KJW

I didn't explicitly say it, but my description implied that the proportionality between speed and separation was for different pairs of particles at a particular instant in time. At different instants in time, the proportionality constant will be different such that the relative velocity between any particular pair of particles will be constant over time.

That is not true. Although there are limitations on how much spacetime we can be directly observe, we can observe enough of it to know that it is real. Bear in mind that "to observe" means "to measure"... one doesn't need to be able to look at a four-dimensional block of spacetime directly with our eyes.

For an explosion from a single point, every particle moving away from that point also moves away from each other at a speed that is proportional to the separation between the particles. Consider two arbitrary particles. Construct the triangle from the positions of the two particles and the origin of the explosion. This is the configuration at time=T. At time=2T, the configuration is a similar triangle of doubled lengths. This means that none of the angles have changed, the two particles are moving away from each other without any transverse motion.

There seems to be confusion about the nature of general relativity, conflating it with special relativity. In special relativity, one speaks of measurements by different observers, leading to such notions as time dilation, length contraction, relativistic mass, etc. General relativity isn't like that. In general relativity, one may consider different coordinate systems. Coordinate systems are not necessarily the perspective of any observer. Therefore, there isn't the question of whether different observers agree about some notion in general relativity. For example, one often specifies a spacetime using a metric. Everyone agrees on the metric, but no one regards the metric as THE metric, because one can coordinate-transform the metric to some other equally valid metric. All coordinate systems are equally valid in general relativity. One can construct a metric based on a particular observer, but this is no more valid than any other metric obtainable via a coordinate transformation, even if such a metric is not based on any observer. Some metrics may be preferable to other metrics for practical reasons, such as manifesting inherent symmetries of the spacetime, but even in such cases, all the possible metrics obtainable by a coordinate transformation are equally valid. In general relativity, one may determine the distance between two points in spacetime along some curve between them. Although the specification of the two points and the curve between them may vary according to the coordinate system, in all coordinate systems the distance will be the same. This is also true in a proper treatment of special relativity. But unfortunately, much of the confusion regarding special relativity occurs because one is not dealing with points in spacetime, but is trying to deal with space and time separately. It is my impression that many people think that motion distorts space and time. The formulae that usually introduce people to special relativity give that impression, but seeing special relativity in terms of four-dimensional Minkowskian spacetime provides clarity to the nature of relativity.

I assume the same is not true for fermions. While I know that particular fermions, such as electrons, have conserved properties that prevent their creation and destruction, can it be said that the property of being a fermion itself prevents their creation and destruction?

Transparent substances such as gases do emit thermal radiation. But because they are transparent, one has the take into account Kirchhoff's radiation law (https://en.wikipedia.org/wiki/Kirchhoff's_law_of_thermal_radiation).

3D geometry doesn't explain gravity. And if there is no need for curved spacetime, why do we have it?

It appears that my analogy has failed. I shall abandon it in favour of an even simpler analogy: Imagine walking along a road that is constant in width. The journey along the road represents the journey through time. It does take time to travel along the road in the analogy, so the analogy is somewhat realistic in this way. At regular intervals along the road, there are lines perpendicularly across the road from kerb to kerb. Each of these lines represent a spatial slice through spacetime such that each spatial slice is at least approximately the same as every other spatial slice, and different points on any given line represent different locations in space. As you walk parallel to the kerb anywhere on the road, you are at the same location in space at different times. The line is one-dimensional and represents the dimension along which the time dilation occurs. In the case of the earth and its gravity, a line on the road represents a vertical line, and the constant width of the road represents a constant height over time. While the road is straight, the distance between the lines along the left and right kerbs is the same. But as the road curves to the left, the distance between the lines along the left kerb will be shorter than the distance between the lines along the right kerb. And as the road curves to the right, the distance between the lines along the left kerb will be longer than the distance between the lines along the right kerb. That is, the road always curves toward the shorter distance along the kerb and away from the longer distance along the kerb. There is a mathematical relationship between the curvature of the road and the relative difference between the lengths along the left and right kerbs. The relative difference between the lengths along the left and right kerbs represents the gravitational time dilation between the two different heights. And the curvature of the road represents the upward acceleration we perceive as we stand on the ground. Although we can see ahead along the road, seeing the curve of the road, this would represent seeing into the future, which we can't do. So, we don't actually see a curved trajectory in spacetime. We perceive it as gravity. It should be noted that there is also time dilation for an accelerated frame of reference in flat spacetime, and that there is the same mathematical connection between accelerational time dilation and acceleration as between gravitational time dilation and gravity, as required by the equivalence principle. However, there is an unavoidable complication between the analogy and the physics it represents that is due to the peculiarity of the geometry of spacetime compared to the familiar Euclidean geometry of the analogy. Thus, the acceleration of a trajectory in spacetime points in the opposite direction to the curvature of the road. That is, on earth we accelerate upward and away from the shorter distances in time, not towards it as in the analogy. The analogy is still correct, the mathematics describing it is the same as the mathematics describing gravity. In familiar Euclidean geometry, the two-dimensional shape that is a constant distance from a given point is a circle. In the case of spacetime geometry, the corresponding shape is a hyperbola, the central point that is equidistant from every point of the hyperbola is the intersection between the two asymptotes. Note that a hyperbola curves relative to the central equidistant point in the opposite direction to a circle. This is the difference between the analogy and the physics it represents. Thus, unlike the trampoline analogy, which is a "lie-to-children" that does not correctly describe gravity, the road analogy actually does explain the relation between time dilation and gravity.

