Non Euclidean Geometry and the Universe

With Critical density ( Omega ), matter distinctions become apparent, when looking at the computerized model of Andrey Kravtsov.



Georg Friedrich Bernhard Riemann 1826 – 1866

Riemannian Geometry, also known as elliptical geometry, is the geometry of the surface of a sphere. It replaces Euclid's Parallel Postulate with, "Through any point in the plane, there exists no line parallel to a given line." A line in this geometry is a great circle. The sum of the angles of a triangle in Riemannian Geometry is > 180°.

To me this is one of the greatest achievements of mathematical structures that one could encounter, It revolutionize many a view, that been held to classical discriptions of reality.

In the quiet achievement of Riemann’s tutorial teacher Gauss, recognized the great potential in his student. On the curvature parameters, we recognize in Gauss’s work, what would soon became apparent? That we were being lead into another world for consideration?



So here we are, that we might in our considerations go beyond the global perspectives, to another world that Einstein would so methodically reveal in the geometry and physics, that it would include the electromagnetic considerations of Maxwell into a cohesive whole and beyond.

The intuitive development that we are lead through geometrically asks us to consider again, how Riemann moved to a positive aspect of the universe?



The activity in string theory and quantum gravity is aimed at developing a quantum theory that incorporates the physics of gravity and is valid down to the smallest length scales, where conventional quantum field theory can no longer be applied. There has been rapid progress in this area in recent years, in part due to work of Princeton faculty and students, and it continues to be a fertile source of research problems.


Friedman Equation What is _pdensity.

What are the three models of geometry? k=-1, K=0, k+1

Negative curvature

Omega=the actual density to the critical density

If we triangulate Omega, the universe in which we are in, Omega_m(mass)+ Omega(a vacuum), what position geometrically, would our universe hold from the coordinates given?

If such a progression is understood in the evolution of the geometry raised in non euclidean perspectives, this has in my view raised the stakes on how we percieve the dynamical valuation of a world that we were lead into from GR?

Facing the frontier of cosmological proportions, we soon meet views as demonstrated in the Solvay meetings where the thought experiments plague the relation of Quantum Mechanics. Even though Einstein held his position about the beauty of GR( it's stand alone feature) in it's own right, did not mean that the efforts to quantization had not been considered by him?

Moving to the non-euclidean realm, set up my thinking in terms of gravitational considerations. Dali's example of the tesserack reveals a deeper understanding of this progression to an non-euclidean view that Dali heightened in this aspect of religiousness and God implication, by demonstrating the Crucifixation paintng that he did. Even Escher in his realization, understood that the royal road to geometry has some road(physics) to travel before it could meet his perspective eye.

Searching for Changes in the Fine-Structure Constant Using Atomic Clocks

Another laser beam is used to make the atoms fluoresce, and the amount of fluorescence is measured as a function of the microwave frequency to plot a "resonance curve". An ultra-precise measurement of time can be made by measuring the frequency of the peak in this resonance curve (see "Atomic clocks" by Pierre Lemonde in Physics World January 2001 pp39-44).



The reason for this post was triggered by what can be found at Lubos Motl's site One will have to watch the interaction and information that is presenting itself in the comments that come, and maybe we can build from this?

When I looked at Glast, it seemed a fine way in which to incorporate one more end of the "spectrum" to how we see the cosmo? That we had defined it over this range of possibilties? How could we move further from consideration then, and I fall short in how the probabilties of how we might percieve graviton exchange of information in the bulk could reveal more of that spectrum? A resonance curve?

If one alters their perspective in terms of resonantial features(using string theory concepts as a quantum mechanical discription of the spacetime), how would we measure these changes in planck epoch ( deeper then gamma ray detection)? Can this be done?

Variation of the Standard Two-Pin-hole "welcher-Weg" Optics Experiment

Results of different measurements of the Ft value in beta decay George R. Welch setting up an optics experiment with graduate student Sophia Ilina Dr. Scully and Dr. Townes, the inventor of the maser and laser (University of California, Berkeley), during the dedication of the Charles H. Townes Reading room Assistant research scientist Daya Rathnayaka with a UHV deposition chamber

Tomorrow at 4 PM, physicist Shahriar S. Afshar, a Visiting Scientist at Harvard University's Physics Department will give a talk entitled Violation of Bohr's principle of complementarity in an optical "which-way" experiment at Texas A&M University.

