B Who Won the 2020 Nobel Prize in Physics and What Did They Discover?

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Didn't see a thread on this.

https://www.nobelprize.org/prizes/physics/2020/summary/

The Nobel Prize in Physics 2020 was divided, one half awarded to Roger Penrose "for the discovery that black hole formation is a robust prediction of the general theory of relativity", the other half jointly to Reinhard Genzel and Andrea Ghez "for the discovery of a supermassive compact object at the centre of our galaxy."
 
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Physics news on Phys.org
I find it a great choice. Penrose was overdue being awarded. I had no clue, who might get the Physics Nobel this year. I didn't expect another cosmology/astronomy prize again this year :-).
 
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vanhees71 said:
I find it a great choice. Penrose was overdue being awarded. I had no clue, who might get the Physics Nobel this year. I didn't expect another cosmology/astronomy prize again this year :-).
It is also interesting. I, personally, wouldn't have expected for them to give an award for such a theoretical work (I mean the Penrose half), one might even say mathematical work.

Speaking of which, what exactly is Penrose's result that the award is based on? Or is it for his whole contribution to the subject?
 
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Phys. Rev. Lett. has made the following papers free to read after the Nobel Committee cited them:

https://doi.org/10.1103/PhysRevLett.10.66
https://doi.org/10.1103/PhysRevLett.14.57
https://doi.org/10.1103/RevModPhys.82.3121

The first two papers are by Prof. Penrose, @martinbn.

Another set of four papers by Reinhard Genzel and Andrea Ghez that have been made free to read:

https://doi.org/10.1103/PhysRevLett.116.031101
https://doi.org/10.1103/PhysRevLett.118.211101
https://doi.org/10.1103/PhysRevLett.122.101102
https://doi.org/10.1103/PhysRevLett.124.081101
 
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You've got to love her enthusiasm! :smile:

https://www.kron4.com/news/national/american-scientist-wins-nobel-prize-for-black-hole-research/

“Today we accept these objects are critical to the building blocks of the universe,” Ghez told an audience at the Royal Swedish Academy of Sciences by phone shortly after the announcement.

Ghez is the fourth woman to be awarded the Nobel Prize for physics, after Marie Curie in 1903, Maria Goeppert-Mayer in 1963, and Donna Strickland in 2018.

“I hope I can inspire other young women into the field. It’s a field that has so many pleasures. And if you’re passionate about the science, there’s so much that can be done,” Ghez said.
 
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Finally! Penrose won a Nobel Prize! Ugh, it would have sucked to see him pass away without winning one.
 
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From https://www.nobelprize.org/prizes/physics/2020/advanced-information/
Schmidt’s discovery prompted Wheeler to reconsider the physics of gravitational collapse and he discussed this with Penrose, who began to think about the problem in late 1964.
...
However, it was far from clear that this could happen in the real world and whether the assumption of spherical symmetry was a prerequisite for gravitational collapse.
...
Penrose set out to analyse the situation without the assumption of spherical symmetry, assuming only that the collapsing matter had a positive energy density. To do this, he had to invent new mathematical methods and make use of topology. The key concept that Penrose introduced was that of a trapped surface.
...
After realizing the power of the idea of trapped surfaces, Penrose proceeded to prove that once a trapped surface had formed, it is impossible, within the theory of general relativity and with a positive energy density, to prevent the collapse towards a singularity (Penrose 1965).
...
...
Penrose’s result is heralded as the first post-Einsteinian result in general relativity. It proves that gravitational collapse cannot be stopped after the trapped surface is formed. One should note that its formation happens at a stage in the collapse when density of matter is not very high. (The supermassive black holes of Michell and Laplace have average densities no higher than those of the Sun or Earth.) A few years later Penrose, together with Stephen Hawking, went further to show that similar results also applied to cosmological singularities (Hawking 1965, Hawking & Penrose 1970). Under reasonable assumptions, a past singularity is inevitable in the Big Bang model. Penrose (1969) wrote a beautiful summary of many of these results.

Penrose, R., 1963, “Asymptotic properties of fields and space-times”,
Physical Review Letters, vol. 10, no. 2. pp. 66–68.
https://doi.org/10.1103/PhysRevLett.10.66

Penrose, R., 1965, “Gravitational collapse and space-time singularities”,
Physical Review Letters, vol. 14, no. 3. pp. 57–59, 1965.
https://doi.org/10.1103/PhysRevLett.14.57.

