The relevance of directions in the cosmos

The fundamental notion of anisotropy is unconventional, but not new. In 1946, G. Gamow pointed out the ubiquitous presence of rotation of successively larger accumulations of matter, such as planets, stars, and galaxies [Gamow, 1946]. The rotation of stars presumably originated from the rotating spiral arms of the protogalaxy that the stars formed from, but what is the origin of the galactic rotation? Gamow proposed that the origin lies in some type of "universal rotation" about some axis along a certain direction in space. This implies that space is anisotropic, or inequivalent with respect to directions. And in 1949, K. Gödel at the Institute for Advanced Study in Princeton, New Jersey, showed that anisotropic solutions of A. Einstein's field equations of general relativity exist [Gödel, 1949]. His solution was equivalent to a rotation of matter relative to what he named "the compass of inertia" of the universe.

In 1982, P. Birch studied the polarization of electromagnetic radiation from various galaxies. Although his data sample was limited, and the statistical method he used was fairly simple, he found indications of a certain type of electromagnetic anisotropy in the data [Birch, 1982]. Birch took this as evidence for a universal rotation. Further work related to anisotropy and rotational asymmetry of the cosmos has followed [Ahluwalia and Goldman, 1993; Kendall and Young, 1984; Sachs, 1989].

Electromagnetic anisotropy could also be the signature of so-called axions or other pseudoparticles interacting with the electromagnetic field of charged particles. Several papers discuss this [Peccei and Quinn, 1977; Sikivie, 1984; Weinberg, 1978; Wilczek, 1978].

In this article, I will show how a new and well documented statistical analysis of a relatively large data set indicates a new type of electromagnetic anisotropy over the largest distance scales in the cosmos. The emphasis will be on describing the data analysis, since at this point, the correct explanation for our anisotropic effect is not known.

In the cosmos, there are many galaxies that emit highly plane-polarized electromagnetic radiation. In 1950, H. Alfven and K. Herlofson predicted that the strong plane-polarization of radio waves emitted from certain galaxies was produced by synchrotron motion of charged particles within such galaxies [Alven and Herlofson, 1950]. Such motion is a high-speed (close to the velocity of light), circular motion, which is the same type of motion imparted to elementary particles in so-called "synchrotrons," which are machines used to study such particles. Alven and Herlofson hypothesized that the charged particles revolved rapidly around a strong galactic magnetic field, emitting higly plane-polarized electromagnetic waves, with their polarization plane being perpendicular to the magnetic field.

Today, astronomers have accumulated a fair amount of data on the plane-polarization of radio waves that have traveled over cosmological distances. Such polarization data is generally quite valuable, since the extraordinarily vast distances involved in such data allow the detection of possible electromagnetic effects that presently would be unmeasurable over Earthly distance scales.

() = ² + .

This linear relationship between and ² is a characteristic feature of so-called "Faraday rotation," and is shown in Figure 1 below, where data from several different galaxies are plotted [Gardner and Whiteoak, 1963]. In its journey through the cosmic expanse, a plane-polarized wave passes through localized regions of space that are filled with magnetized plasmas of charged particles, like ions and electrons. The interaction between a magnetized plasma and the plane-polarized wave produces the rotation of the polarization plane of the wave. This so-called "Faraday effect" is a well-understood physical process.

The handedness and strength of the Faraday polarization rotation depend on the orientation and strength of the magnetic field in the plasma, the plasma density, and the wavelength of the wave. There will always be a component of the magnetic field in the plasma that is parallel to the wave's line of travel. If this component points in the same direction as the propagation direction of the wave, the rotation of the wave's polarization plane will be counterclockwise, as observed from a point on the wave's line of travel where the wave is approaching you. If the magnetic field component points oppositely to the wave's propagation direction, the Faraday polarization rotation is clockwise. The magnitude of the polarization rotation depends on the magnitude of the magnetic field component along the wave's line of propagation, the density of charged particles making up the plasma, and on the wavelength of the wave. Experimentally, it is found that the amount of Faraday rotation is proportional to the square of the wavelength of the wave.

