Differential Geometry - The Language of General Relativity

A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging ultraparallel lines.

Differential geometry is a mathematical discipline that uses the methods of differential and integral calculus, as well as linear and multilinear algebra, to study problems in geometry. The theory of plane and space curves and of surfaces in the three-dimensional Euclidean space formed the basis for its initial development in the eighteenth and nineteenth century. Since the late nineteenth century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. It is closely related to differential topology, and to the geometric aspects of the theory of differential equations. Grigori Perelman's proof of the Poincaré conjecture using the techniques of Ricci flow demonstrated the power of the differential-geometric approach to questions in topology and highlighted the important role played by the analytic methods. Differential geometry of surfaces already captures many of the key ideas and techniques characteristic of the field.

Contents 1 Branches of differential geometry 1.1 Riemannian geometry 1.2 Pseudo-Riemannian geometry 1.3 Finsler geometry 1.4 Symplectic geometry 1.5 Contact geometry 1.6 Complex and Kähler geometry 1.7 CR geometry 1.8 Differential topology 1.9 Lie groups 2 Bundles and connections 3 Intrinsic versus extrinsic 4 Applications of differential geometry 5 See also 6 Notes 7 References 8 External links

Branches of differential geometry

Riemannian geometry

Main article: Riemannian geometry

Riemannian geometry studies Riemannian manifolds, smooth manifolds with a Riemannian metric, a notion of a distance expressed by means of a smooth positive definite symmetric bilinear form defined on the tangent space at each point. Riemannian geometry generalizes Euclidean geometry to spaces that are not necessarily flat, although they still resemble the Euclidean space at each point "infinitesimally", i.e. in the first order of approximation. Various concepts based on length, such as the arc length of curves, area of plane regions, and volume of solids all admit natural analogues in Riemannian geometry. The notion of a directional derivative of a function from multivariable calculus is extended in Riemannian geometry to the notion of a covariant derivative of a tensor. Many concepts and techniques of analysis and differential equations have been generalized to the setting of Riemannian manifolds.

A distance-preserving diffeomorphism between Riemannian manifolds is called an isometry. This notion can also be defined locally, i.e. for small neighborhoods of points. Any two regular curves are locally isometric. However, Theorema Egregium of Gauss showed that already for surfaces, the existence of a local isometry imposes strong compatibility conditions on their metrics: the Gaussian curvatures at the corresponding points must be the same. In higher dimensions, the Riemann curvature tensor is an important pointwise invariant associated to a Riemannian manifold that measures how close it is to being flat. An important class of Riemannian manifolds is formed by the Riemannian symmetric spaces, whose curvature is not necessarily constant. They are the closest to the "ordinary" plane and space considered in Euclidean and non-Euclidean geometry.

Pseudo-Riemannian geometry

Pseudo-Riemannian geometry generalizes Riemannian geometry to the case in which the metric tensor need not be positive-definite. A special case of this is a Lorentzian manifold which is the mathematical basis of Einstein's general relativity theory of gravity.

Finsler geometry

Finsler geometry has the Finsler manifold as the main object of study — this is a differential manifold with a Finsler metric, i.e. a Banach norm defined on each tangent space. A Finsler metric is a much more general structure than a Riemannian metric. A Finsler structure on a manifold M is a function F : TM → [0,∞) such that:

1. F(x, my) = |m|F(x,y) for all x, y in TM,
2. F is infinitely differentiable in TM − {0},
3. The vertical Hessian of F² is positive definite.

Symplectic geometry

Main article: Symplectic geometry

Symplectic geometry is the study of symplectic manifolds. An almost symplectic manifold is a differentiable manifold equipped with a smoothly varying non-degenerate skew-symmetric bilinear form on each tangent space, i.e., a nondegenerate 2-form ω, called the symplectic form. A symplectic manifold is an almost symplectic manifold for which the symplectic form ω is closed: dω = 0.

A diffeomorphism between two symplectic manifolds which preserves the symplectic form is called a symplectomorphism. Non-degenerate skew-symmetric bilinear forms can only exist on even dimensional vector spaces, so symplectic manifolds necessarily have even dimension. In dimension 2, a symplectic manifold is just a surface endowed with an area form and a symplectomorphism is an area-preserving diffeomorphism. The phase space of a mechanical system is a symplectic manifold and they made an implicit appearance already in the work of Lagrange on analytical mechanics and later in Jacobi's and Hamilton's formulation of classical mechanics.

By contrast with Riemannian geometry, where the curvature provides a local invariant of Riemannian manifolds, Darboux's theorem states that all symplectic manifolds are locally isomorphic. The only invariants of a symplectic manifold are global in nature and topological aspects play a prominent role in symplectic geometry. The first result in symplectic topology is probably the Poincaré-Birkhoff theorem, conjectured by Henri Poincaré and proved by George Birkhoff in 1912. It claims that if an area preserving map of an annulus twists each boundary component in opposite directions, then the map has at least two fixed points.^[1]

Contact geometry

Main article: Contact geometry

Contact geometry deals with certain manifolds of odd dimension. It is close to symplectic geometry and like the latter, it originated in questions of classical mechanics. A contact structure on a (2n+1)-dimensional manifold M is given by a smooth hyperplane field H in the tangent bundle that is as far as possible from being associated with the level sets of a differentiable function on M (the technical term is "completely nonintegrable tangent hyperplane distribution"). Near each point p, a hyperplane distribution is determined by a nowhere vanishing 1-form α, which is unique up to multiplication by a nowhere vanishing function:

A local 1-form on M is a contact form if the restriction of its exterior derivative to H is a non-degenerate 2-form and thus induces a symplectic structure on H_p at each point. If the distribution H can be defined by a global 1-form α then this form is contact if and only if the top-dimensional form

is a volume form on M, i.e. does not vanish anywhere. A contact analogue of the Darboux theorem holds: all contact structures on an odd-dimensional manifold are locally isomorphic and can be brought to a certain local normal form by a suitable choice of the coordinate system.

