Back to the energy conservation discussion from a few posts back. It would be extraordinary to discover that we have some major flaws in classical or quantum physics. Both branches of physcis work extraordinarily well - after all, I'm typing on a computer whose operation at the semiconductor level rests firmly on quantum mechanics and our understanding of the atom and Einstein's SR and GR do the same for us at the cosmological level. Interestingly, Lee Smolin makes the point that the Standard Model (our most accurate scientific theory according to him) lies completely within the 'Newtonian paradigm', save for one small almost irrelevant area. In other words, many of the tools used in the standard model to describe energy, force etc existed after Newton, and definitely after Maxwell. We couldn't of course apply them at that point because we had not formulated a model of the atom, which came much later. So, rather than believe there is a fundamental flaw in any of these key fields, maybe the issue is we have to reinterpret them a little differently before we make another breakthrough.
It is correct that the energy density is likely to be changing. The discussion (as instigated by Ethan in my link) is complicated by the fact that dark energy is the major component of the energy density of the Universe.
As we know, it is dark energy that is thought to be responsible for the Universe expanding at an accelerating rate.
Being a cosmological constant (a constant of space), dark energy has a constant energy density irrespective of how much the Universe expands.
Dark energy's contribution to the total energy density increases as the Universe expands because there is more and more space. On the other hand, the total amount of matter and radiation is fixed. As the expansion progresses they become more and more dilute. In this scenario, the total energy density can change over time.
[As an aside, note that the three different geometries that we discussed earlier refer to a matter-only Universe. Including dark energy can cause dramatically different outcomes for the age, geometry, and eventual fate of the Universe, as compared to cases without dark energy.]
Lagrangian Mechanics says objects move between two points in such a way as to minimise the action.
Action is an abstract quantity that describes the trajectory, or path, of an object in time and space.
It shows how the balance of kinetic versus potential energy of an object changes with trajectory.
In the above diagram, the black line indicates the path that involves the least action, representing as it does the best balance of kinetic versus potential energy.
Action is of importance in quantum theory because the only movements of subatomic particles that are possible are actions which are whole-number multiples of Planck's constant.
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Agreed. But in the large scale perhaps and after all, we cant be sure that they tick because although we can observe long distances and time, there are questions around the references (standard candle etc). Also, not to forget, these still not totally reliable observations is only of the observable universe - cosmos is probably larger... So not a "flaw" necessarily when put in context but perhaps, ultimately we probably will end up in an other understanding.
Why would we be so fortunate that we lived in during the time when we finally figured out cosmos inner workings... that would be an insane luck because there is probably yet another couple of 1000 years before we can start to be quite sure we get it... still, good work so far!
(pls excuse the party pooing....)
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Thanks, Mark! Action is indeed defined as an integral over time of the Lagrangian at each point of the trajectory.
I was trying to keep the pesky mathematics out of it so had confined that definition to a link!
Your "minimising the average squared-speed" comment is timely, as I read that action may be thought of as twice the average kinetic of the object multiplied by the time interval between its initial and final position.
I believe you are referring to the principle of least time, or Fermat's principle.
Out of all possible paths that it might take to get from one point to another, light takes the path which requires the shortest time.
Tonight's The Sky at Night TV programme took us to the deepest mine in Great Britain.
There, 1.1 kilometres below the surface, is located the Boulby Underground Laboratory.
https://www.icl-uk.uk/undergroundlab/
I was surprised to learn that the search for Dark Matter has been going on there for over 20 years.
There, 1.1 kilometres below the surface, is located the Boulby Underground Laboratory.
https://www.icl-uk.uk/undergroundlab/
I was surprised to learn that the search for Dark Matter has been going on there for over 20 years.
I based my comment on this Feynman lecture: https://www.feynmanlectures.caltech.edu/I_26.html
In it, Feynman asks how light decides which is the shortest time, or the longest one, and chooses that path.
"What does light do, how does it find out? Does it smell the nearby paths, and check them against each other? The answer is, yes, it does, in a way."
Thought provoking!
In it, Feynman asks how light decides which is the shortest time, or the longest one, and chooses that path.
"What does light do, how does it find out? Does it smell the nearby paths, and check them against each other? The answer is, yes, it does, in a way."
Thought provoking!
I guess this must be linked to some underlying principle that nature will never use more energy than necessary to get the job done, or to move something from point A to point B. If things did not obey this principle then (I'm thinking aloud here) energy would have to be 'organized' against the countervailing drift of entropy. In the end, entropy always wins, but why would something inanimate expend energy to alter the easiest path forward? Answer: it would not.
In the lecture to which I linked, Feynman was referring to geometrical optics as represented by reflection and refraction.
Snell's law of refraction was discovered in 1621, but was no more than an equation (sin i = n sin r, where n is a constant).
In 1650, Fermat found a way of thinking that put flesh on the strange law and made it clear and obvious - his principle of least time.
