Markus Hanke

Spacetime and its geometry are “there” not only in vacuum, but also in the interior of energy-momentum distributions. There is no situation where there is not spacetime, since there is nowhere one can not place rulers and clocks. I still don’t get what the “issue” here is…?

They are the current scientific consensus, and thus the best models we currently have. Take careful note of the word “currently”. Physics, like all sciences, is a process - as new data becomes available to us, the consensus may need to be updated, and occasionally radically reworked (“paradigm shift”, like from Newton to Einstein eg).

It depends what is meant by “precisely”. If you mean exactly, ie with no deviations at all, then I agree that this is probably not possible. In practice though it is often possible to minimize differences such that their effects on the evolution of the system are negligible, at least for some specified period of time.

How about the Vaidya class of black holes? These spacetimes are not asymptotically flat.

Nice way to visualise this +1

You’re absolutely right, and it was meant to be that, I once again forgot the conversion. This is what happens when you don’t do this stuff every day. Thanks for picking up on it 👍 I’m not entirely sure what “to second post-Newtonian order” actually means, but I presume this is an approximation of some kind? The full integral looks elliptic, so there shouldn’t be a closed-analytic form for the exact result.

Oh my, you are absolutely correct! My apologies. I took the expressions for E and L from my personal notes, without realising that they were in natural units, whereas the integral was in SI units. Silly amateur mistake on my side. Let’s try again - we have, this time in SI units, \[E=\gamma c^{2},\ L=\gamma v_{\infty}b\] with \[\gamma =\frac{1}{\sqrt{1-\frac{v_{\infty}^{2}}{c^{2}}}}\] Popping this into the original E-L integral, I get, in slightly different form \[\varphi =\int_{r_{\min}}^{\infty}\frac{\gamma v_{\infty} b dr}{r^2 \sqrt{\gamma^2 c^4 - \left(1 - \dfrac{2GM}{rc^2}\right) \left( c^2 + \dfrac{\gamma^2 v^{2}_{\infty}b^2}{r^2} \right)}}-\pi\] The units should be correct now, with the result being in radians - but perhaps it’s wise if you double check, since I’m doing all this pen-on-paper.

My pleasure. As a little exercise, I’ve reworked the integral to something more explicit (I personally never really liked the notation with E and L), and if I’m not mistaken this is what we get: \[\Delta \varphi = 2 \int_{r_{\min}}^{\infty} \frac{dr}{r^2 \sqrt{\dfrac{1}{b^2 v_\infty^2} - \left(1 - \dfrac{2GM}{rc^2}\right) \left( \dfrac{1 - v_\infty^2/c^2}{b^2 v_\infty^2} + \dfrac{1}{r^2} \right)}} - \pi\] So the deflection angle depends only on initial speed far away, impact parameter, and mass of the central object - as one would expect.

It is: \[\varphi =\int_{r_{\min}}^{\infty}\frac{dr}{r^2 \sqrt{\dfrac{E^2}{L^2} - \left(1 - \dfrac{2GM}{rc^2}\right) \left( \dfrac{1}{L^2} + \dfrac{1}{r^2} \right)}}\] For non-relativistic speeds and weak fields, this reduces to the Newtonian scattering formula. For v=c and massless test particles, you get the Schwarzschild light deflection formula. For strong fields and massive particles, the integral can be evaluated numerically.

This is certainly true for some of us, but one has to remember that autism manifests along a spectrum - some autistics have very profound difficulties with communication, whereas some others might be at a near-neurotypical level in that particular area, but might be really struggling with other things. It’s difficult, if not impossible, to generalise what the “typical” autistic person might be like.

I understand what you are trying to say here, but I’d like to highlight that it is only particular patterns / manifestations that one can improve on, given the right tools and strategies. Autism itself is a physiological difference in the human brain, you cannot snap out of it any more than you can snap out of being pregnant or having cancer. But you can find skilful ways to manage it.

I guess the difference is in the level of intensity - for autistic people the fixation on their hyperfocus can be very powerful, to the point that it is at the forefront of their inner lived experience much of their waking hours, and can often almost look like an obsession of sorts. Eg someone with a hyperfocus on Spongebob Squarepants might own all the relevant media, have SBSP bedlinen und brush their teeth with SBSP-branded toothpaste, while simultaneously knowing everything there is ever to know about SBSP. This can then also "bleed over" to other areas, for example when in a conversation they might inadvertently start to blabber about their hyperfocus ("infodumping") even though the initial interaction was about something entirely unrelated. This is not to say that neurotypical people don't have special interests or expertise in particular subjects, but the difference is in the degree / intensity of how this is experienced. Note also that this in isolation is not a defining indicator for someone being autistic, but it does form a part of a larger list of diagnostic criteria. Indeed.

It is very common for autistic people (at least the high functioning ones) to have areas of special interest, called a hyperfocus, which they get deeply fascinated by and perhaps over time come to know a lot about. Sometimes this can be the stereotypical maths, but it can just as well be LEGO, Marvel superheroes, or SpongeBob SquarePants. So no, not all autistics are maths geniuses - I know a lot of people in the autistics community, and not one of them fits that bill. But many of them are very knowledgeable at something.

