joigus

Yes, I think you're right that depletion is not that important. I was so stumped by the humongously big number that I didn't know what to think. You could try to work it out with the barometric density formula: https://en.wikipedia.org/wiki/Barometric_formula#:~:text=The barometric formula%2C sometimes called,1000 meters above sea level. Both expressions, for pressure and density, are basically the same, \[\rho\left(r\right)=\rho_{0}e^{-V\left(r\right)/k_{B}T\left(r\right)}\] But they are isothermal expressions. The leap of faith I've tried is assuming that you can substitute T by T(r). I'm not sure that would hold. You could always take a look at the wiki derivation and try to work it out for little slices of width dr and constant temperature for each slice. My sub_zero is value at the surface. Maybe that's the ticket.

Not a very useful line of thinking for homework help.

My hippocampus doesn't quite agree with that.

It depends.

Why do you want to augment your brain (or anybody else's)? Why not improve synapse, efficiency of multiplexing, neuron plasticity or ability to regenerate? Brains are not breasts.

Not even wrong.

How do you know consciousness is continuous? What does "continuous" mean? "Continuous" is a sophisticated mathematical concept. A topological set is continuous if it contains its closure. What's the topology? Edit: Sorry. Closed, not continuous. But it's the same really in this context. A continuum must be closed.

Are you suggesting beta decay is playing a role in conscience?

The very fact that you talk about quantum coherence in a mesh of organic tissue at near 310 K proves to me that you are clueless about the meaning of the concept. The fact that you've mentioned gravity in a context where polar electrostatic forces are orders of magnitude bigger proves to me that you are clueless about constants of Nature and dimensional scales. You don't make sense philosophically either: Your room is not you. Your notepad is not your ideas, etc.

No knowing agent has to know anything. Embryonic stem cells make brains, and livers, and lungs, without anybody conscious orchestrating it. It seems to be the current informed opinion that quantum coherence plays no role in brains, human or otherwise. It could be. But it's not likely, to say the least. Quantum coherence cannot be preserved in such a system as a brain. The electron positions are irrelevant for all we know. I'm not aware of any serious model that contemplates quantum mechanics and gravitation as playing a fundamental role in conscience either. You're using a very loose logic. Something like: quantum mechanics and gravity imply entropy, which implies information, which implies brains and conscience. That's not how it works. That much can be said even not knowing how it really works.

I agree that pressure must go up as you go down towards the centre of the Earth. I also agree that temperature is important and must go up (however gradually) as you go down. Up to about 6000 Celsius at the centre. I've made a quick check with the following model: 1) the amount of air within the shaft does not deplete the atmosphere significantly 2) Gravitational field is obtained inside the Earth by applying Gauss's theorem 3) Exponential atmosphere is assumed with T as a function of r (distance from centre of Earth) and V(r) the expression for the interior potential \[P\left(r\right)=P_{0}e^{-V\left(r\right)/k_{B}T\left(r\right)}\] The expression for the interior potential is, \[V_{\textrm{int}}\left(r\right)=G\frac{M_{\oplus}}{2R_{\oplus}}\left[\left(\frac{r}{R_{\oplus}}\right)^{2}-3\right]\] And the model doesn't seem to work. It gives a ridiculously high value for the pressure; much, much bigger than 3.8 trillion psi, which is the value for the centre of the Sun! 😲 To me, this means that at least one of hypotheses 1), 2) or 3) is plain wrong. The most suspect to me is 1). Maybe what would happen is that the whole atmosphere would be swallowed up by the shaft density deficit, and it would cram itself into it, thereby depleting the outer atmosphere. Assumption 1) is implicit in that P0 keeps its current value, even though the whole atmosphere would have been sucked into the shaft. P0 is the 14 psi reference value that @Janus has mentioned. A more realistic model should involve the distinction between crust and mantle (discontinuous layers). But I don't think that would make that much of a difference. Maybe I made a mistake or the simple substitution in exponential atmosphere relation with the temperature as a function of r simply does not hold. Why do you see the density-pressure dependence as a problem, @Martoonsky? What particular model are you trying to apply?

