Archive for September, 2009

September 30th, 2009

Dark Matter Detector

From “A BGO scintillating bolometer as dark matter detector prototype.” By N. Coron, E. García, J. Gironnet, J. Leblanc, P. de Marcillac, M. Martínez, Y. Ortigoza, A. Ortiz de Solórzano, C. Pobes, J. Puimedón, T. Redon, M.L. Sarsa, L. Torres and J.A. Villar. Optical Materials, Vol. 31 Issue 10, August 2009.

Even though dark matter has never been directly detected, that does not bother most physicists. Dark matter is matter that does not interact with barionic matter via the electromagnetic force; it does however, interact gravitationally. Dark matter is a particle (or set thereof) beyond the Standard Model of Physics, so detection would be a great discovery, and powerful purview of science.

Even without understanding exactly what dark matter is, we can still use General Relativity, Newtonian equations of motion, and celestial mechanics to determine projected properties of the dark matter. Dark matter is currently best examined by the behavior of faraway galaxies; which move in ways that can only be explained by a gravitational pull caused by more mass than can be detected using light.

Particle cosmologists state that only around five percent of matter in the universe can presently be detected. They estimate dark matter represents around 20 percent of the universe, with the other 75 percent made up of dark energy, a repulsive force that is causing the universe to expand at an ever-quickening pace.

Coron et al, created a detector called a scintillating bolometer, a crystal so pure it can nearly perfectly conduct the energy generated when a particle of dark matter strikes the nucleus of one of its atoms. (There is some quantum 1/f noise generated, along with phonon radiation.)

prevent interference by cosmic rays, the scintillation bolometer is sheathed in lead and kept underground, under over a kilometer of rock. It’s also chilled to near-absolute zero, the temperature at which all motion stops, and quantum zero energy effects become greatly measurable. Millionths of degrees above absolute zero, it is possible to measure expected changes of a few millionths of a degree Fahrenheit.

As described in a paper published in the August Optical Materials and released online Friday, the bolometer is currently able to distinguish between the vibrations produced by the vibrations of nuclei and spinning electrons. While this is too much quantum noise to detect dark matter, it is orders of magnitude greater in sensitivity then current liquid Xenon detectors, which operate at temperatures of -100ish C.

In order for the bolometer to work reliably, it needs to become even more sensitive, and maintain that sensitivity as it’s scaled up from the 46-gram prototype to a half-ton full scale working experiment.

While this seems insurmountable, I expect that it will be achieved within five to ten years. I, for one, cannot wait for the results from the experiment to be made public, as a positive result will greatly enhance our knowledge of physics beyond the standard model, and a negative result will leave us with more questions to answer.

September 24th, 2009

Musical Universe

Musical Universe now has a logo!

Musical Universe is my two hour internet radio show/ live podcast that goes up every Tuesday night 9pm-11pm CDT; 0200-0400 UTC. Yesterday, I spent half the show discussing alien life in the galaxy, and next week, 09/29/2009, I will be talking about Semiclassical Instabilities in Warp Bubbles. The paper I will be discussing is located here.

The first half of the show will be devoted to Kepler’s laws, and Newton’s Law of Universal Gravitation. It will be mildly mathematical, however, the second half will be extremely mathematical.

Click to Embiggen

EDIT: I forgot to mention you can listen in to the show at riverfrontradio.com.

September 22nd, 2009

Messier Tour (16-20)

Continuing our slow nighttime tour of the enigmatic Messier objects brings us to the next five.

Messier 16

RA: 18h 18m 48s

DEC: -13 deg 49′

Apparent Magnitude: 6.0

M16, the famous Eagle Nebula is a young open cluster of stars in the constellation Serpens, discovered by Jean-Philippe de Cheseaux in 1745-46. Its name derives from its shape which is resemblant of an eagle. It is the subject of a famous photograph by the Hubble Space Telescope, which shows pillars of star-forming gas and dust within the nebula.

