DrRocket

Chargeless and neutral charge are in fact the same thing. There is such as thing as an anti-photon. It is a photon. The photon is its own anti-particle. The photon is an elementary particle. It is not composed of other particles. There is a known symmetry, but it it is not just charge symmetry or time symmetry. It is charge, parity and time symmetry. Charge does not appear during pair production. Charge is conserved in pair production. During the usual pair production process, a nucleus is involved. It is not possible for a single photon to produce a pair -- that would violate conservation of momentum. It is possible for two photons to produce a pair. Rather than just pulling stuff out of the air (or elsewhere) you might want to read a book on elementary particle physics. Introduction to Elementary Particles by David Griffiths is accessible, if somewhat dated (it was written before it was found that neutrinos are massive).

DH is right, but the omission is not much more severe than a simple typo and is readily filled in since it necessary for the problem to make sense. If that is the biggest mistake that you made today, then you are doing all right.

"Invariant" is a precise technical term which means that a quantity takes on the same value in all reference frames. Thus the spacetime interval is an invariant quantity. There is no double-negative involved, if one understands the language. To say that a quantity is "not invariant" has a precise meaning. To say that a quantity is "variant" is meaningless gibberish in the context of relativity. Like ow,l you seem to think that you can critique a scientific theory without first understanding what it says and the language, mathematics, in which it is formulated. That is ridiculous.

In any fixed inertial reference frame things are always pretty "normal". Time dilation and length contraction are merely relations between two distinct reference frames in relative motion. But the reason that you don't perceive time dilation in your own reference frame is NOT because the mechanism of your watch "is also being dilated". That would only make sense if there were such a thing as absolute motion, but there is not. Your watch appears to work normally because your watch is indeed working normally. There is no reason that it should not work normally -- it is at rest in your reference frame. Yes, the same can be said for space. But you didn't say it. There is no such thing as an outside or internal observer. There are only observers tied to specirfic reference frames. You are mixing special and general relativity. The cosmological expansion of space is a phenomena described in models based on general relativity and has nothing to do with relative motion or special relativity. If space were to expand points would appear to move apart to any and all observers. The expansion of space is based on measurements tied to the intrinsic, and invariant, Lorentzian metric of spacetime. It has nothing to do with relative motion or varying reference frames. You took a wrong turn. See above. This is actually a very subtle topic. Even to describe, in general relativity, what is meant by "space" and "expansion of space" is non-trivial and takes significant background in the differential geometry that is the basis of general relativity. If you really want to study this in detail you will need to undertake a serious study of general relativity, on par with the text Gravitation by Misner, Thorne and Wheeler. Though, you might want to start with this thread: http://www.scienceforums.net/topic/33180-cosmo-basics/

In the U.S. one takes a set of exams called the "qualifying exams" early on in order to enter the PhD program. That washes out quite a few. Then one takes the "general exams" just before starting the formal research for the dissertation. When I took the general exam there were about 3 of us who passed that year. We were the first group to have anyone pass in several years. At the time that I was in school, in mathematics the qualifying and general examinations were individual oral exams. Anything that any of the examining professors knew was fair game for a question. I believe that now most of the examinations are conventional written tests. A recent student at my old school tells me that very few students can pass the analysis portion of the tests, so they tend to specialize in other areas. It is relatively rare for anyone to wash out after the general exams, unless they simply can't produce a dissertation. It is very rare for anyone to fail the final examination, which is the "dissertation defense". I know of a couple of very unusual instances in which this happened, but they are pretty extreme examples. However, on the whole, only a relatively small per centage of those who enter graduate school following the BS make it to the PhD. I can also speak to the qualifying examinations in electrical engineering, which I also took. There one took a written exam and could pass at either of two levels -- qualified for an MS or qualified for the PhD program. I qualified at the higher level, and I think a couple of others also did, but most either passed at the MS level or did not pass at all. I don't recall anyone in the PhD program washing out after the qualifier, but there were only a half-dozen or so PhD students in the program. I am sure that this varies with the particular school. I was once told by a professor at a major Eastern U.S. university that he felt it his responsibility to see that any student that he took on would graduate -- to the point of writing the dissertation for him if necessary. I was shocked.

Joe Kennedy acquired the trait of being rich. John, Bobby, and Teddy inherited that trait.

