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It’s a version of the 2nd law of thermodynamics; you would be spontaneously heating an object with a cooler object. i.e. focusing sunlight will never get you above ~6000K

Yes, I enjoy analysing "perpetual motion" machines. I actually came across one which I could not debunk without invoking a principle that I was not previously aware of. The principle I invoked was that a focused image of an object cannot be brighter than the object itself.

Then go get a blog and post it there. This is a discussion forum; we’re interested in what people have to say about science and related topics. LLMs don’t “make points.” They regurgitate plausible-sounding dialog based on their training.

You should see/talk to someone who knows what they are doing; there are professionals who deal with this and “tips about being happy” is a rather simplistic (and wrong) summary of what mental health professionals do. By phrasing it this way you are soliciting medical advice, and nobody here is qualified to dispense it.

From: https://bigthink.com/starts-with-a-bang/10-scientific-truths-unpopular-2025/ (December 9, 2025) Scientific truths remain true regardless of belief. These 10, despite contrary claims, remain vitally important as 2025 draws to a close. 1.) 2024, the latest full year on record, saw the highest CO2 levels and the highest average temperatures since we first began tracking them. 2.) Interstellar interlopers are real, and while we found a new one (only the third ever) in 2025, they are still not aliens. 3.) We broke the record for most distant galaxy ever found but still haven’t spotted the first generation of stars. 4.) Earth’s orbit has a finite “carrying capacity,” and if we exceed that, such as with megaconstellations of satellites, it will inevitably lead to Kessler syndrome. 5.) The germ theory of disease is real, and vaccination is the safest, most effective strategy to combat these deadly pathogens. 6.) SARS-CoV-2 led to COVID-19 in humans as the result of a natural, zoonotic spillover event, not as the result of a leaked pathogen from a Wuhan Lab in China. 7.) The Universe’s expansion is still accelerating, the Hubble tension remains an important puzzle, and the much-publicized evidence we have is insufficient to conclude that dark energy is evolving. 8.) “Passing peer review” doesn’t make a scientific study true; it just means the study is robust enough that it’s passed the “start line” for consideration by the community. 9.) We’ve found evidence for organics on Mars (again), but still have no good evidence for life on any planet other than Earth. 10.) You still need to know science in order to do it; “vibe science” is nothing more than AI slop. See https://bigthink.com/starts-with-a-bang/10-scientific-truths-unpopular-2025/ for details on each of the 10 scientific truths that somehow became unpopular in 2025.

Interesting chinese optical illusion

It’s practical considerations, but the Heisenberg uncertainty principle would be a limit even if experimental precision were improved.

Before getting into this sort of masturbation, a start would be for AI developers to recognise the intellectual property rights of the authors whose material their robots scrape off the internet. AI is by its nature an intellectual parasite. So it's a bit rich to suggest that AI should be given IP rights as if it were an author.

Okay. Let's have a look at your numbers. I'm guessing you just pulled that estimate from where the sun doesn't shine, but let's run with 70, a full 2 SD below the (US?) average. So simply meeting par, the Nigerian American community must have been drawn from on or around the 97.5% mark on the IQ bell curve. But since the 2nd generation exceed all others by some margin, they must either be ~15 points higher IQ than their parents (a possibility you have already rejected), or their parents were drawn from on or around the 99.85% mark on the Nigerian bell curve. Now 0.15% of the Nigerian population 25 years ago would be about 170,000 individuals or about double the Nigerian American population. So effectively, you're saying that the US immigration managed to lure half of the top 0.15% of the population to emigrate? From a country where top 1% of the country enjoy a standard of living in Nigeria far higher than anything they could afford in the US? And how did they achieve this remarkable feat? A selection process that correlates only weakly with IQ and would pass at least 10% of the population for lottery participation. Your numbers fail at the first glance of real scrutiny. As numbers made up in a dishonest attempt to justify deep prejudice always do. LOL

I sometimes write a poem and this is one I wrote some time ago - it seems appropriate. Being In aimless drift, the countryside I wander and where I roam my eyes perceive in wonder arrays of life within my sphere of being. Which makes me glad to be among the living. But was it chance that brought me into being, this heady mix of thoughtfulness and feeling; elation in the marvel of existence within these many forms of life’s persistence? And why should I among these forms of living think that my life was special in the giving? A mote of dust that wanders in infinity; a moment’s glow within endless eternity.