Complex and Kähler geometry

Complex differential geometry is the study of complex manifolds. An almost complex manifold is a real manifold M, endowed with a tensor of type (1,1), i.e. a vector bundle endomorphism (called an almost complex structure)

, such that J² = − 1.

It follows from this definition that an almost complex manifold is even dimensional.

An almost complex manifold is called complex if N_J = 0, where N_J is a tensor of type (2,1) related to J, called the Nijenhuis tensor (or sometimes the torsion). An almost complex manifold is complex if and only if it admits a holomorphic coordinate atlas. An almost Hermitian structure is given by an almost complex structure J, along with a riemannian metric g, satisfying the compatibility condition g(JX,JY) = g(X,Y). An almost hermitian structure defines naturally a differential 2-form ω_J,g(X,Y): = g(JX,Y). The following two conditions are equivalent:

1. N_J = 0 and dω = 0,
2. 

where is the Levi-Civita connection of g. In this case, (J,g) is called a Kähler structure, and a Kähler manifold is a manifold endowed with a Kähler structure. In particular, a Kähler manifold is both a complex and a symplectic manifold. A large class of Kähler manifolds (the class of Hodge manifolds) is given by all the smooth complex projective varieties.

CR geometry

CR geometry is the study of the intrinsic geometry of boundaries of domains in complex manifolds.

Differential topology

Differential topology is the study of (global) geometric invariants without a metric or symplectic form. It starts from the natural operations such as Lie derivative of natual vector bundles and de Rham differential of forms. Beside Lie algebroids, also Courant algebroids start playing a more important role.

Lie groups

A Lie group is a group in the category of smooth manifolds. I.e. beside the algebraic properties this enjoys also differential geometric properties. The most obvious construction is that of a Lie algebra which is the tangent space at the unit endowed with the Lie bracket between left-invariant vector fields. Beside the structure theory there is also the wide field of representation theory.

Bundles and connections

The apparatus of vector bundles, principal bundles, and connections on them plays an extraordinarily important role in the modern differential geometry. A smooth manifold always carries a natural vector bundle, the tangent bundle. Loosely speaking, this structure by itself is sufficient only for developing analysis on the manifold, while doing geometry requires in addition some way to relate the tangent spaces at different points, i.e. a notion of parallel transport. An important example is provided by affine connections. For a surface in R³, tangent planes at different points can be identified using the flat nature of the ambient Euclidean space. In Riemannian geometry, the Levi-Civita connection serves a similar purpose. More generally, differential geometers consider spaces with a vector bundle and a connection as a replacement for the notion of a Riemannian manifold. In this approach, the bundle is external to the manifold and has to be specified as a part of the structure, while the connection provides a further enhancement. In physics, the manifold may be the spacetime and bundles and connections correspond to various physical fields.

Intrinsic versus extrinsic

Initially and up to the middle of the nineteenth century, differential geometry was studied from the extrinsic point of view: curves and surfaces were considered as lying in a Euclidean space of higher dimension (for example a surface in an ambient space of three dimensions). The simplest results are those in the differential geometry of curves and differential geometry of surfaces. Starting with the work of Riemann, the intrinsic point of view was developed, in which one cannot speak of moving 'outside' the geometric object because it is considered as given in a free-standing way. The fundamental result here is Gauss's theorema egregium, to the effect that Gaussian curvature is an intrinsic invariant.

The intrinsic point of view is more flexible. For example, it is useful in relativity where space-time cannot naturally be taken as extrinsic (what would be 'outside' it?). With the intrinsic point of view it is harder to define the central concept of curvature and other structures such as connections, so there is a price to pay.

These two points of view can be reconciled, i.e. the extrinsic geometry can be considered as a structure additional to the intrinsic one. (See the Nash embedding theorem.)

Applications of differential geometry

Below are some examples of how differential geometry is applied to other fields of science and mathematics.

• In physics, three uses will be mentioned:
  • Differential geometry is the language in which Einstein's general theory of relativity is expressed. According to the theory, the universe is a smooth manifold equipped with pseudo-Riemannian metric, which described the curvature of space-time. Understanding this curvature is essential for the positioning of satellites into orbit around the earth. Differential geometry is also indispensable in the study of gravitational lensing and black holes.
  • Differential forms are useful in the study of electromagnetism.
  • Differential geometry has applications to both Lagrangian mechanics and Hamiltonian mechanics. Symplectic manifolds in particular can be used to study Hamiltonian systems.
• In economics, differential geometry has applications to the field of econometrics^[2].
• Geometric modeling (including computer graphics) and computer-aided geometric design draw on ideas from differential geometry.
• In engineering, differential geometry can be applied to solve problems in digital signal processing^[3].
• In probability, statistics, and information theory, one can interpret various structures as Riemannian manifolds, which yields the field of information geometry, particularly via the Fisher information metric.
• In structural geology, differential geometry is used to analyze and describe geologic structures.
• In computer vision, differential geometry is used to analyze shapes.^[4]