Therein lies a significant problem. If you don't have a fairly good idea of what interests you before you enter graduate school, you may find that the road is a bit rocky. While vascillation is the norm for undergraduates, graduate school tends to be a bit more focused. It is not impossible to change directions in graduate school (I did it myself) it is much more difficult than changing majors as an undergraduate (where it is not at all unsual for a student to make several changes before finally getting a BA or BS). If you are in a good graduate program you will find it FAR more intense than what you saw as an undergraduate. A good graduate course would cover the amount of material that you see in a typical undergraduate science class in the first two weeks, in greater depth and with greater demand on understanding. You will find the work very interesting, but very challenging and with little time for "findng yourself" in the broad sense. You might want to consider delaying entrance to graduate school and instead getting some experience with a job in an area of potential interest. Do some reading on the side, and when you have a somewhat better idea of specific areas of interest then pick a school on the basis of that new-found understanding. If you have a better idea of what really interests you, then your chances of success will be better. While it is unusual for Masters candidates to wash out, it is quite common for PhD candidates not make the grade.

The Drake Equation is basically rubbish. It is a product of rather obvious factors, not one of which are known or can be estimated on the basis of known scientific principles or available empirical data. Hence it has zero actual predictive power. While some progress has been made in understanding star formation and in locating a few extra-solar planets the associated factors in the Drake equation can be at best educated guesses, the remaining factors (all of which much be known with precision for any useful prediction) are total mysteries. It makes for interesting science fiction, but is useless in real science.

I doubt that anyone whose behavior is materially shaped by video games or even movies was all that thoughtful to start with, and, "studies" aside I rather doubt that there is in reality much effect. It certainly doesn't hurt the prospects for funding of a "social scientist" to study such things and publish sensational conclusions. On the other hand, I am,at this moment, engaged in instructing people as young as 12 years of age (I have had and am required by law to accept even younger students, but don't have any at the moment) in the safe use of firearms in the ethical harvesting of game animals. I have done this for many years and have no evidence of any anti-social violence resulting from the training of young people in this application of real guns. In fact I have ample quantitative evidence that this (state cetified and sponsored) educataion has resulted in a very large reduction in accidents. While there may be some correlation between video games and violent behavior I suspect that the causative factor may be found in the home environment that produces the availability of the games and encourages participation -- lax parenting rather than the games themselves. The contrast between violent kids raised on video games and responsible kids who are familiar with and use real guns is too strong for me to accept these "studies" at face value.

Your link doesn't seem to be working. There is nothing special about 4D or higher with respect to the notion of lines and their intersection. As I said, given two lines in n-dimensions you can always find a 3-dimensional subspace that contains those two lines, so anything that you care to say about two lines can be said and studied in dimension 3 without losing anything. There are topological questions -- for instance the Poincare conjecture, now the Poincare theorem -- that are relatively easily solvable in higher dimensions (though Smale received the Fields Medal for solutions in dimension 5 and above), but very difficult in dimension 3 and 4 (Friedman received the Fields Medal for the solution in dimension 4 and Pereleman was awarded, but did not accept, the medal for the solution in dimension 3). There is also the issue of things like "exotic differentiable structures" that arise only in special dimensions -- and 4 happens to be special for this particular problem. But to even understand the problem you need to understand what is meant by a differentiable structure and the difference between a homeomorphism and a diffeomorphism. There are references on topology in dimension 4. The Geometry of Four-Manifolds by Donaldson and Kronheimer and Topology of 4-Manifolds by Friedman and Quinn are two such references. However, these are very specialized and advanced graduate texts and not for the faint of heart. Unless you have extensive education in topology and geometry you are likely to find these books unreadable. As far as the question of lines in dimension 4 and above there are no specific references simply because the problem reduces, rather trivially, to dimension 3 and that is covered in any decent text on analytic geometry or calculus.

