Category: Astrophysics

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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December 24th, 2009

Voyager is still working

The Voyager probe has exited the Solar System!

Even though Voyager 2 reached the edge of the solar system back in 2007, the consensus amongst astronomers is that it has not actually reached interstellar space yet. Voyager 2 is still stuck in the heliosheath, the boundary between the effective range of the sun’s wind, and the interstellar medium. At this point, the probe can is busy examining the complex interplay between the medium, and the solar wind. Amongst the things currently discovered, Voyager has determined the heliosheath is misshapen, it can be compressed, depending upon the interstellar wind strength, and the solar wind helps to protect the sun’s planets from high energy cosmic rays, stray particles, and other interstellar matter.

Now, another fun fact can be added to what the venerable Voyager 2 has discovered: It has discovered an interstellar cloud with a strong magnetic field. More specifically, it discovered the magnetic field.

This cloud, called the Local Fluff, is a thirty light year cloud of heated Hydrogen and Helium, surrounded by supernova remnants. The shock waves from the supernova remnants should have either dispersed, or crushed the cloud, but still it persists. Because the Solar System is plunging through the cloud, Voyager 2 has easily detected a magnetic field from it. This field, with a strength of 4 to 5 microgauss, is strong enough to hold the Local Fluff together against the supernova bits trying to rend it to pieces. This field also puts pressure upon the heliosphere, causing it to collapse and distort in a giant, interacting gas-wind cosmic dance.

This data from Voyager 2 will also allow astronomers to see how other forces effect the heliosphere, and what implications they have for the future of human space travel. Not bad work for a forty plus year old probe initially designed to take pictures.

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December 20th, 2009

Musical Universe Supplimental Podcast 2

Hello again, and welcome!

I have here another Supplimental Podcast for my radio show, Musical Universe. Musical Universe is a live internet radio show done every Sunday night on riverfrontradio.com, from 9pm to 11pm CST. (UDT -6) The show is about Astronomy, Astrophysics, Space science, Science in general, Educational Science, and Astrophysical Engineering.

The show, much like this podcast, is done with no script, and in one take. The podcast is five and a half minutes long, and I do apologize about the sudden change in sound quality about a minute in. It gets better. I hope you enjoy the podcast!

If you cannot see a music player, make sure that riverfrontradio.com is not blocked at your location.

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December 10th, 2009

New Hubble HUDF

The Hubble Ultra Deep Field 2009 is a four day exposure of the Hubble Ultra Deep Field of 2004, which was an eleven day picture of a starless, apparently empty region of space 1/15th the size of the (full) moon. Of course, when the picture was developed, it turned out that the empty region of space was not so empty after all.

Click to see the picture. The link leads to the HubbleSite page, where you can view the picture at a resolution of your liking.

A couple of beautiful things I would like to note. In the upper left corner, we find a diffracted star (the white object with a red, green, and blue halo, and crossbeams through it). Near this star, we find a diffracted red object, which looks like a star, but is slightly pinkish. This is a quasar, a black hole eating matter and spewing out energy. To the right of the quasar, we find a deeply red dot with no discernible shape. This is a very early galaxy, with an age of thirteen billion years or more. Up, and to the left of the diffracted star, we see a spiral galaxy with only one massive arm, indicating a new galaxy that has recently undergone an interaction with a massive object.

In the top center of the image, we find not one, but two pairs of interacting galaxies. As we move to the right from our galaxies, we find lots of blue streaks. These streaks are clusters of hot, massive stars forming in galaxies that are too dim to see. The stars are so hot, bright, and massive, they drowned out the light from the rest of the galaxy, forcing us to guess at their shape.

Almost dead center in the image, we see a stunning, well formed barred spiral. Contrast this to the still forming, dim spirals nearby. Our barred spiral appears to be a recent galaxy, at least by the pictures standards.