While I cannot discount a genetic racial or inherited component to intelligence, keep in mind that a newborn's brain doubles in size during the first year, and is 90% of adult size by age 5. "This rapid development is driven by the creation of billions of neural connections that are strengthened by positive experiences like responsive caregiving, nurturing interactions, and a stimulating environment. A healthy environment, good nutrition, and consistent routines are crucial for building a strong foundation for future learning, problem-solving, and emotional well-being" I would think this is the main cause of, or lack of, intelligence. This OP is not about race; stop trying to make it so.

@jalaldn I have added the observer movement back and forth for you: Fair point; I kept it simple. New version posted.

A wise person once posted the following to me. Welcome back the ghost of Bishop Berkely. https://en.wikipedia.org/wiki/George_Berkeley It's might be interesting to you.

Ugh, that looks like something one might see in a medical textbook on venereal disease.😆

This giant of biology has recently passed at the age of 97, leaving behind a coloured legacy. What's the verdict? Gifted pioneer of scientific inquiry or scheming, bigoted plagiarist? There is no middle ground. https://www.irishtimes.com/science/2025/11/19/an-irish-perspective-james-watson-1928-2025-the-dna-titan-with-a-downside/

Yes YouTube seems to be becoming progressively enshittified. One specially annoying new feature of the YouTube app ( as opposed to the browser version) is that the algorithm gives you different results each time you enter a given search criterion. I use it to practice singing and often can’t get back to the recording I found last time. It insists on giving you a new selection. And the bloody ads get longer and more intrusive. Now I find if you stop at the end for more than 30 secs or so, a bloody ad will automatically start playing, even though you haven’t touched anything.

I have three young Nigerian nephews living in my Lagos apartment, and take an interest in their schoolwork. They go to quite a good school by Nigerian standards, but both their tuition and teaching materials fall short of those I received in an obscure Yorkshire farming village 60 years ago. And way short of those of my children. The Flynn Effect suggests that this 60-year+ lag will penalise them by oto 18 IQ points. So that figure of 70 is rather vital to your argument. When I challenge it, you disown it and run away. Your dishonesty is clear for all to see. It's a reasonable approximation when the gradient flattens out at the tail of the curve. To balance this, I deliberately omitted the detail that over 60% of the Nigerian population, dim and gifted alike, are below the age of 25 and therefore ineligible for the visa lottery. Your available 'pool of talent' is really only 0.06% of the population: ie less than the Nigerian American population. I certainly wouldn't. But you're the one who claimed that these high IQ types were the ones with the drive and ambition to be successful in their endeavours. You can't have it both ways: and whichever you pick now will obliterate your own arguments elsewhere. Go argue your case with the US Census Bureau. The critical population under discussion is 2nd generation Nigerian Americans. Their parents must have emigrated from Nigeria roughly ten years either side of 2,000 when the total population was ~122 million. You're clutching at straws here. Learn to read. I credited your immigration process with filtering out up to 90% of applicants. Don't forget, I know quite a few people who emigrated under this scheme. Some quite closely. Such as my best ever junior process engineer. We still keep in touch. As I said, irony wouldn't even give you a hair parting.

How do you define a “speed of time”? This seems like a fairly meaningless concept to me. Can you give an unambiguous mathematical definition for this? I should note here that in non-flat space times, energy-momentum is conserved everywhere locally, but there is no global law of conservation.

xeno = alien, g = general intelligence. That's my working assumption. Thanks for that little gem of knowledge. Though irony tends to go way over the heads of these people. Yes, very strange. But you have to admit, those odd little dances they do can be quite impressive. Natural rhythm?