It is commonly assumed that straightness in curved spacetime is a modified form of straightness and not a true form of straightness. And the Christoffel symbol in the formulae for straightness doesn't help to dispel that view. But along any spacetime trajectory in an arbitrary spacetime, a Minkowskian coordinate system can be applied, and in that local coordinate system, a straight line is truly straight in the ordinary sense. A spacetime trajectory that is straight in spacetime may appear curved to us because we are assuming a flat spacetime. In other words, we have made an error in our assumptions about the spacetime, the apparently curved trajectory being the manifestation of this error. If one considers the orbit of a planet around the sun to be a trajectory in spacetime, where the distances in the time direction are very large compared to the distances in space, then the trajectory of the planet in spacetime doesn't appear to be very curved at all. Also, if the trajectory of the planet in spacetime is projected onto a three-dimensional space, then the trajectory in the three-dimensional space will be curved. But that curvature is because all consideration of the time dimension has been removed.

Doesn't thermal equilibria imply ergodicity? Statistical thermodynamics is based on this assumption. It is often said that one can't assign a temperature to a single particle, and that temperature is a statistical property. But I have shown how temperature can be assigned to a single particle while maintaining the statistical nature of temperature. The ensemble is the particle (over time). I'm not the one who started the discussion on assigning temperature to a single particle. Oddly enough, @martillo rejected my suggestion in favour of something that is not going to work.

So, the psychiatrists are in on it? Who else is in on it?

I see no one thought to check this... but I did. This is what I got: [math]1.02 \text{ milliarcsec} = 1.02 \times \dfrac{1}{1000} = 0.00102 \text{ arcsec}[/math] [math]0.00102 \text{ arcsec} = 0.00102 \times \dfrac{1}{60} = 0.000017 \text{ arcmin}[/math] [math]0.000017 \text{ arcmin} = 0.000017 \times \dfrac{1}{60} = 2.83 \times 10^{-7} \text{ degree}[/math] Continuing further: [math]2.83 \times 10^{-7} \text{ degree} = 2.83 \times 10^{-7} \times \dfrac{2 \pi}{360} = 4.95 \times 10^{-9} \text{ radian}[/math] Calculating in radians allows you to calculate without the explicit use of trigonometric functions.

This simply doesn't follow. The IDE is not universal, particularly for higher pressures By "universality", I meant universal with respect to different gases, including gases that differ in their heat capacity. I had already said that I view the ideal gas law as a low-pressure limit, so "higher pressures" don't apply. And even if one does consider the van der Waals equation, the difference compared to the ideal gas law are non-zero volume of the molecules and intermolecular forces. Neither of these relate to the rotations of molecules or vibrations within molecules. Why are you fixated on particle collisions? Particle collisions are irrelevant to the point I was making. A bit of algebraic rearrangement from what? We are talking about gases... simply place it in a sealed borosilicate glass flask. Why are you mentioning "processes"? Processes, are also not relevant to the point I was making. This seems to be where the error in your interpretation of what I said occurs. I never said that energy doesn't transfer between all the degrees of freedom. In particular, I never said that energy doesn't transfer between translational and non-translational degrees of freedom. I also never said that if the temperature is changed, that this only affects the translational degrees of freedom. What I did say is this: That the same ideal gas law applies to argon, nitrogen, carbon dioxide, water, and ethane proves the point I was making. That all these gases have different heat capacities, particularly at higher temperatures, also proves the point I was making. I never mentioned "causation". The word I used was "contribute". Ultimately, what I said rests on a definition of "temperature". That's why I mentioned a gas thermometer earlier. Thermometers measure temperature and so must have a definition of temperature in their design. And gas thermometers are based on the ideal gas law, thus defining temperature in terms of the translational motion only.