Penrose, R., 1969, “Gravitational collapse: the role of general relativity”,
Nuovo Cimento Rivista Serie, vol. 1. p. 252.
( reprinted as
Penrose, R. “Golden Oldie”: Gravitational Collapse: The Role of General Relativity.
General Relativity and Gravitation 34, 1141–1165 (2002).
https://doi.org/10.1023/A:1016578408204
)(Other references in the quoted document of possible interest:

Hawking, S. W., 1965, “Occurrence of singularities in open universes”,
Physical Review Letters, vol. 15, no. 17. pp. 689–690.
https://doi.org/10.1103/PhysRevLett.15.689.

Hawking, S. W., and Penrose, R., 1970, “The singularities of gravitational collapse and cosmology”,
Proceedings of the Royal Society of London Series A, vol. 314, no. 1519. pp. 529– 548.
https://doi.org/10.1098/rspa.1970.0021

Hawking, S. W., 1972, “Black holes in General Relativity”,
Communications in Mathematical Physics, vol. 25. pp. 152.
( https://projecteuclid.org/euclid.cmp/1103857884 )

Hawking, S. W., 1975, “Particle creation by black holes”,
Communications in Mathematical Physics, vol. 43, no. 3. pp. 199–220.
https://doi.org/10.1007/BF02345020. ( https://projecteuclid.org/euclid.cmp/1103899181 )

)

Other references of possible interest:José M. M. Senovilla, David Garfinkle
"The 1965 Penrose singularity theorem"
https://arxiv.org/abs/1410.5226
https://iopscience.iop.org/issue/0264-9381/32/12
Just posted...
 
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So, more accurately it is for the singularity theorem, not that "..black hole formation is a robust predictions of the theory of general relativity..".
 
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  • #10
martinbn said:
It is also interesting. I, personally, wouldn't have expected for them to give an award for such a theoretical work (I mean the Penrose half), one might even say mathematical work.

Speaking of which, what exactly is Penrose's result that the award is based on? Or is it for his whole contribution to the subject?
I watched the announcement on youtube. The 1/2 prize for Penrose was given for the fact that he established mathematically the inevitability of the existence of black holes from the General Theory of Relativity. The point was that it's not only for the special case of spherical symmetry (Schwarzschild black hole) but also for less symmetrical. Penrose then checked it for the (then brand new) Kerr solution of a rotating black hole and also gave a general proof for the inevitability of singularities in GR. I think it's a well deserved prize.

Of course, that's a theoretical argument only, and the other 1/2 prize thus was given to Genzel and Ghez, who established the existence of a supermassive object in the center of our galaxy. To the best of our knowledge that can only be a black hole, and thus the evidence is pretty overwhelming. They observed the motion of stars close to it for decades. It's also a well-deserved Nobel for them.
 
  • #11
vanhees71 said:
he established mathematically the inevitability of the existence of black holes from the General Theory of Relativity

But, strictly speaking, that isn't what the singularity theorems establish.

The singularity theorems establish that, if certain conditions are met and a trapped surface is present, then a singularity must also be present.

However, a black hole is not defined by whether a trapped surface is present, but by whether an event horizon is present, i.e., whether there is a region of spacetime that is not in the causal past of future null infinity. The singularity theorems say nothing about that.

It is true that, if the energy conditions are met and a trapped surface is present, there must be an event horizon at or outside the trapped surface; but the theorem that establishes that is not one of the singularity theorems.

To be clear, none of this means that Penrose didn't deserve the prize; he did.
 
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  • #12
I am a bit surprised that the observational scientists didn't get more of the prize. For instance Penzias and Wilson did not share the prize with Dicke , Peebles, or Wilkins, or any other theoretician related to the CMB discovery, though they did share it with Pyotr Leonidovich Kapitsa for his unrelated work in low temperature physics, according to Wiki.
 
  • #13
The prize is not awarded by figuring out what, then who. It is closer to who, then what.
 
  • #14
pervect said:
I am a bit surprised that the observational scientists didn't get more of the prize. For instance Penzias and Wilson did not share the prize with Dicke , Peebles, or Wilkins, or any other theoretician
Historically I believe the maximum number of awardees is three. Five is more than three...so it may have been that simple.
 