The Eagle Nebula is part of a diffuse emission nebula, or H II region, which is catalogued as IC 4703. This region of active current star formation is about 6,500 light-years distant. The tower of gas that can be seen coming off the nebula is approximately 57 trillion miles (97 trillion km) high.

The brightest star in the nebula has an apparent magnitude of 8.24, easily visible with good binoculars.

Messier 17

RA: 18h 20m 26s

DEC: -16 deg 10′ 36″

Apparent Magnitude: 6.0

M17 is known as both the Omega Nebula, as well as the Swan Nebula; is an ionized hydrogen cloud in the constellation Sagittarius. It was discovered by Philippe Loys de Chéseaux in 1745. It is located in the rich starfields of the Sagittarius area of the Milky Way.

The Omega Nebula is between five and six thousand light-years from Earth and it spans some 15 light-years in diameter. The cloud of interstellar matter of which this nebula is a part is roughly 40 light-years in diameter.

A cluster of 35 stars lies embedded in the nebulosity and causes the gases of the nebula to shine due to radiation from these hot, young stars.

Messier 18

RA: 18h 19m 54s

DEC: -17 deg 08′

Apparent Magnitude: 7.5

M18 is an open cluster of stars in the constellation Sagittarius. It was discovered by Charles Messier in 1764 and included in his list of comet-like objects. From the perspective of Earth, M18 is situated between the Omega Nebula (M17) and the Sagittarius Star Cloud (M24). It is rather unimpressive, and sadly, easily missed.

Messier 19

RA: 17h 02m 37.69s

DEC: -26 deg 16′ 4.6″

Apparent Magnitude: 7.5

M19 is a globular cluster in the constellation Ophiuchus. It was discovered by Charles Messier in 1764 and added to his catalogue of comet-like objects that same year.

M19 is the most oblate of the known globular clusters. It is at a distance of about 28,000 light-years from the Solar System, and is quite near to the Galactic Center, at only about 5,200 light-years away.

Messier 20

RA: 18h 02m 23s

DEC: -23 deg 1′ 48″

Apparent Magnitude: 6.3

M20 is also known as the Trifid Nebula, a beautiful ionized hydrogen gas cloud located in Sagittarius. Its name means ‘divided into three lobes’. The object is an unusual combination of an open cluster of stars, an emission nebula, a reflection nebula, and a dark nebula. Viewed through a small telescope, the Trifid Nebula is a bright and colorful object, and is thus a perennial favorite of amateur astronomers, professional astronomers, and the public at large.



September 19th, 2009

Messier Tour (11-15)

Here, we continue our brief tour of the Messier objects.

Messier 11

RA: 18h 51.1m

DEC: -06 deg 16′

Apparent Magnitude: 6.3

Messier 11, also known as the Wild Duck Cluster is an open cluster in the constellation Scutum. It was discovered by Gottfried Kirch in 1681 and Charles Messier included it in his catalogue in 1764.

The Wild Duck Cluster is one of the richest and most compact of the known open clusters, containing about 2900 stars. Its age has been estimated to about 220 million years, about 5% of Sol’s age. Its name derives from the brighter stars in the cluster that form a triangle which could represent a flying flock of ducks.

Messier 12

RA: 16h 45m 14.52s

DEC: -01 deg 56′ 52.1″

Apparent Magnitude: 7.68

M12 is a globular cluster in the constellation of Ophiuchus. It was discovered by Charles Messier on May 30, 1764.

Located roughly 3 degrees in the sky from the cluster M10, M12 is about 16,000 light-years from Earth and has a spatial diameter of about 75 light-years. M12 is rather loosely packed for a globular cluster, and was once thought to be a tightly concentrated open cluster.

M12 is expected to burn out quickly, as a study in 2006 showed that it contains very few low mass stars. Low mass stars live much longer then high mass stars, with high mass stars burning out in a few million years.

Messier 13

RA: 16h 41m 41.44s

DEC: 36 deg 27′ 36.9″

Apparent Magnitude: 5.8

M13 also called the Hercules Globular Cluster is a globular cluster in the constellation of Hercules. M13 was discovered by Edmond Halley in 1714, and catalogued by Charles Messier on June 1, 1764.