Interesting question, but too loosely defined, as @iNow points out. One very important thing you must clarify in my opinion is whether you are considering developmental biology. You can't just practice surgery, or implants technology in order to do that. Brains develop slowly, and the growth process monitored by enzymes, I surmise, is very important in the forming of such a complex organ as the brain.

I would just add "degrees of certainty" to your list, which I just assume you've thought about before. How you grade concepts in a meaningful and useful way is another matter, of course. I'm trying to get closer to precisely this concept you mention. Namely: What would be a good criterion for a "philosophy of science"?

Why do you mention spherical coordinates in relation to time? Have you got something in mind? I've got something in mind, but you go first.

This connection is fleshed out by a theorem (I forget the name now) that relates time-ordering of operators with expected values in the vacuum for normal-ordered products. Irrespective of technicalities, this strongly suggests that time in getting in the way because of its being very very deeply entrenched in our language, or sequential alphabet, or what have you. I hope it's clear what I mean, although I recognize that what I mean is difficult to make explicit in terms of language and symbols.

A must-read for this post. +1

True. +1

Well, I was going for minimal conditions for a time to be possible to define (from a purely mathematical POV, so no operationalism, no clocks at this stage). Another example to add to the ones you provide in which this technique wouldn't quite work are chaotic systems, because for chaotic systems there aren't nearly enough integrals of motion to reduce the dynamics to a clear trajectory that can be pictured as an implicit relation between your dynamical variables of which a one-parameter could be deduced. To me, once you have defined (in clear-enough cases) this one parameter that suggests to you a sequencing of events, you still have the enormous freedom to choose which particular parametrization corresponds to your clocks, how different sets of clocks relate to one another, etc. That would be problematic if one photon were the only thing that exists in the universe (you could always define an affine parametrization for a photon which is what people do to describe the geodesic equation for photons, but the interpretation of such parameter as a time is another matter). In the scenario that I'm talking about, there are more things, and the coordinates of the photon could in principle be included in the aforementioned implicit equations fi(xmatter,pmatter,xphoton,kphoton)=0, so a one-parameter sequencing for the whole system would be possible to define. Now, on the proviso that this sequencing can serve as the minimal condition for a time to be definable, and taking into consideration your caveats about clocks (the question would be pending of what re-parametrization of this emergent parameter to use so that it corresponds to our physical clocks), there's still the subtle matter that it's defined from a metric or pseudo-metric. And these are always based in physics on quadratic forms, and thereby you must chose an orientation for the sequencing parameter to run along. Here I think you took me a little bit too seriously. Mind you, I said: So I was more cautious than you seem to suggest, and just expressing a feeling. This feeling has been seeded through many moments when studying physics, but the strongest one by far is QFT. In QFT you start from very neatly defined state operators in terms of a given coordinate time, write down the Heisenberg evolution equation, and formally solve it in terms of creation and annihilation operators in momentum space. Because Dyson's formula imposes on you a time ordering, you get a sequence of products. For one of these terms, e.g.: \[a\left(a^{+}\right)^{2}aa^{+}a^{3}a^{+}a\] What's the next thing you do before you engage in any calculation at all? Well, you re-define your exact solution to be "better represented" by the normal ordering: \[:a\left(a^{+}\right)^{2}aa^{+}a^{3}a^{+}a:\overset{{\scriptstyle \textrm{def}}}{=}\left(a^{+}\right)^{4}a^{6}\] If nothing else, to me, that very strongly suggests that there's something deeply problematic about time. It may be possible that sequencing of concepts might have consequences, logical consequences, that we simply cannot get rid of within our present system. I'm not saying that every mathematical statement we make has time impregnated in it, but I'm saying that it may well be that we are quite incapable of totally escaping the non-trivial consequences of having a sequential language. Don't pay too much attention to what I'm saying. Maybe I'm just sounding people out about my deepest intellectual insecurities, that's all.

