve, but that's not Riemannian curvature. It's based on a moving reference frame embedded in a flat space of 3 dimensions. The Riemannian curvature of a curve is 0. That's because Riemannian geometry is intrinsic: not embedded in a higher-dimensional space. Language and intuitive notions by themselves are misleading: A curve is not curved, a plane is not curved (makes sense, but it's true nonetheless), and a cilinder is not curved. A cone is not curved, but at a point, where it has infinite curvature. I can't tell which one is your curvature. And I don't think that's a good thing. It all sounds like you lack a basic understanding of curvature. A historical note: Einstein found his famous equations only after a long correspondence with David Hilbert, and consultation with mathematicians, and from there he postulated a certain combination of the 20 curvatures of space-time (reducing them to just 10) to be proportional to the different components of energy-momentum densities. This checks with 4(4+1)/2=10 independent components of the energy-momentum tensor in dimension 4. He achieved that only after several faltering attempts. If my memory serves, it took him the best part of a decade to come up with that after his initial intuitions. He later proved that in the limit of low velocities and weak gravitational fields you get Newton's law of gravitation. He predicted light to be bent by gravitational fields. Lastly, he did not coin the name "Einstein field equations". It was other people who named them after him. To my mind, it belittles the genius of Einstein when people try to emulate or better this mind-blowing feat based on some loose notions and a couple of graphs. Don't take this personally but, at what point are you going to decide it's time to go back to the drawing board? To be more precise: pseudo-Riemannian, because of the difference in sign for time and space in the metric.

But irreversibility is a premise of the OP. Also, entropy increase doesn't only happen because of heat transfer. Certainly, entropy change will happen whenever there is heat transfer. But it will also happen in situations in which there is irreversible work involved, like eg when you stir the fluid with a blender or a rapid fan. The fact that irreversible work leads to entropy increase is betrayed by the fact that, after a while, the temperature increases, as this irreversible work is quickly converted to heat. If you wait a couple of seconds, say, and measure the temperature, you'll see that it's decreased (expansion) or increased (compression), and it's no longer possible to know whether this change in temperature has come from irreversible work or from heating with, eg, a Bunsen burner --or from cooling through a wall. It's only that, in this particular example, it's much easier to calculate the work without involving energy at all, thus reducing the calculational work to a minimum. Now, what you seem to be demanding from the OP is to: 1) Express the entropy as a function of V, T, and the number of molecules; or perhaps P, V, and the number of molecules, if it's a P,V,T,N system. 2) Calculate the values of S1 and S2, given that S is a state function --it sure is--, and compute it. Now, step 1) isn't elementary. It certainly can be done, and could be needed in more complicated irreversible processes. But for expansion against a fixed external pressure it's more simply done with the method that the OP proposed. I meant "without involving entropy at all".

That's not how curvature is defined. Do you know what curvature is?

What does this have to do with a metric in curved space-time? And how could energy-momentum be "depleted" when the particle picks up speed?

Thank you. It seemed too far-fetched to me too. You're right, material from erosion must be orders of magnitude more sizeable for both reasons you point out.

I once heard claims that even such subtle effect as changes in the moment of inertia of the Earth due to seasonal leaf shedding in deciduous forests can have a measurable effect in the Earth's rotation. I don't know how much truth is behind such claims, or whether it's in the milliseconds. Could that be true? It seems like small potatoes in comparison with motions in the mantle, for example.

Exactly! They key fact is your observation that, \[ \frac{\boldsymbol{f}\left(\boldsymbol{r}\left(t\right)\right)}{\left\Vert \boldsymbol{f}\left(\boldsymbol{r}\left(t\right)\right)\right\Vert }=\frac{\boldsymbol{r}'\left(t\right)}{\left\Vert \boldsymbol{r}'\left(t\right)\right\Vert } \] You only have to dot-multiply by \( d\boldsymbol{r}\left(t\right)=\boldsymbol{r}'\left(t\right)dt \), remember the definition of the norm, and you're there. I hope that helps.

