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.