On further reflection, after reading Studiot's comments (+1) carefully, and to correct myself: This definition, while it's very widely used theoretically and very solid, is not universal; and in particular is not useful to assign space coordinates to distant planets. Also, it had nothing to do with the OP's confusion. Sorry, it was late, my reasoning was obfuscated by the phrasing of the question, and consequently I wasn't very helpful.

I see. And what about the switching frames? Did you understand that part?

Nice post. +1. And nice picture. Dinosaurs are a long way away for this solar system.

Exactly! I want the OP to think: How do I measure separation of distant things? By measuring times. Observers cannot read space directly. How can distant "news reporters" at rest with respect to us liaise with us? It takes some time to realise that this is necessary. I hope this sketch clarifies the meaning of coordinates in SR: \[t=\frac{1}{2}\left(t_{1}+t_{2}\right)\] \[x=\frac{c}{2}\left(t_{2}-t_{1}\right)\] So everything, spacial coordinates included, is based on observation times. I'm not sure if that's what confusing you, @can't_think_of_a_name. It looks to me like that may be it. Once it's clear that you've got all of space-time mapped with these x's and t's, you do what Swansont is telling you and apply Lorentz transformations. But you don't get information back from distant events by sending someone back that "switches" inertial frames. It's a matter of sending light rays back and forth what gives meaning to the coordinates, not people on a spaceship with the news. I hope that helped and had to do with the source of your confusion. +1. Thank you. That got me confused too.

I'm having difficulty understanding what you mean. Do you understand how you assign coordinates to events in SR? I don't know what you mean when you say "switching" frames. Jumping from one to another? That's not how you measure time intervals and distances in SR. It's by sending light rays and receiving them back, and then kind of "triangulating", but taking into account that the signal speed is c no matter what inertial frame you're in. Let me ask you a question: How do you picture in your mind that Alice knows Bob has already arrived in planet 2?

Intuitions can mislead you (and they will if you pay no heed to experimental evidence and logical consistency). Aristotle thought that you needed a physical action on a body to make it move at constant speed. This was a wrong intuition that lasted for centuries. Theories (Galilei, Newton) showed us the first inkling that Aristotle's intuition is not correct. Space navigation has proven beyond any doubt that Aristotle's intuition was wrong. Eise has pointed out somewhere (I don't seem to find it now, maybe it was another post) that once you study physics you get better at developing your intuitions and rendering them less naive. When you study quantum mechanics for years, when you watch interference patterns in a laboratory, you eventually get to develop a feel for what physical systems do. And they don't do what you intuitively would expect. There doesn't seem to emerge a picture of a decision-taking anthropomorphic god in that world. Maybe he created the universe and has been in absentia forever after.

Well, I may be wrong. Maybe this notion I have that time is inevitably present in anything we say, even in formal mathematics, is somehow misled. In some demonstrations, e.g., two different results of previous lemmas used in order to prove a theorem could be used in sequence but in interchangeable order. So there may be many flaws or grey areas to what I'm saying. But, if that were verifiable in some sense; namely, that no matter what kind of logical/mathematical argument you make, you cannot help that your concept of time has some bearing on it, contaminates it, then it could well be that time itself is a formidable obstacle to formulate any system of ideas in complete (all encompassing) and completely consistent (free of contradictions) way. IOW, it could be that you can "solve the world" except for this nagging presence of a sequencing parameter that we call time, because it's a constraint of conscience itself, not because it's an especially important (distinguished) variable of the universe. This connects roughly with some comments that @Markus Hanke has made before. I wouldn't go as far as to distinguish a "logical time" and a "reference frame" time. That would be uneconomical to say the least. As to "subjective time"... Well, we must agree that the perception that time goes the same way consistently (in orientation at least) for everybody involved is pretty persuasive.