The argument still stands. Orthonormal is a particular case of orthogonal. Orthonormal=orthogonal and normalised. More specifically, a gravitational singularity is a region of spacetime in which any components of the Riemann curvature tensor become infinite. Read carefully @Markus Hanke's previous post. You do not invent the properties of the metric. You postulate the other (non-gravitational) fields. Then you obtain the energy-momentum tensor. Then you symmetrise it (with techniques like, eg, Belinfante's symmetrisation technique), because the canonical energy-momentum tensor is generally non-symmetric, and the source of the gravitational field must be symmetric in the space-time indices. Then you postulate boundary conditions, as Markus told you. Then you solve for your metric. Having done all that, you're still not home-free, because the particular coordinates that you use to solve for the metric can have false singularities, ie, singularities of your coordinate map that are not physical. So you must obtain the Riemann tensor and try to identify the singularities there. You have a lot of ground to cover still before you can meaningfully talk about your singularity. I hope the comments here you find helpful. The metric is not gauge-invariant. It's the Riemann tensor that's gauge invariant. This is in close analogy to electromagnetism. The vector potential in EM doesn't really give you the physics (except for the Aharonov-Bohm effect or the "holonomy" of the field). Infinitely many vector potentials give you the same physics. It's Faraday's tensor plus the holonomy which gives you the complete physics of electrodynamics. There is only one Faraday tensor (the E's and the B's) that define the physics. Gravity displays remarkable mathematical similarities to EM. It has a huge gauge arbitrariness. In modern GR we say space-time is not defined by a metric, but by an equivalence class of infinitely many metrics, all gauge-equivalent to each other. The matter is even more subtle. Sometimes you find a coordinate map that solves the equations. But the map has singularities of itself --fictitious. Then you introduce a change of coordinate maps that fixes the coordinate singularities. Example: Kruskal-Skezeres coordinates being a well-known example.

OK I have to come clean at this point and confess I do not know what "being more primitive" means in mathematics. I think it was Poincaré who tried to base everything mathematics in terms of group theory. Another attempt of basing maths on something "primitive" was Felix Klein's Erlangen program to unify geometry. Category theory seems to be another attempt at building a really primitive branch of maths. "Primitive" meaning something like "least number of assumptions." [?] Perhaps "primitive" means theory A can be based on concepts derived from theory B, but not the other way around, and therefore theory B being more "primitive" than theory A? I'm not sure of what mathematicians mean when they say they're trying to refer things to something more primitive.

What is a "pofinage"? And how can a basis remain orthogonal irrespective of the metric? The metric is what tells you whether a basis any given set of vectors is orthogonal or not. How can rest (which is an observer-dependent concept) be associated with a singularity? Does a frame change remove the singularity? Infinite value of what? Singularities in the metric are meaningless. It's only singularities in the Riemann tensor that are physically significant. Also, a 3pi angle points in the negative x direction, not in the negative y direction. I think you meant 3pi/2. Whatever x and y mean. There we go again.

Beatifully made, very motivational, educational videos on paleontology, geology, and the entire history of the Earth: The extra code just takes you to the list: https://www.youtube.com/watch?v=0_a_xU2KQdE&list=PLoOkB6QkDW_S7fdqmfW-_lwS_Kp_uwfeo

No metric to be seen.

Perhaps the most primitive mathematical "operator" is that of a relation?: Any well-defined connection between two elements of a set. aRb = a is related to b by means of relation R within a set A and a relation within a set is any binary pairing, that is, any subset R of AxA So a is related to b by relation R if (a,b) is in R (we write \( aRb \) ) and a is no related to b by R if (a,b) is not in R (we write \( a{\not}R b \) For that you kind of must have set theory first, so... Perhaps "belongs", as an external operation (between elements and sets). If you think about it, equivalence relations are just a particular example of relations in general. If aRb, then bRa If aRb and bRc, then aRc and aRa always And equivalence relations are at the basis of our categorical thinking. But this is so abstract that my head hurts. So I guess what I'm saying is the Cartesian product.

\[ {\not}R \]

Markus had said it already. Thanks for elegantly pointing that out.

Let's not forget the tall-poppy syndrome, which could be a factor in this too.

Yes, it has all the hallmarks of the Dunning-Kruger effect. I wonder whether there could be a survival component to this cognitive bias, as the effect is so common. After all, a modern scientist can afford months of agonizing about whether they got it right. Under more stringent survival conditions, being self-assertive no matter what may have played a role in decision-taking.