See also

• Integral geometry
• List of differential geometry topics
• Glossary of differential geometry and topology
• Important publications in differential geometry
• Important publications in differential topology
• Basic introduction to the mathematics of curved spacetime
• Affine differential geometry
• Projective differential geometry
• Noncommutative geometry
• Synthetic differential geometry
• Abstract differential geometry
• Discrete differential geometry
• Analysis on fractals

Notes

1. ^ It is easy to show that the area preserving condition (or the twisting condition) cannot be removed. Note that if one tries to extend such a theorem to higher dimensions, one would probably guess that a volume preserving map of a certain type must have fixed points. This is false in dimensions greater than 3.
2. ^ Paul Marriott and Mark Salmon (editors), "Applications of Differential Geometry to Econometrics", Cambridge University Press; 1 edition (September 18, 2000).
3. ^ Jonathan H. Manton, "On the role of differential geometry in signal processing" [1].
4. ^ Mario Micheli, "The Differential Geometry of Landmark Shape Manifolds: Metrics, Geodesics, and Curvature", http://www.math.ucla.edu/~micheli/PUBLICATIONS/micheli_phd.pdf

References

1. Wolfgang Kühnel (2005). Differential Geometry: Curves - Surfaces - Manifolds (2nd ed. ed.). ISBN 0-821-83988-8.
2. Theodore Frankel (2004). The geometry of physics: an introduction (2nd ed. ed.). ISBN 0-521-53927-7.
3. Spivak, Michael (1999). A Comprehensive Introduction to Differential Geometry (5 Volumes) (3rd Edition ed.).
4. do Carmo, Manfredo (1976). Differential Geometry of Curves and Surfaces. ISBN 0-13-212589-7. Classical geometric approach to differential geometry without tensor analysis.
5. Kreyszig, Erwin (1991). Differential Geometry. ISBN 0-48-666721-9. Good classical geometric approach to differential geometry with tensor machinery.
6. do Carmo, Manfredo Perdigao (1994). Riemannian Geometry.
7. McCleary, John (1994). Geometry from a Differentiable Viewpoint.
8. Bloch, Ethan D. (1996). A First Course in Geometric Topology and Differential Geometry.
9. Gray, Alfred (1998). Modern Differential Geometry of Curves and Surfaces with Mathematica (2nd ed. ed.).
10. Burke, William L. (1985). Applied Differential Geometry.
11. ter Haar Romeny, Bart M. (2003). Front-End Vision and Multi-Scale Image Analysis. ISBN 1-4020-1507-0.

External links

• B. Conrad. Differential Geometry handouts, Stanford University
• Michael Murray's online differential geometry course, 1996
• A Modern Course on Curves and Surface, Richard S Palais, 2003
• Richard Palais's 3DXM Surfaces Gallery
• Balázs Csikós's Notes on Differential Geometry
• Modern Differential Geometry for Maple
• N. J. Hicks, Notes on Differential Geometry, Van Nostrand.
• MIT OpenCourseWare: Differential Geometry, Fall 2008

Sex in Space

I think the picture says it all, but prose follows:



Vanna Bonta's concept for the "2suit" garment includes Velcro strips, zippers and diaphanous inner material that would be designed for intimacy in the near-weightless environment of space.

by Alan Boyle Science editor
msnbc.com

LAS VEGAS — Having sex in the weightlessness of outer space is the stuff of urban legends and romantic fantasy — but experts say that there would be definite downsides as well.

Spacesickness, for instance. And the difficulty of choreographing intimacy. And the potential for sweat and other bodily fluids to, um, get in the way.

"The fantasy might be vastly superior to the reality," NASA physician Jim Logan said here Sunday at the Space Frontier Foundation's NewSpace 2006 conference. Nevertheless, Logan and others say the study of sex and other biological basics in outer space will be crucial to humanity's long-term push into the final frontier.

"Sex in space is not just a good idea, it's survival," said Vanna Bonta, a writer who blends romance with space travel and quantum physics in the novel "Flight."

Sex in the space environment has long been a source of rumor and speculation: Several years ago, one author claimed that NASA had conducted a study of sexual behavior during a space shuttle mission, sparking a quick round of denials. Today, NASA follows something of a "don't ask, don't tell" policy on the subject — leading Logan to stress that he was not representing the space agency at Sunday's panel discussion.

The subject is coming to the fore again now for several reasons — including next month's publication of a book by Laura Woodmansee titled "Sex in Space," as well as billionaire Robert Bigelow's plan to host research into animal propagation on his commercial space modules.

After all, sometime in the next decade Bigelow Aerospace envisions putting a hotel complex in orbit, "where people will probably be recreating and having sex," Bonta said.

Germany and England

Once upon a time there was a Celtic tribe called the Britons. There were noble and tough, but by now they have been mated/raped out of existence, and that happened a long time ago. A thousand years ago it was the French, before them the Norse (called Vikings when they went to war), but before them in the 400's were three Germanic tribes, called the Angles, the Saxons, and the Jutes, who pretty much had their way with Britain.

