joigus

No. It's essentially the same complaint. People often confuse the dumbed down idea, the metaphor, the motto, with the real idea. Physicists come up with these metaphors to help themselves, and others, remember, understand, and suggest: Antiparticles are ordinary partlcles going backwards in time. Virtual particles carry interactions between real particles. Black holes evaporate. Etc. Those are not the ideas. They are motivational instruments. You can't go from the motivational instrument to the actual theory without filling in the details. That's what I meant.

You're living dangerously here. Time propagates with respect to "something else". And what might that be? And how does it relate to everything else that's known? Feynman bitterly complained about ideas like this: Maybe time is not continuous. Maybe there are other dimensions. Maybe spacetime is a fractal... Yes. But how does it relate to everything else we know? What the auxiliary hypothesis that gets you out of this arbitrariness/circularity? I'm not saying that everything we define must be directly measurable. In quantum mechanics, we speak of the wave function, which is not an observable of the theory. But we have ways to check that it's a useful and relevant mathematical construction. In the case of QM, it's Born's hypothesis that the odds of something happening can be calculated from the wave function as quadratic functions of this wave function. Also interference patterns, etc.

Ok. I will try to think about this more carefully tomorrow. Or more likely, over the weekend. But flatness of the universe, as commonly expressed in cosmology books, seminars, etc., refers to spatial flatness. Not to Doppler effect. Doppler effect has to do with ratios d(tau)/dt; that is, proper time over coordinate time. It's, for lack of a better word, some kind of "time curvature". Flatness of the universe refers to flatness of the spatial sections of it. Because the question stemmed from a post on "velocity of time", and you mention Doppler effect, I think there might be the rub. Does that make sense to you?

Why do you say Kruskal has been "pushing the equations"? The equations give a catalogue of exact solutions, and Kruskal didn't touch Einstein's equations to do what he did. He just re-arranged the solution. He re-expressed a well-known exact solution by analytic continuation (maximal analytic extension). He introduced a change of charts, which is singular at the point where the initial chart was singular. Sure enough, if you plug in the solution in Einstein's equations again, but in terms of Kruskal coordinates, it still satisfies them. Are you saying it doesn't? I agree with you that the actual singularities probably point to a region were GR alone probably is not the whole story, so they're not "real", if you will. I totally agree with, No, it doesn't. Schwarzschild's solution looks the same for all times. In fact, the solution is unrealistic, among other things, due to this. It was always there and it never grows. The fact that there is a time doesn't imply the metric is "dynamic". It doesn't display collapse, it doesn't display accretion, it doesn't display evaporation. But Markus will explain this better, I'm sure.

Oh, I think Bohr's pessimism will never be rebutted as a factual constriction of Nature. But Bohr has been a little bit obscure to me at times when trying to explain why. But we're getting off the tracks here, because I don't think he ever gave much thought to black holes. Although, as Markus knows well, and you probably do too, there may be a connection between complementarity and BH's through the EPR = ER principle.

As it stands today, quaternions are used as an alternative representation of the Dirac equation. The 4 components would correspond to the 4 components of the Dirac spinor, which are interpreted as both spin states of electron and positron. I'm not aware of any full-fledged formulation of QED based on quaternions. I don't know much more about this idea. As to octonions... mmm. I don't know. They are non-associative, which is, right off the bat, quite strange to represent physics. But who knows.

Yeah. Some nifty definition is needed here. I'm not saying it's impossible to define a "speed of time" in a meaningful way. Cosmology comes to mind. The second quality standard that a definition worth the name has to satisfy is being useful. Otherwise, speed of time = rate of change of time with respect to time = change of time / change of time = 1

Forgive my ignorance but, what exactly is the "speed of time"? You cannot define things in a vacuum. How do you measure that speed? In order to measure a speed, you need a quantity changing. And then you need something else changing in a way regular enough from the perceptual POV to serve as a standard clock. What is the standard clock against which time is seen to change?

Superstring theory purports to generalise the standard model. In the SM neutrons and protons, as well as mesons, are not elementary, but aggregates of quarks and gluons. So strings would correspond to quarks, gauge bosons (photons, W, Z and the graviton).

