Markus Hanke

Great thanks +1 PS. I accidentally hit the downvote button instead of the upvote one (touchscreen)…I tried to correct it, but it’s now displaying an upvote in red color. Not sure what that’s about, but it’s definitely meant to be an upvote.

The word “apple” is just an arbitrary label in a particular language (English) - just precisely what such labels refer to is usually given by common consensus of the speakers of said language, and that consensus is usually rooted in ordinary everyday experience, and organically emerges from there over time. In English, when people speak of an “apple”, they refer to a piece of fruit that is sharply delineated from its environment - an apple can be on a table, on a plate, hanging from a tree branch, be located in my backpack, can be held in my hand etc etc. It is a label that is to some degree independent from its external context, so all these differing instances of the same fruit can be called “apple”. Of course you can decide that “apple” should refer to the fruit itself plus the “topmost layer” of atoms on a table. The problem here is of course that you then need different labels to refer to this situation in different contexts that don’t involve table tops - for example, if the fruit hangs on a tree, you can no longer call it “apple”, because there’s no “top-most layer of atoms on a table” (what does this even mean?) present there. So it wouldn’t be an “apple”, but must carry a different label instead. Also, the aforementioned layer of atoms wouldn’t be a “layer of atoms” anymore, but “part of an apple”, whereas the layer immediately underneath (let’s assume they can be neatly separated), would still be “atoms”. If you don’t delineate labels carefully, things become messy quite quickly. But to make a long story short - labels carry no physical significance, so their choice is entirely arbitrary, so long as the labelling scheme is internally self-consistent. This is why you can use a completely different language to talk about the same physical situation.

It does, thank you I did notice though that you haven’t mentioned Schaum’s Outline in your literature list - is the omission deliberate, ie is there something about the text I should be weary of? I’m just asking because I happen to have that text in my possession already - I haven’t read it yet, but it looks good at first glance.

Yes, thank you I’m clear about index symmetries…it was rather about that strange notation where indices are written vertically aligned one atop the other. Turns out it’s just sloppy notation.

Ok, that clarifies it! I was simply wondering if that was just sloppiness, or whether there is any significance to the notation. Thank you

Ok, so basically this is just a sloppy way of writing them?

Yes sure - but if both indices are written in one vertical line, as in \(B^{\mu}_{\nu}\), how do you know which is which?

Ok, thank you As it happens I have both Schaum and Synge/Schild in my personal library, but haven’t read either one of them yet. Just leafing through them, they both look pretty good. Still, it will be helpful to see what other recommendations people here might have!

Thank you joigus. Unfortunately though this isn’t quite what I meant, but the fault is entirely mine - I didn’t make my question clear enough, and the additional clarification I just added came too late for your post, and somehow got tacked onto Lorentz Jr’s post instead of the OP. It’s all gone a bit messy here. Apologies for causing you lots of typesetting work. My question was actually about vertical alignment of indices - how does \(B^{\mu}_{\nu}\) formally relate to (eg) \(B{^{\mu}}{_{\nu}}\), with the emphasis on vertical alignment or lack thereof of tensor indices? Or is there perhaps no difference at all?

Yes, I have indeed - but only to the extent of ordinary exterior calculus in the context of GR, as presented in (eg) MTW. I’d definitely like to deepen my understanding and skills in the area of geometric algebra, since it is a very powerful formalism; many aspects of physics can be cast into that language. But that’s a future project - at present I need to recap my tensor calculus. But thank you for the links, I really do appreciate that +1 PS. I think I need to add some clarification to question (1) in my OP (which, for some reason, it won’t let me edit?). I’m good with raising and lowering indices, thus the relationship between the latter two notations \[B{^{\mu}}{_{\nu}}=g^{\mu \alpha} g_{\nu \beta}B{_{\alpha}}{^{\beta}}\] isn’t the problem. What I’m wondering about is specifically the notation where two upper and lower indices are vertically aligned; thus I’m wondering how the above relates to \(B^{\mu}_{\nu}\).