With an apparent magnitude of 5.8, M13 is barely visible with the naked eye on a very clear night. Loated about 25,000 light years way, and with a diameter of about 145 ly, it appears about 23 arc minutes wide and it is readily viewable in small telescopes. Nearby is NGC 6207, a 12th magnitude edge-on galaxy that lies 28 arc minutes directly north east. A small galaxy, IC 4617, lies halfway between NGC 6207 and M13, north-northeast of the large globular’s center.

Due to it’s massive size, and large number of stars, Astronomers sent a message from the Arecibo telescope in 1974, designed to communicate the existence of human life to hypothetical extraterrestrials. With a higher star density, the chances of a life harboring planet with intelligent life forms, were higher.

Messier 14

RA: 17h 37m 36.15s

DEC: -03 deg 14′ 45.3″

Apparent Magnitude: 8.32

M14 is a globular cluster in the constellation Ophiuchus. It was discovered by Charles Messier in 1764.

At a distance of about 30,000 light-years, M14 easily contains several hundred thousand stars. At an apparent magnitude +7.6 it cannot be observed with the naked eye, except under the very best conditions, but it can be easily observed with binoculars. Medium-sized telescopes will show some hint of the individual stars of which the brightest is of magnitude 14.

M14 is about 100 light-years across. The shape of the cluster is decidedly elongated.

Slightly over 3 degrees southwest of M14 lies the faint globular cluster NGC 6366.

Messier 15

RA: 21h 29m 58.38s

DEC: 12 deg 10′ 00.6″

Apparent Magnitude: 6.2

M15 is a globular cluster in the constellation Pegasus. It was discovered by Jean-Dominique Maraldi in 1746. At an estimated 13.2 billion years old, it is one of the oldest known globular clusters.

M15 is about 33,600 light-years from Earth. It thus has an absolute magnitude of -9.2. To the amateur astronomer Messier 15 appears as a fuzzy star in the smallest of telescopes. Mid to large size telescopes will start to reveal individual stars, the brightest of which are of magnitude +12.6.

Messier 15 is one of the most densely packed globular clusters known in the Milky Way galaxy. Its core has undergone a contraction known as ‘core collapse’ and it is hypothesized to have a central density cusp with an enormous number of stars surrounding what may be a central black hole.

Messier 15 contains 112 variable stars, at least 8 pulsars, including one double neutron star system, M15 C. Moreover, M15 houses Pease 1, one of only four planetary nebulae known to reside within a globular cluster.

Your Ad Here

September 18th, 2009

Messier Tour (6-10)

Continuing our tour of the Messier objects, we begin with M6.

Messier 6

RA: 17h 40m 6s

DEC: -32 deg 13m

Apparent Magnitude: 4.2

Personally, I do not find this cluster to be very impressive. It does contain a massive amount of large blue stars, and an orange giant, which implies it is very young, as star clusters go.

Messier 7

RA: 17h 53.9m

DEC: -34 deg 49′

Apparent Magnitude: 3.3

This cluster, often called the Ptolemy cluster, is said to have been viewed by the Greek astronomer Ptolemy. This cluster is easily visible, even in less then idea conditions, being a magnitude 3.3 brightness, and near the tail end of the constellation of Scorpio.

Messier 8

RA: 18h 03m 37s

DEC: -24 deg 23′ 12″

Apparent Magnitude: 6.0

M8 is a giant interstellar cloud in the constellation Sagittarius. It is classified as an emission nebula and as an H II region. M8, also known as the Lagoon Nebula, was discovered by Guillaume Le Gentil in 1747 and is one of only two star-forming nebulae visible to the naked eye from mid-northern latitudes. A fragile star cluster appears superimposed on it.

The Lagoon Nebula is estimated to be 4,100 light-years from the Earth with an actual dimension of 110 by 50 light years. Like many nebulae, it appears pink in time-exposure color photos but is gray when seen through binoculars or a telescope, human vision having poor color sensitivity at low light levels.