I wasn't sure what xenog was short for, unless it was xenogenesis, as in the Octavia Butler novels. Which would be, at this point, ironic given Butler's ethnic origins. Anyway, one can actually use your hillbilly reference to observe that no one has come up with a conjecture that rural folk of Scots-Irish descent in Appalachia have consistently shown lower IQ scores for many generations due to their ethnic origins. Funny how that works, eh? Please cite your sources for claiming fluid intelligence is not influenced by such interventions. This is not what I have seen. Especially where we are talking of early pre-K learning options. This is the fallacy of Argument from Incredulity. Many systemic biases have persisted long after legal/political remedies were put in place. See my comment about Appalachia. Long persisting lower scores there haven't led you or anyone else to posit a genetic lower intelligence in Scots-Irish and English ancestry. Hmm.

Never knew until today that Curved Air named themselves after a Terry Riley album. O my, Sonya Kristina...

The photographer that took the photo that the model was trained on? More seriously; difficult to provide a generally applicable answer. It may differ between regions, product lines, license etc. Short answer, for a few services I am familiar with, you own the output the AI generates. Issues may involve if the AI outputs very similar images for several users or if output includes other copyrighted material. That is a good suggestion since the model may have limitations of analysing its own output. A prompt "rotate 90 degrees" may cause issues if the model does not manage to grasp what to use as a reference. @Externet this is a case where prompting have limited success; there may not be many similar constructions in the training data. If the AI tool allows you may experiment with uploading references or modifying the image in an image editor and then submit it to AI for "cleaning". Example: I rotated the turbine manually in an image editor. No need to be precise; there are artefacts. Submit the above image to AI, prompting something like "just clean up this picture a little bit" Result: (If more control is required there are the concepts of control nets and similar, but that may require local installations, configurations etc. I'll can provide some links if interesting.) Note @Externet from the title I got the impression you wanted to use ChatGPT; the cleaned up image above was created using ChatGPT, model GPT 5.1

What do think is an appropriate response for a person, knowing they don't fully understand what, is said to them to make ? To repeat a previous response, thinking they are the only correct person on the planet, or to ask can you explain what you mean in more setail so that I can fully understand. ?

Relating back to solipsism, I find that great music offers a profound rebuttal to solipsism (not like it needs one, but having that on a deep emotional level is something). Funny how you scratch the surface of a science forumite and so often find a poet and/or musician. (Jazz improviser here) A poet like Shelley can use words that are about certain feelings and intuitions, but where we go from that is beyond those words. Music can also be about something but it can also just be something. In that case, the art IS the experience rather than about or reflecting experience. Is a C9 chord "heavenly," as some say, i.e. is it about heaven? No it just is - the experience is what it is. You're not reproducing some other experience, you're just having one. (Ok, wildly off topic and I'm ready for my manacles, moderator)

Nice! I've always found it easier to express my feelings through music than words. This old effort is a setting of Shelley's "Ozymandias" and expresses a similar sense of humility before the vast and eternal glories of existence. https://soundcloud.com/seth_of_lagos/fantasia-on-ozymandias-for-brass-quartet

You and your kind. In what way? Our local word is 'oyinbo' which literally means 'peeled'. Would that make more sense to you?

There’s a list of historical determinations of the AU in this link. The ones in the 1600s tended to be low, by as much as 40%. The best ones from that era are still off by around 7.5% https://en.wikipedia.org/wiki/Astronomical_unit There would be errors in Jupiter’s position as well, so the distance to Jupiter would have errors, and there would be limits on the timing precision and accuracy.

