hansen

To be a bit pedantic: the identification of tensors with multilinear forms requires finite dimensions (or reflexive topological spaces).

Physicists and philosophers of science give those terms different meanings. E.g. what a philosopher of science would call a theory is called model by physicists. What physicists call a theory is more a broad framework in which more concrete models can be build.

From the view of a philosopher of science string theory isn’t even a hypothesis as a hypothesis needs to be testable

Spacetime is extremely stiff or equivalently gravity extremely weak. Only these extreme events result in detectable gravitational waves.

I doubt there are many users out there who care if someones analytics stuff breaks.

The standard (but rather heavy and difficult) is

Abraham, Marsden: Foundations of Mechanics

More gentle is

Vladimir Arnold: Mathematical Methods of Classical Mechanics

I don’t think the typical “change of variables” definition is bad. You take the derivative of L along the fiber of the tangent bundle. If the derivative is non-singular it defines an isomorphism in each point of the tangent space with the cotangent space. And that’s the important thing, going from the tangent bundle to the cotangent bundle. Now we can use all the beauty of symplectic geometry

Mathematicians messed up here… A manifold with boundary is not a manifold. But a manifold is a manifold with boundary (the empty set).

Gravitational waves are disturbances in a background spacetime, i.e. you linearize the Einstein equations around a given solution. The analogy is less bad as one might initially think

I always thought it's more about caching

The problems in the classical theory are easily understood. The charge and current densities of point particles are not smooth functions but distributions (think of the Dirac δ-“function”). If they act as the sources of the EM field the EM field itself becomes singular. Now if you try to solve the full Maxwell equations including the backreaction of matter & radiation fields you would have to multiply distributions which is ill defined.

There are similar problems in the quantum theory but the divergences are less severe and can be dealt with in a systematic way. Most physicist believe they will totally disappear in some more fundamental underlying theory. From a mathematicians point of view there is the hope that at least some QFTs are finite and the divergences are just an artifact of the construction & pertubation theory.

Given the trajectories of charges it will tell you what the electromagnetic field will be, and given the electromagnetic field it will tell you how charges will move. Unfortunately, these two parts of the theory seem to be incompatible, and the theory will not tell you how fields + charges will evolve in time.

The Problem is not coupling charged matter fields to the EM field. You get a well defined set of coupled and (now) non-linear PDEs. The problems arise when you try to model point charges. Then the theory is plagued by infinities that originate in the infinite charge and current densities.

The infinities in QED are actually far less problematic than those in the classic theory. They just seem to be more problematic because you cant (approximately) ignore the backreaction of EM and matter fields.

And all those Emacs jokes don't work anymore

If you want Cᵏ to be a normed space you do

|f| = ∑ᵢ₌₀ᵏ sup|f⁽ⁱ⁾|.

On open domains you can also use the topology that forces uniform convergence on all compact subsets. But this will only give you a metric space, no Banach space (but you’ll include unbouded functions). This is needed for studying Brownian motions with an unbounded time domain.

Regarding the other question: If you take Cᵏ⁻¹ as a the co-domain differentiation will be continuous. IIRC to get unbounded linear maps defined on the whole Banach space you need the axiom of choice, you won’t be able to write one down.

The problem with differential operators is that they are usually only defined on a dense subset of the domain, and there they are not bounded. E.g. in quantum mechanics the space of states is L² but all the interesting observables are differential operators. You can weasel out of this situation by defining them on a subset of “physical states” (e.g. smooth wave functions of rapid decay). But they aren’t continuous anymore (the spectrum is unbounded). But on Hilbert spaces everything mostly works out fine. Physicist usually ignore those technical problems and still don’t make mistakes.

there is enough experimental evidence that is anomalous to the theories

Huh? I have more the impression that Λ-CDM & SM are working just too damn good. I’m not aware of any observation that hints that we need something completely different than GR & local QFT.

But “settled” and “well established” doesn’t mean that humans will never find a more fundamental theory. Classical mechanics is settled, well established and correct as it was in Newtons times. Now we just better understand its limits and when to use different models.

GR, QM & QFT are correct and won’t go away. At some point in the future we will have a better understanding of their limits too. But that won’t make them wrong.

The fact that we don't know and that our current suite of theories is incomplete and hence could very well be wrong in both minor and major ways should be driving us onward into understanding the universe around us. It really is not a problem if we don't understand. It should mean that we strive to learn more.

