KJW

What the text is saying is that the equilibrium constant for the reverse reaction is the inverse of the equilibrium constant for the forward reaction. So, it should not be: Kc = 1/Kc but: Kcreverse = 1/Kc In the text, I can't see whether they distinguished between the equilibrium constants of the forward and reverse reactions, but it would be wrong if they did not.

Because that is the stoichiometry of the chemical reaction.

No. Multiplying a number by a prime number only increases the number of factors by one, the prime number you multiplied the number with. You do know that the factorisation of a number into primes is unique up to order, don't you?

One thing I'd like to point out to @HbWhi5F is that everything in thermodynamics is about the system and everything that happens to the system. For example, suppose one has some gas in a cylinder with a piston, and one presses on the piston to compress the gas. The work done on the gas depends on the pressure of the gas inside the cylinder and the decrease in volume of the gas inside the cylinder. All well and good. But now release the piston and allow the gas to expand. The work done by the gas now depends on the pressure of the environment. That's no good because the environment is outside of the system which everything is about. So what one has to do is make the pressure of the environment equal to the pressure inside the cylinder at all times. But if the two pressures are exactly equal, there will be no compression or expansion of the gas. So to make the gas compress, one makes the pressure outside the cylinder infinitesimally greater than the pressure inside the cylinder; and to make the gas expand, one makes the pressure outside the cylinder infinitesimally less than the pressure inside the cylinder. Because the two pressures differ by only an infinitesimal amount, they can be regarded as equal. Now the work done by the expanding gas depends on the pressure inside the cylinder (because it is equal to the pressure outside the cylinder) and the increase in volume of the gas inside the cylinder. Now everything is about the system. Also, note that expansion and compression of the gas inside the cylinder are the reverse of each other. Also note that to go from compression to expansion (or from expansion to compression), one only needs to change the pressure outside the cylinder from infinitesimally greater than to infinitesimally less than the pressure inside the cylinder (or from infinitesimally less than to infinitesimally greater than the pressure inside the cylinder). In either case, only an infinitesimal change in pressure is required to reverse the process. Processes that are reversed by applying an infinitesimal change are called reversible processes. Reversible processes are very important in thermodynamics because thermodynamic quantities such as enthalpy and entropy are defined in terms of them. You may have noticed that only in the case of the reversible compression and expansion of the gas inside the cylinder that you get back all of the work that you put into it. In the case of the irreversible expansion, because the pressure outside the cylinder must be less than the pressure inside the cylinder (in order for the gas to expand), the work done by the expanding gas must be less than the work done on the gas when it was compressed. So, what happens to the compression work not done by the gas during its irreversible expansion? It gets converted to heat. The heat increases the temperature of the gas inside the cylinder. Now let the temperature inside the cylinder cool to the temperature it was before it was compressed. The state of the system (its pressure, volume, and temperature) is now the same as it was before the compression. But the total work of the system is not the same. Neither is the total heat of the system. That is, by performing some process of changing the state of the system and eventually returning the system to the original state, neither the work nor the heat of the system return to the same value. It is said that neither work nor heat are functions of state. A function of state is a function that returns to the same value when the state variables return to the same values after traversing any path in the space of states. However, reversible processes do ensure that quantities do return to the same value after the system returns to the same state. Thus, thermodynamic quantities such as enthalpy and entropy are functions of state.