There is something I think called the Nobel syndrome, where Nobel price winners go off on the deep end once they get it. Curie was a notable exception (as she went on to win a second price, instead). But I also meant that hi grew up in a different time, where gender and racial differences were just accepted as facts and are therefore less inclined to review information that counters it, as a good scientist should. The issue is often that scientists might inadvertently provide credibility to such notions, though they are way outside of their expertise. As I mentioned before, Rushton (whom you seem to cite here) has been pretty much outdated and he created a body of literature who tried to link IQ with race. It was fairly prominent in the 90s I would say. A key issue even then was that he was a psychologist and tried to invoke biological concepts (such as reproductive strategies) which he clearly only poorly understood. His contemporaries already questioned some of the results based on their own studies (there has been a quite some exchange with Nisbett, for example). And in the later works Rushton increasingly seemed to slice and dice his data to accommodate his view while dismissing other studies, which created some bad blood with his colleagues. There is a huge spinoff, many questioning suitability of standardized IQ tests in various contexts (there was a paper from Wicherts discussing it in the Sub-Saharan context). The long and short of it is that certain psychometric measures considered to be universal, are not. Only once certain environmental components are fulfilled (e.g. nutrition, basic schooling, stable environment) do this measures become comparable. One especially strong correlated was also found in vocabulary development (which in turn is associated with education). In multiple studies using IQ test without vocabulary tests the IQ gap pretty much vanished (studies from the 2000s). But even some of the basic assumptions you have mentioned are not quite correct. For example: I believe that is what Jensen and/or Rushton have repeated in their papers. Yet in Flynn's work back dating back to the 90s have already talked about the diminishing gap, which Rushton later tried to argue away (which in turn lead to a whole slew of related discussions). The point is that Rushton's work keeps cementing a hard delineation, something that is not found by most other researchers and, importantly, makes little biological sense, from our understanding of population genetics. Even if disregarding ongoing gene flow, Rushton's hypothesis was delineated among black as the lowest racial IQ group, whites in-between and Asians the highest. Yet, studies have shown that folks closer related to the Asian's, such as Indigenous Americans, as well as Asians with low socioeconomic status, have similar scores as black folks. So from first principles, the biological argument was already weak, but there was a fairly recent work (I forgot the author, but could dig it out) using genome wide associated studies based on the 1000 genomes using a range of cognitive tests have failed to find any genetic links. Ultimately there is a huge body of lit that disputes this rather old claim, bolstered by improvements in our understanding of genetics. On the other side, we have Rushton and a few other researchers who not only claim a racial element, but even a hard racial delineation, essentially re-using their own arguments they came up with, when we knew less. Reviewing the full body of literature, this hard delineation is simply not supported and should at this point (or really, since around 2000) should not be taken as fact. It goes a fair bit into old men's pet theory territory (and I know a fair bit about old men and their scientific theories).

A pretty good explanation of the origins and predictions of IQ testing, including the 'corruption' and mis-use by Americans and NAZIs, Notice that all the 'predictors' that IQ tests provide are all correlations, not causes. I wonder how a stable and advantaged childhood and teen-age years, with good schooling, would correlate with the metrics of success ( education, income, wealth, etc. ) identified in the video. Maybe the most important correlation is how responsive caregiving and nutrition, nurturing interactions, and a stimulating educational environment affect IQ scores.

It’s a mirage from the hot air right above the track refracting the light because it’s at a lower density than the air away from the track. The air is also turbulent, so the refraction is varying in time, blurring the image https://en.wikipedia.org/wiki/Mirage Scroll down to “heat haze”

Living in a location with hot summer days, I often see mirages while driving along a road. Mirages are an example of total internal reflection. When light passes from a medium of low refractive index to a medium of high refractive index, the angle of refraction is less than the angle of incidence. But when light passes from a medium of high refractive index to a medium of low refractive index, the situation is reversed, and the angle of refraction is greater than the angle of incidence. That means there are angles of incidence less than 90° such that the angle of refraction would be greater than 90°, and at such high angles of incidents, the light is reflected instead. This is known as total internal reflection. When the surface of the ground is heated by the summer sun, the refractive index of the air just above the ground is less than the refractive index of the air higher up, forming the right conditions for total internal reflection to occur.

Yes, that’s because the distance between Jupiter and the observer changed. So the light path got longer.Your diagram fails to show this basic point.

There is no 'killer' argument against solipsism, but there are a few considerations that make it a useless position. The 'outer reality' simply goes its own way, uninfluenced of what you would like it to be. So even if solipsism would be true, there is no difference in how 'the world' behaves. So you could just as well believe reality is 'out there' with the same kind of existence as you self. It is also an easier way to understand that other minds exist, and that scientific established facts are valid for everybody. As somebody once said: reality is that which stubbornly refuses to go away, how much you would like it to be different. Here is a funny quote from Bertrand Russell: I also must point you to a category error: you also experience your head with your senses, so it is in the outer world too. So what is in there also falls under the 'solipsist verdict'. The only thing a solipsist can say consistently is that the whole word is in his mind, and that only this mind exists.