Messier 9

RA: 17h 19m 11.78s

DEC: -18 deg 30′ 58.5″

Apparent Magnitude: 8.42


    M9 is a globular cluster in the constellation of Ophiuchus. It was discovered by Charles Messier in 1764.

    M9 is one of the nearer globular clusters to the center of the Milky Way Galaxy with a distance of around 5,500 light-years from the core. Its distance from Earth is 25,800 light-years.

    At about 80′ to the northeast of M9 is the dimmer globular cluster NGC 6356, while at about 80′ to the southeast is the globular NGC 6342.

    Messier 10

    RA: 16h 57m 0.899s

    DEC: -04 deg 05′ 57.6″

    Apparent Magnitude: 6.4

    M10 was discovered by Charles Messier on May 29, 1764, who described it as a “nebula without stars”, but later study revealed it as a globular cluster of thousands of stars, situated apparently in the constellation of Ophiuchus.

    M10 has an apparent diameter about two-thirds of the apparent diameter of the Moon. Viewed through medium-sized telescopes it appears about half that size, as its bright core is only 35 light-years across. M10 has a spatial diameter of 83 light-years and is estimated to be 14,300 light-years away from Earth.

    Your Ad Here

September 16th, 2009

A tour of the Messier objects

The Messier objects are a series of deep sky objects that are not stars, or comets, as cataloged by French astronomer Charles Messier, in collaboration with his assistant Pierre Mechain. This list of objects were compiled in a book, “Catalog of Nebulae and Star Clusters” published in 1771. (Of course, the book was originally French.)

So, I am going to go ahead and create a list of the Messier objects, as well as their celestial coordinates, and some notes about the object.

Messier 1

RA: 05h 34m 31.97s

DEC: 22 deg 00′52.1″

Apparent Magnitude: 9

This object is also known as the Crab Nebula, a supernova that was seen on the Earth in 1054. Since the Crab Nebula is only 3/2 a degree off the ecliptic, that is, the plane of the Sun-Planet systems, it can can be used to explore the planets as they occult the nebula.

Messier 2

RA: 21h 31m 27s

DEC: -00 deg 49′ 24″

Apparent Magnitude: 6.3

At 13 Gy (13 thousand million) years old, and 175 light years wide, this giant globular cluster is one of the oldest star clusters in the Milky Way galaxy. It contains 150,000+ stars, and is highly elliptical.

On a dark night, this star cluster can often be just seen by dark adapted eyes.

Messier 3

RA: 13h 42m 11.23s

DEC: 22 deg 21′ 31.6″

Apparent Magnitude: 6.3

This globular cluster has a half million stars and an apparent magnitude of 6.2, making it visible to dark adapted eyes.

Messier 4

RA: 16h 23m 35.41s

DEC: -26 deg 31′ 31.9″

Apparent Magnitude: 7.12

This cluster is near Antares, a bright star, and at 75 light years across, it is about as wide as the full moon. This cluster can be resolved in medium and high resolution telescopes.

This cluster was the first to have individual stars resolved, as well as the first to have white dwarfs IDed. In 1995 the HST found a white dwarf binary with a pulsar companion and a planet. At 13 Gyr, this planet is obviously old, and metal poor.

Messier 5

RA: 15h 18m 33.75s

DEC: 02 deg 04′ 37.7″

Apparent Magnitude: 6.65

This cluster should not be confused with the much smaller and fainter Palomar 5, which is nearby. M5 should be barely visible under good seeing conditions. It is easily visible with binoculars, or a small telescope.

This cluster is a massive 165 light years across, with a gravitational attractive radius of 200 ly. It is 13 Gyr old, making it one of the oldest clusters in the Milky Way.