Please, do tell us. I gave you an early alert that you need to up your game. This is a good chance for you to start making some sense. Getting on the nerves of people is certainly no way to push your arguments forward. I find your first statement surprisingly bold. QM and GR have proven to be extremely difficult to reconcile so far, if not impossible. The only ways I've heard of to make sense of quantum corrections to all levels and include gravity are superstrings, LQG, and MHV amplitudes. MHV makes next-to-impossible calculations actually doable. The problem is you lose track of explicit Lorentz invariance and locality. And, of course, you need to devise experimental techniques to ramp up the energies of the experiments to Planck scale. Please, tell us also about that one necessary characteristic for a sensible theory of quantum gravity. For dramatic effect, you can use the spoiler function, like this, Q: What is the one characteristic that a quantum theory needs? A: I'm looking forward to your illuminating answers.

Good point. I forgot to mention this, which is essential, I think, to understanding why in irreversible processes it doesn't work that way.

Human interest: The human potential to imagine, to invent, fill in the missing details, and propagate false belief plays out in many different ways. But generally there is a big human motivation behind all these stories. I've seen this craze go on and off for many years now. I'm a child of the sixties. The amount of books, films, magazines etc sold is a factor that shouldn't be taken lightly. How the brain works: There are rigorous scientific studies that go to prove that our memory does not work at all like a video camera, which is what our intuition tells us. Our memory edits these impressions in the hippocampus and makes up a story according to different "interests" that may be convenient to different purposes or internal needs. https://www.sciencedaily.com/releases/2014/02/140204185651.htm#:~:text="Your memory reframes and edits,editor and special effects team. People even talk to each other and "reconfigure" their impressions, correcting them with "data" from other witnesses into a narrative that they feel must have been "what really happened." See next point. Collective memory: Think about this: Many people in the past believed in centaurs, fairies, angels, leprechauns, giants, dragons etc. And now consider this example (with a possible explanation): Accounts of centaurs, IMO, probably had their origin several thousand years BCE from real facts, when the first peoples to domesticate horses swept across the Eurasian steppe in East-West direction. The first agriculturalists who saw this must have been terrified by these warriors (the Yamnaya), and haunted by visions of horses whose business end was a human torso with an axe in one hand. They had no previous experience of anything like that. The first accounts probably included some phrasing like "half horse, half human." By the time all peoples of Europe are used to the sight of warriors riding horses, they understand what they see now, but the initial story of these hybrid creatures already has a life of its own. You can now re-edit the story --socially-- and make these centaurs benefactors of the human kind, spreading good will; or you can turn them into demons, or whatever the wishful thinking of the times takes them to be. Reason and evidence are paramount. You cannot build objective knowledge only from witness accounts.

Pressure is not constant in an irreversible expansion. The pressure that you've got there is not a state variable of the system that's expanding; it's the external pressure of the environment, which is constant. Thereby the subindex "ex" in the formula for work. In fact, during an irreversible expansion, the pressure of the system is not even well defined. It's only the external pressure what's well-defined as a thermodynamical variable.

Oh. Is that for my benefit? Thank you. The straw man argument is even discussed in the guidelines of these forums. You may want to have a look at those. Thanks for the news update. Do they punch people in the face too? Maybe kangaroos are behind it. Here's another idiom that you may be interested in: jumping to conclusions. Other names for the fallacy that's operating behind it are "just so" stories, ad hoc arguments or associative logic. From Google: Carl Sagan has a beatiful example in his book (and TV series) Cosmos of how even scientists in a not to remote past used similar arguments to surmise that Venus must be populated by dinosaurs. The logical fallacies are plain to see for anybody who's familiar with logical fallacies. Humans have been inventing outlandish explanations for unexplained phenomena for at least 11'000 years. It's all a recurring theme: Beings from another level of existence visit us and affect our lives. I'm not saying that there aren't phenomena that cannot be explained by current science. There are. I'm not even gonna touch your argument about parallel universes. Back to you.

You really need to up your game. There is definite proof that kangaroos exist. There is not even a clue that intelligent civilisations from another planet are visiting us. Are you familiar with the concept of a straw-man argument?

Playing straw man, are we? You really have to up your game here.