I have some questions here, which I’m hoping someone might be able to help with. I’ve spent the last few years focussing on other things in my life, so I’m afraid I’ve lost touch with the some of the basics - I’ve recently attempted to once again put pen to paper and actually work out some GR tensor calculus practice problems from scratch by hand, and…let’s just say it didn’t go so well 😕 1. Notational question - assume we are working in the context of GR, ie we are on a semi-Riemannian manifold endowed with the Levi-Civita connection and a metric. What is the actual significance of the vertical alignment (or lack thereof) of indices on tensors and spinors? In other words, what is the actual difference between the following three notations (let B be a rank-2 tensor), if any at all? \[B_{\nu }^{\mu } \ vs\ B{_{\nu }}^{\mu } \ vs\ B{^{\mu}}_{\nu}\] 2. I need to really revise and - above all - practice my tensor calculus index gymnastics, but I’m having trouble finding a suitable text that actually focuses on the mechanics of index manipulation, rather than abstract definitions and proofs (which is what you most often get in GR texts). Does anyone here have recommendations? What I am specifically after is something not too high-level that goes through the various concepts in tensor index manipulation, provides worked examples, and then gives exercises to work through. The relevant chapters of MTW actually are good in that regard (they’re on a level I can follow easily enough), but I think the material is presented too concisely and quickly - I’m looking for something that introduces it more slowly and in more detail, including worked examples, and gives many more exercises of varying levels of difficulty to do. I understand the concepts involved reasonably well if I see them written down in an equation, I just need much more practice in actually using them in a pen-on-paper kind of way - which is an entirely different skill set. So I’m after something that really drills home the mechanics of index manipulation through worked examples and exercises. Any suggestions, anyone? TIA.

\[B_{\nu }^{\mu } \ vs\ B{_{\nu }}^{\mu } \ vs\ B{^{\mu}}_{\nu}\]

Have you ever actually worked in the area of addiction recovery or homelessness? Yes, that’s my vocation. I am simply attempting to point out that your understanding of this issue is inadequate, because you cannot simply equate addiction with physical dependency. It’s a far more complex issue, and continuing to ignore this basic fact will not be helpful in developing effective policies - which is ultimately what we all want. I disagree. We have been criminalising drug use and waging a “war on drugs” for at least the past 40+ years, to no avail whatsoever. If anything, the problem is now far worse than it ever was, despite the heavy-handed approach of authorities in the US and elsewhere. To give another example, I have just spend 1+ year in Thailand, and they have mandatory death sentences if you are caught with more than a certain amount of drugs on your person. It’s also common practice there to force addicts into “reeducation camps”. The result? The place remains awash with drugs of all kinds - if you think the problem is bad in the US, it’s far far worse in Thailand, by orders of magnitude. Clearly, you won’t dissuade people from using by threatening them with punitive measures, or putting them forcibly through detox programs. There is not a single data point (that I am aware of) that supports the efficacy of such an approach, but plenty of data to suggest it doesn’t work. This has been the standard in many jurisdictions around Europe for quite some time. Again, it did not solve the problem - the drug problem in many places in Europe is still bad. No one here said anything about not punishing people who have committed crimes. Of course, if someone commits a crime they need to be held accountable, irrespective of whether they are addicts or homeless or whatever else. What you are suggesting though is something quite different - you want to forcibly commit people into camps purely on suspicion that they might at some point in the future commit a crime, solely based on their status as being homeless and/or addicts. Preventative incarceration, is what I’d term this - please don’t try to window-dress this as “helping the addicts”, because that is deeply disingenuous. I’m sorry, but this is simply not ok. Luckily I have enough trust in our democratic institutions to be reasonably sure that such a thing will not happen anytime soon - even if there was data available to show that this would actually solve the problem, which of course there isn’t. Personally though I must say I am quite horrified that anyone would even suggest such a thing in all earnestness. I am German by birth, and at one time not too long ago a government of my country sent people into camps based on their ethnicity, race, sexual orientation, political conviction, and even mental/physical health status. We all know how that turned out. Do we as human beings really forget so quickly? If I was to suggest a policy it would be roughly along the lines of: 1. Take drug consumption off the streets by providing safe, supervised and hygienic injection and usage facilities - harm reduction as a first step! 2. Address the problem of homelessness through policies that directly tackle the issue of poverty, income inequality, and lack of social mobility. So long as you facilitate an economic system where large numbers of people work full time jobs and yet remain near or under the poverty line, your drug problem isn’t going to go away, like ever. 3. Make substances available to those addicts who need them in a controlled and safe fashion, as part of a public health program - this stops the flow of money to drug cartels, cutting off much of the large-scale organised criminality that flourishes around addiction. Once addicts are within a public health network, it will be easier to help them with further therapeutic measures 4. Provide proper education around drugs to our kids - “just don’t take them” evidently doesn’t cut it! 5. Completely decriminalise possession of small quantities for personal use This is neither exhaustive nor complete, just a rough outline. All in all, I’d advocate a radical shift away from a punitive towards a public health approach - simply because the punitive approach has already proven itself to simply not work. Only a fool would continue to do the same thing over and over, and expect different results somehow. So I stand by what I said earlier - a complete paradigm shift is needed, because the current paradigm has failed us, and quite badly so.