Afshar has done a variation of the standard two-pin-hole "welcher-Weg" optics experiment, in which he demonstrates that wave interference is present even when one is determining through which pinhole a photon passes. This result is in direct contradiction to Neils Bohr's Principle of Complementarity, which would require in the quantum world that when one is measuring particle properties [formerly read "measuring quantum properties" -KC], all wave interference phenomena must vanish. Afshar's trick is to find the location of the minimum points of wave interference, place one or more wires at these minimum points, and observe how much light is intercepted when one is determining the pinhole through which the photons passed.

It has been widely accepted that the rival interpretations of quantum mechanics, e.g., the Copenhagen Interpretation, the Many-Worlds Interpretation, and my father John Cramer's Transactional Interpretation, cannot be distinguished or falsified by experiment, because the experimental predictions come from the formalism that all such interpretations describe. However, the Afshar Experiment demonstrates in an interaction-free way that there is a loophole in this logic: if the interpretation is inconsistent with the formalism, then it can be falsified. In particular, the Afshar Experiment falsifies the Copenhagen Interpretation, which requires the absence of interference in a particle-type measurement. It also falsifies the Many-Worlds Interpretation which tells us to expect no interference between "worlds" that are physically distinguishable, e.g., that correspond to the photon's measured passage through one pinhole or the other [the word "measured" added 4/28. -KC].

The Transactional Interpretation, on the other hand, has no problem in explaining that Afshar results. "Offer waves" from the source pass through both pinholes and interfere, creating a condition in which no transactions to the wires can form. Therefore, no photons are intercepted by wires, as Afshar observes. The quantum formalism makes the same predictions.

On this basis, it appears that two of the major interpretations of quantum mechanics have been falsified and should be relegated to the waste basket of physics history. The Transactional Interpretation, which involves a forward/back in time handshake, is one of the few (perhaps the only) interpretation(s) left standing after the Afshar test.

Yay for the home team!

(See also the Power Point presentation for my dad's Hal Clement memorial lecture at Boskone; Google also has an html cache of the Powerpoint presentation.)

LET ME KNOW if anyone reading this attends Afshar's talk or attended a similar one he gave at Harvard recently.

TRACKBACKS: My MT trackbacks don't work. One of these days I'll figure out why. But this entry has received really a lot, so here is a link to Technorati's listing of links to this post. Boingboing also blogged it, but the post has already slid off their main page.

SEE UPDATE, 5/10/04.

No Royal Road to Geometry?

All those who have written histories bring to this point their account of the development of this science. Not long after these men came Euclid, who brought together the Elements, systematizing many of the theorems of Eudoxus, perfecting many of those of Theatetus, and putting in irrefutable demonstrable form propositions that had been rather loosely established by his predecessors. He lived in the time of Ptolemy the First, for Archimedes, who lived after the time of the first Ptolemy, mentions Euclid. It is also reported that Ptolemy once asked Euclid if there was not a shorter road to geometry that through the Elements, and Euclid replied that there was no royal road to geometry. He was therefore later than Plato's group but earlier than Eratosthenes and Archimedes, for these two men were contemporaries, as Eratosthenes somewhere says. Euclid belonged to the persuasion of Plato and was at home in this philosophy; and this is why he thought the goal of the Elements as a whole to be the construction of the so-called Platonic figures. (Proclus, ed. Friedlein, p. 68, tr. Morrow)

It was interesting to me that I find some thread that has survived through the many centuries , that moves through the hands of individuals, to bring us to a interesting abstract world that few would recognize.



While Euclid is not known to have made any original discoveries, and the Elements is based on the work of his predecessors, it is assumed that some of the proofs are his own and that he is responsible for the excellent arrangement. Over a thousand editions of the work have been published since the first printed version of 1482. Euclid's other works include Data, On Divisions of Figures, Phaenomena, Optics, Surface Loci, Porisms, Conics, Book of Fallacies, and Elements of Music. Only the first four of these survive.