Yes, pressure and temperature are statistical quantities. I never said that non-translational motion had no effect on each individual collision. But the universality of the ideal gas law is a testament to the statistical independence of pressure and temperature on non-translational motion. Only in one very special case: that of a theoretical constant volume thermodynamic process. Why are you saying that constant volume is a theoretical very special case? Can you elaborate on this?

An alternative is to consider a gas thermometer. The temperature defined by such a device is based on the kinetic energy of the translational motion of the gas and does not include non-translational motion that may occur in the gas. Thus, one could replace the gas with a different gas that has a different heat capacity, based on the universality of the ideal gas law.

No, I simply applied it (although I had not read the Wikipedia article). What I consider to be my idea is that an arbitrary isotropic radiation wavelength distribution has a minimum and a maximum temperature (defined by the application of Kirchhoff's law) from which can be defined a maximum thermodynamic efficiency for the extraction of work from that radiation (a physical realisation of the two temperatures).

Strictly speaking, I mean the Weyl tensor, traceless Ricci tensor, and Ricci scalar. One can combine these in a multitude of ways, but these three are the true algebraically distinct curvature tensors.

Ok, I'll leave you to it. 🙂

Photosynthesis is made up of two sets of reactions: light reactions and dark reactions. The light reactions are associated with the absorption of light to drive the synthesis of two short-term energetic substances as well as the production of oxygen from water. The dark reactions use the two short-term energetic substances from the light reactions to produce sugars from carbon dioxide. The light reactions naturally require light, but the dark reactions do not occur in the dark even though light is not actually used in the dark reactions. Instead, light acts as a regulator of the dark reactions. What this means is that the synthesis of sugars (or at least those synthesised from carbon dioxide rather than from other sugars) does not occur in the dark. Thus, for growth to occur in the dark, it would seem that the sugars required would have to have already been synthesised during the daytime.

Actually, the reason I chose to read about photosynthesis was because of this thread. I wanted to find out if the dark reactions only occurred in the dark, as suggested by your post. But I found that instead of the dark reactions also occurring in the light, they actually only occur in the light.

No, you can't decouple translational from non-translational motion, but the point I'm making is that the energy that is in the non-translational motion doesn't contribute to either the pressure or the temperature of the (ideal) gas, except if you try to remove it.

Curvature is a two-dimensional notion. It requires a surface to be curved in both directions. So, a sphere is curved but a cylinder is not. In higher dimensions, curvature can also be considered in terms of sectional planes. However, in two dimensions, there is only one type of curvature; in three dimensions, there are two types of curvature; and in four or more dimensions, there are three types of curvature.

If pressure is independent of non-translational motion, and temperature is directly proportional to pressure, then temperature is independent of non-translational motion. One thing that was never made clear throughout this discussion is what temperature is. I felt that temperature was being conflated with energy. But I will agree that my view was somewhat perverse, and I'm not even sure how it arose. It is correct for gases, but questionable for solids and perhaps liquids as well. Each degree of freedom has an energy of ½kT. Therefore, temperature can be regarded in terms of the energy of a single degree of freedom. And the equipartition theorem ensures that the single degree of freedom can be any of the degrees of freedom, not just translation. This seems to be the argument against me. My argument was basically that it isn't all the degrees of freedom that contribute to temperature because it is only one degree of freedom whose energy specifies the temperature. But translations are quite ubiquitous, and the three translation modes always occur together. So instead of temperature being specified in terms of one degree of freedom with energy ½kT, temperature could be specified in terms of the three translational degrees of freedom with energy 3/2kT. And given that only translations contribute to pressure, it would seem that translations are quite special compared to the other degrees of freedom. However, I am also aware that temperature emerges as a Lagrange multiplier in statistical thermodynamics, so I'm not entirely wedded to the ideas I expressed in this thread.

I was reading about photosynthesis in Wikipedia earlier today, and it said that the "dark reactions" (the reactions that in effect reduce CO2 to simple sugars) do not occur in the absence of light.

No, that's not how gravity operates. The two wheels represent two clocks at different locations in space. The two wheels rotate at the same angular rate, representing simultaneity. But each wheel rolls different distances on the road, representing different proper times. The different proper times at different locations in space is the time dilation that causes the path to curve. The road represents a two-dimensional spacetime where width is space and length is time, and this spacetime need not be curved, for example in the case of an accelerated frame of reference in flat spacetime. In other words, the analogy doesn't only describe true gravity caused by matter, but also artificial gravity caused by an accelerated frame of reference. The equivalence principle ensures that the same analogy applies to both. The rubber sheet analogy is wrong because the curvature of the rubber sheet is not a correct representation of what causes gravity.