Yes - this often happens when people (consciously or unconsciously) adopt scriptural literalism as their mode of engagement with their chosen tradition, ie when they fail to distinguish the word from the message. When you allow this to happen, you essentially make it impossible for yourself to meaningfully consider other sources of information or modes of enquiry, resulting in a more or less filtered/distorted/one-sided view of the world.

The evidence for the existence of a gravitational attraction between masses is so overwhelming that it is entirely beyond any contention. Using the Cavendish setup, you can even perform experiments that show you this right at home in your living room. I recommend you give it a try - you can purchase DIY kits for this right off the Internet at a cheap price. Some of these come with spheres made of different materials - metal, stone, high-density plastics etc -, and some are even enclosed in vacuum chambers and Faraday cages, so that you can confirm yourself that these effects are not the result of any electromagnetic interactions, air movements etc. Gravity is very much real, and it is very easy to show that it is.

It depends largely on how exactly you define “actual present moment”, because the meaning of this is not at all clear. We are remembering a subset of whatever mental objects were present in consciousness at the instant the memory refers to. These mental objects are the result of the reality-model that the mind has constructed - this model takes as its input sensory data (“perception”), memories, and previously accumulated habitual processing patterns, which are then filtered and further processed in certain ways, and eventually output as mental objects that we can direct our attention to and be aware of. Phenomenological reality is a mental construct. It’s kind of like a graphical user interface on a computer - the operating system takes external inputs (user interactions, information stored in RAM, ROM and other storage media), processes these, and renders them graphically on a screen, printer, or other output device. The basic inputs and information storage units do not themselves contain GUI elements such as letters, windows, menus, dialog boxes etc etc (they are just electric potentials), but it turns out that these higher-order representations make for a useful “reality-model” for a computer to interact with its user. The specific form such reality-models take are strongly influenced by teleological concerns, ie the reason why the computer was constructed in the first place. But all of this aside, the basic issue is this - our memories contain what is by and large a structured and ordered sequence of remembered mind-moments. We remember perceptions from our childhood, and we remember our teenage years, and we recognise that one came before the other - in other words, we never remember the future, only the past. If the original input data does not already contain such a temporal ordering (ie there is no present moment), then why are our memories temporally structured in a self-consistent manner? A similar argument can also be made for any kind of non-sentient recording device that records a structured sequence of information based on external inputs. If there is no concept of present moment, it is very difficult to explain the emergence of this temporal structure.

What you suggest seems like an obvious solution, but unfortunately it does not and cannot work. Addiction is much more complex than just being a physical dependency on something. Yes, you could (ethical concerns aside for now) round them all up, put them in a camp, and forcibly put them through physical detox - the trouble with this is that it doesn’t actually address the underlying issue at all, because the dependency is in large part of a psychological, social and systemic nature. No one wakes up one morning and decides “I’m going to become a homeless addict…seems like a cool career choice!”. That’s not how it works. Most long-term addicts are in this situation because of multiple factors connected to their social environment, upbringing, past trauma, etc etc, many of which they have little or no control over. These are all complex issues that are not easily nor quickly fixed. It’s a common mistake to think that people remain addicts purely because of their physical dependency, and if we kick the physical dependency they cease to be addicts - that’s quite simply not true at all. So as for your proposal - you take them to your camp, forcibly put them through detox, and at some point will have to let them out again to re-join their families and social environments. What do you think happens then? I can pretty much guarantee you that within days or weeks almost all of them will be right back on their drug of choice, with perhaps the odd exception. Why? Because the underlying reasons for why they have begun to use substances in the first place have not been addressed. Addiction is a symptom of an underlying disease, not really the cause itself - just putting people through detox is like giving painkillers to a cancer patient; it alleviates the symptoms for a little while, but it doesn’t cure the disease. People don’t start off using because they are physically dependent, but for other reasons. It’s those initial reasons that need to be addressed. You cannot help an addict who doesn’t want to be helped - the impulse must always come from him/herself. People have to be ready to change, before therapy has any chance of success, and even then the relapse rates are still high. Forcing people into a treatment they are not ready for does not work. I don’t know if there are actual studies to show this (there probably are), but everyone who has ever actually worked with addicts knows that this is a basic fact. BTW, rounding up addicts and forcing them into rehab camps is what the Taliban in Afghanistan tried to do. Needless to say, it didn’t work. But it makes for an interesting case study if you want to research into it. So as for your proposal - it certainly has political appeal to those who don’t know much about drug addiction, but ultimately it does not and cannot work. It would just create a revolving-door kind of situation with people going in and out of camps, and the ones who ultimately profit will be the dealers and cartels, as always. Until we begin to treat homelessness and addiction as the social and health issue which it is, and stop criminalising something that the victims have little or no control over, no progress can be made on this problem. Criminalising the addicts and waging a “war on drugs” has never once worked, does not work now, and never will work. A complete re-think is needed.