Of interest, is that some line of departure from the classical defintions, would have followed some road of developement, that I needed to understand how this progression became apparent. For now such links helped to stabilize this process and the essence of the departure form this classical defintion needed a culmination reached in Einstein's General Relativity. But long before this road was capture in it's essence, the predecessors in this projective road, develope conceptual realizations and moved from some point. To me, this is the fifth postulate. But before I draw attention there I wanted to show the index of this same projective geometry.

A theorem which is valid for a geometry in this sequence is automatically valid for the ones that follow. The theorems of projective geometry are automatically valid theorems of Euclidean geometry. We say that topological geometry is more abstract than projective geometry which is turn is more abstract than Euclidean geometry.

The move from the fifth postulate had Girolamo Saccheri, S.J. (1667 - 1733) ask the question?

What if the sum of the angles of a triangle were not equal to 180 degrees (or p radians)?" Suppose the sum of these angles was greater than or less than p. What would happen to the geometry we have come to depend on for so many things? What would happen to our buildings? to our technology? to our countries' boundaries?

The progression through these geometries leads to global perspectives that are not limited to the thread that moves through these cultures and civilizations. The evolution dictates that having reached Einstein GR that we understand that the world we meet is a dynamical one and with Reason, we come t recognize the Self Evident Truths.

At this point, having moved through the geometrical phases and recognitions, the physics of understanding have intertwined mathematical realms associated with Strings and loop and other means, in which to interpret that dynamical world called the Planck Length(Quantum Gravity).

Reichenbach on Helmholtz

The Six Men and the Elephant

I'd like to embrace all those who are attempting to speak in regards to quantum gravity. This is a common poem, that reflects the efforts and differing perspectives at trying to descibe a world that is very much in the understanding below the planck epoch.

If we acknowledge the structural integrity that we assign strings and loops this then can embrace solidified positions, to the understanding that we may differ in those same positions. Logically the perspectives are even displayed very nicely in this poem? Wall, Spear, Snake, Tree, Fan, and Rope.

What taste comes to mind, and we have this same inabiltiy to express the intangible yet we have engaged mathematically something very real?

The Blind Men and the Elephant
John Godfrey Saxe (1816-1887)

It was six men of Indostan
To learning much inclined,
Who went to see the Elephant
(Though all of them were blind),
That each by observation
Might satisfy his mind.

The First approached the Elephant,
And happening to fall
Against his broad and sturdy side,
At once began to bawl:
"God bless me! but the Elephant
Is very like a WALL!"


The Second, feeling of the tusk,
Cried, "Ho, what have we here,
So very round and smooth and sharp?
To me 'tis mighty clear
This wonder of an Elephant
Is very like a SPEAR!"

The Third approached the animal,
And happening to take
The squirming trunk within his hands,
Thus boldly up and spake:
"I see," quoth he, "the Elephant
Is very like a SNAKE!"

The Fourth reached out an eager hand,
And felt about the knee
"What most this wondrous beast is like
Is mighty plain," quoth he:
"'Tis clear enough the Elephant
Is very like a TREE!"

The Fifth, who chanced to touch the ear,
Said: "E'en the blindest man
Can tell what this resembles most;
Deny the fact who can,
This marvel of an Elephant
Is very like a FAN!"

The Sixth no sooner had begun
About the beast to grope,
Than seizing on the swinging tail
That fell within his scope,
"I see," quoth he, "the Elephant
Is very like a ROPE!"

And so these men of Indostan
Disputed loud and long,
Each in his own opinion
Exceeding stiff and strong,
Though each was partly in the right,
And all were in the wrong!

Bubble Nucleation

Based on the no boundary proposal, I picture the origin of the universe, as like the formation of bubbles of steam in boiling water. Quantum fluctuations lead to the spontaneous creation of tiny universes, out of nothing. Most of the universes collapse to nothing, but a few that reach a critical size, will expand in an inflationary manner, and will form galaxies and stars, and maybe beings like us.