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  • #15
PeterDonis said:
It is true that, if the energy conditions are met and a trapped surface is present, there must be an event horizon at or outside the trapped surface; but the theorem that establishes that is not one of the singularity theorems.

Who showed that?
 
  • #16
atyy said:
Who showed that?

I haven't been able to find a specific reference. I think the theorem is mentioned in Wald, but I don't have my copy handy to check.
 
  • #17
atyy said:
Who showed that?
I think that is not proven yet. Unless you take weak cosmic censorship for given.
 
  • #18
Just to make it clear I was just being pedantic, I do think that he deserves the award. In fact I think he deserves any of the existing maths/physics awards.
 
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  • #19
Yesterday I tried to read the "theory" part of scientific background of the Nobel Prize:
https://www.nobelprize.org/uploads/2020/10/advanced-physicsprize2020.pdf

Could someone with better knowledge/expertise on the topic explain to me the following:

As I understood from the article, Lifshitz and Khalatnikov proved that a singularity cannot occur in a realistic model, because there is no spherical symmetry for an arbitrary distribution of matter (the "because" part is what I inferred).
However, later Penrose later proved that it could occur (given positive energy density)? Aren't the two statements exclusive? Or under which conditions was Lifshitz and Khalatnikovs' conclusion correct?

I also struggled a little to understand the definition of trapped surfaces... "all light rays orthogonal to the surface converge when traced toward the future":
By orthogonal to the surface they meant the vectors pointing out from the surface? And how is the direction of the "future" defined? Is it referring to something like a "closed" light cone?
 
  • #20
ChrisVer said:
Yesterday I tried to read the "theory" part of scientific background of the Nobel Prize:
https://www.nobelprize.org/uploads/2020/10/advanced-physicsprize2020.pdf

Could someone with better knowledge/expertise on the topic explain to me the following:

As I understood from the article, Lifshitz and Khalatnikov proved that a singularity cannot occur in a realistic model, because there is no spherical symmetry for an arbitrary distribution of matter (the "because" part is what I inferred).
However, later Penrose later proved that it could occur (given positive energy density)? Aren't the two statements exclusive? Or under which conditions was Lifshitz and Khalatnikovs' conclusion correct?
They didn't prove it, they thought it was true, but they were mistaken.
 
  • #21
martinbn said:
They didn't prove it, they thought it was true, but they were mistaken.
It is that I can't have access here at the moment (apart from the abstract) which I think is the cited work by the prize editors:
https://inspirehep.net/literature/8888
 
  • #22
ChrisVer said:
I also struggled a little to understand the definition of trapped surfaces... "all light rays orthogonal to the surface converge when traced toward the future":
By orthogonal to the surface they meant the vectors pointing out from the surface? And how is the direction of the "future" defined? Is it referring to something like a "closed" light cone?
This is more dificult to imagin, you need the maths. Think of a sphere that flashes in an instant. Some of the light rays will go outward, some will go inward. The outgoing ones will diverge, the ingoing will coonverge. For a trapped surface you need both to converge.
 
  • #23
Here is a link to Hawking's chapter's of the pop science/ bit of math book "The Nature of Space and Time" that was coauthored with Penrose:

https://arxiv.org/abs/hep-th/9409195

This has the first chapter that introduces the singularity theorems, but it doesn't have the chapter written by Penrose on the "Structure of Spacetime Singularities" which introduces, amongst other things, the cosmic censorship hypothesis.
 
  • #24
I thought it was a maths result because the singularity theorems prove (that under physically reasonable conditions) there exist geodesics that come to an end - incomplete geodesics. It is usually assumed that the characteristic of a singularity is that the curvature divergences. However, this is not exactly what the singularity theorems imply.

I think there has been progress in the direction of addressing this issue. I don't know if this might be a reason why it has taken so long for Penrose to receive a noble prize. I don't know the status of this field.
 
  • #25
julian said:
I think there has been progress in the direction of addressing this issue. I don't know if this might be a reason why it has taken so long for Penrose to receive a noble prize. I don't know the status of this field.

The simple answer is that you can't really get the prize for purely theoretical work; the predictions of the theory must be well established and have been confirmed by experiments. This is one reason why the prize is shared with experimentalists.
I suspect recent observations of black holes by e.g. LIGO and VIRGO; and the fact that the observations seem consistent with theory is another reason for why the prize was awarded now.
 