The constant in Equation 1 is generally different for different galaxies. It is called the "Faraday rotation measure" of a galaxy, and represents the strength of the polarization rotation of waves emitted by the galaxy. Its magnitude depends upon the magnetic field strength and the electron density along the line of sight from the galaxy to Earth. Note that conventional Faraday rotation does not account for the angle , the orientation of the polarization plane of the wave at = 0, as found by extrapolation of the Faraday rotation lines in Figure 1. represents the orientation of the polarization plane before the Faraday mechanism rotates it. , as opposed to , is the angle we work with, since it represents the part of the polarizational data that does not involve the known Faraday effect.

Figure 1. The observed orientation angle of the polarization plane of a galactic radio wave varies linearly with the square of the wave's wavelength, ². This dependency is shown here for various radio galaxies. ² is plotted along the x-axis, and is plotted along the y-axis. The y-intercepts are the angles , which remain after the Faraday rotation effect is subtracted out of the data (Taken from Fig. 1 in the paper by Gardner and Whiteoak, 1963).

The polarization data we analyzed consisted of the angle that labels the observed, "Faraday-compensated" orientation of the plane of polarization of radio waves emitted by 160 galaxies. These polarization orientations are quite meaningless by themselves however, unless they are compared with some other similar, physical characteristic of their respective galaxies. Since all of the observed galaxies were elliptical in shape, astrophysicists measured the orientation of the axis of elongation - also called the "major axis" - of the galaxies for this purpose. This orientation is specified as an angle . The two angles are illustrated in Figure 2 below. They are by definition restricted to the be between 0 and 180, since an angle greater than 180 is superfluous. = 191 for example, represents the same orientation of the polarization plane as = 11.

Figure 2. The angles and are measured by astronomers to pinpoint the orientation of the observed major axis of a galaxy ( ) and the orientation (after Faraday rotation has been subtracted) of the observed polarization plane of a wave emitted by the galaxy ( ). E is the electric field of the polarized wave, and oscillates within the polarization plane of the wave (The polarization plane is by definition the plane that contains the line of travel of the wave and the electric field of the wave).

The measurement uncertainties were less than 5 for and typically 5 for . In our analysis, is a central parameter - it represents a possible rotation of a wave's polarization plane that is not explicable in terms of the Faraday effect. The frequency of the radio waves emitted from the 160 galaxies in our data set varies, but typically spans a range of 1 to 3 GHz. Their visual magnitudes are between 8 and 23.

Positional information on galaxies abounds [Burbidge and Crowne, 1979; Spinrad, Djorgovski, Marr and Aguilar, 1985]. As part of our total data set, we recorded the positional coordinates of the 160 galaxies above that we had found polarization data for. The positional coordinates of galaxies are given in the astronomical literature as the redshift, declination and right ascension of a galaxy. We used the redshift z to compute the distance r to a galaxy from the expression appropriate for a universe of "critical average mass density," namely

where h₀ = (2/3) (10^-10 years^-1), and h is the Hubble constant.

Right ascension and declination are so-called "equatorial coordinates" for specifying spatial directions in the cosmos. The equatorial coordinate system in astronomy is analogous to Earth's cordinate system of latitudes and longitudes. Declination corresponds to latitude, and right ascencion corresponds to longitude. A direction of 90 declination points along Earth's polar axis toward the North pole (extending beyond Earth to outer space), 0 declination points somewhere along Earth's extended equatorial plane, and -90 declination points along Earth's polar axis toward the South pole. Positive declination values refers to directions in the northern celestial hemisphere, while negative declination values refers to directions in the southern celestial hemisphere. Right ascension values are celestial latitudes, running from 0 to 24 hours. The direction of 0 declination and 0 hours right ascension points from Earth to the point in space where the Sun's ecliptic intersects Earth's equatorial plane at Spring equinox.

It is worth noting that the positional distribution of the 160 galaxies we studied is not uniform over the sky. Rather, the majority of the galaxies come from the northern sky, since most of the world's radio observatories are located in Earth's northern hemisphere.

(1) galaxies - since the angles are part of the polarization properties of the waves emitted by the galaxies,
(2) electromagnetism - since the waves are electromagnetic waves, and
(3) space - since the waves travel through immense distances, and from all directions in the universe.