And Hawking radiation was predicted on the basis of quantum field theory on curved spacetime. That is in fact a rather shaky foundation and it is not certain that Hawking radiation is actually a viable phenomena -- it is thought to be viable but there is no experimental evidence on only the noted shaky theoretical foundation. To really answer the question of Hawking radiation will require the previously mentioned theory that unifies quantum theory and general relativity. In point of fact the pop-sci explanation of Hawking radiation involvind virtual particles becoming real particles at the event horizon is just pop-scie nonsense. In quantum field theory on curved spacetime one loses the very concept of a particle. It is in fact not really a well-formulated theory (and that says rather lot since even ordinary quantum field theory is in fact not well-formulated from a mathematical perspective, but on curved spaceetime things are taken to a new level of imprecision).

That equation, as corrected, is just an application of conservation of energy. Note that this is not really mathematics. It is physics.

I think you will find that the terms vary with the school. I know of schools at which the astronomy PhD is the same as the physics PhD, the only distinction being the research topic chosen and the advisor. Others have separate departments, but one might well find astrophysics being handled in the astronomy department. I don't know of any serious distinction at this point in time (as opposed to a century or so ago) between astronomy and astrophysics as far as ongoing research is concerned. Certainly hobbiest astronomers are not astrophysicists, but at the professional level I doubt you can find many non-physicist astronomers. I am not sure what you mean by "space research". That could cover a lot of territory, from cosmiology to stellar astrophysics to planetary geology to astrobiology. As with most interdisciplinary topics, you would do well to get a firm foundation in a fundamental science, and then apply that foundation to the interdisciploinary topic that interests you, likely as part of a team with diverse training. Bottom line -- find a school and a faculty member who conduct research in an area that interests you. The fit between you, the school, and the specific faculty are more important that terminology.

Everythiing that I have said is correct, and mathematically rigorous. I would be most concerned if I had your respect, as you have repeatedly made statements that have no basis whatever and have demonstrated lack of understanding of mathematics and inability with logic. Your assertion that you have had trouble with "academics" speaks volumes, though I have not been in academia for some time. Nevertheless you should probably listen to what those academics have told you. Go learn some mathematics and stop making absurd statements.

As always, I recommend following your interests. If you are doing something that truly intersts you, you are likely to be good at it. That said, if you are thinking in terms of how industry would view such a degree then I can say a couple of things, which might seem contradictory. 1) Industry tends not to understand non-traditional degrees very well. Thus you may well receive a cool reception in and industrial interview. 2) Industry doesn't really care all that much about what it says on a diploma, so long as it is not fraudulent, if they believe that the holder of the degree can make a significant contribution to the enterprise. So, if you can convince them that you can do something that will be of value to them, then they will hire you. So, your problem will be in convincing someone that you can contribute to the company. That would probably be easier with a traditional degree that says "engineering" or "computer science", but if you really know what you can do and can explain it to an interviewer you can overcome the initial problem of lack of understanding of your degree. Once you have been hired what counts is what you do, and no one will ever remember or care what the diploma said. Academia is a different ball of wax. But even there what you produce is more important that what your diploma says. Eugene Wigner had a degree in chemical engineering -- and a Nobel Prize in physics.

You are missing the point. For the moment let's not worry about the sign ambiguity that arises because of the squares. With that understanding we restrict attention to the case where x,y,z are all non-negative. So the equation [math]x^2+y^2+z^2=a^2[/math] defines a 2-sphere of radius a in 3-space. We restrict attention to the portion of that sphere in the octant where all of the variables are non-negative. In that octant you can think of the equation as defining y implicitly as function of x and z, or defining x implicitly as a function of y and z or defining z implicitly as a function of x and y -- BUT not all three at the same time. Thus you cannot sensibly talk about [math]\frac{\partial y}{\partial x}\frac{\partial z}{\partial y}\frac{\partial x}{\partial z}[/math] becuse the objects that are purportedly being multiplied are functions that do not have a common domain. You can juxtapose the symbols, but the implied multiplication does not make sense, and that is why the "chain rule" does not appear to work here. It all comes back to the simple fact that derivatives and partial derivatives are defined in terms of functions, not equations. You can take derivatives of functions related by an equation, but you do need to have functions, not simply variables, to start with,and those functions must have a common domain. In this case you don't have that situation and the statement that [math]\frac{\partial y}{\partial x}\frac{\partial z}{\partial y}\frac{\partial x}{\partial z} = \left(-\frac x y\right) \left(-\frac y z\right) \left(-\frac z x\right) = -1[/math] is not meaningful. It is just a bunch of meaningless symbols being pushed around. I have seen similar things being done with symbol pushing sans meaning in physics texts before. Sometiime there are compensating errors or alternate, but correct, logic chains that can result in the final answer being valid -- and sometimes not. Clearly you don't know what I am talking about. But that is not surprising since you don't know what you are talking about either This is rather fundamental. A function from a set A to a set B is a subset of the cartesion product AxB such that if (a,x) and (a,y) both belong to that subset then x=y. If (a,x) is a pair in the function then we commonly write x=f(a). An equation is simply a statement that two thing are in fact the same. Your opinion on the superficiality of the difference is incorrect and evidence of abject ingnorance concerning the most important definition in all of mathematics, that of a function.