I believe that each of us has a personal moment in time that we can call “now.” We experience reality from our own point of view, always with a slight delay—light needs a full second to travel 300,000 km, after all. Everyone carries their own internal clock, their own “now,” observing the world from a unique position in space and time. When I take an action, no one can witness it in truly absolute real time. The distances between us are essentially small differences in space-time; we can never occupy exactly the same place at exactly the same moment. Fortunately, in everyday life this hardly causes any problems. 😉 The point I want to make is that what we perceive as reality is always a glimpse of the past. Our sense of “now” is, in a way, an illusion—an internal construction created and projected by the brain.

As I understand it, one of the objections to attempting to correlate "race" (i.e. skin colour) with IQ is that skin colour is no more fundamental to the human genome than say eye colour or the presence or absence of ear lobes. So it would be a bit like correlating cancer incidence with TV ownership.

We don't, actually, because this is a science forum not a Far Right garbage conspiracy theories forum. White males who get demoralized because brown people and women turned out to be smart at science too are responsible for their own biases and mood disorders. If that causes them to stomp their feet and blow off study sessions, then maybe they're too fragile to do science. Best leave it to people who know how to work hard.

There is a biological aspect to this and it's perfectly normal and natural. We have a "minds eye" how we view, experience and remember our world. We update it with new experience like a journal. It is part of our and many animals survival. Totally normal.

Not only providing the data, but a careful interpretation of the precise elements of it. This is, unfortunately very common. A researcher I worked with, which is a powerhouse in the area of crystallography had similar stories about her advisor, whose career was built significantly based on her "fiddly" work with advanced their field significantly. Based on the stories, he was aware that, too, a he took her in as a postdoc, but torpedoed job offers she got for faculty positions. To some degree, yes. But in sciences it lagged quite a bit behind. I am fairly confident that in the 60s the issue persisted. Mayer, who won a joint Nobel was delegated to a research assistant for much of a career, despite advancing the field significantly. That being said, to me the sexism is just an addition to the story, I very much prefer a careful investigation and analysis of generated data, rather than presenting the most exciting and attention-seeking one. I will add that coming from the experimental field, I am of course somewhat biased. But even theoretical models tend to have to pass either independent validation or have some sort of mathematical proof. Eh, the economy was rebuilt before the crash. And Hitler's recovery was built on war economy deficit spending, which is not really rebuilding but just taking on a huge credit with the hope of making the occupied nation pay. In Watson's case, at least from my point of view, Franklin's paper was much better as it more closely followed the available data. At least from my reading I see two papers, one loud and brazen and the other careful and meticulous. Yet only one gets all the credit.

No of course it doesn't. But the very fact that you have bothered to state this shows that you have not understood the method of deduction. Again of course, but so what ? You are not considering the appropriate information. Today we know that the Earth is moving at an average speed of 2.575 million km/earthday In Roemer's day they knew the average period of Io's orbit of Jupiter to be around about 1.8 earthdays from Cassini's observations, taken over a complete earthyear. In that time, the Earth will have travelled about 4.5 million km. They did not know this because they did not know Earth's speed accurately enough. But Roemer realised that there was a big difference depending upon which direction that 4.5 million km were measured. Along the line of the Earth's path or perpendicular to it. This in turn makes a difference, not to the Io's period of obit, but to when Io appears and disappears. Perpendicular earth travel makes no difference to when Io appears or disappears. But when Earth is travelling ( as near as possible ) directly towards Jupiter that period appears to be shortened by 15 seconds and When the Earth is travelling directly away from Jupites (six months later) the period appears lengthened by 15 seconds So there is a variation in the apparent period of 30 (+/-15) seconds over a full year. Now it is very important to understand what this information will tell you and what it won't. The time variation from one Io cycle to the next will be nothing like this number because the Earth is still travelling towards or away from Jupiter. It is the comparison with the average that matters and where it shows up. So Roemer did not have a good radius for the Earth's orbit and records that have survived do not show an actual lightspeed calculation. He estimated that when the Earth was travelling directly towards or away from Jupiter the light had a corresponding shorter or longer distance to travel roughly equal to the radius of the Earth's obit, because of the geometry of the situation. So the peak to peak variation corresponds to the diameter of the Earth's orbit and the deviation to the radius. Does this help ? Actually I realise that I didn't put that very well. I should have said along the direct line between Earth and Jupiter or perpendicular to it. In other words when earthspeed is perpendicular to that line it make no difference to the transit time for light as the distance between the two does not vary over a couple of days. But when earthspeed is along the direct EJ line the distance Earth travels will be additive or subtractive from the EJ distance, that light has to travel, therby changing its travel time (slightly).