The images produce here of bubble formation are most pleasing to me, about what could have emerge from that early universe. If stringy components were evident and cosmic clumping rvealed as in previous post then how would such images lead to bubble nucleations as stringy cosmological patterns?

For such ideas to emerge in thinking there had to be a time when such conditions were conducive to bubble nucleation? Such energy considerations had to provide for these considerations to emerge so. How so?

First-order phase transitions (illustrated below) occur through the formation of bubbles of the new phase in the middle of the old phase; these bubbles then expand and collide until the old phase disappears completely and the phase transition is complete.



During a first-order phase transition, the matter fields get trapped in a `false vacuum' state from which they can only escape by nucleating bubbles of the new phase, that is, the `true vacuum' state.





G -> H -> ... -> SU(3) x SU(2) x U(1) -> SU(3) x U(1). Here, each arrow represents a symmetry breaking phase transition where matter changes form and the groups - G, H, SU(3), etc. - represent the different types of matter, specifically the symmetries that the matter exhibits and they are associated with the different fundamental forces of nature

In order for such thinking to produce the cosmos then we would have to understand its early conditions.

Physically, the effect can be interpreted as an object moving from the "false vacuum" (where = 0) to the more stable "true vacuum" (where = v). Gravitationally, it is similar to the more familiar case of moving from the hilltop to the valley. In the case of Higgs field, the transformation is accompanied with a "phase change", which endows mass to some of the particles.

The Cosmic String

It was important for me to recognize the microcosmic view, could have been united with cosmological proportions, and revealled structural integrity to the way in which the universe was formed. If such views were manifesting from planck Epoch realizations, then it really was kind of easy to see how such conceptualizations might have revealled themselves in stringy ways? I have deferred from brane considerations( this is most interestng to me of holographical proportions that such branes could intersect themselves to coordinated references?) at this point to explore this unusual unversal explanation.





This figure shows the SDSS spectrum of a quasar at a distance of 12 billion light years. The middle panel shows the complete spectrum. The upper panel is an expanded view of the region of the spectrum affected by the filaments of gas whose clumping is the focus of the present study. Each of the hundreds of dips in the spectrum corresponds to a different parcel of gas along the line of sight between the quasar and the Earth. This is schematically shown in the lower panel, which indicates a line of sight through a simulation 30 million light years across of the distribution of gas in the universe. The clumpiness of the gas is determined by, among other things, the constituents of the universe, including dark matter, dark energy, and massive neutrinos. Renyue Cen of Princeton University carried out the simulation.

A lot of times it seems the general concept is far fleeting from educated minds that in a laymen's way I wanted to put forward a generalized view from information gathered. I hope this summation is correct?












Scharf and Mukherjee's new research compared a catalog of 2,469 galaxy clusters with the Compton database. Using sophisticated statistical techniques, they showed that the sky surrounding the most massive clusters was systematically brighter in gamma rays than other regions.

"The more massive the cluster (and greater the gravitational potential), the brighter the gamma-ray halo," said Mukherjee. "The enhancement observed was very similar to that predicted by the Loeb-Waxman theory."

Without some dynamical realization of that early universe it wouldn't make much sense not to consider the deepening revelation of how we percieve that Window on the Universe? Kip Thorne's early introspective views have become interesting ways in which to interpret that same universe, with Ligo and other means. From Wheelers day, it was important to understand this developement, along with Bells Theorem.

How could we not have considered the temeprature values of that same early universe and not wondered about the nature of supergravity associated with these energy considerations? How else might we understand resonance curve?

The Man Who Knew Infinity:

A Life of the Genius Ramanujan
by Robert Kanigel



Srinivas Ramanujan (1887-1920)In the past few decades, we have witnessed how Ramanujan's contributions have made such a profound impact on various branches of mathematics. The book, "The man who knew infinity", by Robert Kanigel reached out to the general public the world over by describing the fascinating life story of Ramanujan. And now, in the form of a play, the public is made aware, once again, of this wonderful story. This is a very impressive play and I had the pleasure of seeing it with Prof. George Andrews, the world's greatest authority on Ramanujan's work and on partitions.