September 15th, 2009

Beyond Special Relativity

Beyond SR

Special Relativity deals with transformations from one frame to another, when both frames are moving with a constant speed on both frames. If we want to include any acceleration, we have to move beyond special relativity, and ascend to general relativity. GR is beyond the scope of this article, however, understanding it is important for any physicist, so I hope to cover the basics of GR in another article soon. In the meantime, I recommend reading “An Introduction to Relativity, Cosmology, and Gravitation”, by Ta-Pei Cheng, an absolutely delightful book on relativity, cosmology, and gravitation. Once that book has been digested, I recommend “Gravitation”, by Wheeler, Misner, and Thorne.

Bibliography and Links

Dr. Cheng’s webpage at UMSL
His book at Amazon
His book at Alibris
Gravitation at Amazon



September 14th, 2009

Relativity and Light

Doppler Shift and of Light Aberration

In the K frame, very far from the origin (that is, (0,0,0), let there be a source of electrodynamic waves, which at the origin of the K frame may be represented by:
E_{x_i} = E_{{x_i}0} \sin \Phi, and B_{x_i} = B_{{x_i}0} \sin \Phi for i=1,2,3 (that is 1=x\hat{x},2=y\hat{y},3=z\hat{z}).
where
\Phi=\omega\left\{t-\frac{1}{c}(x\hat{x}+y\hat{y}+z\hat{z})\right\}

Here E_{{x_i}0}, B_{{x_i}0} are the amplitudes of the electric and magnetic fields, and \hat{x}\hat{y}\hat{z}, the normalized direction-cosines of the wave-normals. We will now transform these equations into the moving frame K'. Remember, K' has speed v.

Applying the transformation equations for electric and magnetic forces, and those for spacetime coordinates, we obtain (derive this for yourself):

E_x' = E_{x0} \sin \Phi', E_y' = \gamma (E_{y0} + \beta B_{z0}) \sin \Phi', E_z' = \gamma (E_{z0} + \beta B_{y0}) \sin \Phi',

B_x' = B_{x0} \sin \Phi',B_y' = \gamma (B_{y0} + \beta E_{z0}) \sin \Phi',B_z' = \gamma (B_{z0} + \beta E_{y0}) \sin \Phi', and

\Phi' = \omega \gamma(1-\beta\hat{x})\left\{ \tau -\frac{1}{c}(\frac{x-\beta}{1-x\beta}x'\hat{x} +\frac{y}{\gamma(1-x\beta)}y'\hat{y} +\frac{z}{\gamma(1-x\beta)}z'\hat{z})\right\}.

Frequency

From the equation for \omega' it follows that if an observer is moving with velocity v relative to an infinitely distant source of light of frequency \nu, in such a way that
the angle between the source-observer line and the velocity vector of the observer is real, which we denote as \phi, the frequency \nu' of the light perceived by the observer is given by:

\nu' = \nu\frac{1-\cos\phi\cdot \beta}{\sqrt{1-\beta^2}}.

This is Doppler’s principle for any velocities whatever. When \phi=0 the equation assumes the more commonly seen form \nu'=\nu\sqrt{\frac{1-v/c}{1+v/c}}.

Note that, in contrast with the customary view, when v=-c, \nu'=\infty.

If we call the angle between the direction of our electromagnetic ray in K' and the connecting source-observer line \phi', the equation for \phi' assumes the form \cos\phi'=\frac{\cos\phi-v/c}{1-\cos\phi\cdot v/c}.

If \phi=\frac{1}{2}\pi, the equation becomes simply
\cos\phi'=-v/c

Wave Amplitudes

Let us now find the amplitude of the waves, as it appears in the moving frame K'. If we call the amplitude of the electric or magnetic force in K or K’ A or A' respectively, we obtain
{A'}^2=A^2 \frac{(1-\cos\phi\cdot \beta)^2}{1-\beta^2}

which simplifies into
{\rm A'}^2={\rm A}^2\frac{1-v/c}{1+v/c} if \phi = 0. (Prove this fact.)

It follows from these results that to an observer approaching a source of light with the velocity c, this source of light must appear of infinite intensity.