It is true that the reality we see is in fact an event happening within your head. Your eyes receive light like antennas pick up a radio signal. Your brain converts this signal into "live pictures". So in some way we are kind of watching a movie inside our head. I do have my own ideas about this concept of reality but the edge between philosophy and physics might cross each other here. I really do not know where to start..

Yes, it's called "Solipsism". There is also "Solipsism syndrome".

In my mind it is in some ways even worse. When I read his paper as an undergrad, I could not shake the feeling that I just didn't get it. The paper was fairly short, didn't really show any data and the "only" remarkable thing about it, is that he was actually right. However, in my mind this is not how science should work. We have to show data and demonstrate that the conclusion you arrived at is the only one possible. That is what well designed experiments are for. Or, at minimum present the possible answers based on the data you were able to generate and discuss those. Yet the original paper by Watson on Crick doesn't really do that. It mostly proposes a model, based on one possible interpretation, based on the data Franklin generated. I really never reconciled that feeling until at one point I was dabbling in crystallography and dug out Franklins' criminally paper (which was cited only a fraction of Watson and Crick's paper). There I found clear as day that the resolution they had was simply insufficient to clearly rule out confirmations other than the B-form. Franklin in her paper makes it really clear and proposes the right-handed double-helix as one of the possible configurations, which IMO was the right call. IOW Watson and Crick they had the right idea but didn't actually put the work in to provide evidence. They just postulated it and happened to be right. And this is one of the things that is a problem in science, that folks that are considered "towering" can make calls without having the evidence (or even putting their own time into it) and are declared heroes when they happen to be right (and often just conveniently forget the times they weren't). Other folks, especially women, or other in the old boy's club OTOH have to work harder for less recognition. It took me way too long to realize what I didn't like about the paper that has been lionized and that realization has sufficiently soured my view on Watson as a scientist. While I never met Watson, I have met folks who were a visiting scholar in his lab. Suffice to say that there was nothing that really there that could have improved my view on him as a person, either, even before the allegations were widespread. In a broader sense, it is emblematic for the desire of folks, including scientists, to have visible rockstar researchers. Lionizing those rarely benefits science and more often than not it takes away oxygen (and funding) away from those that are doing steady work without overselling it.

Exactly ... Quantum Gravity. The exact same new Physics KJW proposes that might allow for electrons to be quantum ( not classical ) Black Holes. It's expected ) though not certain ) that as Planck scale is approached, gravity becomes comparable in strength to the other forces and can no longer be ignored. There is a further expectation that all the fundamental forces are unified at that scale. Most theories, including Sting Theory and LQG, favor the unification of all fundamental interactions.

Why not both? Most people are a mixed bag. Newton was by all accounts an unpleasant man, Einstein was sexually unfaithful, Mozart had a scatological sense of humour….. This notion that famous people must be pigeonholed as either saints or devils seems very naïve to me.

According to the Wikipedia article, "Rømer's determination of the speed of light", it wasn't the occultation of Io that was used for the time measurements, but the eclipse of Io by Jupiter's shadow. And it was both immersion, when Io suddenly disappears into Jupiter's shadow, and emergence, when Io suddenly reappears from Jupiter's shadow, that was used (immersion and emergence cannot be observed from Earth for the same eclipse because one or the other will be hidden by Jupiter).

