timo

To say it in the words of a famous physicist: Defining terms as they happen to suit your idea helps a lot in that (c.f. Swansont's reply).

That's what the FAQ hopefully will explain. I suggest the keywords "big bang" AND "black hole". But you are free to think out of the box and try something else.

Apart from an arbitrary amount of random buzzwords I could throw at you (how about "tunneling between branes"? I have no idea what a "brane" is, though): Why would there be a problem in the first place? I do think it is indeed best to think that the reason is that particles are point-like and only resist being squeezed together by various forces (electrostatic repulsion being the most known; effective force caused by the fermionic occupancy statistics probably being the most relevant close to the BB). If you accept that picture (which may come with a lot of problems) then compressing your stuff is just a question of pushing it together strong enough. This then certainly allows to start in such a compressed state. Airbrush: The early universe scenario is not the same as or (mathematically) similar to the black hole scenario. I'm sure you'll find plenty of FAQs explaining that if you do a forum- or Google-search.

Don't waste time on it if you've got better things to do. It's just a math proof for a statement I think I've never needed; no physics to learn there.

I thought he was the janitor on sfn ...

AzurePhoenix and insane_alien are not moderators on sfn. SFN moderators are marked as such (even though I don't see why it would be a problem if some were not). AP and IA might be moderators on the chat (in fact they quite likely are if IA says so). That's a different issue. But I think moderators on the chat are flagged there, too - at least in most chat clients. Scruffy? Who's that?

I don't see how your approach is going to help you. You need a statement about the entries of U, the [math]U_{ij}[/math]. So at some point you must either separate out a single matrix entry (sketched in post #7) or obtain a set of equations of these entries (post #4 and #5 are two such approaches) that allow to make the desired statement. Unless it is important that you do this homework assignment I don't think I should give any more hints; there's probably not much learning involved in writing down a proof when you were told all of the steps. And don't worry if you didn't get the communication between Andrew and me - you were not supposed to.

The funny thing is that all of these three statements can either be correct or wrong, depending on how you interpret them. In the mainstream answer, the key to your problem is the 2nd statement: a gravitational field influences a body in a way that does not depend on the mass. You even have that in classical mechanics: a gravitational potential [math]\phi[/math] creates causes a force [math] \vec F = -m \nabla \phi [/math] on an object with mass m. This causes an acceleration [math] \vec a = \vec F / m = - \nabla \phi [/math], i.e. an acceleration that is independent of the mass. If you take the intermediate step seriously then you run into problems when m=0, but you could imagine forgetting about forces and just make [math] \vec a = -\nabla \phi[/math] the basic principle of gravitational action. Indeed, forces as you might know them are usually not used in modern physics (i.e. new fundamental physics developed in the last 100 years). In Relativity it actually is the case that you (sometimes) make the extension of [math]\vec a = -\nabla \phi[/math] the basic equation for the impact of a gravitational field on an object and don't run into any special case for m=0, then. That's of course only the formal dropping of an intermediate principle; the justification comes from seeing that nature indeed seems to follow that rule.

My suggestions works reasonably well if you write it down in index notation. My first idea was to select a single [math]U_{ij}[/math] by sandwiching the matrix with two base vectors (of the basis the matrix is written in) and then compare it to the result you get when the base vectors are are represented as a combination of eigenvectors (thereby indeed using [math]|\lambda |=1[/math]. You might call that elegant but I'm not sure it is faster - I didn't even try if it works.

Just to be sure here: you know what the exponential of a matrix is, right? your tex problem was caused by an extra \right] at the very end: [math]e^{A}B e^{-A} = B + \left[ A, B \right] +1/2 \left[ A, \left[ A, B \right] \right] + \ldots + 1/n \left[ A, \left[ A, \ldots \left[ A, B \right] \ldots \right] \right] + \ldots [/math] EDIT: Oh, and I'm not sure if "Hilbert space commutators" is technically correct. The Hilbert space is the vector space of the states. What you commute here is operators on that space, not elements of the space itself.

I was hoping that Moo had an idea how to do the proof once she had proven that [math]|\lambda |=1[/math]. If that was not the case then here's a hint how I'd solve it: [math]U^* U = 1[/math] by definition and [math]U^* U[/math] contains the sum of several products of the [math]U_{ij}[/math] (and their conjugates). But you (=Moo) should first try your idea, in case you have one.

If [A,B]=C, then AB can be rewritten as C + BA. My point here is: if you rewrite a product using a commutator relation, then it would be replaced by two addends, not gain a factor ([math]i\hbar[/math]) as it seems in your case.

EDIT: Ah well, just do some work and follow ajb's proposal (assuming it works out). Just in case that is not clear: showing that [math] |\lambda |=1 [/math] is not what the question asked for.

It's not the opposite, it's the reverse (direction of thought). You're asking about the effect. I'm saying that if it has an effect then the effect is "visible" at a point for which no prediction exists in the first place (and what do you want to compare the effect with, then?).

I find the reverse direction much more interesting: If a non-measurable effect could in principle have an impact over some large time span that directly means that you can't even know if earth will still be orbiting sun the way it does now - irrespective of this additional force.

What's the point in adding an extra variable that by definition has no effect? Or asked the other way round: what would the effects we realize be?