For me, it's the Many Worlds Interpretation. I'm not nearly as comfortable with quantum mechanics as I am with general relativity. I'm not sure that a "correct" quantum theory is even tractable. I like MWI because it provides a natural explanation of intrinic randomness. The mathematics of the Born rule points naturally to the MWI by removing wavefunction collapse and allowing all the eigenstates to prevail. Also, in the high-dimensional Hilbert space of classical-scale quantum physics, arbitrarily chosen vectors are almost certainly orthogonal, which not only provides an explanation of one aspect of the Born rule, but also why classical states don't exhibit interference, including why only a single eigenstate outcome is observed. A difficulty of MWI is explaining the nature of wavefunctions because all the "worlds" seem naturally to be weighted classically. Could it be that you are trying to visualise this as if from outside of spacetime? No, spacetime curvature is intrinsic and doesn't need any higher-dimensional space in which it is embedded. I tend to visualise spacetime as a rectangular grid overlay in which the distances between the nodes do not necessarily obey Pythagoras' theorem. I first became convinced of spacetime curvature when I saw the metrics of seemingly random coordinate systems over a flat space, and noted that these ostensibly didn't look any different from random metrics. But the random metrics will almost certainly not be flat. Inspecting the metrics themselves, unless they are especially simple or familiar, one can't tell if they are describing a flat spacetime, one has to obtain the Riemann tensor to determine this. In other words, given the infinitude of flat spacetime metrics, why not go all the way and consider the infinitude of all spacetime metrics. Anyway, I tend to regard spacetime curvature like the Mercator projection map of the world. I am somewhat ambivalent about spacetimes with non-trivial topologies. I have considered removing the entire interior of the Schwarzschild blackhole using a wormhole metric obtained from the Schwarzschild metric by a coordinate transformation. This raised an interesting question concerning the role of complex coordinates in physics.

I can't accept that spacetime curvature cannot be measured directly. However, I do accept that measuring it may be more problematic than the mathematics suggests. For example, I know that local spacetime is Minkowskian. Therefore, to measure spacetime curvature requires that one measure spacetime over an extended region, which seems to conflict with the mathematical notion that spacetime curvature exists at each point. The smallness of the magnitude of spacetime curvature certainly makes spacetime curvature difficult to measure with all but the most precise instrumentation, but I see this discussion as about measuring spacetime curvature in principle rather than the practical limitations of current technology. That would depend on what you say. If you stick with empirical facts, you will be on solid ground. But I think it is important to be able to explain those facts. And that involves a certain amount of risk because one is no longer in the comfort zone of empirical facts. But I think having a good grounding in both physics and mathematics would help one to avoid the pitfalls that many people in the Speculations forum make.

BTW, if one considers how both the energy and frequency of a beam of light changes as it moves away from a gravitational source, one can derive the proportionality of energy and frequency. In other words, the proportionality of energy and frequency does not come from quantum mechanics. However, the proportionality constant h for a single photon does not emerge from this and must come from quantum mechanics. Well, it wasn't very long ago that I said I rejected the statement "a map is not the terrain", so I tend to accept mathematical descriptions of reality as truth about reality. I see no problem with imbuing physical reality with mathematical properties. And I see no problem with inferring truth about reality such as the reality of spacetime. Reality is clearly four-dimensional to me. Any description of a physical field will include time as well as the three dimensions of space as the domain. That the four-dimensional space has a metric also seems natural, particularly considering the principle of relativity. That the local metric is Minkowskian can be justified by the need for time and space to be distinct. That the global spacetime is curved seems obvious on the basis of how unlikely it is for it to be everywhere flat.

Why doesn't it Imply a universal force of attraction between objects with mass? (Still trying to understand the arguments.) The notion of gravity as a force contradicts itself: If you shine a beam of light upward, the gravitational force on the beam causes its frequency to decrease (the decrease in frequency is itself an empirical fact). This decrease in frequency is directly proportional to the frequency of the beam. This implies that the gravitational redshift is a time dilation. The time dilation implies that the spacetime is curved. The curvature of spacetime contradicts that the gravitation is a force. It should be noted that when being accelerated upward in the absence of gravitation, there are the same forces experienced as there are in gravity, and the same time dilation, but no mass causing the attraction.

Perhaps. But I will look at the mathematics before I look at who wrote it. And yes, I've read stuff by Kip Thorne. That is obvious for me now. But don't take that to mean that I don't know what I'm talking about. What I'm saying is that I have a different mindset. Are you a physicist? What am I supposed to search for? I feel the need to point out that I've already read a lot about general relativity. And not pop-science books, actual textbooks, including textbooks on Ricci calculus. And I even perform mathematical derivations of my own. Yep. That is an empirical fact. And why do things fall to the ground? It's because we are in an accelerated frame of reference. But people all over the globe can say the same thing. Everyone is accelerating away from the surface of the earth. And that implies that the spacetime around the earth is curved because there is no global frame of reference in which everyone's acceleration is zero. What makes you think I'm angry?

