MJ kihara

What am trying to say is that at any point in a manifold it can be just be a point of another manifold...it depends on how they are connected and to the extend it's possible....as you have said 'In both cases they look locally flat. But globally these situations differ.' locally the point becomes almost indistinguishable from a point on the flat manifold and a point on circular manifold. Back to my point of discussion...we are deriving Riemann tensor from parallel transporting a vector in this case a parallelogram loop....it's a loop or in the shape of parallelogram loop coz every other place in the loop is a point arranged, joined,fused( I don't know more better term to use) to form the parallelogram, therefore, the basis vectors has to close for lie bracket to be zero and since to move from one bases vector of parallelogram to there should be no gap for flow curves to close...that point of closure where [X,Y] =0 is what am interested in and what am saying is that, that point can be a point in another manifold... alternatively when we subtract the covariant derivates along the basis vector we remain with [X,Y] lie bracket...this lie bracket can be a point in another manifold. Am trying to introduce another local manifold...where principle of uncertainty exist....into local patches that are globally related as you have stated,where Riemann tensor matters. Minkowski is tangent to this 'another local manifold '.... To extract information from this ' another local manifold' Minkowski becomes a limiting factor. True. Am also trying carefully not to enter into the rabbit hole of mathematics where I keeping digging and can't get out.

It's seems we are getting each other wrong...to be clear make a very(infinitely )thin wire into a circle,then another one flat(straight) which extend, let's say infinitely...put the straight(flat) wire on top of the circle then put an ant on it then restrict it to move on forward or backwards...how will it detect the circle below? It's moving on the straight wire.

All the same they simplify the math and leads to the correct answer. Derivative of Christoffel symbols in one direction reduces to zero...when then done in another extra direction it doesn't vanish.....this appears to indicate that when the degree of freedom increases it become more sensitive in extracting information from the manifold, I'm basing this argument on proving second Bianchi identity at a point. By introducing some constraints, and cycling the lower indices adding up the available degrees of freedom Riemann tensor in a given direction reduces to zero as indicated in the second Bianchi identity. What is the importance of lie bracket in all of this,why bother adding it up in Riemann tensor formula?

Am not disputing this fact..using Riemann normal coordinate at point p the metric tensor becomes the identity matrix and Christoffel symbols equals zero...that is the point am referring to... isn't used to prove second Bianchi identity? Am not disputing the established math am just trying to use it to achieve what am thinking. If you evaluate a straight line on top of a circle ...you won't detect the curvature of the circle below the line using Riemann tensor,as you have said,it reduces to Mankowski metric unless you are aware there is a circle...e.g you put an ant and restrict it to move forward either way on the line it won't realize there is a circle below it,the circle is the example of a hidden curvature..where the circle meets the line becomes just a point in the line..hope am not wrong with that reasoning.

Yes,a manifold perceived to be flat. In my assumptions am using a parallelogram loop. The lie bracket am talking about,simply defined as per Wikipedia... the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns to any two vector fields X and Y on a smooth manifold M third vector field denoted [X,Y]. The gap between the two basis vector of parallelogram to make a complete loop for a smooth movement of parallel transported vector on a flat manifold it's taken to be zero as per Riemann tensor. What I mean is the smooth movement without detecting a gap [X,Y]=0...the 'smoothness' is the one am referring to as uniformity of information,that is,information of X axis side of parallelogram to Y axis side of the parallelogram...assuming the vector being parallel transported is aware, it could not know it has been transported from one basis vector(sides of the parallelogram) to another, the information its gaining is uniform. If it's a point in the manifold Christoffel symbols reduces to zero and the metric tensor becomes the identity matrix if am not wrong.hope the idea am trying to present make sense.....this point is part of the curvature of the manifold....to visualize...if the Riemann tensor is able to detect just the point of this manifold without gaining further information i.e it's just sliding on the tangent plane of the manifold....to it,the manifold will just be flat while there is the 'hidden curvature' that is the rest of the manifold. This point that am explaining about,is the one am talking about being a point between the the basis vector of the parallelogram when it closes..if I were to use a drawing to illustrate this I would draw a parallelogram and at the further end where basis vector close,at that point,draw a circle below it.

Am suggesting a situation where when parallel transporting a vector in this case on a parallelogram loop lie bracket becomes zero meaning the manifold is perfectly flat...yet it's due to uniformity of information between the end points of the two vectors of the parallelogram because the rate at which we are obtaining this information is limited to the speed of light...am trying to look for a better way to explain this..

When the measurements are at a point in the manifold the Christoffel symbols reduces to zero...i think it depends to the extend and the knowledge we have or we can gain concerning the manifold and it's curvature,for instance, the observable universe is said to be 5%...also to prove if the vector coincide ,it's practically performed,this measurements is limited to the speed of light.its highly possible for the vectors to coincide while there is hidden curvature... information is critical and how we are obtaining it. Riemann geometry I understand it preceded GR, however,lie bracket is an important part of it and when the loop is completely closed it's taken to be zero,since am looking at away of unifying gravity and quantum mechanics/QFT and given GR is the best framework of how gravity behave I think this is the appropriate point to join the two theories. I think the laws of physics should be universal and if you are parallel transporting a vector,,even if it's in a thought experiment or done mathematically on a sheet of paper we need to consider those laws...like the constancy of speed of light if not so appropriate reasons should be given to explain otherwise. Am trying to unify GR with quantum effects ..for instance am stating that divergences at zero point in GR that appear to be non renormalizable are just normal issues of quantum fluctuations that are as a result of rich world inside a graviton....if we equate all this to information, graviton represent the threshold at which this information(hidden curvature) inside the graviton leaks to the outside curvature that becomes detectable by GR i.e a point in the manifold. Being infinite dimensional, I think it will depend on the properties of the graviton topology that is also related to quantum fluctuations.

