February 1st, 2010

Differentiantion of 1-forms

In General Relativity, which relies on differential geometry and tensor calculus, a quick way to do coordinate free calculus is to use differential forms. A differential k-form, that is a form of degree k, is a smooth section of the k-th exterior power of the cotangent bundle of a smooth manifold M.

As examples, a differential 0-form is a smooth function on M, where a differential 1-form is the dual to a vector field on M. If we let U be an open set on \mathbb{R}^n, then there exists some smooth function f on U, which we define to be the differential 0-form. Given a vector field v on \mathbb{R}^n, for each v, there exists a directional derivative \partial_v f, which is the directional derivative in the usual sense, that is, if v=e_j is the jth coordinate vector then \partial_v f is the partial derivative of f with respect to the jth coordinate function

By their very definition, partial derivatives depend upon the choice of coordinates: Given two coordinate systems x^n and y^n, the transform between them is simply:

\frac{\partial f}{\partial x^j} = \sum_{i=1}^n\frac{\partial y^i}{\partial x^j}\frac{\partial f}{\partial y^i}

Since any vector v is a linear combination \sum v^j e_j of its components, df is uniquely determined by d_f p(e^j) for each j and each p\in U, which are just the partial derivatives of f on U. Since the coordinates x^n are themselves functions on U, and so define differential 1-forms dx^n. Since \frac{\partial x^i}{\partial x^j} = \delta^{i}_{j}, the Kronecker delta function, it follows that

df = \sum_{i=1}^n \frac{\partial f}{\partial x^i} dx^i.

The meaning of this expression is given by evaluating both sides at an arbitrary point p: on the right hand side, the sum is defined “pointwise”, so that

d f_p = \sum_{i=1}^n \frac{\partial f}{\partial x^i}(p) (dx^i)_p.

Remember, since f is an arbitrary smooth function on the dual manifold, we can define, and use, it pointwise. More generally, for any smooth functions g_i and h_i on U, we define the differential 1-form \alpha = \sum_1 g_i dh^i pointwise by coordinates as \alpha = \sum_{i=1}^n f_i d x^i for some smooth functions f_i on U.

The second idea leading to differential forms arises from the following question: given a differential 1-form \alpha on U, when does there exist a function f on U such that \alpha = df? The above expansion reduces this question to the search for a function f whose partial derivatives \frac{\partial f}{\partial x^i} are equal to n given functions f_i. For n>1, such a function does not always exist: any smooth function f satisfies \frac{\partial^2 f}{\partial x^i \partial x^j} = \frac{\partial^2 f}{\partial x^j \partial x^i}

so it will be impossible to find such an f unless \frac{\partial f_j}{\partial x^i} - \frac{\partial f_i}{\partial x^j}=0 \forall i,j.

The skew-symmetry of the left hand side in i and j suggests introducing an antisymmetric product on differential 1-forms, the wedge product, so that these equations can be combined into a single condition \sum_{i,j=1}^n \frac{\partial f_j}{\partial x^i} dx^i \wedge dx^j = 0

where dx^i \wedge dx^j = -dx^j \wedge dx^i.

This is an example of a differential 2-form: the exterior derivative d\alpha of [/latex]\alpha= \sum_j=f_j dx^j[/latex] is given by d\alpha = \sum_{j=1}^n d f_j \wedge dx^j = \sum_{i,j=1}^n \frac{\partial f_j}{\partial x^i} dx^i \wedge dx^j.

Differential forms can be multiplied together using the wedge product, and for any differential k-form α, there is a differential (k+1)-form dα called the exterior derivative of α.

Thus, I hope to have convinced you that differential forms, the wedge product and the exterior derivative are independent of a choice of coordinates. Consequently they may be defined on any smooth manifold M. If this makes you uncomfortable, you can reintroduce coordinates. One way to do this is cover M with coordinate charts and define a differential k-form on M to be a a family of differential k-forms on each chart which agree on the overlaps. However, there are more intrinsic (read: modern) definitions which make the independence of coordinates manifest. See the modern idea of tensors for a good idea what coordinate free geometry can do, and the intrinsic power of dealing with objects in a coordinate free space.

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January 29th, 2010

A life for science

Yesterday, 28 January 2010 was the day that NASA paused to remember all those whom have fallen in their pursuit of human ambition.

There are no great limits to growth, because there are no limits to the human imagination, intelligence, and sense of wonder.

It takes a lot of gumption to put your trust into the workings of a barely controlled man-made directed explosion, much more so if that explosion is attached to unique, ahead-of-the-curve technology in the most sophisticated piece of equipment ever built. However, the crews of Apollo 1, Challenger, and Columbia had done just that. They placed their complete and total trust into a mechanical monstrosity designed and built by the greatest minds and the lowest bidder, respectively. They placed their very being into the hands of friends and colleagues, knowing full well they were living and working at the edge of human knowledge, the point in which the unknown becomes known; the edge of science. With science, ignorant darkness begets declaratory light, knowledge transcends incomprehension, and the unknown is peeled back to reveal the grains of knowledge hidden deep within the beauty of Nature.