That's an argument from authority. Physicists do have a tendency to approximate physics for the sake of ease of calculation. I'm not a physicist. I'm not interested in approximations. I would rather see an exact expression, even if it is of an idealised universe. When you use an approximation, you may miss out on some salient point. For example, you mentioned gravitational waves. It is commonly believed that gravitational waves carry energy-momentum. No, they don't! (But that's another discussion for later.) The point is that the treatment of gravitational waves as an approximation is flawed. This doesn't say anything. Is he talking about the tetrad formalism? In that formalism, the metric is everywhere Minkowskian, but that doesn't make the spacetime flat because one now has an object of anholonomy to deal with and where the spacetime curvature resides. And yet, when I drop a pen, it falls to the ground.

I don't necessarily see these as proof of spacetime curvature, which is why I didn't mention them, but focused instead on spacetime geometry. I already mentioned the Pound-Rebka experiment. This actual experiment plus a number of very reasonable assumptions that could be the result of experiments yet to be performed would provide data that would prove the existence of spacetime curvature as a physical quantity. By the way, I don't need to determine all the independent components of the curvature tensor, one non-zero component will suffice. You said I can't directly measure the geometry of spacetime, and I asked why not. You told me what has been measured but didn't answer why I can't measure the spacetime metric. The EFEs is a system of PDEs whose solution is the metric tensor field corresponding to the measured energy-momentum tensor field source term on the right-hand side of the equation. This is part of the theory and not what I'm talking about. If something can be physically measured, even if it involves some calculation based on a definition, it's real. Thus, because the Riemann curvature is mathematically defined in terms of the metric tensor, measuring the metric tensor makes the Riemann curvature physically real.

Why not? Well, I did actually say in the beginning that the curvature of spacetime is as real as anything else. So, if you prefer to diminish the reality of everything else, then that doesn't conflict with what I said. I was merely making the distinction between an abstract theoretical notion and a physically measurable notion, with spacetime curvature being the latter rather than the former.

That's irrelevant to the point I'm making, which is that spacetime curvature is a physically real quantity (or at least as physically real as anything else). Just because we don't understand what happens inside a black hole does not mean that spacetime curvature isn't physically real. However, I am making a distinction between the theory of general relativity and spacetime curvature. I'm not actually claiming that general relativity is correct. For one thing, I know that general relativity is a classical theory that doesn't describe the quantum domain. But spacetime curvature exists as a physical quantity quite independently of general relativity. The mere fact that spacetime curvature can be measured proves its existence as a physically real quantity. And my use of the word "proves" highlights the distinction I make between a theory and physical quantity. Until we discovered general relativity, we did not know what caused gravity. We knew that two masses in close proximity tended to be attracted to each other, but we did not know why. The attraction is physically real as are the forces associated with gravity. General relativity didn't change that. But before general relativity it was implicitly assumed that spacetime is flat, and therefore we were missing the key to explaining the nature of gravity, and had to invoke a mysterious gravitational force. But we now know that spacetime curvature physically exists and that this provides a genuine explanation of gravity. But even if our knowledge of gravity does improve in the future, this is not going to remove spacetime curvature any more than the attraction between masses and the forces associated with gravity were removed by general relativity.

I don't know why he would say that. The curvature of the three-dimensional space surrounding the earth is very small and perhaps too small to be measured. Time dilation is magnified by the speed of light, but even that requires precision measurements. It's not clear to me that there have been sufficient measurements to actually prove spacetime curvature, but the Pound-Rebka experiment plus some very reasonable assumptions do indicate spacetime curvature beyond doubt in my mind.