All these is compensated by cosmological constant in Einstein field equations,esp if we consider ricci tensor to be zero...the cosmological constant and metric tensor will equate to Einstein constant * energy momentum tensor. My thinking is that vacuum energy density is related to/is the energy content of graviton,as a side note, "I intent to include all these thinking in the second edition of the book,"the reason for this, is that to prove second Bianchi identity one way is to solve the equation at a point where Christoffel symbols reduces to zero and the metric tensor becomes the Kronecker delta,from my earlier arguments in the thread..where I stated in two dimensions, limit of a curvature is a straight line and a point...having linked that point to a graviton...I can confidently state that gravity is renormalizable by introduction of a graviton,,,,,,and THAT the descripancy between observed cosmological constant and theoretical prediction of zero point energy is due to the rich world inside the graviton that makes divergence to be persistent at zero point.

Am thinking that the loop on which the vector is being parallel transported Is infinitesimal and given the speed of light in a vacuum is huge...the result will appear as the vector has been moved instantly I.e 0.0000000 lots of zero that makes lie bracket taken as zero to give the required answer..what I mean is that the parallel transported vector will end up conciding with the initial vector,therefore,detecting no curvature and since Riemann tensor is the bedrock of GR that effect is carried all the way to Einstein field equations. Implications of these is that GR is unable to detect zero point energy and quantum fluctuations on the background.

It has been a while,some clarification from what I asked about the interior of a manifold, Einstein tensor is obtained from ricci tensor which is a contraction of Riemannian tensor....while finding the Riemannian tensor the vector is parallel transported across a parallelogram loop which is assumed to be perfectly closed meaning it's lie bracket is zero ....is this not against special theory of relativity? I.e you can't parallel transports a vector instantly...the lie bracket I think should be approximately zero but not exactly zero ..therefore, Riemann tensor used in GR will leave some information not captured.

How? 'Faith' am not getting it..hhh..whose faith? Does it keep changing?

It depends with how you comprehend the information....to what extent do you think you know about yourself?.....you can read a 'whole book' only to make a wrong summary about it. Back to yourself;

Reinventing the wheel...it might be the same wheel but a more efficient one... probably not... Maybe a different perspective...to be superficial.....self.... Cambridge dictionary -the set of someone's characteristics, such as personality and ability, that are not physical and make that person different from other people. It seems to me(self) to be someone's summary... 'written' by yourself or by others...we get summaries after processing a lot of/ or less of information. This summary(self) I regard it as the projection of one's information by yourself or by others basing on the information they have about you.

This sounds interesting 🤔.

'...seperate from the rest of the universe.'..???? What do you mean by that? It seems you are suggesting the universe is just a mental projection of some kind.

They don’t interact I expected the same answer, therefore,the source of confusion..how do you know about something else without interacting with it? That’s not really how wave functions work Really? How does it work in a nutshell?

We are not getting each other correct...I meant the link between consciousness and many worlds interpretation....as in Oxford dictionary; link..a relationship between two things or situations, especially where one affects the other. No, each world has one universal wave function. If multiple worlds had the same wave function they would be in identical states. That’s contrary to the notion of the many-worlds premise. We are talking about the same thing only that it has been paraphrased differently.

Since you have read the book....I earlier asked you to offer the link,many worlds interpretation and consciousness...your response

I haven't read the book... synopsis from Wikipedia..In this book, Carroll examines the reasons why people misunderstand quantum mechanics and advocates a version of the many-worlds interpretation, while objecting to the views often grouped together as the Copenhagen interpretation. ....many world interpretation supports a universal wave function..if am not wrong, that means any other wave function Is independent;that means no collapse to the other...if so, there are many worlds for each wave function....to me these increases confusion...how do this 'many worlds' interact? Me am of the opinion that there might be one universal wave function that collapsed to many wave functions(not many worlds)...these many wave functions interact through wave function collapse...after all,so far it's generally accepted that it was a single Big bang. Therefore I regard this.. as an over statement .

How is it related to consciousness, what's the link?

Have you ever considered quantum fields to be an emergent phenomenon?..... The aurora of consciousness...mmmm...

Humans,trees and 'stone'...not all is equal...+1. We like( us humans)comparing everything with ourselves..so egocentric.

Oxford dictionary.....The state of being aware of and responsive to one's surroundings.... Get the key words.."aware"...and..."responsive"... We can have as many as possible definitions but I think all of the will narrow down those two terms. I think one part is already solved...."responsive=interaction=wave function collapse"....the other part "aware" is somehow debatable.

It's high time we develop a fundamental theory that considers consciousness as a cornerstone...otherwise machines will out smart us as we are busy defending our status quo...it would be a pity for humans to wait for AI to develop TOE...with time the machine will see the common sense we refuse to recognize..anyway what a wonderful conversation from Deepseek... personally I have so far refrained from chatting with AI bots as I think they are at the stage of harvesting whatever information they can get from humans and developing human mental maps... afterwards I don't what they will become!

A solution is a solution because it can be arrived at from different perspective/methodologies....it's becoming clearer to me we might already had a solution,from an explanatory point of view...this is after trying to compare my approach, others approach;of course author's approach(OP) and explanations from Various sources such as this.. https://www.google.com/url?sa=t&source=web&rct=j&opi=89978449&url=https://www.youtube.com/watch%3Fv%3Dbl_wGRfbc3w&ved=2ahUKEwjCgqy36qSKAxXFfqQEHaMtO70QwqsBegQIIBAF&usg=AOvVaw135bqe_R6OrGVjR7q_uUE5