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  • #26
I read the background paper by the Nobel committee and saw these beautyfull graphs of the star's position on an ellipse. What I was wondering: The black hole does not seem to be in one of the focal points, not even on the axis. On the other hand, I would have expected the rotation of the perihelion to be visible.

PS: I just found out that the rotation of the perihelion is only about 0.15 degrees per revolution. So still too small to be visible in the plot.

And here is a detailed answer why Sgr A appears not to be in a focal point of the elliptical orbit:
https://astronomy.stackexchange.com...tar-s2-to-sgr-a-not-appear-to-be-near-the-foc
 
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  • #27
Penrose deserves the prize IMO, even though his other theory about consciousness having something to do with quantum gravity is a bit crazy (unless it's meant to be a prank that shows how some people will believe almost anything when it comes from someone seen as an authority).

I remember some professor here in Finland saying in an interview that the only actual genius he's ever seen in person is R. Penrose.
 
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  • #28
BWV said:
Another tell that physics is mostly over?

Remarks along these lines belong in General Discussion, not here. Please keep this thread focused on the actual content of the work for which the prize was awarded, not on general opinions about the state of physics.
 
  • #29
Moderator's note: A number of off topic posts have been deleted.
 
  • #30
I think Hawking's and Ellis' "the large scale structure of spacetime" is a good place to look for more clarifications. I'm currently trying to make sense of it.
 
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  • #31
ShayanJ said:
I think Hawking's and Ellis' "the large scale structure of spacetime" is a good place to look for more clarifications. I'm currently trying to make sense of it.
If you get past the mathematics. It is the place to fill in some gaps of Wald, I believe.
 
  • #32
ShayanJ said:
I think Hawking's and Ellis' "the large scale structure of spacetime" is a good place to look for more clarifications.

Yes, Hawking & Ellis is the classic reference for global methods in GR and the singularity theorems.
 
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  • #33
I'm very happy for Sir Roger as he was always one of my favorites. Some may not be aware that Penrose, along with his father Lionel, invented some classic optical illusions such as the Penrose Stairs and the impossible triangle which were subsequently used by the Dutch graphic artist M.C. Escher in some of his most famous prints such as Ascending and Descending.

https://www.escherinhetpaleis.nl/escher-today/ascending-and-descending/?lang=en


Penrose, L. S. and Penrose, R., 1958. Impossible objects: A special type of illusion. British Journal of Psychology, 49, pp.31-33.

https://onlinelibrary.wiley.com/doi/abs/10.1111/j.2044-8295.1958.tb00634.x
 
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  • #35
bob012345 said:
I'm very happy for Sir Roger as he was always one of my favorites. Some may not be aware that Penrose, along with his father Lionel, invented some classic optical illusions such as the Penrose Stairs and the impossible triangle which were subsequently used by the Dutch graphic artist M.C. Escher in some of is most famous prints such as Ascending and Descending.

Penrose, L. S. and Penrose, R., 1958. Impossible objects: A special type of illusion. British Journal of Psychology, 49, pp.31-33.

https://onlinelibrary.wiley.com/doi/abs/10.1111/j.2044-8295.1958.tb00634.x
I remember watching his interview with Joe Rogan and being mind-blown when he used Escher's Circle Limits to elucidate hyperbolic geometry. Does anyone where I can find a more technical explanation of what he was talking about starting at 1:01:00?

 
  • #36
I like his (semi-) popular book, Road to Reality.
 
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  • #37
vanhees71 said:
I like his (semi-) popular book, Road to Reality.

I never got the chance to read his "Spinors and Twistors" classic. Does anybody know what twistors are and what they useful for?
 
  • #38
Amrator said:
I remember watching his interview with Joe Rogan and being mind-blown when he used Escher's Circle Limits to elucidate hyperbolic geometry. Does anyone where I can find a more technical explanation of what he was talking about starting at 1:01:00?
I think he might be talking about his ideas put forth in one of his latest books titled Cycles of Time where he also discusses a 'conformal' view of the universe as he does in this interview.

https://cds.cern.ch/record/1381231/files/9780099505945_TOC.pdf

https://www.penguinrandomhouse.com/books/129417/cycles-of-time-by-roger-penrose/
 
  • #40
robphy said:

To remove the guy on the left, paste this into the browsers developer console, lol.