Science isn’t always simple, or safe. No-one intended for the men and women of NASA to die, but they all knew it would be a risk. Everyone knew that with complex systems, there were many unknowns, and although we, as scientists and engineers, can account for many of them, there are many more that simply cannot be comprehended. These unknowns are quantified as risk, and as such, as pioneers, we have to take those risks. We take them, and learn from them. We take them, and wrestle with them. We take risks, because the payoff is so great. The pursuit of truth is littered with the bodies of brave men and women, scientists and common folk alike. It is a tragedy when people die, and so, to you, the memories of all whom have died in the pursuit of science; to you, the crews of Apollo 1, Challenger, and Columbia, I raise my glass.

The crew of the space shuttle Challenger honored us by the manner in which they lived their lives. We will never forget them, nor the last time we saw them, this morning, as they prepared for their journey and waved goodbye and ’slipped the surly bonds of earth’ to ‘touch the face of God –Ronald Reagan

While I am not old enough to remember either Apollo 1, or Challenger, I do remember the Columbia accident quite clearly. I was in a Red Cross CPR class through the Boy Scouts of America, and like many of the people in the class, it was nothing more than a refresher course. So, I was talking to the instructors and other students, all old friends, when one of the fellow students walks in, looking quite shell-shocked.

“Did you hear that Columbia blew up?”, she asked. We all looked at each other, waiting for the punch-line, which never came. “Yeah, it disintegrated in the air over Texas, and it is presumed everyone aboard is dead.” With that, the instructors turned on the TV, and we watched the news briefly, then began the class. I was unable to concentrate on the material, just going through it by rote, not really paying attention, or even caring. My mind was with the Columbia, wondering if there were any survivors, what happened, and why the shuttle had failed to protect its precious cargo of flesh and knowledge.

Despite the setbacks in the NASA program, despite the difficulties, despite the risk, I still want to be an astronaut.

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January 28th, 2010

The Little Rover that Did

From the aptly named Space.com comes a wonderful, yet sad, story about the fate of the stuck Martian rover Spirit.

In May 2009, Spirit became stuck in a Martian sand pit while driving about on the Red Planet’s southern hemisphere. The soil that the rover became mired in is exceptionally soft, providing no traction for the stuck wheels, and hence, no help in pulling the plucky solar powered rover out of its predicament. Due to wheel malfunctions that left Spirit with only four working wheels, Spirit has been driving backwards, combined with the low power generated by the Martian dust upon the solar panels, left Spirit unprepared for the sand pit it drove into.

Despite months of extraction attempts, Spirit only became more mired in the dastardly pit, with hope of a successful extraction fading with every failed attempt. In January 2010, scientists at NASA finally declared that Spirit will just stay embedded in the sand pit, becoming a stationary research probe, unlike its still-roving twin, Opportunity. When I found out the news, even though it was fully expected, I was still sad, for the days of Spirit are now numbered.

With no movement to help clean its solar panels, Spirit will have to rely on the wind to scour its surface and give it the power boosts it needs to survive on the planet. Scientists are now trying to work Spirit into a great parking position on Mars, trying to maximize the angle between the probe’s solar panels and the sun, in hopes of keeping it alive during the upcoming harsh Martian winter.

Even stationary, there is still months, if not years, of science the probe can do, and scientists are proud of the work it has done thus far. Spirit, and Opportunity, landed on Mars in January 2004 for a three month mission, although six years later, they are still working hard, performing science on another planet. It will be a sad day for all when Spirit has no more to give, and returns its last ping; becoming nothing more than a monument to humanity as it finally dies on the surface of the alien planet, Mars.




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January 18th, 2010

Brief History of Particle Discoveries

Like all good physics programs, particle physics has kept several different generations of physicists employed. To remain employed, one must succumb to the publish or perish disease currently infesting physics. And of course, to publish, one must discover.

Here, I present a list of particles discovered by physicists, along with some useful information about the particles. This list is not intended to be exhaustive, as there are more particles in nature than useful politicians in Congress. One thing to note is that I only discuss the list of elementary particles discovered, to list all the baryons and mesons made from them is going to have to wait for another blog post.

Particle Year Mass (MeV/c2) Spin Antiparticle Comments
Electron 1897 0.511 1/2 Positron Also known as beta ray
Alpha particle 1899 big 0 anti-helium A doubly ionized Helium nucleus
Photon 1900 0 1 self 1
Proton 1919 938.272 1/2 anti-proton
Neutron 1932 939.566 1/2 anti-neutron
Anti-electron 1932 0.511 1/2 electron 2
Muon 1937 105.7 1/2 anti-muon 3
Pion 1947 134.98 0 Self Predicted in 1935 by Yukawa
Kaon 1947 497.65 0 Self First strange particle discovered
Anti-proton 1955 938.272 1/2 Proton
Electron neutrino 1956 under 2.2×10-6 0 Anti-electron neutrino 4
Up, Down, and Strange Quark 1969 1.5, 3.5, and 70 1/2 for all Anti-up, anti-down, anti-strange Deep elastic scattering in protons led to this discovery
J/Psi meson 1974 3096.916 1 self Showed existence of charm quark
Upsilon meson 1977 9460 1 self 5
Gluons 1979 0 1 self Gluons mediate strong force, and confine quarks.
W and Z bosons 1983 8039.8 (2.3), 9118.76 (21) 1 self Mediate weak force. W bosons violate parity.
Top quark 1995 1731 (13) 1/2 top antiquark 6