Some studies of the and angles have promted astronomers to propose a so-called "two-population hypothesis" to explain the observed values of and . This hypothesis asserts that there are two populations of galaxies out there: one population in which a galaxy's observed radiation has its polarization plane oriented approximately parallel to the major axis of the galaxy ( | - | = 0 ), and one in which a galaxy's observed radiation has its polarization plane oriented approximately perpendicular to the major axis of the galaxy ( | - | = 90 ) [Clarke, Kronberg, and Simard-Normandin, 1980]. But these conclusions were based on a very small subset of the galaxies that have polarization measurements taken on them. The full data set available indicates that the observed polarization plane orientation relative to the major axis take all possible values, so one really need to assume that several "populations" exist, in order to explain the data. This is admittedly not a very satisfying explanation.

As an alternative to the invokation of arbitrary of ad-hoc galaxy populations, we asked ourselves whether there is a pattern in the and angle data that can be explained by some general, unifying relationship. We were not able to find any relationship that could arise from any obvious, conventional physical theory, as discussed toward the end of this article. We therefore decided to explore unconventional relationships - relationships that, if present in the data, would force Physics to move to new frontiers.

This mode of scientific investigation is extraordinarily useful, since it contains in it the seeds for scientific improvement. If one always studied phenomena within the framework of conventional science, the chance of hitting upon something new that could not be explained within our present understanding would be minimal. And if one never observed contradictions to a theory - because one was too complacent to care to look for them - the theory could never be improved upon.

One unconventional relationship we investigated was whether the measured values for the and angles of a galaxy depended on the direction of the line of sight to the galaxy on the sky. More specifically, we investigated whether the observed angles can be reproduced by assuming that the polarization plane of a wave emitted by a galaxy is initially oriented at a fixed angle relative to the galaxy's major axis, and then undergoes a rotation (as specified by an angle ) that depends on the wave's direction of travel. My Ph. D. dissertation [Nodland, 1995], and also to some degree our article in Physical Review Letters [Nodland and Ralston, 1997], present theoretical calculations which predict a specific mathematical form for such a rotation. These calculations are quite involved, so I will mention them only briefly toward the end of this article. The final result of the calculations is that, to first order in ^-1, the rotation angle for a particular galaxy is given by

= (1/2) ^-1 r cos ,

where is the angle between a fixed direction "s" in space and the line of sight to the galaxy. r is the distance to the galaxy, and (with units of length) is a constant of proportionality, representing the "size" of the rotation. It is clear that any dependency of on the sky angle , such as Equation 3, would be an indication of anisotropy (inequivalence of directions) in the behavior of electromagnetic wave propagation through the universe. Figure 3 illustrates the relationship given by Equation 3.

Figure 3. Equation 3 says that a rotation of the polarization plane of a plane-polarized electromagnetic wave from a galaxy depends on the angle between a fixed spatial direction s, and the propagation direction k of the wave. E(1) and E(2) are the electric field vectors (which define the orientation of the wave's polarization plane) of the wave at two different points on the wave's line of travel.

We handled the ambiguity by making the fairly reasonable assumption that any anisotropic polarization rotation must be small. This assumption, and the requirement that the rotation be signed, led us to restrict the angles to have values between - and + only. In order to calculate a value for a polarization rotation , one must assume some initial orientation of the polarization plane at the galaxy. Our choice was the simplest possible - that the polarization plane was initially oriented parallel to the galaxy's major axis.

These relatively simple conditions allow the data analysis to be manageable. One may assume other initial orientations, or a larger range for , but such conditions are not fundamentally different from the simple conditions we employed, and would not fundamentally change results. With the simple conditions above, two possible rotations of the initial polarization plane of a wave emitted from a galaxy will reproduce the observed polarization plane orientations. One rotation is positive (⁺) and one is negative (^-), as seen in Figure 4.

Figure 4. The positive rotation ⁺ or the negative rotation ^- of a polarization plane initially oriented along the major axis of a galaxy will produce the observed orientation of the polarization plane. Part (a) shows the rotations when - 0, and part (b) shows the rotations when - < 0.