So why can you remember it afterwards?

I myself am an ordained monastic in a Buddhist tradition of contemplative forest monks and nuns. The impulse to follow a contemplative and spiritual life - irrespective of what specific form this may take - need not at all run counter to science, as many who were simultaneously scientists and spiritual seekers have shown throughout history.

No, it’s silly because - as I have already shown you - it directly follows from the principle of least action (as well as other principles, as pointed out by several contributors here), so it makes no sense to claim that it is “wrong”.

It depends what you mean by “problem”. You were asserting in your OP that F=dp/dt isn’t the correct formulation of Newton’s 2nd law, so I tried to provide you with the bigger picture of where this relation comes from. My point was to show you that it isn’t just made up (and thus potentially wrong), but that it follows from more general principles. It’s essentially a special case of the principle of least action. F=ma then is in turn a special case of F=dp/dt - it works well for many simple systems (which is why we all learn it in school), but its domain of validity is limited. F=dp/dt is more general, and also works in some cases where F=ma doesn’t. And then of course there are systems where neither of these work (as joigus has pointed out), and then you have to use the full Lagrangian formulation of mechanics. This isn’t to say that F=dp/dt can’t be derived in other ways too, which is what studiot is trying to show you. Either way, just claiming that F=dp/dt is wrong and must always be replaced by F=ma - which is what you seem to be saying - is silly, because that’s manifestly false.

Newton’s second law in the form F=dp/dt (which is equivalent to F=ma iff m=const) isn’t just made up, but follows directly from the principle of least action, and is thus mathematically derivable in a self-consistent manner. I suggest for starters you might take a look here. Your claim that this is “wrong” somehow is thus pretty meaningless.

No, it wouldn’t. This is purely a SR problem, and all it involves is comparing the lengths of the two world lines, which is a straightforward (though not always easy to evaluate) line integral: \[\tau =\int _{C} ds=\int _{C}\sqrt{\eta _{\mu \nu } dx^{\mu } dx^{\nu }}\] wherein C is the path in question. Now, even without having to consider any specific numbers, we can work out the differential ageing between the two. Suppose we have two events in spacetime, A and B, which we connect via two different paths (the two twins) - we assume for simplicity that the two events have the same spatial coordinates, and differ only in time; this would correspond to (eg) one twin remaining stationary and inertial somewhere, and the other one travelling away and returning again, so that they both start off together at rest, and finish together at rest. This is the simplest possible twin scenario. Let’s use a coordinate system that has its origin at event A, and a metric with sign convention (+,-,-,-). The inertial twin then simply traces out a world line of length \[\tau_{1} =\int _{A}^{B} ds=\int _{A}^{B}\sqrt{\eta _{00} dx^{0} dx^{0}} =\int _{A}^{B} d\tau\] The other twin, the one that is travelling away and returning, on the other hand traces out a world line of length \[\tau_{2} =\int _{A}^{B} ds=\int _{A}^{B}\sqrt{\eta _{\mu \nu } dx^{\mu } dx^{\nu }} =\int _{A}^{B}\sqrt{( d\tau )^{2} -( dx)^{2}}\] wherein dx depends on the spatial path of that travelling twin. Since we know from the extremal value problem in variational calculus that the inertial traveller always maximises the path length between given events (using this sign convention in the metric), and since the two paths in this scenario are not identical, we can immediately conclude that it must be the case that \[\tau _{1} >\tau _{2}\] This also implies that the path of the travelling twin can never be a geodesic, so it cannot be everywhere inertial.

! Moderator Note Thread moved to the ‘Speculations’ section, since this isn’t mainstream physics.

No, that’s a common misconception. GR only comes into play if there is non-negligible gravity involved; the case of acceleration in a flat spacetime is handled perfectly well by SR. We already have plenty of direct evidence from particle accelerator runs that accelerated particles behave exactly like SR says they do. Perhaps the best demonstration of this is Fermilab’s “Muon g-2” experiment, where unstable muons are introduced into the ring and accelerated to a \(\gamma\) of ~30. And as expected, their average life time really increases by the expected amount as compared to a reference sample that remains stationary in the lab frame. And this is only one example - literally every accelerator run we do demonstrates the reality of time dilation and length contraction, in exactly the way SR predicts it will.