And so the Brits' land went from being Britain, to The Land of the Angles, Saxons and Jutes, to Angle-Saxon Land, to Angle-land, to Ang-land, to Eng-land, then the hyphen was eventually and blessedly dropped.

And then there was 1919, when this postcard was written by a noted German Physicist to his mother, Pauline:

Dear Mother, joyous news today. H.A. Lorentz telegraphed that the English expeditions have actually demonstrated the deflection of starlight from the Sun.

... Albert Einstein, postcard to his mother, September 27th, 1919

Remember that when England faces Germany today in the World Cup. Their history is inexorably intertwined. It has been said that if The Industrial Revolution had not started in England in the 1750's, it would have started in Germany 60 years later. Indeed, the Germans did pick up on the Industrial Revolution quite fast, which only encouraged more competition thus improving the wealth of both countries markedly.

I am looking forward to today's game, but the ultimate result matters not to me. I root for both teams, but mostly I root for good sportsmanship. It should be a great game, and better than USA-Ghana, it is to be hoped. May the better cousins win.

And never forget the Albert Einstein - Sir Arthur Eddington connection. All that did was simply change the world. Today's Cup match, in comparison, is simply a game.

The Phil Warnell Page

Here are the first five articles (of 23) at Phil Warnell's excellent Philosophy of Science website, What is Einstein's Moon ?

Just five, consider it a tease and if like what you read I'd suggest visiting Phil's site directly.

Phil is a rock solid no-BS philosopher whose foundational thinking has helped me immensely. It's always a pleasure to meet someone of like mind. Thank you, Philip.

1. How and Why?

Well I guess the first thing to answer is why the blog and why dedicated to science and philosophy? That in itself is a bit of a story. The thing is I have always thought of the world and my relation to it in such terms. Ever since I was young I was one of those wonder people. That is I would wonder about this and wonder about that. It always seemed strange that here I was, born into this ponderous world preconstructed for me to observe and I didn’t have a clue what it was, how it was and why it was. Well when you think like this you are unavoidably lead to science and philosophy. The pursuit of the “what” questions and the “how” questions are things us mortals try to understand through a method called science. The “why” questions are attempted to be discovered through philosophical analysis and consideration. Many may wonder why science does not try to tackle it all. Well at one time this was truly the case. The term philosopher is Greek for “lover of knowledge“. In fact still today when someone receives a PhD in any of the sciences he is awarded a Doctorate in Philosophy. Of course, many of you know this. This doesn’t provide an answer. Well let’s take a look at a standard dictionary definition of philosophy taken from answer.com it reads:

“Investigation of the nature, causes, or principles of reality, knowledge, or values, based on logical reasoning rather than empirical methods“.

Still confused? Well the key words here are “empirical methods” Philosophy in the modern definition shuns empirical methods or in other words testing it’s validity by way of predictions against what we see in the world. This was not always the case.

Plato for instance did not exclude physical testing and at the same time warned us about where and when it was appropriate. He concludes in his Allegory of the Cave:

” Whereas our argument shows that the power and capacity of learning exists in the soul already; and that just as the eye was unable to turn from darkness to light without the whole body, so too the instrument of knowledge can only by the movement of the whole soul be turned from the world of becoming into that of being, and learn by degrees to endure the sight of being and of the brightest and best of being, or in other words, of the good.”

Plato here reminds us that true understanding incorporates both methodologies not only the empirical method but also by what he refers to as the “good”, which in the context of what I am talking about is the “why”. He implies that to have any success that both must be considered jointly . In the modern world these things have grown to become separated. The next question here is of course is “how and “why”. So that’s what this whole blog will be dealing with, my understandings and questions of the “how” and the “why”.

2. Influences

Before we go more into the "hows" and the "whys" that are the focus of this blog one might ask what my influences have been in all this? To be truthful there are many. Too many to simply synopsize in a single entry. What I can say is that I have been fortunate enough to have a mentor in all this for several years. Now this mentorship is not the usual one. Many consider a mentor as someone who takes you by the hand to lead you to conclusions that they already have. This is not what my mentor has been for me. My mentor has been a friend , guide, confidant and critic, one that recognized my interest and encouraged me to explore it thoroughly and consider as many of the options that one can . My mentor also had me examine my current positions and thoughts in terms of their soundness. This person for me is Douglas L. Hemmick PhD . He holds a Doctorate of Philosophy who's specialty is Quantum Foundations. You will see a link to his web site listed on this page. One might say Quantum Foundations, what is that? Well to be fair I think it better for you to click the link and see what Dr. Hemmick has to say himself about this. What I will tell you is that the study of his subject brings one closer to the realization that we simply can't separate the "how" questions from the "why" questions. Now a curious thing is that the vast majority of his fellow physicists don't feel that the "foundations" as a line of research has much relevance from a scientific standpoint. Many feel his area of endeavor is better described as metaphysics and is not true physics. Well curiously enough, if one limits themselves to the narrow modern definition that I set out in my last post, that would be true.

So is this current position of not mixing the hows with the whys a valid one? What we will find when we examine this closely is that it is a position that has evolved over time. Also, to be accurate, it is not a position that is totally universal or static. In the main though it suggests that science is only to address the "how" questions and philosophy the "whys". More importantly, the sciences in general, particularly physics, in some sense doesn't feel that "why" is a valid question in relation to the understanding of our world. On the other side, main stream philosophy has evolved into something that is homocentric, with man at the centre where the "how" questions are considered somewhat unimportant. Is it not strange that the pursuit of understanding has found itself in this seemingly paradoxical state?

