joigus

Interesting... Thank you. I wasn't aware of gravastars.

I'm looking forward to that description. This swapping of the radial coordinate with the time coordinate is a general feature of horizons, though, AFAIK. Back in the sixties, Penrose and Hawking proved singularities to be an inevitable consequence of GR spacetimes, and formulated a conjecture of 'cosmic censorship', that there are no naked singularities ever; they're always hidden behind a horizon, with this funny swapping going on. But horizons appear from the POV of a far-away observer, from the POV of the free-falling observer nothing funny is going on. Looking at the maths, though, the falling observer is doomed. There is a point on their future worldline that means 'the end of space-time' for them. And it's in their future, because the coordinate that monitors time, from the perspective of the far-away observer (the one that carries the negative sign in the metric), is in their radial approach coordinate, rather than the 't' one that describes time outside. I know it's strange, but that's what the maths say. Even leaving aside the problem of evaporation, there are many problems with the Schwarzschild solution. An important one being that it is past-eternal. And astrophysical black holes presumably are not. Don't forget the Schwarzschild solution is static. The way you deal with that in GR is by directly deleting this freak eternal past, modelling a spherical shell of collapsing matter, and patching up the solutions à la Penrose. My approach to any physical problem is very mathematically-biased, but that's the only thing I can do with my toolkit. Yes, I am aware. But Laplace's 'black holes' were very different. Laplace realised that there can be objects that are capable of swallowing any light trying to escape from them, as a result of an escape-velocity calculation, but back then the speed of light was not considered to be the universal limit that today we know to be. From the Laplacian perspective, there's no problem in those photons being detected outside the 'horizon' if given proper initial data. They can't reach spatial infinity, that's all. Gravitational waves open up an invaluable tool to understand black-hole collisions and the like, but the physics community seems to agree that further theoretical clarification of what goes on in them is necessary. Are the words 'in them' even meaningful? In a way, all about a BH is coded in its surface. Some people like to think they are bridges to other space-times. My hunch is that they signal to a limitation of the description and that the interrelationship between gravity and gauge theory must be understood much better than it is today. Easier said than done, though.

The singularity of a BH lies in the future, rather than at the centre. So it's a time, not a position, from what I know. Now, you can call that the centre, for convenience, but it's a time, not a radius. Time and radius change roles when you cross the event horizon. That's what the maths says. What does that mean? I don't know. There are many things about black holes that I would like to understand better. Is the Schwarzschild black hole anything to go by, or is it just a freak of the equations of relativity for being so unrealistically simple? The only thing I can say is that theorists keep discussing them and the role they play in physics, including giant ones, microscopic ones that may exist, etc. There is no unanimous agreement about them. That's all I can say. The best thing about black holes is probably that they create conflict in our theories. I hope that means that research in black holes will usher in the next revolution in physics, but not much is certain about them except one thing: astrophysical black holes do exist.

I like this sentence. We should never underestimate the power of rephrasing the basics.

Love these tunes... There's something about waltz. And there's something about folk songs... Nightmarish tho.

Loved it.

It is not strictly necessary to excel at maths in order to have a good idea. Historically, Faraday was a perfect example of this.. Although it helps knowing your maths. But you need to understand how the ideas of physics relate to each other.

There is absolutely no significance whatsoever in the ratio between a kilo and a Coulomb. Same reason why the length of my nose divided by the mass of my head has no significance whatsoever. But I can define units o mass, length and time so that aforementioned ratio happens to be 9×109 or whatever other value I find convenient. Such is the nature of \( \varepsilon_0 \).

Bound state? You do have Yukawa couplings in the SM. Are you changing the SM? Your question isn't focused on what the Higgs field does? Can you 'hear' yourself? The Higgs field was summoned into physics because of what it does. Why else would we have a Higgs field? To make physics more spicy?

Saw it a couple of days ago. Almost spooky!

Force is related to momentum, and KE is a function of momentum: essentially (momentum)2/mass.