The coordinate r is not the radial coordinate. That much I, for one, will concede. For starters, Schwarzschild's r in the interior maps time, not space. But even outside it doesn't have to be, even though the temptation to call it so is very strong. If you want an argument that shows this very clearly, consider this: Scharzschild coordinates separate the angular factor so that the total solid angle gives 4*pi. Well, that cannot be. And the reason is that in a space with (positive/negative) spatial curvature, there must be an angular (deficit/excess). Of that, Scharzschild's coordinates tell us nothing. Here's the thing: Coordinates in GR are meaningless. Scharzschild's coordinates are only meaningful at spatial infinity. What you do in GR is pick out a set of coordinates that suits you to solve the equations, then do an analytic extension that smooths out all the singular points of the coordinate chart, then you identify the real singularities (infinite curvature). Finally, you discuss causality and the like in terms of the best set of coordinates(Kruskal-Skezeres, conformal, with Penrose diagrams, etc). Mind you, if your favourite coordinate chart that you used to solve the equations was "contaminated" with spurious singularities, you will have to introduce a singular change of charts that undoes the damage. It's not you, it's not me, it' generations of physicists that have been confused by the mirage of their coordinates charts. GR is a lanscape to tread very carefully.

As exchemist said, neurons are too big to be affected by a single atomic event. When an action potential is triggered, many atoms are involved. On the other hand, although cosmic rays hit our bodies constantly, keep in mind: Not all DNA is being constantly transcribed and translated. A big part of it is not being expressed, and never does during an organism's lifetime. Many genes code for IF, THEN clauses in the genetic code: If such environmental factor is present, then synthesise such and such protein. Also, there are specialised enzymes that detect changes in DNA and correct them constantly to a precision that's about 1 part in 109 --if I remember correctly. Finally, modified cells produce chemicals that act as stress signals, resulting in the removal of the cell. Among the suite of enzymes that cells produce, several of them are in charge of telling the cell when to divide (mitosis). If the part of DNA that does this job is damaged, it's certainly possible for a cell to start dividing indefinitely. That's what happens when carcinogenic mutations take place. It's perhaps worth noting that cosmic rays are not the main source of high-energy radiation that we're exposed to. Rocks are probably a much more important factor. For genetic mutations to be inherited, they must affect sexual cells (gametes).

That's what I meant. There is no interference pattern. There is interference, of course, but we wouldn't be any the wiser about it by watching just one electron. Thank you, because I think the point is important. Sometimes brevity is not your best friend.

The second person would see the electron land somewhere on a screen and wouldn't notice anything remarkable at all. No interference, because it's just one electron. It's been done, and that's what happens.

I'm not sure what strings you're referring to here. There are hadronic strings and superstrings. Hadronic strings are a model to represent bound states of quarks joined by gluons (making up the body of the string). It takes at least two quarks to make a hadronic string. Or --most likely-- you mean superstring theory (really tiny strings of Planck size). Then every elementary particle (electrons, quarks, photons...) would be a string. As a quark is, as far as we know, elementary (point-like, no internal structure), every quark would be a string within the framework we call superstring theory.

This should go in Speculations, and wild ones at that.

This is so important, and so often forgotten...

I would agree with that. This is what Einstein didn't like: IOW, whatever objective limits to knowledge. You absolutely cannot know what's behind this curtain. It would be very interesting to know what Einstein and Gödel talked about when they took those long walks by the campus of Princeton. Here's the mathematician who proved that there are things in maths that are not for us to know, and the physicist who always dreamed everything could be understood. Then Hawking and Penrose found out that singularities are (almost?) inevitable, but always hidden behind horizons. It's an interesting pattern. As @MigL correctly said, they sometimes draw a curtain on the remote past (cosmic singularity), and sometimed towards the future (BHs). If I'm allowed to take sides, I would borrow Einstein's optimism, that we will understand more if we come up with the right mathematical mapping of maths to physical reality. In that direction, there's a very interesting formulation of GR --called Plebanski's formulation-- some developments of which strongly suggest complex numbers. It's one thing to know that you shall not pass, but understanding why, and a very different thing just to know that you shall not pass. Then you just look silly.