3. Does Science Dismiss the Big Question, "Why" ?

Well now to begin where I left off I made a sweeping statement that modern science not only avoids the “why” questions but goes further to profess that such questions are not appropriate within the discipline. It goes even further to proclaim that such questions will not expand the quest for human understanding of the natural world. Many may say this is a outlandish statement and further where would I get such an idea. Well you don’t have to go far to find support for this.

As a example I quote Lisa Randall from a interview that appeared in this month’s Discover magazine. Professor Randall is a leading theoretical physicist and expert in particle physics, string theory, and cosmology. Her current research is focused on a aspect of string theory that suggests that our three dimensional universe may be only a part of a larger multi-dimensional one. This is all in pursuit of what is commonly and might I add improperly referred to as “The Theory of Everything”. Ms. Randall is currently the most quoted and cross referenced physicist in the world. I would contend that this qualifies her as being a representative of modern science, its thinking and its views. When Professor Randall was asked:

“Will physics ever be able to tackle the biggest questions—for instance, why does the universe even bother to exist?”

She responds with:

”Science is not religion. We're not going to be able to answer the "why" questions. But when you put together all of what we know about the universe, it fits together amazingly well. The fact that inflationary theory [the current model of the Big Bang] can be tested by looking at the cosmic microwave background is remarkable to me. That's not to say we can't go further. I'd like to ask: Do we live in a pocket of three-dimensional space and time? We're asking how this universe began, but maybe we should be asking how a larger, 10-dimensional universe began and how we got here from there.”

She then is asked:

“This sounds like your formula for keeping science and religion from fighting with each other.

” She then responds:

“A lot of scientists take the Stephen Jay Gould approach: Religion asks questions about morals, whereas science just asks questions about the natural world. But when people try to use religion to address the natural world, science pushes back on it, and religion has to accommodate the results. Beliefs can be permanent, but beliefs can also be flexible. Personally, if I find out my belief is wrong, I change my mind. I think that's a good way to live.”

So as you can see the lines have been drawn. First, Professor Randall admits that science does not even attempt to answer the “why” questions and then proclaims such questions are not relevant to understanding the natural world. She considers such questions the purview of religion. Now as we know religion can be seen and considered within the wider view as philosophy. I think if we pushed Professor Randall further she would agree with this extension. Now don’t get me wrong, I’m not saying that Professor Randal’s ideas are silly for I respect and admire what she does and how she strives to further our understanding of the natural world. I’ve read her new book - Warped Passages- and even attended a recent public lecture she gave. I’m simply making the point the this is what the general view is. So then, is it true that the “why” questions are beyond what one can expect of human understanding? It appears this is what science thinks. But how has science arrived at this? More importantly is it correct? Also, have all modern scientists thought this way? Well this is what we will continue to explore.

4. Can "Why" Be a Valid Question?

Now that we have established that modern science rejects the “why” question as something that will lead to increased human understanding of our natural world, some might tend to agree. You might say what possible benefit can one attain by asking the question in the first place? Well the answer could be simply that by asking such questions we end up with answers that suggest a greater truth. The reverse sometimes is the case as well in that by asking the question “ how” we are lead to or given the answer to the question “why”. When this happens the “why” then expands and adds validity to the answer and suggests that it is true. You might say can I give such an example?

One such example is when Charles Darwin went out on his famous voyage aboard the Beagle to visit many parts of the world and returned with a great collection and documentation of life’s species. In the course of studying them he observed slight variations within many species. Variations such as the shape and length of the beaks of finches he had found on the Galapagos Islands. First he asked himself, “how” could these small variations have occurred. The answer given was that small variations occurred by random mutation. The next question is “how” did these mutations persist. The answer put forth was that these mutations would only persist if such change made the species more viable. In other words it gave the individual an advantage to live. This advantage to live instilled in such individuals a better chance to propagate and have this change passed on to the next generation. When this change is passed to following generations then this would increase the survivability of that subset of the species that inherited them. So what am I getting at here? Well by asking those two “how” questions and proposing the answers Darwin was also given the answer to a “why” question. That “why” question is why is life so varied and ever changing. The answer suggested by Darwin's inquiry was that it is natures strategy for the survival and continuance of life.

I would argue that because Darwin’s “how” questions lead to the answer of this “why” question is what gives the theory of evolution such appeal. In other words suggests that it is true. It is also what makes it so reprehensible to many. What do I mean by this? Remember now what I contended has happened to the pursuit of human understanding. I said that it had divided into two camps. One being science that explores and answers the “how” questions and philosophy which is to explore and answer the “why” questions. Well here in the course of asking “how” Darwin had also answered “why”. In other words he had crossed the line that has been drawn between the two disciplines.

5. Making Jello a la Darwin

My last post ended where Darwin discovered as the “why” answer that natural selection was nature's strategy for the survival and continuance of life. You might inquire, do all “why” questions lead to such ground breaking conclusions and deep understanding ? The answer of course is no. We could ask “how” did the chicken cross the road? The answer being by putting one foot in front of the other. Then you might say “why” did the chicken cross the road? Well I’ll let you answer this question. The point is the question in itself, on its own does not have the power. It is where it is asked. When it is applied to the inquiries of the natural world it can lead to profound insight. The problem being that it is not asked very often by modern science.