Really, really strange animal. Pink fairy armadillo, culotapado (hidden ass) or pichiciego (which I dare not translate). It has unique features among mammals, can use its tail and legs as a tripod, and its muscles reach the end of its extremities. https://en.wikipedia.org/wiki/Pink_fairy_armadillo

Maybe you don't see it this way, but everything I've seen so far upstream of this thread is help. You first need to understand what the Higgs field does in the standard model. The job description of the Higgs field is to provide mass for particles that, for some fundamental reason, shouldn't have one. The first class is gauge bosons that are found to be massive (W+, W-, and Z0 of the weak interaction). The other (quite important class) is charged fermions (not Majorana fermions, provided they exist). That includes quarks, and all leptons (electron, tau, muon...). It's not through strong coupling, as you've been told; it's through spontaneous symmetry breaking. In a manner of speaking, the particles get 'dressed' with a mass term. It's not at all like a coupling. How does your Gauss-law-based 'Higgs' do that? How does the symmetry get broken? How does it even work as a Higgs field? The Higgs field enters the physics through a potential, but it's a potential in the Higgs-field variable itself, not the space variable, as in Gauss' law. Etc. You need to understand basic physics. Let alone quantum field theory, and how the vacuum operates in that theory.

(my emphasis) SOL* * Smirking out loud

That's not how thermodynamics or heat transfer works. A process is exo or endothermic irrespective of what the temperature of the thermal bath is. Also wrong that high-energy states 'naturally transition' to lower energy states. That depends on the temperature. And force has very little, if anything, to do with heat transfer. Starting with the fact that equations of heat transfer are irreversible, while mechanics equations are reversible --except when friction is involved. There's nothing about what you've said so far that shows any understanding of how physics works, IMO.

That is incorrect. It's the Aristotelian mistake. Aristotle thought that in order to have something move you need a force. That's now how Nature operates. You need a force in order to have something change its motion.

Gauss' law has nothing in the way of spontaneous symmetry breaking, which is what the whole idea of the Higgs mechanism is based on. You need an equation with a continuous symmetry with particular solutions that break that symmetry. I haven't thought about it, but I don't think Gauss' law can accomplish that, nor have I seen it discussed anywhere.

What is c? The speed of light? Oh, OK. No, that math doesn't even start to talk about speed of light. As said by Studiot, the quantum wave function is a very different thing. For starters, it's a function, while \( \varepsilon_{0} \) and \( c \) are constants. Another thing is that \( \varepsilon_{0} \) is not really a constant of Nature. Rather, just an artifice in the choice of units for electric charge. Stick to Heaviside-Lorentz units and there's no \( \varepsilon_{0} \). It disappears!

You could call it an error. I prefer to see it as a limitation of the ideas of electrostatics when you consider charges smaller than one electron's charge, and --directly related to that-- the fact that, at some point, you need to replace classical electrodynamics with quantum electrodynamics. You cannot consider an electron as made up of elementary charges smaller than e. Capacitance is perhaps not the best way to see it, because it's not a fundamental quantity. It doesn't make a lot of sense considering an electron as infinitely many infinitesimally-small charges adding up to give an electron. You would have, I guess, to assume what the capacity of an 'incremental electron of charge dq' is. For a point-like electron (you haven't specified what spatial distribution of charge you're thinking of) it's best to use the expression, \[ \frac{\varepsilon_{0}}{2}\int\left|\boldsymbol{E}\right|^{2}dV=\int_{0}^{\infty}\frac{e^{2}}{32\pi^{2}\varepsilon_{0}r^{4}}4\pi r^{2}dr \] which gives you nonsense, \[ U=-\frac{e^{2}}{8\pi\varepsilon_{0}}\left(\frac{1}{\infty}-\frac{1}{0}\right)=\infty \] So, considering an electron as being made up of infinitely many (smaller) incremental charges doesn't make sense. You would like to consider a finite electron. In that case you're going to have to face even worse problems --mostly having to do with the fact that this electron cannot consistently be considered as a rigid object, nor has anybody found a way to make it elastic and be consistent with relativity --Poincaré and others tried very hard. That's why we use quantum mechanics when things get so small. And also forget about capacitance, which is a highly-derived concept. I hope that was helpful. Sorry, I misread this.I thought you meant dq<=e, which is what you need. If you integrate from zero you should add smaller charges than e, which is what Studiot is pointing out, in a way.

No, that must have been before I 1st came here. Teaching methods interest me a lot.

Lame joke on my part. I know how much you love books. You actually mentioned the human factor. I couldn't agree more.

No. You need to say what these things are.

Wonderful complement to what I was saying

Now I know @studiot is going to hit me hard. What I mean is: Look out for teachers that are putting out their material online, a lot of them for free. When the drive to learn synchs with the drive to teach, something wonderful happens, and knowledge is passed on. If the teacher does well, and the student does well, elements of criticism and analysis of how ideas are built and corrected/improved are also transferred in the process. Khan Academy --that @Phi for All already mentioned--, Stanford, MIT courseware, and a long etc. You've got plenty of university material out there. And my all-time favourite, PI (Perimeter Institute): Psi Online