You mean gravitational waves ;)

Thanks everyone, this is all valuable, in particular the fact that it is the boundary conditions that impose the precise form of the solution, rather than the equation itself. Which I of course knew before, but hadn’t thought about deeply enough. I’m currently investigating the differential forms formalism for all this, which I find very valuable too for building geometric intuition.

I’m noticing this exact pattern on the side of many users as well. They’re being told by their AI that they’ve stumbled across some amazing idea or another, and then the AI actively reinforces this to keep the user hooked - irrespective of whether there is actual value in the original idea or not. Because let’s face it, this is the only reason these tools even exist in the public domain - to get the user to spend as much time on them as possible, same as social media. It’s a business model, not some altruistic public service. AI isn’t designed to champion objective truth, it’s designed to increase screen time.

I’m not so sure about this actually. If you equip a properly designed robot with a suitably trained AI or AGI , I think this is precisely what will happen. It’s just a matter of time. In Japan they already have restaurants entirely staffed by robotic waiters.

The following may be an unpopular opinion, but I’ll say it anyway. When I went to school, we spent the first five years of math education doing pretty much nothing else but pen-on-paper arithmetic. Addition, subtraction, multiplication, division…over and over and over again, with increasingly large numbers and more decimal points. Since I left school 30+ years ago, never even once was I in a situation where I in fact had to do pen-and-paper arithmetic, at least to the best of my recollection. I’m unsure if now I even remember how to do it. The reality is that we live in the Information Age, and it’s a skill that’s basically never needed anymore. Of course one needs to have an understanding of what those operations mean, but being proficient in working out 6537.45/765.44 by hand on paper has kind of lost its relevance, IMHO. I think it’s enough to spent at most a year on this. On the other hand, had we gone further on the upper end, beyond single-variable calculus, perhaps into differential equations, calculus on manifolds etc, it would have been very helpful. Just my personal opinion. You can crucify me for it :)

I think the best case scenario would be that it strengthens one’s ability to ask the right questions. I for example don’t work in academia, but I have a strong interest in physics. I often play around with ideas in my head which require some mathematical investigation. Unfortunately I don’t have access to advanced CAS such as Maple, and in GR calculating stuff with pen and paper is generally cumbersome and error-prone, especially when it’s not something you do every day for a living. I can do it, because I’ve taught myself how to, but I often make silly mistakes. So nowadays I offload the cumbersome stuff to AI, and just focus on the overarching ideas (caveat - AI does get maths wrong, so one needs to check!!!). That requires me to consider carefully what questions to ask, and how the answers fit into an overall context. So I think AI might (!!!) ultimately help to focus better on the bigger picture, by automating the cumbersome details, just as calculators helped us focus on concepts rather than manual arithmetic. But again, one has to think about the answers one gets, because they are often flawed, meaningless, or straight out wrong. But of course, it really depends much on how people use it in practice. There are no straightforward answers. One thing is for sure though - AI is here to stay.

This is extremely helpful - I knew about the general analytical solution in terms of arbitrary functions, but hadn’t thought about what it actually means. So the above is an important piece of the puzzle. I will think about this some more. But already now, thanks so much to you all, you are very helpful, as always!

An important piece of the puzzle, thank you! Yes, that’s where I’m stuck - I’ve only ever seen the equation used as a wave equation, but I never got the geometric intuition as to why it is specifically waves, as opposed to something else. Interesting, thanks! I’ll have to think about this a little more, before I can comment. These operators are linear, but the dynamics of GR are not, so the field equations needed to be something a little more complex. Good point! But again - why waves in the first place? But thanks everyone for the inputs :) It still hasn’t quite “clicked” for me yet, so do keep it coming if you can!

I’m trying to develop a geometric intuition about what the d’Alembert operator actually signifies. Specifically, I’m looking for a geometric intuition as to why equations of the form \[\square =0\] have waves as solutions, as opposed to something else that “lives” on the light cone. I can see where the light cone comes in, and I also understand why analytically/algebraically the solutions to this PDE are waves; I’m just missing a geometric intuition as to where these “waves” come from, if that makes sense. I’m a very visual thinker, so having such an intuition is always really helpful to me. Any takers?

No it is not. Rs of the sun is not the same as Rs of earth, for example. It depends on the mass of the body in question, as well as the relative strength of gravity. Again, the gravitational potential depends on both the mass of the body as well as the relative strength of gravity. The potential function of the sun isn’t the same as that of earth. I don’t know what “measurement” has to do with the basic fact that not all bodies share the same potential. The orbit itself depends on M and G. So why do you observe those quantities, orbits etc to be different for different bodies? What is it about those bodies that makes them different? You asked me before what I think of all this. I’m sorry to say that the only fitting word that comes to mind regarding your reasoning here is “bizarre”, especially after this last reply of yours. For my part, I’m not interested in investing more time in this, but I wish you all the best.

Did you not just say that your solution depends on neither G nor M, yet the above substitution explicitly introduces both of those quantities? How do you find this scale quantity a? This, again, explicitly depends on both M and G. And it implicitly assumes you are in a spacetime that has a time-like Killing vector, and is asymptotically flat, or else no concept of gravitational potential exists. How do you find this quantity? And where does this come from?

How am I mistaken in that G and m don’t appear in the vacuum equations?