The more I read about Ramanujan I was attracted to the idea of him being able to predict outcomes devised in some general mathematical way that seemed easy to him, but in the case of the Taxi Cab and Hardy's question, what was the nature of the taxi cab's number?

When looking through this infomration I come across interesting perspectives about such a man whose culture was far away from the society of currents math and physics. Who developed logically, through his own study. This is a intersting case to me of what could be brought to a world steep in mathematical structures. CRanks or not, whohave found joy in peering into a world that few would cosider in their day to day lives.

No account of Ramanujan is complete without the Taxi Cab episode. There is a charming scene of Hardy meeting Ramanujan in a hospital, and when Hardy mentions that he arrived by the taxi numbered 1729, Ramanujan immediately points out that 1729=10^3+9^3=12^3+1^3, the smallest positive integer that can be expressed as a sum of two cubes in two different ways. Of course, the Ramanujan taxicab equation x^3+y^3=z^3+w^3 yields Fermat's equation for cubes by setting w=0, but it is to be noted that the taxicab equation has positive integer solutions, whereas Fermat's does not.

There is something deeper here that has caught my attention. The Harmonical nature permeates my thoughts, about the extra dimensions and how we could look at the bulk?

If such thinking went beyond the two points and our focus was drawn to the space in between, what would such a nature exemplified by such harmonical attributes? Would it not have made one wonder how the world would seem in ways that our current perceptions are not accustomed too? How would Ramanujan fit here, as a emergent property of strings?

When strings vibrate in space-time, they are described by a mathematical function called the Ramanujan modular function.26 This term appears in the equation:27

[1-(D - 2)/24]
where D is the dimensionality of the space in which the strings vibrate. In order to obey special relativity and manifest co-variance), this term must equal 0, which forces D to be 26. This is the origin of the 26 dimensions in the original string theory.

In the more general Ramanujan modular function, which is used in current superstring theories, the twenty-four is replaced by the number eight, making D equal to 10.28

In other words, the mathematics require space-time to have 10 dimensions in order for the string theory to be self-consistent, but physicists still don’t know why these particular numbers have been selected.

I was interested in how this man came to think and I place this here for consideration from another poster.

Dick Well, in Ramanujans case we have some clues.

He spent his teenage years studying and mastering one book, on analytical function theory and analytical number theory (which are joined at birth). The format of the book was step by step: it started with (x-y)(x+y)=x2 - y2. You proved that and then came a slightly harder theorem of the same kind, in which the proof uses the first theorem, and so on one result building on another up to the state of the art as it was when the book was published (1880s). The book was intended to prepare or "cram" Cambridge students for the math exam known as the Tripos.

Ramanujan became not just familiar with the math in this book, it became his environment.

A recent author has suggested that math ability derives from the brain abilities used in social understanding. Think of living in a tribe or small town where "everybody knows everybody". By growing up in such an environment you know not only everyone else's name, but their preferences and personal characteristics. You are freely able to think what so-and-so and such-and-such would talk about if they had a conversation. And it is proposed that mathematicians have this same ability, only with the abstract things they think about and discuss, rather than people.

And so Ramanujan's "town" was the complex number system and the various peculiar things that could happen there. This was the focus of his imagination throughout his growing up and it is scarcely surprising that he was able to see relationships that more lazily prepared mathematicians (including great ones like Hardy) could not.

You don't have to postulate extra dimension, the depth of the human capabilities is sufficient.

As you can see Dick rejected the idea that such information could have settle in any mind, from a fifth dimensional consideration, and the extra dimensions, as he has stated. It just made sense to me, that solid things, had other information that it concealed. The harmonical nature was not only limited to the numbers and oscillations?

The Planck Epoch to now, contained interesting information. Should we find some structure to contain it all, and yet, find that this structured world is not limited to such structures alone? It was just a pattern, and one of many?

Mathematicians and Physicists Do is Perhaps Like Sculpture

"Science is that human activity in which we aim to show towards nature that respect that in a democracy we endeavor to show towards each other."

Smolin then goes on to write something interesting here. Many would have interpreted the discusion between Peter Woit's Blog and Lubos Motl's Blog, that rebels exist, but something more define I'd say in the way Smolin describes it. Arun too helps with a interesting link, that makes one wonder about the nature of numbers

But again in response to above quote, Smolin continues here below.