Why has there been no mention of Fermat's little theorem in this thread? [BTW, I have a personal interest in Fermat's little theorem because I independently discovered this theorem 33 years ago. At the time, I thought I had discovered something truly remarkable. Fortunately, it wasn't too long afterwards that I found the theorem called "Fermat's little theorem" in a mathematics dictionary. I discovered it by exploring the decimal expansion of 1/p for prime number p, investigating the length of the repeated sequence of digits. For example, for p=7, the repeated sequence 142857 has length 6 and 142857x7=999999.]

No, spacetime curvature is not merely an abstract theoretical notion. It is a measurable physical quantity. I've said on a number of occasions that the gravity with which we are familiar, including artificial gravity, is caused by time dilation. Time dilation is a measurable physical quantity, and by measuring how time dilation varies over the space surrounding the earth, one can prove that the spacetime surrounding the earth is curved. Thus, a correct theory of gravity must account for spacetime curvature.

The curvature of spacetime is as real as anything else.

Time dilation is already built into the formula for the redshift and blueshift of light. Indeed, the twin paradox can be explained entirely in terms of the redshift and blueshift that is observed by the two twins. This also provides a clear explanation of the asymmetry between the travelling twin and the stay-at-home twin.

No it doesn't. You seem to consider determinism/indeterminism as all-or-nothing notions. You've not taken into account how indeterminism scales. For example, the Heisenberg uncertainty principle: ∆x∆p ≥ ℏ/2 shows that the product of uncertainty of position and momentum can be as small as ℏ/2. Although for subatomic particles, this is relatively quite large, for ordinary macroscopic objects, it is negligible. Thus, even though quantum physics may be indeterministic, classical physics is essentially deterministic. Another example is that the standard deviation of the mean decreases relative to the standard deviation of the random values themselves as the square root of the number of values increase. Generally speaking, the relative randomness of a system decreases as the scale of the system increases.

I assume that this is also directed to me even though it doesn't address my particular point. I know that strictly speaking, Bell's theorem only forbids local realism, implying that either non-locality or non-realism can satisfy quantum mechanics. You have chosen non-locality over non-realism. The problem is that non-locality and non-realism are not on equal footing in terms of their ability to satisfy quantum mechanics. The violation of Bell's inequalities by quantum mechanics directly indicates a requirement of non-realism, whereas non-locality is merely a loophole that enables you to clutch at straws in your belief that quantum superposition does not exist. I would argue that a loophole is not sufficient, and that Bell's theorem really forbids both local and non-local realism. You would then need to explain precisely how non-locality enables realism to violate Bell's inequalities.

My biggest problem with AI, especially in the future, is that it will simply be just another way to widen the gap between the haves and the have-nots. I do have a question, though: Do you think that AI is just another technological advance like computers or machines, or do you think that AI is sufficiently different to be especially dangerous to society? By this, I mean that in the past there have always been fears that various technological advances will render a range of jobs obsolete. But in the end, society adjusted to the new technology, and the fears have largely been unjustified. However, is the same true of AI, or is AI of such a nature that the fears are truly justified?

It's worth noting that to an observer outside the blackhole, nothing ever actually falls into the event horizon, not even the collapsing star itself.

Non-locality is irrelevant. Realism, whether local or non-local, satisfies Bell's inequalities and therefore fails to satisfy quantum mechanics. It should be noted that the context of Bell's theorem in this discussion is not "spooky action at a distance" but rather the nature of quantum states. That is why non-locality is irrelevant and why realism is significant.

While non-locality can address causality, it doesn't address determinism. You claim that particles have definite values for all of it's properties. But this implies that Bell's inequalities are satisfied, contrary to the results of various experiments that test Bell's inequalities. The violation of Bell's inequalities by quantum mechanics indicates the violation of realism, the notion that measurement outcomes are well defined prior to and independent of the measurements. However, because quantum states are described by wavefunctions, realism need not be satisfied by quantum mechanics.

There are two problems with this: 1: If a system is deterministic in an unobservable higher dimension, but not in observable reality, is it really correct to say that the system is deterministic? 2: The notion of determinism in an unobservable higher dimension seems like a hidden variable theory. Hidden variable theories have been invalidated by Bell's theorem.