The question now is how did we get to this point where science and philosophy have parted company? There are actually a few ways to look at this. If you asked a scientist of today they would most likely say that it is because science has become so specialized and complex that the philosophers are not able to keep up with it all and therefore cannot meaningfully contribute. This however is a dodge, for it avoids why the scientists themselves are not looking at their work from a philosophical perspective. Many scientists are of the opinion that to pose the “why” question suggests motive or intent plays a role in the nature of our world. I’m going to be bold here for I am convinced the vast majority of scientists believe that there is no motive or intent or to put it another way a scheme of nature. Many may point to Darwin’s Theory which I just outlined and say it shows a process where random is a element and this suggests that there is no scheme to nature. Equally, many philosophies, some of which are described as religions rail at the very same point for they believe that this also indicates there is no scheme in nature and so therefore it must be wrong. Despite their opposed views, on this point to they both agree. But are they correct to think this way?

To examine this let’s draw an analogy between nature's process for the survival and continuance of life to making a bowl of jello. When you make jello you take a bowl you put in some water plus jello powder along with some sugar. Then you stir, after which you put the whole lot in the refrigerator for a while and wait for it to set. Now in Darwin’s theory nature takes a world (the bowl) adds in primitive life along with a ever changing environment (the ingredients) then allows some random changes (the stirring), then waits for some time (sitting in the frig) and there you have it, life as it is recognized today. So what then is there in Darwin’s theory that is so convincing to many scientists and so unacceptable to some philosophers? Well it’s this random aspect. Both groups consider that random process shows that there is no scheme to nature and yet what is the act of stirring when we made the jello. Most would say it was a quick and easy method to get all the ingredients evenly mixed or distributed. In truth though you accomplished this by use of a process that evokes random. Now what did nature do? To assure that life was given the benifit of trying out numerous possibilities in terms of making it viable, nature invokes a scheme that involves random as part of the process. Now I would ask both the scientists and the philosophers, is it logical or reasonable to restrict nature in such a way? If this were true then they should also insist that I couldn’t make jello properly unless I purposely and accurately located all of the jello powder and the sugar within the water. If one looks at it this way it doesn’t make much sense. There is more to be said about this random aspect to nature and its implications for science and philosophy. This however I will leave for future posts.

End. Almost.

Here's a picture of what Phil looked like in the early spring of his life (it's only midsummer now Phil, plenty of time for us both, the world ain't done with us yet). Heck, I looked like that too. Thank goodness for memories!

Zidane's Revenge: Slovakia 3 Italy 2

I hope the Italians now know that saying bad things about a Legend's sister isn't nice.

Slovakia spanked the Italians something fierce in South Africa today (Slovakia was up 2-0 and 3-1 at two points), though it must be noted the Italians (physically) beat the crap out of a few Slovaks at the end.

No repeat Italy, sorry about that. You and your dirty sister-dissing ways are through for the '10 Cup.

Johannesburg has many flights to Rome, or so I hear.

A tip for Italy: next time, bring a better goalie/keeper, whatever. I hope that helped.

Aw, I'm just kidding. I love Italy and Italians fine. I grew up in a mostly Italian-American town, complete with old real Italians-from-Italy grandmothers in damn near every home, and they're wonderful people. Also, I used to be in theater, so I appreciate those with great acting chops like the Italian soccer team.

But I'm half Slovak, so I had to root for my cousins.

I must celebrate by making the old country family recipe of Kapustu tonight, basically kielbasa, sauerkraut, bacon, flour, onions, and no shortage of salt and pepper, served up with Gulden's brown mustard and a heap of well-buttered mashed potatoes. Yum.


"You said WHAT about my sister, Materazzi ?!" ... Zinedine Zadane, pre-head butt, World Cup Finals, France v. Italy, 2006

From Wiki:

"Popular belief has it that the Italy coach asked Materazzi to say what was said to Zidane to spark a reaction as during their days in Serie A together, the two often had conflicts."

Christ. So the cesspool stinks from the head down, eh?

Let's end on a positive note. We close with a picture of Iveta Radicova, whom many suspect will be the next Prime Minister of Slovakia. Sheesh, she looks like my sister!

The Etymology of "Physics"

Physics

Etymology: Latin physica, plural, natural science, from Greek physika, from neuter plural of physikos of nature, from physis growth, nature, from phyein to bring forth.

That was fun, lets try another word:

Phenomenology

Pronunciation: \fi-ˌnä-mə-ˈnä-lə-jē\

Function: noun

Inflected Form(s): plural phe·nom·e·nol·o·gies

Etymology: German Phänomenologie, from Phänomenon phenomenon + -logie -logy

Date: circa 1797

1 : the study of the development of human consciousness and self-awareness as a preface to or a part of philosophy
2 a (1) : a philosophical movement that describes the formal structure of the objects of awareness and of awareness itself in abstraction from any claims concerning existence (2) : the typological classification of a class of phenomena b : an analysis produced by phenomenological investigation

I don't like that definition, it's rather incomplete, but I don't expect anyone at Merriam-Webster to understand Science.

What would Wiki do? What does Wiki say?