Einstein abhorred singularities. If my history's right, he thought that his equations must be modified somehow in order to eliminate singularities when the fields become too strong. I think he always clung to the idea that everything wrong with his equations had to do with unifying GR with electromagnetism. He spent many years blissfully ignoring the development of new kinds of interactions (weak, strong). As well as quantum mechanics. But that's another story. I think that deep down inside, he was a firm believer --with Dirac-- that the mathematical beauty of the theory will tell you what to do next.

I hope I'm not splitting hairs here in the eyes of the beholder --I know I'm not. But: Not exactly. They're not models; they're predictions of a theory. Models are assortments of ideas to fit some --previously known-- and desired properties. Predictions, OTOH, are completely unexpected features of a previously known set of ideas. Einstein didn't believe in BH's. That's one of two times he was wrong in thinking he got it wrong.

Hi. Welcome to the forums. The v you see there is not x/t. x/t is the ratio between the x-coordinate of a point and the t-coordinate moving in an arbitrary way (mind you, not necessarily at a constant speed), while v is the rate at which O' moves with respect to O. I hope that clarifies the question. Is that what's troubling you?

You make some good points that the problem has other possible alternative treatments, probably more realistic, and that the real problem is more involved. But I'm not sure that using the Navier-Stokes equation would be the best approach for, eg, freshman or sophomore students. We don't know the student's level, so... It seems he's been exposed to the --probably simplistic, granted-- formula of Pex(V2-V1). Because that's the recipe that he's going to be responsible for in his assignments and exams, I suppose, trying to draw a simple intuitive reasoning of how it works and why, to a first approximation is, IMO, the way to go here. There is a million-dollar prize for just proving the existence and uniqueness of the Navier-Stokes. Generations of mathematicians have failed at solving it in general. On the other hand, H, S are quite abstract in comparison to P, V, which are a lot more intuitive. Should we start teaching gas behaviour by the Van Der Waals equation? It's certainly closer to how real gases behave. There's the key to phase transitions, the triple point of water, and so on. But generations of students have come to understand gases first with the concept of an ideal gas for a reason. As to using PVk, I'm all in favour of it for general quasi-static processes, when the system has a P. When the system is out of equilibrium and adjusting to a new equilibrium, I'm not sure it's the right approach from the conceptual point of view. Unless you know of a way to derive an effective P and an effective k from the Navier-Stokes equation in out-of-equilibrium situations. If such method exists, I'm not aware of it and I would love to learn about it to the extent of my abilities, which are quite limited, I must say.

Thank you. You don't need to explain yourself. You're very welcome. The fact that professional scientists have thought in similar terms indicates that the idea is not silly at all. You did express it in a non-standard way, though, and I was a bit confused.

Quite the opposite: https://www.livescience.com/9090-religion-people-happier-hint-god.html Life becomes easier when you are a believer, especially if you believe in whatever your neighbours declare to believe. Try to be a Christian in Yemen! Then you fit in, you don't think about ethical problems, because you've got it all written down for you, and you have a picture of a fancy wonderland where to go when your time is up on this planet. What's harder, as anything that's worth anything in this life, is facing things objectively and without bias. And learning from the process. That's incomparably harder, and brings incomparably more good to this world.

This is known not to be the case. The reason is that in order to have chaotic systems in classical mechanics, you need very little: 1) High sensitivity to initial conditions 2) Mixing of trajectories (so-called topological mixing), which means that any particular "patch" of possible initial conditions ends up --through evolution-- covering all the possible space of possible dynamical states (phase space). Any dynamical system that satisfies non-linear equations (which means any dynamical system to all intents and purposes) is non-linear. Even linear systems, for more than 2 degrees of freedom (number of independent coordinates necessary to describe them) is chaotic too. On the contrary, quantum systems are inevitably non-chaotic, as the Schrödinger equation is always linear (the superposition or addition of two possible motions is also a possible motion). There is a connection between both, though, which manifests itself through quantum scarring. Also, indeterminism in large chaotic systems does not come from the quantum. Quantum fluctuations are negligible for planetary motion, yet the 3-body problem already displays chaos, even though quantum mechanics can be safely ignored in that context. So for all we know, even if the world were classical --and not quantum-- the slightest complexity in the dynamics would imply chaotic behaviour.

I don't understand the question. Chaos is a qualitative property. It's not a number. What do you mean it "fluctuates"?