I’m not objecting exploring new speculative models. I’m worried about presenting highly speculative ideas as facts or probable solutions to the public. A lot of the pop-sci articles you read these days are very misleading. And IMHO Rovelli, Green, Susskind, Smolin and the like aren’t doing anyone a favor with their pop-sci books.

For example, ask a lay person about BB and they will tell you it's an explosion, in the classical sense

Give her Weinberg’s book to read and she’ll come to a different conclusion. Of course those books won’t give you a deep understanding of cosmology but that doesn’t mean they have to be completely wrong. Stephen Weinberg’s The First Three Minutes is my favorite example of pop-sci done right.

All our accepted model of nature assume that in some sense time exists. But in general relativity it doesn’t make much sense to speak of a single event. In the same way it doesn’t make much sense to speak of a single point in the euclidean plane (you can only localize one point relative to another one).

So some people (like Rovelli) think one should try to formulate relativity only in terms relations of events. This might be relevant in a more fundamental theory of spacetime but on a classical level it’s irrelevant. You can measure the geometry of spacetime in our solar system w/o disturbing it. The gravitational field of e.g. some satellite is negligible to the gravitational field of planets and the sun.

Yeah, science isn’t about being useful and relevant in day to day life. It’s an important part of human culture and scientist should at least try to communicate their findings to general public (which is paying for it!).

I partially agree with the first comment that it isn’t really helpful to popularize highly speculative – and thus probably false – ideas. But well established but still “weird” discoveries like GR & QM are interesting on their own and have huge implications on our understanding of nature. You don’t need a science degree to be fascinated by black holes, the big bang and quantum weirdness.

Sure, but regular quantum mechanics does the same thing. It's under-constrained and allows for too many possibilities.

No it doesn’t. There are a few good guiding principles that make it possible to build models that turn out to be an accurate description of nature. These guiding principles don’t manifest themselves in a set of axioms of a mathematical theory nevertheless they are there. And they are part of the theory, just not set in stone like the underlying mathematical structure. But that doesn’t matter at all as long as you are doing science and not math.

Take classical EM, measure the masses of the electron and atomic nuclei, and write down a suitable Hamiltonians. Viola, you have an explanation of almost every phenomena that’s relevant in our daily life and way beyond. Similar principles exist in elementary particle physics and lead to standard model.

Because the author is a well respected scientist and quite talented science writer & journalist.

https://en.wikipedia.org/wiki/Ethan_Siegel

their [EU] first ruling is that basically every website is not allowed to use ads to pay itself.

Of course they won’t. This is absurd.

They might be stupid but almost every website is even more stupid. Almost none offers a way out of spying.

The GDPR requires users to be able to say no

Even better, it requires the use to say “yes”.

If it's anything like the cookie consent,

It won’t. It will just replace the common “no one reads but clicks” TOS. And the user can change her mind anytime she wants.

The amount of "no clue what this is" among non technical people I know is 100%.

If you can’t explain a non engineer or scientist how personal is collected and used it’s probably not a bad idea to outlaw this practice.

But the EU pats itself on the back cause they're tackling privacy issues. It's a joke.

It’s certainly not enough but a step in the right direction.

Seems to come down to having to protect users from their own lack of understanding.

I had the impression that this is rather clearly regulated by the GDPR. A user has to consent to each use of her data. And you have to explain the use in an understandable way, no legalese. Just make a list where you explain in simple words how you want to use the data and add a checkbox to each item (default not checked). I don’t see how this could hurt any ethical business model.

The startup could just respect their users privacy. It's only a matter of decency. Most people don't have much problems with it.

In Germany ads must be distinguishable from other content. If a human can spot the difference it shouldn’t be too hard to automate that process.

You shouldn't use the term enhancing user experience. It’s a synonym for doing something against the interest of the user.

You can get close with a built in browser setting.

And you can still keep cookies from specific sites by defining an exception. They should better document this feature. For me it would be a good default behavior. There aren’t that many sites where I want to stay logged in.

enhance the user experience

Sorry, but this made me laugh.

whats the alternative to 3rd party cookies?

Don't use a 3rd party service.

It's a bit a matter of definition or interpretation but I would count the LIGO observations as direct observations of black holes. But as others pointed out there's lots of evidence for their existence.