I don’t see any Einstein’s rings yet, but I will keep looking. An Einstein’s ring in an image this old will not only provide a deep look into the very beginnings of galaxies at the beginning of galactic formation, but it may also provide a peek at matter distribution beyond our Hubble Volume.

Pictures in a later post.



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November 29th, 2009

Zombie Galaxies say (Dust) Graaaains

Phil Plait has a really neat post on the evolution of spiral galactic bulges by means of dwarf cannibalization here.

It really is a neat piece of work, in which the astronomers in question took a look at a galactic cluster called Terzan 5 and discovered that it contains two populations of stars, an outer shell of older (red) stars, and an inner core of newer (blue) stars. This is interesting, yet disturbing, because if Terzan 5 is the remnants of a galaxy, then it is one pretty messed up galaxy. Assuming it is a dwarf galaxy, we can rule out that it is not a dwarf irregular, since they have no distinct shape, and Terzan 5 is spherical. We can rule out that it is a dwarf spiral for the same shape mismatch reason, and we can rule out that it is a dwarf elliptical or dwarf spheroidal, since it shows evidence of recent star formation.

Terzan 5 is probably an ultra compact dwarf galaxy, yet is shows a unique feature not known (AFAIK) to UCDs. The newer core of stars surrounded by a shell of older stars. Although this is a relatively unknown area of study, I would presume that ultra compact galactic nuclei contain a heterogeneous mixture of old and new star populations due to stellar interactions throughout their rather small core. Dr. Plait argues that the dual star populations imply that Terzan 5 is a dwarf core galaxy, but I am of the opinion that the younger stars should be on the outside, not the inside.

The one thing we both agree on is that the fact that the iron in stars in Terzan 5 matches the amounts of iron in stars in our galactic bulge. This is indicative of interactions within the two objects, although I am not convinced that it is because the Milky Way stripped Terzan 5 of its outer stars and added them to the bulge. It is a tantalizing mystery, one that can only be sure to add to the mystique of galactic evolution.

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November 26th, 2009

Japanese Neutrino Experiment T2K

From Astronomy.com and Symmetry Breaking, comes the next chapter in Japanese neutrino physics, the Tokai-to-Kamioka (T2K) experiment.

T2K is a neutrino beam generated by a particle accelerator directed at a target clear across the country of Japan, and the target, fourteen iron and scintillation boxes designed to pick up the neutrinos. The neutrinos are generated by a 30 GeV proton beam slamming into a carbon target to produce a short lived particle called pions. The pions decay while travelling through a helium filled space, producing neutrinos. Charged pions decay via the weak interaction (W^{+}) to form a muon and a muon neutrino. The muon then decays to a positron, electron neutrino, and a muon antineutrino.

\pi^+ \rightarrow \mu^+ + \nu_\mu and \mu^+ \rightarrow e^+ + \nu_e + \bar{\nu}_\mu

Thus, the majority of our neutrinos are expected to be muon neutrinos, alongside their antineutrino counterparts, assuming we start with positively charged pions. Since neutrinos oscillate between their three flavors, some of the neutrinos are expected to change into other neutrino forms, while their antineutrino counterparts change into other antineutrino forms. The differences between the amount of neutrino and antineutrino oscillations will help narrow down the differences between matter and antimatter.

From their press release comes evidence that T2K has had three neutrino hits, in line with expectations. Indeed, you can see one of the hits directly here. I look forward to seeing what kind of useful physics comes out of this experiment, since antimatter and matter differ on a fundamental level, and this experiment will root out some of those deep differences with fundamental particles. Since we are dealing with neutrinos, it will also help identify physics beyond the standard model, our current best theory for dealing with particle physics.



Reading the press release carefully, one will note that Symmetry Breaking has copied the press release verbatim, a kind of dick move on their part. Not the worst thing they could have done, since they at least attribute the press release, and link it, but they could have at least given their analysis on it. Your thoughts?

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Posted in Astrophysics, Particle Physics, Physics | 1 Comment »

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