Energy Transformation of Light

Since the energy of light per unit volume is \frac{A^2}{8\pi}, we have to regard \frac{A'^2}{8\pi}, by the principle of relativity, as the energy of light in the moving system. Thus the ratio of in motion to at rest energy is given by $\frac{{A'}^2}{A^2}, if the volume of a light complex were the same, measured in bot K and K'. This is not the case.

If \hat{x}\hat{y}\hat{z}are the orthonormal direction vectors in our stationary system, no energy passes through the surface elements of a spherical surface moving with the velocity of light:

(x-ct\hat{x})^2+(y-ct\hat{y})^2+(z-ct\hat{z})^2=R^2

We may therefore say that this surface permanently encloses the same light complex. We inquire as to the quantity of energy enclosed by this surface, viewed in the K' frame, that is, the energy of the light complex relatively to K'.

In K' the spherical surface is an ellipsoidal surface, the equation for which, at \tau =0, is
(\gamma x'-\gamma x' \beta \hat{x})^2+(y'-y\gamma x'\beta\hat{y})^2+(z-\gamma x' \beta\hat{z})^2=R^2

If V is the volume of the sphere, and V’ that of this ellipsoid, then by a simple calculation
\frac{V'}{V}=\frac{\sqrt{1-\beta^2}}{1-\cos\phi\cdot \beta} = \gamma^{-1}(1-\cos\phi\cdot\beta)

Thus, if we call the total energy enclosed in the above mentioned volume E when it is measured in K and E’ when measured in K', we obtain
\frac{E'}{E} = \frac{{A'}^2 V'}{A^2 V} = \frac{1-\cos\phi\cdot \beta}{\sqrt{1-\beta^2}}

which for \phi=0, reduces to
\frac{E'}{E} = \sqrt{\frac{1-\beta}{1+\beta}}

Be sure not to confuse this total energy E with the electromagnetic field E (\vec{E}).

It is remarkable that the energy and the frequency of a light complex vary with the state of motion of the observer in accordance with the same law. (Or is it remarkable? Find out from which postulate this fact stems.)

Pressure on Perfect Reflectors

Now let the co-ordinate plane x'=0 be a perfectly reflecting surface, at which the plane waves considered above are reflected. We must find the pressure of light exerted on the reflecting surface, and the direction, frequency, and intensity of the light after it is reflected.

Let the incidental light in frame K be defined by A, \cos\phi, \nu. Viewed from K' the corresponding quantities are

    • A' = A \frac{1-\cos\phi\cdot\beta}{\sqrt{1-\beta^2}}
      \cos\phi'= \frac{\cos\phi -\beta}{1-\cos\phi\cdot\beta}
      \nu'= \nu\frac{1-\cos\phi\cdot \beta}{\sqrt{1-\beta^2}}
  • In K', the reflected light is simply:

    • A''= A'
      \cos\phi''= -\cos\phi'
      \nu''= \nu'
  • Finally, by transforming back to K, we obtain for the reflected light

    • A'''= A'' \gamma(1+cos\phi''\cdot \beta) = A \gamma^2 (1-2\cos\phi\cdot\beta + \beta^2)
      \cos\phi''' = \frac{\cos\phi'' +\beta}{1+\cos\phi''\cdot\beta} = \frac{(1+\beta^2)\cos\phi - 2\beta}{1-2\cos\phi\cdot\beta +\beta^2}
      \nu''' = \nu''\gamma(1+\cos\phi''\cdot\beta) = \nu\gamma^2{1-2\cos\phi\cdot \beta +\beta^2}
  • In K the energy incident upon a unit area of the mirror in a unit of time is \frac{A^2(c\cos\phi-v)}{8\pi}. The energy leaving the same is \frac{A'''^2(-c\cos\phi'''+v)}{8\pi}. The difference between these two expressions is the work done by the pressure of light in the unit of time. If we denote this work as the product Pv, where P is the pressure of light, we obtain
    P=2\cdot\frac{A^2}{8\pi}\frac{(\cos\phi-\beta)^2}{1-\beta^2}

    By letting v<<c, we can perform a Taylor approximation to the first order, and we get P=2\cdot\frac{A^2}{8\pi}, which is in agreement with experiment.