You say a lot of things. Most of them are wrong. That is evidence of nothing more than a failing on your part. One more time -- partial derivatives are defined for functions, not equations. If your equation does not at least implicitly define a function then there is no meaning to a partial derivative.

The concept of a line works in a vector space, or affine space, of any dimension, including infinite-dimensional spaces. The styatement made in thevdiscussion at your linked site is just plain wrong. But any two lines can always be found in a 3-dimensional subspace so there is not a lot to be gained from going to the higher dimensions to study two lines at a time.

All science is either physics or stamp collecting. – Ernst Rutherford

That is great site for those with a strong stomach and a weak mind. In other words it is a somewhat useful mnemonic that one ought to follow up with rigorous reasoning since the mnemonic can get you into trouble on occasion. The derivative is not a ratio in the rigorous sense (outside of nonstandard analysis), but as a limit of ratios that reasoning is sometimes useful, so long as you don't get carried away. The problem here is that [math]x^2+y^2+z^2=a^2[/math] does not define a function of x, y, and z but only gives one in terms of the other when you fix the third variable. Thus without further explanation [math]\frac{\partial y}{\partial x}\frac{\partial z}{\partial y}\frac{\partial x}{\partial z}[/math] is not really meaningful as they cannot all be defined under a single set of assumptions -- partial derivatives are defined for functions, not equations.

I am a mathematician. If what you say is true then it appears to me (since 4pi r^2 is the area of a sphere and T^4 is part of the Stephan-Boltzman radiation law) that the equation arises from physics rather than mathematics and that it has something to do with radiative heat flux. But to be certain it would help rather a lot if you had some sort of reference to where you found the equation since any fool with a pencil can write down a bunch of symbols. the symbols a, ε, and δ are a bit mysterioius when taken out of context. If you had used [math]\epsilon \sigma[/math] rather than εδ that would be consistent with the way most authors write the Stephan-Boltzmann law.

Good schools need good students as much as good students need good schools. GPAs in this day of grade inflation are hard to judge, as is the contribution of secondary authors of multi-author papers, but if you have strong recommendations from strong faculty members you should have no trouble. It is more important that you find someone who does high quality research in an area that interests you and in which you have aptitude than it is to be enrolled in a "name" school. What wil count later on is the quality of your research. That said, name schools have big names for a reason -- and that reason is strong faculty doing first-rate research. So, the chances of finding a strong faculty member doing cutting edge research in an area that may interest you are very good at those name schools.

z Surely you could use the water to brew beer and drink that. With that modification it sounds like a good plan.

The laws of physics as known to Newton and Maxwell arre still valid today and are quite adequate for the technology that is relevant to your OP. Those laws may have been extended by quantum electrodynamics and relativity, but within their domain of validity they remain as valid as ever and will in the future. Physics does not overturn laws so much as revise and extend them to every more general settings; e,.g. the very small, the very fast, regions of extrarordinary gravitation, etc. But for normal conditions the classical physics of 100+ years ago remains valid and in fact is the basis for the vast majority of engineering. You can fool some of the people all of the time and all of the people some of the time, but you cannot fool all of the people all of the time -- Abraham Lincoln

And lest folks get the impression that all is known, the no one has yet actually derived the residual strong force among nucleons from the quantum chromodynamics that describes the strong interaction involving quarks and gluons. Nevertheless, it is believed that the theory, in principle, should be able to descrive the residual strong force. Note that not only does the interaction between quarks not drop off with distance, it actually increases with distance, which serves to reinforce the fact that it takes enough energy to separate quarks any significant distance to create a new pair.