Footnotes

  • 1 Also known as gamma radiation, it was originally discovered in 1895 as X-rays, but was only successfully identified as electromagnetic radiation in 1900.
  • 2 Dirac’s relativistic quantum equation predicted this particle would appear, and have the same mass as the electron, but opposite charge. Dirac reasoned it was actually the proton it was predicting, however, the anti-electron was found experimentally in 1932. Despite not predicting it, Dirac still shared in the Nobel prize awarded for its discovery.
  • 3 With a long half-life, the muon can form chemical bonds in atoms.
  • 4 The neutrino was proposed by Wolfgang Pauli in 1931 to save the idea of conservation of energy, one of the most fundamental tenents in physics. During beta decay, it was found that some energy was escaping in an unknown form, so Pauli proposed the neutrino, an unknown particle with no mass and no charge, but containing energy. Neils Bohr favored the idea of getting rid of the conservation of energy instead. Luckily for physics, the neutrino was found, allowing us to keep the conservation of energy intact.
  • 5 Showed bottom quark properties. Feynman immediately predicted top quark existed.
  • 6 This particle is heavier than gold, and the last discovered quark. Some beyond-the-standard-model models predict a fourth generation of quarks, but none have been discovered.

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January 12th, 2010

Change in posting style, frequency

Gentle readers,

After noticing that my blog has become little more than a repository of recycled links, I have decided to go in a new direction. Rather then just re-post the same news everyone else is, I am going to take fewer stories, and go further in depth with them. I will be reading arXiv papers, and newsgroups, and reporting on what I find, as well as my analysis thereof.

This means that my posting frequency will drop, and posting will be sporadic. On the other hand, the posts will be of higher quality, and original material. I am also going to try and cut down on the number of ads served, and instead try and concentrate on the quality of my writing.

Until next time,

Gordon

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January 11th, 2010

Spiked Genitals, Ouch

A LiveScience article recently posted talks about the way fruit flies hook up.

In their case, “hooking up” really means hooking up. The male fruit fly has spikes that cover their genitals that help them hook on to unwilling female fruit fly partners during the copulatory act.

To see what the spikes are used for, researchers ablated them on virgin fruit flies, then introduced the hobbled males into nests of females. When the males tried to hook up with unwilling females, they essentially slid off the (bucking) women, unable to maintain a hold. However, when they hooked up with willing partners, they were able to engage in sex, and keep on their partners; in addition, their insemination rates were comparable to their non-hobbled counterparts.

Further studies are planned.

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January 10th, 2010

Neanderthals were pretty too!

From The Beeb comes a story about the discovery that Neanderthals were capable of symbolic thinking. Although I think that it was known beforehand that they were capable of such higher order thinking, as well as having a language, culture, complex tools, and great manly strength.

The discovery in southern Spain was of Neanderthal makeup kits, with foundation, metallic makeups, and dark pigments inside of shells that doubled as jewelery. This makeup was dated to ten millennium before Homo Sapiens had first contact with Homo Neanderthalis, thus dispelling the notion that Neanderthals were stupid, brutish peoples. With makeup, they were able to make themselves look different, as well as use it for decoration, much as modern humans do today.



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January 9th, 2010

Climate change

This absolutely stunning photo of the U.K. taken by NASA demonstrates the effects of global warming.

One of the most impressive effects from global warming is that weather becomes more extreme, with colder winters and hotter, drier summers. With so much of the world in a cold snap, this is an excellent example of extreme weather manifesting itself. As a complex nonlinear system, small nudges in initial conditions can cause huge changes in outputs, in this case, temperatures.

For a simple, non-mathematical introduction, feel free to check out the Wikipedia articles on Chaos theory, and the Butterfly effect. One should note that Chaos theory (the article) mainly describes deterministic chaos theory, and although there are deterministic parts to the Earth’s environment, there are also non-deterministic components that throw a monkey wrench into calculations into the Earth’s future.

Despite the non-deterministic components, enough is understood about the mathematics of chaos and the climate to make qualitative predictions about the future and evolution of earth, and its climate.



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January 8th, 2010

SOFIA flies

From this article comes news that the self guiding, infrared, plane mounted telescope SOFIA has begun in flight testing.

SOFIA, short for Stratospheric Observatory for Infrared Astronomy, is a giant telescope attached to a plane. The telescope searches infrared wavelengths at an altitude of at least 15,000 feet, above the safe veil of water vapor ensconced about the Earth. It is far cheaper to launch and maintain a plane then it is to launch a telescope into space; although vibrations are expected to put a damper on things.

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January 5th, 2010

AAS meeting

The newest American Astronomical Society meeting will be taking place in D.C. January 3rd through the 7th. If you are in the area, there i an open lecture being given by Brian Schwartz that will talk about the role of the scientist as portrayed by various media. It promises to be quite an interesting lecture.

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