It says this:

Phenomenology may refer to:

• Phenomenology (architecture), based on the experience of building materials and their sensory properties
• Phenomenology (archaeology), based upon understanding cultural landscapes from a sensory perspective.
• Phenomenology (particle physics), the part of particle physics that deals with the application of theory to high energy experiments
• Phenomenology (philosophy), a philosophical method and school of philosophy founded by Edmund Husserl (1859 – 1938)
• Phenomenology (psychology), used in psychology to refer to subjective experiences or their study
• Phenomenology (science), used in science to describe a body of knowledge which relates empirical observations of phenomena to each other
• Phenomenology of Perception, the magnum opus of French phenomenological philosopher Maurice Merleau-Ponty
• Phenomenology of religion, concerning the experiential aspect of religion in terms consistent with the orientation of the worshippers
• The Phenomenology of Spirit by G. W. F. Hegel (1770 – 1831)
• Existential phenomenology in the work of Husserl's student Martin Heidegger (1889 – 1976) and his followers

Phenomenology (science) is rather important. Without it, we don't get Quantum Mechanics therefore all these wonderful toys, as The Joker would say.

It's also important (currently) regarding the upcoming conference to be held in 3 weeks at Nordita in Sweden, here:

Physics Today (Top 10 items)

1. Agilent helps scientists explore the nature of matter ---Sponsorship (Tue Jun 1 8:00 am)
   High-energy particle accelerators are helping researchers investigate questions about the origins of the universe. Typical experiments involve controlled collisions between either intersecting particle beams or a beam and an atomic-scale target....
2. High Energy Density Physics (Tue Jun 1 8:00 am)
   The novel, mysterious, and controversial behavior of matter at high pressure involves the interplay of electromagnetic, statistical, quantum, and relativistic physics.
3. Magnetic field reconnection: A firstprinciples perspective (Tue Jun 1 8:00 am)
   Recent satellite missions and computer simulations of charged-particle dynamics in Earth's magnetosphere are helping unravel the mysteries behind the breaking and reforming of magnetic field lines and the concomitant acceleration of electrons...
4. James Franck: Science and conscience (Tue Jun 1 8:00 am)
   In World War I, Franck helped his native Germany develop gas-warfare defenses. Three decades later he urged the US, his adopted country, to tread carefully with an even more terrible weapon.
5. Focus on improving transmission electron microscopes starts to pay off. (Tue Jun 1 8:00 am)
   Two isotopes of the newly discovered element decay to give nine more previously unobserved nuclei.
6. Yoking real and virtual cells confirms theory of cochlear amplification (Tue Jun 1 8:00 am)
   Elastic coupling between cells in the inner ear enhances the hearing of amphibians, reptiles, birds, and mammals.
7. Focus on improving transmission electron microscopes starts to pay off (Tue Jun 1 8:00 am)
   The latest advance is the chemical identification of closely spaced, lightweight atoms.
8. Physics update (Tue Jun 1 8:00 am)
   
9. Megalasers to pulse in several new EU countries (Tue Jun 1 8:00 am)
   As the world celebrates 50 years since the invention of the laser, a European facility approaching exawatt power is expected to stimulate new research areas and communities.
10. DOE begins rationing helium-3 (Tue Jun 1 8:00 am)
    As the extent of the shortage becomes clear, an interagency task force is giving scientific users priority, but some say the material is not available at any price.

Quantum Hall effect

The quantum Hall effect (or integer quantum Hall effect) is a quantum-mechanical version of the Hall effect, observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall conductivity σ takes on the quantized values

where e is the elementary charge and h is Planck's constant. The prefactor ν is known as the "filling factor", and can take on either integer (ν = 1, 2, 3, .. ) or rational fraction (ν = 1/3, 1/5, 5/2, 12/5 ..) values. The quantum Hall effect is referred to as the integer or fractional quantum Hall effect depending on whether ν is an integer or fraction respectively. The integer quantum Hall effect is very well understood, and can be simply explained in terms of single particle orbitals of an electron in a magnetic field (see Landau quantization). The fractional quantum Hall effect, however, is more complicated, and its existence relies fundamentally on electron-electron interactions.

Contents 1 Applications 2 History 3 Integer quantum Hall effect – Landau levels 4 Mathematics 5 See also 6 References 7 Further reading

Applications

The quantization of the Hall conductance has the important property of being incredibly precise. Actual measurements of the Hall conductance have been found to be integer or fractional multiples of e²/h to nearly one part in a billion. This phenomenon, referred to as "exact quantization", has been shown to be a subtle manifestation of the principle of gauge invariance. It has allowed for the definition of a new practical standard for electrical resistance, based on the resistance quantum given by the von Klitzing constant R_K = h/e² = 25812.807557(18) Ω.^[1] This is named after Klaus von Klitzing, the discoverer of exact quantization. Since 1990, a fixed conventional value R_K-90 is used in resistance calibrations worldwide.^[2] The quantum Hall effect also provides an extremely precise independent determination of the fine structure constant, a quantity of fundamental importance in quantum electrodynamics.