For example, the light ray above does not go through the glass block in a straight line as indicated by the dotted line, but instead it decreases the time spent in the block by changing direction as it enters.
Snell's law of refraction was discovered in 1621, but was no more than an equation (sin i = n sin r, where n is a constant).
In 1650, Fermat found a way of thinking that put flesh on the strange law and made it clear and obvious - his principle of least time.
For example, the light ray above does not go through the glass block in a straight line as indicated by the dotted line, but instead it decreases the time spent in the block by changing direction as it enters.
A more general mathematical interpretation of general relativity has been suggested here.
https://pure.tue.nl/ws/portalfiles/portal/326418342/20240524_Heefer_hf.pdf
https://pure.tue.nl/ws/portalfiles/portal/326418342/20240524_Heefer_hf.pdf
I've been following (among other things in the thread) your search for the nova. I've read a few news articles, but this one really explained things for me regarding how they operate and lifetimes and such. I'm pleasantly surprised to see a mainstream article that gives as much description as this one does. Over the decades I've read mention of novae when reading about astronomy, but never remembered anything other than they're temporarily bright stars. Supernovae, on the other hand, are the ones that not only shine brighter, but produce the heavier elements, making them "more important" in the universe.
The link text on cnn.com says "Explosive star event will create ‘once-in-a-lifetime’ sight any day now. Here’s how to see it"
https://www.cnn.com/2024/06/11/science/nova-new-star-nasa-scn/index.html
I have rarely wasted my precious time (which is far more productively spent T CrB Nova hunting, weather permitting...) so much as in reading this incomprehensible nonsense about Finsler Geometry, which seems vaguely related to Hilbert's notoriously vague 23rd Problem on calculus of variations.
I do however find that geometer Paul Finsler was (like myself) a gifted amateur astronomer, and had several comets named after his binocular discovery. Perhaps a Nova might be named after me soon? I can then have it mentioned on my tombstone.
https://mathshistory.st-andrews.ac.uk/Biographies/Finsler/
An interestingly extreme elliptical orbit:
Another of his claims to fame is the Hadwiger-Finsler Inequality relating the sides of a triangle to its area, which may or may not be a world-shattering discovery:
It sounds like an episode of The Big Bang Theory, frankly, And aside of that, surely Hero of Alexandria who also invented the steam engine had the last word on this:
I certainly enjoyed that one as a young geometer since it related to non-Pythagorean triangles, as I am sure you did too.
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I read that violation of Lorentz invariance may happen from the perspective of quantum gravity.
(Lorentz invariance states that the laws of physics remain the same for all observers moving with respect to each other within an inertial frame.)
Apparently, Finsler geometry provides a good tool to study Lorentz violation.
Down that path of study lies such wonders as 'doubly special relativity' and 'very special relativity'.
https://en.wikipedia.org/wiki/Doubl...,a minimum length scale (the Planck length).
https://en.wikipedia.org/wiki/Very_special_relativity
What fun!
I understand that Finsler Geometry is a generalization of Riemannian Geometry, sacrificing something "Quadratic", whatever that means.
I suppose this is like Riemannian Geometry generalising good ol' Euclidian Geometry, sacrificing angles of a triangle adding up to Pi Radians or 180 degrees. But beyond my pay grade too!
It crossed my mind to try this for an equilateral triangle, side length 1, and it was the equality solution, as I half-expected. Area being 1/4 Root (3).
Apparently Comet Finsler is now down near the Southern Cross, and about 100 AU away.
https://www.spacereference.org/comet/c-1924-r1-finsler
Here's something I have never seen before, and probably you haven't either:
Those are the Trojan asteroids (L4 and L5) near Jupiter, predicted by M.Lagrange, no less. WOW!
https://www.spacereference.org/category/jupiter-trojans
If you click on the top right symbol on the image, you get the full calculation. Here supposedly 200 years from now:
Something has gone HORRIBLY, HORRIBLY wrong there, IMO. It shouldn't evolve like that. The computer cannot solve the n-body problem!
I suppose this is like Riemannian Geometry generalising good ol' Euclidian Geometry, sacrificing angles of a triangle adding up to Pi Radians or 180 degrees. But beyond my pay grade too!
It crossed my mind to try this for an equilateral triangle, side length 1, and it was the equality solution, as I half-expected. Area being 1/4 Root (3).
Apparently Comet Finsler is now down near the Southern Cross, and about 100 AU away.
https://www.spacereference.org/comet/c-1924-r1-finsler
Here's something I have never seen before, and probably you haven't either:
Those are the Trojan asteroids (L4 and L5) near Jupiter, predicted by M.Lagrange, no less. WOW!
https://www.spacereference.org/category/jupiter-trojans
If you click on the top right symbol on the image, you get the full calculation. Here supposedly 200 years from now:
Something has gone HORRIBLY, HORRIBLY wrong there, IMO. It shouldn't evolve like that. The computer cannot solve the n-body problem!