JavaScript:
$(".s9e-miniplayer-inactive").each(function() {
    $(this).append( "<div style='position: absolute; top: 25%; left: 0px;height : 50%; width : 50%;background-color:rgb(37,37,37); z-index:10'></div>" );
});
 
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  • #41
Do you think Hawking would have been a co recipient with Penrose for his own work with black holes if he were still alive?
 
  • #42
Thecla said:
Do you think Hawking would have been a co recipient with Penrose for his own work with black holes if he were still alive?

This thought has crossed my mind.
I'm not sure how things would have played out.
Penrose (in the interview above) suggests that if Hawking radiation were detected, Hawking would have won one earlier. I haven't followed the history closely but I think Hawking applied Penrose's methods (used for black holes) to the whole universe (i.e. cosmology).

With a three-person limit, who among the three awardees would be excluded?
(There is also the possibility that only the astronomers get the award.)
 
  • #43
robphy said:
This thought has crossed my mind.
I'm not sure how things would have played out.
Penrose (in the interview above) suggests that if Hawking radiation were detected, Hawking would have won one earlier. I haven't followed the history closely but I think Hawking applied Penrose's methods (used for black holes) to the whole universe (i.e. cosmology).

With a three-person limit, who among the three awardees would be excluded?
(There is also the possibility that only the astronomers get the award.)
If you are willing to wait about 50 years you can see the records of nominations of the years Hawking would have likely been nominated if he had been. Now you can see all the nominees and nominators up to 1966 here;

https://www.nobelprize.org/nomination/archive/
 
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  • #44
bob012345 said:
If you are willing to wait about 50 years you can see the records of nominations of the years Hawking would have likely been nominated if he had been. Now you can see all the nominees and nominators up to 1966 here;

https://www.nobelprize.org/nomination/archive/
Interesting... although the following is off-topic from this year's prizes

In 1957, three of many nominees ( https://www.nobelprize.org/nomination/archive/list.php?prize=1&year=1957 ) were
Subrahmanyan Chandrasekhar ( 1910-1995 ), Tsung-Dao Lee ( 1926- ), Chen Ning Yang ( 1922- )... all first-time nominees.

There was a famous story of these three
http://www-news.uchicago.edu/releases/99/990715.chandra-facts.shtml
(bolding mine)
Chandra’s commitment to teaching was legendary. In the 1940s, he drove 200 miles round trip each week from Yerkes Observatory in Williams Bay, Wisc., to the University to teach a class on stellar atmospheres. One day he insisted on driving from Yerkes to teach the class despite a heavy snowstorm. Chandra ended up teaching a class of only two that day. The two students––Tsung Dao Lee and Chen Ning Yang––won the 1957 Nobel Prize in physics, obtaining the distinction even before their professor.
According to the database, Chandrasekhar was nominated in 1957 and 1962 [and maybe others not available yet] and finally won in 1983.

Interestingly, from a quick scan of https://en.wikipedia.org/wiki/List_of_Nobel_laureates_in_Physics
it appears that all of the physics laureates from the 1960 have passed away. Gell-Mann (1969 prize) passed away in 2019. However, Lee and Yang (who were in their 30s when they won the 1957 prize) are still living.

Lee and Yang won in 1957 for the theory of Parity Violation [published in 1956], which was shown experimentally by
https://en.wikipedia.org/wiki/Chien-Shiung_Wu in 1956.
Wu (1912-1997) was nominated in 1958, 1959, 1960, 1964, 1965 [and maybe more]
https://www.nobelprize.org/nomination/archive/show_people.php?id=10859 but she never won.
An interesting article on Wu at https://physicsworld.com/a/credit-where-credits-due/
 
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  • #45
robphy said:
Interesting... although the following is off-topic from this year's prizes

In 1957, three of many nominees ( https://www.nobelprize.org/nomination/archive/list.php?prize=1&year=1957 ) were
Subrahmanyan Chandrasekhar ( 1910-1995 ), Tsung-Dao Lee ( 1926- ), Chen Ning Yang ( 1922- )... all first-time nominees.

There was a famous story of these three
http://www-news.uchicago.edu/releases/99/990715.chandra-facts.shtml
(bolding mine)

According to the database, Chandrasekhar was nominated in 1957 and 1962 [and maybe others not available yet] and finally won in 1983.