History

The integer quantization of the Hall conductance was originally predicted by Ando, Matsumoto, and Uemura in 1975, on the basis of an approximate calculation which they themselves did not believe to be true. Several workers subsequently observed the effect in experiments carried out on the inversion layer of MOSFETs. It was only in 1980 that Klaus von Klitzing, working with samples developed by Michael Pepper and Gerhard Dorda, made the unexpected discovery that the Hall conductivity was exactly quantized. For this finding, von Klitzing was awarded the 1985 Nobel Prize in Physics. The link between exact quantization and gauge invariance was subsequently found by Robert Laughlin. Most integer quantum Hall experiments are now performed on gallium arsenide heterostructures, although many other semiconductor materials can be used. The integer quantum Hall effect has also been found in graphene at temperatures as high as room temperature.^[3]

Integer quantum Hall effect – Landau levels

In two dimensions, when classical electrons are subjected to a magnetic field they follow circular cyclotron orbits. When the system is treated quantum mechanically, these orbits are quantized. The energy levels of these quantized orbitals take on discrete values: , where ω_c = eB/m is the cyclotron frequency. These orbitals are known as Landau levels, and at weak magnetic fields, their existence gives rise to many interesting "quantum oscillations" such as the Shubnikov-de Haas oscillations and the de Haas-van Alphen effect (which is often used to map the Fermi surface of metals). For strong magnetic fields, each Landau level is highly degenerate (i.e. there are many single particle states which have the same energy E_n). Specifically, for a sample of area A, in magnetic field B, the degeneracy of each Landau level is N = g_sBA / φ₀ (where g_s represents a factor of 2 for spin degeneracy, and φ₀ is the magnetic flux quantum). For sufficiently strong B-fields, each Landau level may have so many states that all of the free electrons in the system sit in only a few Landau levels; it is in this regime where one observes the quantum Hall effect.

Mathematics

Hofstadter's butterfly

The integers that appear in the Hall effect are examples of topological quantum numbers. They are known in mathematics as the first Chern numbers and are closely related to Berry's phase. A striking model of much interest in this context is the Azbel-Harper-Hofstadter model whose quantum phase diagram is the Hofstadter's butterfly shown in the figure. The vertical axis is the strength of the magnetic field and the horizontal axis is the chemical potential, which fixes the electron density. The colors represent the integer Hall conductances. Warm colors represent positive integers and cold colors negative integers. The phase diagram is fractal and has structure on all scales. In the figure there is an obvious self-similarity.

Concerning physical mechanisms, impurities and/or particular states (e.g., edge currents) seem to be important for the 'integer' effect, whereas in the fractional quantum Hall effect the Coulomb interaction is considered as the main reason. Finally, concerning the observed strong similarities between integer and fractional quantum Hall effect, the apparent tendency of electrons, to form bound states of an odd number with a magnetic flux quantum, i.e. composite fermions, is considered.

See also

• Fractional quantum Hall effect
• Hall effect
• Hall probe
• Graphene
• Quantum spin Hall effect

References

1. ^ Tzalenchuk, Alexander; et al. (2010). "Towards a quantum resistance standard based on epitaxial graphene". Nature Nanotechnology 5 (3): 186 - 189. doi:10.1038/nnano.2009.474. http://www.nature.com/nnano/journal/v5/n3/abs/nnano.2009.474.html.
2. ^ conventional value of von Klitzing constant, NIST
3. ^ Novoselov, K. S.; et al. (2007). "Room-Temperature Quantum Hall Effect in Graphene". Science 315 (5817): 1379. doi:10.1126/science.1137201. PMID 17303717. http://arxiv.org/abs/cond-mat/0702408.

Further reading

• Ando, Tsuneya; Matsumoto, Yukio; Uemura, Yasutada (1975). "Theory of Hall Effect in a Two-Dimensional Electron System". J. Phys. Soc. Jpn. 39: 279–288. doi:10.1143/JPSJ.39.279.
• Klitzing, K. von; Dorda, G.; Pepper, M. (1980). "New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance". Phys. Rev. Lett. 45 (6): 494–497. doi:10.1103/PhysRevLett.45.494.
• Laughlin, R. B. (1981). "Quantized Hall conductivity in two dimensions". Phys. Rev. B. 23 (10): 5632–5633. doi:10.1103/PhysRevB.23.5632.
• Yennie, D. R. (1987). "Integral quantum Hall effect for nonspecialists". Rev. Mod. Phys. 59 (3): 781–824. doi:10.1103/RevModPhys.59.781.
• Hsieh, D.; et al. (2008). "A topological Dirac insulator in a quantum spin Hall phase". Nature 452 (7190): 970–974. doi:10.1038/nature06843.
• 25 years of Quantum Hall Effect, K. von Klitzing, Poincaré Seminar (Paris-2004). Postscript. Pdf.
• Quantum Hall Effect Observed at Room Temperature, Magnet Lab Press Release [1]
• J. E. Avron, D. Osacdhy and R. Seiler, Physics Today, August (2003)
• Zyun F. Ezawa: Quantum Hall Effects - Field Theoretical Approach and Related Topics. World Scientific, Singapore 2008, ISBN 978-981-270-032-2
• Sankar D. Sarma, Aron Pinczuk: Perspectives in Quantum Hall Effects. Wiley-VCH, Weinheim 2004, ISBN 978-0-471-11216-7
• A. Baumgartner et al.: Quantum Hall effect transition in scanning gate experiments, Phys. Rev. B 76, 085316 (2007), doi:10.1103/PhysRevB.76.085316

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Categories: Hall effect | Condensed matter physics | Quantum electronics | Spintronics | Quantum phases

From Wikipedia