    Conclusion

    All problems in the optics of moving bodies can be solved by the method herein employed. What is essential is, that the electric and magnetic force of the light which is influenced by a moving body, can be transformed into an inertial frame at rest relative to the body. By this means all problems in the optics of moving bodies will be reduced to a series of problems in the optics of stationary bodies.



    September 13th, 2009

    Electrodynamics and Relativity (Part 2)

    Transformation of the Maxwell-Hertz Equations when Convection-Currents are Taken into Account

    Starting from the equations \frac{1}{c} \partial t E_{x_i} + \rho u(E_{x_i}) = \epsilon_{ijk}\partial_{x_j}B_{x_k}

    where \rho=\nabla\cdot\vec{E}

    denotes 4\pi times the density of electricity, and u(E_{x_i}) the velocity-vector of the charge. Since the electric charges are such things as ions, electrons, and protons, these equations are the electromagnetic basis of the Lorentzian electrodynamics and optics of moving bodies.

    Since we know these equations are valid in the frame K, we can transform them, with the equations given in this post and this one, to the system K'. We then obtain the equations
    \frac{1}{c}\frac{\partial E_x}{\partial \tau} + \rho' \nu(E_{x'_i}) = \epsilon_{ijk}\partial_{x'_j}B_{x_k}

    where \nu_{x'} = \frac{u_x - v}{1- u_{x}v/c^2}, \nu_{y'} = \frac{u_x}{\gamma(1- u_{x}v/c^2)},\nu_{z'} = \frac{u_x}{\gamma(1- u_{x}v/c^2)}

    and
    \rho' = \partial_{x'_i}E_{x'_i} = \gamma (1-u_{x}v/c^2)\rho

    Since it follows from the theorem of addition of velocities the vector \nu = (\nu_x', \nu_y', \nu_z') is nothing else than the velocity of the electric charge, measured in the K' frame, based on the covariance of the electromagnetic equations we have the proof that, on the basis of our kinematical principles, the theory of the electrodynamics of moving bodies is in agreement with the principle of relativity.



    September 12th, 2009

    Electrodynamics and Relativity (Part 1)

    Electrodynamics

    Let us have a stationary frame K where Maxwells equations hold. Remember, Maxwell’s equations in free space are written:

    \frac{1}{c}\epsilon_{ijk}\partial_{x_j}E_k = -\partial_{t}B_{x_i}
    \frac{1}{c}\epsilon_{ijk}\partial_{x_j}B_k = \partial_{t}E_{x_i}
    or, more familiarly, as \vec{\nabla}\times\vec{E} = -\frac{\partial \vec{B}}{\partial t} and \vec{\nabla}\times\vec{B} = \mu_{0}\epsilon_{0}\frac{\partial \vec{E}}{\partial t}

    If we apply to these equations the the frame transformations, by looking at the electromagnetic processes in the frame K', moving with the velocity v, we obtain the equations (in component form)
    \frac{1}{c}\frac{\partial E_x}{\partial \tau} = \frac{\partial}{\partial y'}\left\{\gamma\left({B_z}-\beta{E_y}\right)\right\} -\frac{\partial}{\partial z'}\left\{\gamma\left({B_y}+\beta{E_z}\right)\right\}
    \frac{1}{c}\frac{\partial}{\partial \tau}\left\{\gamma \left({E_y}-\beta{B_y}\right)\right\} = \frac{\partial B_x}{\partial x'} - \frac{\partial}{\partial z'}\left\{\gamma\left({B_y}-\beta{E_y}\right)\right\}
    \frac{1}{c}\frac{\partial}{\partial \tau}\left\{\gamma\left({E_z}+\beta{B_x}\right)\right\} = \frac{\partial}{\partial x'}\left\{\gamma\left({B_x}+\beta{E_z}\right)\right\} - \frac{\partial B_x}{\partial y'}
    \frac{1}{c}\frac{\partial B_x}{\partial \tau}= \frac{\partial}{\partial z'}\left\{\gamma\left({E_y}-\beta{B_z}\right)\right\} - \frac{\partial}{\partial y'}\left\{\gamma\left({E_z}+\beta{B_y}\right)\right\}
    \frac{1}{c}\frac{\partial}{\partial \tau}\left\{\gamma\left({B_y}+\beta{E_z}\right)\right\} = \frac{\partial}{\partial x'}\left\{\gamma\left({E_z}+\beta{B_y}\right)\right\} -\frac{\partial E_x}{\partial z'}
    \frac{1}{c}\frac{\partial}{\partial \tau}\left\{\gamma \left({B_z}-\beta{E_y}\right)\right\} = \frac{\partial E_x}{\partial y'} - \frac{\partial}{\partial x'}\left\{\gamma \left({E_y}-\beta{B_z}\right)\right}