Interestingly, from a quick scan of https://en.wikipedia.org/wiki/List_of_Nobel_laureates_in_Physics
it appears that all of the physics laureates from the 1960 have passed away. Gell-Mann (1969 prize) passed away in 2019. However, Lee and Yang (who were in their 30s when they won the 1957 prize) are still living.

Lee and Yang won in 1957 for the theory of Parity Violation [published in 1956], which was shown experimentally by
https://en.wikipedia.org/wiki/Chien-Shiung_Wu in 1956.
Wu (1912-1997) was nominated in 1958, 1959, 1960, 1964, 1965 [and maybe more]
https://www.nobelprize.org/nomination/archive/show_people.php?id=10859 but she never won.
An interesting article on Wu at https://physicsworld.com/a/credit-where-credits-due/
Interestingly (to me at least) is I found a little popular book by Yang called 'Elementary Particles' from 1962 and it has an Escher drawing on the cover used to illustrate relevant symmetries in physics. So, C.N. Yang ties to M.C. Escher which ties to Penrose which ties to the topic. Well, almost...

https://www.abebooks.com/first-edit...ort-History-Discoveries-Atomic/17986105339/bd
 
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  • #46
dextercioby said:
I never got the chance to read his "Spinors and Twistors" classic. Does anybody know what twistors are and what they useful for?
Twistors are twisted lines in a complex projective space which directly result from a unique and quite conventional geometric mapping from spacetime points of the worldlines of lightrays. In other words, doing physics using twistors in twistor space essentially is just a mathematical reformulation of standard physics, in the same vein of Hamiltonian mechanics just being a mathematical reformulation of Newtonian mechanics using different mathematical tools.

The main drive behind the idea however is that the twistor space point of view, if taken as more fundamental than the spacetime view, might automatically suggest a unique unification of GR with QT, but this has not yet been fully achieved yet; bluntly put, this has been achieved for linearized GR, but the full nonlinear Einstein theory has not yet been fully recovered in the twistor framework. This has been an unsolved problem for over 50 years now which has halted twistor theory to actually develop into a new physical theory; on the other hand, twistor theory as a mathematical theory/method is well established, especially in the theory of differential equations.

Quite recently however there has been significant progress in the development on the physics side of twistor theory. This happened following a suggestion by Atiyah to Penrose that a natural way to overcome the longstanding issue with recovering nonlinear Einstein theory is by shifting the focus within the theory away from complex projective twistor space onto the non-commutative twistor quantum algebra and then to utilize non-commutative geometry to recover a new space.

In this manner, one should be able to recover the fully nonlinear Einstein theory in a novel kind of twistor space directly from the twistor algebra which inherently has cohomological wave functions and so automatically solve the long-standing problem of unifying QM with GR in a mathematically unique way; suffice to say, actually carrying this out to full completion requires Fields Medal level skill.
 
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  • #47
bob012345 said:
So, C.N. Yang ties to M.C. Escher which ties to Penrose which ties to the topic. Well, almost...
At least not repeatedly...or would that be repeatably...
 
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  • #48
bob012345 said:
So, C.N. Yang ties to M.C. Escher which ties to Penrose which ties to the topic.

hutchphd said:
At least not repeatedly...or would that be repeatably...

Well, ...

Yang Escher Penrose.png
 
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  • #49
2020 Nobel Lectures in Physics
Streamed live on Dec 7, 2020

Tune into watch the 2020 Nobel Lectures in Physics:
Black Holes, Cosmology, and Space-Time Singularities
Roger Penrose, University of Oxford, UK

A Forty Year Journey
Reinhard Genzel, Max Planck Institute for Extraterrestrial Physics, Garching, Germany and University of California, Berkeley, USA

From the Possibility to the Certainty of a Supermassive Black Hole
Andrea Ghez, University of California, Los Angeles, USA

Roger Penrose
&t=5m09s introduction to Roger Penrose
&t=8m18s Roger Penrose


I had to include a
screenshot of the other speakers
because
the youtube link thumbnail
seems set.

Reinhard Genzel
&t=39m07s introduction to Reinhard Genzel
&t=40m11s Reinhard Genzel
1608144188469.png



Andrea Ghez
&t=68m52s introduction to Andrea Ghez
&t=70m12s Andrea Ghez
1608144277872.png

 
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