MigL

OK. So, in the case of the x co-ordinate line segment, drawn on x-y co-ordinate graph paper, where the intrinsic separation between points doesn't change when curled to a circle, but the extrinsic separation does, which is the manifold that ... Is it the line segment or the graph paper ?

Thanks for the explanation KJW; excellent as always. I still have a few questions ... Consider a line segment 20 units long with point A at x=5 and B at x=15. We now curl that segment such that the two ends identify with each other into a circle. The distance between the points A and B doesn't change; it is still 10 units. This is what I would term intrinsic curvature, and separation. Now draw that same line segment on a sheet of graph paper. When the segment is straight the distance between the points A and B is similarly 10 units. But upon performing the equivalent curling operation as before ( into a circle ), the distance between the points A and B is no longer 10 units, but approx. 20/3.14 units, as it depends on both, x and y co-ordinates. I would term this extrinsic curvature, and separation. What am I missing ?

Maybe I've got it wrong, but 'intrinsic', to me, means that the separation between two points on a manifold is based solely on the parameters of those points. IOW, if two points are specified as ( k1, l1, m1 ) and ( k2, l2, m2 ), then the intrinsic separation is along the manifold defined by k, l, and m. The introduction of the parameter n, to the manifold, would imply you can have an extrinsic separation as there is now an 'embedding' dimension, n . My apologies if I've used incorrect terminology.

Don't usually watch random YouTube videos, but there are some channels I follow. Most are Military aviation related. Some that may be of interest to Forum members are 'Pax Americana' and 'Perun', which deal with Military policy, and for the 'scientists' among us, 'Physics Explained' and 'Veritasium'.

In the US money took the form of gold and silver pieces until private Banks started issuing paper currency, essentially IOUs, to 'represent' the real moneys, during the 1700s. The Federal Reserve itself only started issuing paper 'greenbacks' in 1861, essentially IOUs again, to enable the Government to finance the Civil War. They are now giving us digital IOUs, as money is not based on gold reserves anymore, so they can more easily use our actual money.

I assure you no 'slagging' was intended on my part; String Junky, on the other hand, can be awfully mean 😄.. Just intended as a gentle reminder of something I'm sure you already know 🙂.

Is this a random observation, or are you referring to some specific events ? Could you maybe elaborate ?

I read last week where German politicians have have floated the notion of Canada joining the EU. It makes little sense for countries in Europe to still send Billions to Russia for natural gas ( and to fund their war against Ukraine ) and then send money to Ukraine to fight off the Russians. With a pipeline to the East coast, Canada could possibly supply most of Europe's NG needs, as well as an abundance of other natural resources, while dealing with relatively sane people ( at least until we get invaded by the US ).

A Billion dollars to join the Board of Peace ? When the Mob does it, they have the decency to call it 'protection' money.

Yeah, I know what you meant. It is the use of the instrument that matters, even a 'lowly' Casio.

Particle spin is intrinsic angular momentum, and any momentum is a 'source' of Gravity. Is that what you meant ?

Music is not an instrument; it is a composition. You would be surprised what a former member of Depeche Mode could do with a Casio in 1982, along with Alison Moyet. And then, 40 years later, with an orchestra ...

People have believed much more weird ideas. Even that an egotistical, but insecure moron can be voted in as President, and he will do what's best for the country.

Time always passes at the same rate in everyone's own frame. However the very fact that expansion increases the time base of an emitted light signal, resulting in red shift or a 'stretching' of the signal's wavelength,, means that relational time between distant events will vary. this effect is not due to gravitational effects because, as you say, global curvature is very nearly flat.

So, we're not going to see a Trump beachfront resort in Gaza ?

You realize that Galactic North and South are purely arbitrary distinctions ? It is an accepted convention that keeps everyone 'on the same page'.

I can see such a need for places where young people need to move away from their elderly parents ( as has been mentioned about China, and Southern Italy which I'm familiar with ) in order to find employment, but why are their families abandoning their parents, or sending them into retirement homes in places like the US and Canada ? Parents deserve better than an app; their families should care for them.

Ahh, fond memories of Hollerith pinch card machines from the mid 70s. And a stack of cards about 6 inches high, each with one line of FORTRAN IV code, solving for the energy levels of the Helium atom by numerical methods. Tedious and frustrating, but I would gladly do it every day again, if I could be 21 years old again.

My app would activate every few days. I'm at work 12 hours, for 7 days every 2 weeks. Actually closer to 13 hours counting travel time. Most phones have 'health' apps or motion detectors; if no activity is detected by the phone in 12 hours, it should message your contact. Then again, I've been known to sleep for 12 hours as well ...

I thought Klein-Gordon was virtually un-useable, and the Dirac Equation replaced it. See here ... ( since you like his videos also )

From a QM standpoint, if all motion ceases at absolute zero, quantum particles have no momentum and their position is fixed. An impossibility according to Heisenberg. In Quantum Field Theory, this video does a better job than I could of both, QFT fundamentals, and minimum allowed energy. ( and I like his videos )

Energy prices have shot up ( along with everything else ) under the leadership of His Orangeness. In Western New York, and others on the Eastern seaboard, they buy Ontario's and Qjuebec's surplus Hydro at tariff-free, reduced prices. Swansont, and many others in Western New York, didn't vote for Trump, and I feel for them otherwise, I'd say "cut off their Hydro'. ( we can't do that anymore with our oil, because now he's stolen Venezuela's reserves )

And how does entropy reset itself when everything collapses to the same temperature ? Will it need to have a quantum fluctuation restart the cycle ?

😃 😄 🤣

Saaw CotG in the late 70s ( my teens ). Sure, I was a skeptic also. But it did make me think. Thank you E VD.