EDIT: Seeing Mr. Skeptics post: I should probably have looked up the term "brownian ratchet" first . I was assuming you were talking about a switchable ratchet potential (like this one http://elmer.unibas.ch/bm/index.html) for some reason. You can probably ignore the rest of this post. OLD POST: I think you should try to formalize why they should be, first (and what the 2nd law of TD is - I'll assume the non-decrease of entropy in a closed system). So on first sight you'd think that it's a directed transport process so you take a ratchet potential over some bounded interval, put particles reacting to the potential in there at random positions, and then just wait for them to diffuse to one edge of the interval. In practice, you'll have to switch the potential on and off to achieve that. So the system is not closed, so a sorting of the particles is not excluded by that what I think you meant with the 2nd law (I really dislike giving numbers to concepts - who had this silly idea?!?). So well: why do you think these ratchets would violate the 2nd law in the first place?

I'm not completely sure I understand what you mean. How do you think humans communicate? Do you mean the underlying physical principles which would be electromagnetism (e.g. gesture and mimic are transported via optics) and sounds (voice). Or do you mean a more "esoteric" (the term is not meant to express disrespect here; I just have not better term for it) level, i.e. that humans at least partially engage in communication consciously with some goal in mind? In the former case: different parts of the atoms have different electrical charges and mobilities. This causes various effects like sharing more mobile parts (electrons) between different atoms which causes them to stick to another (and form molecules). The molecules then can interact with other molecules e.g. by repelling each other when they come too close or by creating small electric fields around them which are felt by other molecules. That's just examples (and very rough ones). There is a huge variety of effects and whole fields are dedicated just to singe aspects of this (and we've not even reached the size of a cell, there). In the latter case: Atoms are usually not considered conscious. Humans usually are. Somewhere between atoms and humans, the transit must occur, leaving room for an arbitrary amount of wild and incompetent speculations. Since I don't want to participate in these right now and since you didn't really ask for that: It is not common to give human traits to atoms. I don't think it is helpful. To take a more everyday example: It is not very helpful to think that things fall down because earth wants them to stay close. It is common to think that gravitational attraction between earth and the objects happens to be a property that they simply have. Same way, atoms just consist of the various charged parts with the exact properties that causes them to form more complex structures.

Pretty much the same way a water wave interferes with itself.

A ray of light has a certain momentum. When the ray of light is absorbed by earth, it must take up this momentum so that the total momentum is conserved. A transfer of momentum can be considered the effect of a force working: the sunlight pushes earth outward on its orbit around the sun. The reason the number is so large compared to the shuttle is because the shuttle force is concentrated on a relatively tiny area (the size of the shuttle shape). The force caused by sunlight given is the sum over half of earth's surface (the part being lit) - a tiny force per area sums up to a reasonable value that way. For the actual meaning or significance of the sunlight force: I doubt there is one. The force is 10^14 times smaller than the gravitational attraction (*), so it's simply irrelevant for earth's orbit around the sun. For solar energy, the relevant parameter would be the amount of power, not the force Power and exerted force of a light ray are closely related, of course. But force in the sense of pushing things seems pretty irrelevant for solar energy applications to me. Perhaps you should ask on the discussion page of the article what the idea behind adding this seemingly-pointless value was (my guess is "we had no better example for this slot"). (*): I have not bothered checking the numbers from the Wikipedia article, but simply assumed they are correct.

My statements were never supposed to be offensive. You said that the reason you want to become a physician is because you want to be creative. I told you that I do not believe that a physician's job is about creativity - in fact, that I would seriously dislike it if my physician was creative instead of "acting upon the system he/she memorized". You are free to blame the forum members about how nasty everyone is towards you if it makes you feel better. But perhaps consider the following points: - Wonder why I focus on "working as a physician is not creative"? It's the only reason you gave for wanting to go into that direction. I dunno if it is your only reason -I doubt it-, but it's not exactly everyone else's fault if you're not providing anything else. - How about just telling us where exactly you see the creativity in being a physician? Seriously, that's much more helpful (for you) than if I or Mr Skeptic tell you that there is not much creativity, there. I don't actually appreciate being put into a drawer with iNow. I've put some thoughts into what I wrote and I don't exactly feel that you appreciate it. And you are in absolutely no position to judge my intelligence.

And I strongly disagree with the notion that in medicine you are not "being bound by a system". I seriously hope that there's few fields where you are tied to the well-tested routine as closely as there. Not sure how it looks in Psychiatry; they might have the advantages that the patients cannot complain (i.e. sue) properly. That's just a prejudice, though. I'm not exactly worried that you want to become Mengele junior . But I think your view might be more inspired by Dr. House than by reality. That said, I'm not a physician myself, so all I can offer as expertise is having a bit more life experience than you (and having worked two years as an ambulance driver).

If you want to do something creative instead of "memorizing a system and acting upon it" and have a minimum sense of responsibility then you should stay away from treating humans at all.

I don't really see how that answers my question - is it a "no"? Note that I was not asking what time dilatation is. Btw.: I think the time dilatation effect is stronger when deeper inside a gravitational field, not weaker.

Nitpick questions like "what is the 'intensity of strength'? " aside: your point is that time and space cannot be separated (whatever that may mean) because the passing of time depends on the location?