    where \gamma=\frac{1}{\sqrt{1-\beta^2}} and \beta = \frac{v}{c}

    Now the first principle of relativity requires that if the Maxwell equations for empty space hold in frame K (that is K(x,y,z,t)), they also hold in frame K' (that is K'(x',y',z', \tau); that is to say that the vectors of the electric and the magnetic force \vec{E} and \vec{B}, satisfy the following equations in K':
    \frac{1}{c}\epsilon_{ijk}\partial_{x'_j}E_k = -\partial_{\tau}B_{x'_i}
    \frac{1}{c}\epsilon_{ijk}\partial_{x'_j}B_k = \partial_{\tau}E_{x'_i}

    Evidently the two systems of equations found for system K and K' must express exactly the same thing, since both systems of equations are equivalent. (Hence, why I showed them in tensor notation. For the component notion above, compare to the tensor F_{\alpha\beta} given in Maxwell’s equations.)

    Separating, and solving, we have

    • E_x' = E_x, E_y' = \gamma (E_y - \beta B_z),E_z' = \gamma (E_z + \beta B_y)

        B_x' = B_x,B_y' = \gamma (B_y + \beta E_z),B_z' = \gamma (B_z - \beta E_y)

      It is a common exercise to derive these equations yourself. Indeed, I encourage the interested reader to do so. Deriving them will yield great insight into relativity.

      And now, I quote Einstein himself

      As to the interpretation of these equations we make the following remarks: Let a point charge of electricity have the magnitude one when measured in the stationary system K, i.e. let it when at rest in the stationary system exert a force of one dyne upon an equal quantity of electricity at a distance of one cm. By the principle of relativity this electric charge is also of the magnitude one when measured in the moving system. If this quantity of electricity is at rest relatively to the stationary system, then by definition the vector (x, y, z) is equal to the force acting upon it. If the quantity of electricity is at rest relatively to the moving system (at least at the relevant instant), then the force acting upon it, measured in the moving system, is equal to the vector (x’, y’, z’). Consequently the first three equations above allow themselves to be clothed in words in the two following ways:

      • If a unit electric point charge is in motion in an electromagnetic field, there acts upon it, in addition to the electric force, an electromotive force which, if we neglect the terms multiplied by the second and higher powers of v/c, is equal to the vector-product of the velocity of the charge and the magnetic force, divided by the velocity of light. (Old manner of expression.)
        If a unit electric point charge is in motion in an electromagnetic field, the force acting upon it is equal to the electric force which is present at the locality of the charge, and which we ascertain by transformation of the field to a system of co-ordinates at rest relatively to the electrical charge. (New manner of expression.)
    • The analogy holds with magnetomotive forces. We see that electromotive force plays in the developed theory merely the part of an auxiliary concept, which owes its introduction to the circumstance that electric and magnetic forces do not exist independently of the state of motion of the system of co-ordinates.

      Furthermore it is clear that the asymmetry mentioned in the introduction as arising when we consider the currents produced by the relative motion of a magnet and a conductor, now disappears. Moreover, questions as to the seat of electrodynamic electromotive forces (unipolar machines) now have no point.



  • Your Ad Here