I must confess that I have never been able to find its source, but I have been thinking a lot about this quote recently. I hope I may be excused for using it unattributed. The stance of respect seems to me the necessary companion to the stance of the rebel, for respect signifies that we live out our lives inside an intricately structured and enormously complicated world, containing among myriads of other living creatures, many individuals like ourselves. For us human beings, the world we find ourselves in is comprised of nature, imagination and society. Science, art and politics are the ancient crafts by which we seek to understand and define our situation in these worlds. The stance of the rebel comes from the discovery that there is much in these worlds which is unacceptable. The stance of respect arises from another discovery, that to change the world requires that we acknowledge that each of our lives is but a brief moment in the vastly complicated networks of relationships that comprise our shared worlds.

It was important to me to try and find the sources of inspiration in terms of the inconsistancies of quantum mechanics and general relativity.


The first "attempt to combine the quantum theory with the theory of gravitation," which demonstrated that "in order to avoid an inconsistency between quantum mechanics and general relativity, some new features must enter physics," was made by Bronstein in 1935. That the Planck mass may be regarded as a quantum-gravitational scale was pointed out explicitly by Klein and Wheeler twenty years later. At the same time, Landau also noted that the Planck energy (mass) corresponds to an equality of gravitational and electromagnetic interactions.
Theoretical physicists are now confident that the role of the Planck values in quantum gravity, cosmology, and elementary particle theory will emerge from a unified theory of all fundamental interactions and that the Planck scales characterize the region in which the intensities of all fundamental interactions become comparable. If these expectations come true, the present report might become useful as the historical introduction for the book that it is currently impossible to write, The Small-Scale Structure of Space-Time.

Maybe the new book should be written by Lubos?

Quantum Gravity

Here is one of two methods that help explain. The next post will follow tomorrow if I have time. The complexity of the pictures involved is linked down below in Fig 15-17. This will give some generalizations that I had been looking too, to comprehend the model of strings and its geometrical discriptions.

Continuity



Topology is the branch of mathematics concerned with the ramifications of continuity. Topologist emphasize the properties of shapes that remain unchanged no matter how much the shapes are bent twisted or otherwise manipulated.

Such transformations of ideally elastic objects are subject only to the condition that, for surfaces, nearby points remain close together in the transforming process. This condition effectively outlaws transformations that involve cutting and gluing. For instance a doughnut and a coffee cup are topologically equivalent. One can be transformed continuously into the other. The hole in the doughnut will be preserved as the hole in the handle of the coffee cup.










Topology becomes an important tool in superstring when it is treated as quantum mechanical object. This branch of mathematics is concerned with smooth, gradual, continuous change of geometric shape. For example, a square can be continuously deformed into a circle by pushing in the corners and rounding the sides. The essential rule is that no new hole can be created in the new form by tearing. Some topological equivalent objects are shown in Figure 15-17.

Unfortunately I lost the link to this quote and if someone could remember seeing this, I hope you will let me know.

We expect that the divergences of quantum gravity would similarly be resolved by introducing the correct short-distance description that captures the new physics. Although years of effort have been devoted to finding such a description, only one candidate has emerged to describe the new short-distance physics: superstrings. Vibrational modes

This theory requires radically new thinking. In superstring theory, the graviton (the carrier of the force of gravity) and all other elementary particles are vibrational modes of a string (figure 1). The typical string size is the Planck length, which means that, at the length scales probed by current experiments, the string appears point-like.

The jump from conventional field theories of point-like objects to a theory of one-dimensional objects has striking implications. The vibration spectrum of the string contains a massless spin-2 particle: the graviton. Its long wavelength interactions are described by Einstein's theory of General Relativity. Thus General Relativity may be viewed as a prediction of string theory!

This highlighted print tells us a lot, about the higher dimensional values assigned to spacetime as being a result. If we were to entertain the holographical consideration of these higher spaces manifesting into the spacetime curvature, that we have come to know and love, then we have indeed not only used Klein to travel to the fifth dimension but have come back home, to what GR represents for us a sa tangible?