***CORRECTION*** This post is a correction of the following text in the post about circular orbits of the Schwarzschild metric: The above contains an error of sorts. I mishandled the [math]r[/math] and [math]\theta[/math] coordinate variables. Specifically, I substituted the location of the object in the rotating coordinate system before obtaining the partial derivatives of [math]g_{tt}[/math]. Anticipating the correct result for [math]a_r[/math] (and [math]a_\theta[/math]) of the object, the error was confined to the above and did not affect the remainder of the post. Below is obtained the acceleration components [math]a_r[/math] and [math]a_\theta[/math] for an arbitrarily located stationary object in the rotating coordinate system. Only then is the particular location of the object substituted. [math](ds)^2 = \Big(c^2 - \dfrac{2GM}{r} - \omega^2 r^2 \sin^2\theta\Big) (dt)^2 - \Big(1 - \dfrac{2GM}{c^2 r}\Big)^{-1} (dr)^2 - r^2 (d\theta)^2 - r^2 \sin^2\theta (d\varphi)^2 - 2\omega r^2 \sin^2\theta\ d\varphi dt[/math] [math]g_{tt} = c^2 - \dfrac{2GM}{r} - \omega^2 r^2 \sin^2\theta[/math] [math]a_r = -\dfrac{c^2}{\sqrt{g_{tt}}} \dfrac{\partial \sqrt{g_{tt}}}{\partial r} = -\dfrac{1}{2} \dfrac{c^2}{g_{tt}} \dfrac{\partial g_{tt}}{\partial r} = \dfrac{\omega^2 r \sin^2\theta - \dfrac{GM}{r^2}}{1 - \dfrac{2GM}{c^2 r} - \dfrac{\omega^2 r^2 \sin^2\theta}{c^2}}[/math] [math]= \dfrac{\omega^2 R - \dfrac{GM}{R^2}}{1 - \dfrac{2GM}{c^2 R} - \dfrac{\omega^2 R^2}{c^2}}\ \ \ \ \ \text{at}\ \left\{\begin{array}{} r=R &;& dr=0\\\theta = \dfrac{\pi}{2} &;& d\theta = 0\\\varphi = 0 &;& d\varphi = 0 \end{array}\right.[/math] [math]a_\theta = -\dfrac{c^2}{\sqrt{g_{tt}}} \dfrac{\partial \sqrt{g_{tt}}}{\partial \theta} = -\dfrac{1}{2} \dfrac{c^2}{g_{tt}} \dfrac{\partial g_{tt}}{\partial \theta} = \dfrac{\omega^2 r^2 \sin\theta \cos\theta}{1 - \dfrac{2GM}{c^2 r} - \dfrac{\omega^2 r^2 \sin^2\theta}{c^2}}[/math] [math]= 0\ \ \ \ \ \text{at}\ \left\{\begin{array}{} r=R &;& dr=0\\\theta = \dfrac{\pi}{2} &;& d\theta = 0\\\varphi = 0 &;& d\varphi = 0 \end{array}\right.[/math]

Yes, there are tests for prime numbers. Methods like the trial division test, Sieve of Eratosthenes, and modern algorithms such as the Miller–Rabin test can determine whether a number is prime.

It's been my experience that talking on the phone vs talking to a passenger seems much different. In trying to decide why it felt that way to me I concluded it is because the person on the phone is not sharing the driving experience with me and can distract me by continuing to talk when I need to concentrate on driving. A person in the car with me recognizes when I am dealing with a situation that requires my full attention, and thus quits talking for a moment.

If Earth orbited around Jupiter at the moon's present distance from Earth, then it would orbit Jupiter once every 1.43 days. Assuming it starts with its present rotation rate, then Jupiter would cross the sky every 3.33 days. However, it would be undergoing nearly 27,000 times the tidal forces it does now, and this would likely drive a great deal of geological heating and tidal braking, and tidal locking would be the eventual outcome. It wouldn't be a pleasant place to live.