From Figure 4, we see that the mathematical expressions for ⁺ and ^- in terms of and are

⁺ = - if - 0,
⁺ = - + if - < 0,
^- = - - if - 0,
^- = - if - < 0.

A rotation given by Equation 3 is either positive or negative depending on the angle . Furthermore, depends on the direction of the fixed direction s, and the direction to the galaxy for which is computed. To allow to be either clockwise or counterclockwise, we therefore computed it from the galaxy's , , and values according to the natural assignment

= ⁺ if cos 0,
= ^- if cos < 0,

where ⁺ and ^- are computed from the galaxy's and values according to Equation 4.

R = (N x_i y_i - x_i y_i)
{ [ N x_i² - ( x_i)² ]^1/2 [ N y_i² - ( y_i)² ]^1/2 }^-1.

The assignment in Equation 5 of positive angles when r cos is positive, and negative angles when r cos is negative, necessarily introduces artificial linear correlations into the data set, because two quadrants of the data plane of and r cos are excluded. This causes the values of the computed correlation coefficients R_data to be too high. In addition, the spatial non-uniformity of the galaxies' distribution over the sky may have an artificial effect on the value of a correlation coefficient.

Because of this, the actual value of R_data is not very informative. What one needs to do is to compare the correlation coefficient R_data of the true data set (r_i cos _i, _i) with the correlation coefficient R_rand of a data set that has random values computed in the same "correlation-producing" way as that of the real data, and that has the same "non-uniform" r_i cos _i values as those of the true data set.

We achieved this by computing the correlation coefficient R_rand of the set (r_i cos _i, [_{i, rand}, _{i, rand}]), where [_{i, rand}, _{i, rand}] are obtained by substituting random major axis angles (_{i, rand}) and polarization angles (_{i, rand}) into Equations 4 and 5. We drew _{i, rand} and _{i, rand} from uniform, random distributions, consistent with the fact that the _i and _i in the observed data set are also uniformly distributed. To account for the non-uniformity of the galaxy distribution, we did not randomize the r_i cos _i part of the data.

In order to make a reliable comparison with R_data, one needs to calculate a large number of R_rand values by repeatedly drawing _{i, rand} and _{i, rand} angles to produce several sets (r_i cos _i, [_{i, rand}, _{i, rand}]), and then calculating R_rand for each set. In this way, we computed 1000 "random" correlation coefficients R_rand to be compared with the true correlation coefficient R_data. The comparison consisted of determining the fraction P of the 1000 R_rand values that equaled or exceeded the corresponding R_data value. In statistics, this fraction is called a "P-value," and the method of computing it by randomizing the data and computing a large number of R_rand values, is called a "Monte Carlo method." P estimates the probability that the correlation R_data arises from random fluctuations in the data. An other way to state this is to say that P estimates the probability that a correlation given by Equation 3 does not exist in the data.

We computed R_data, the corresponding 1000 R_rand values, and the corresponding P-value for the set of pairs (r_i cos _i, _i) for over 400 trial orientations of the direction s, which systematically covered all directions in space. As one varies s, all the _i change, as well as the _i in the set. In conformity with astronomical conventions, we specified s in terms of a declination angle and right ascension angle, which are described in the previous section. One should again note that the coefficients R_rand are not computed from truly random galaxy sets, as explained above. They were computed because the value of R_data doesn't tell us how significant a correlation (as given by R_data) is. Only P can tell us how significant the correlation is. Figure 5 summarizes the computational procedure described in this section.

Figure 6. The plot of inverse P-value versus spatial direction s (given in terms of their declination and right ascension) for the full data set of 160 galaxies shows that 1/P is large only for s in the region s = (declination, right ascension)_s = (-10 20, 20 hours 2 hours) compared to all other directions for s. This indicates anisotropy in the data. P is the fraction of galaxy sets with randomized major axis (_{i, rand}) and polarization (_{i, rand}) angles that yielded a linear correlation coefficient R_rand of the set [r_i cos _i, (_{i, rand}, _{i, rand})] greater than or equal to the linear correlation coefficient R_data of the actual set [r_i cos _i, (_i, _i)].

To explore this indication of anisotropy, we selected the galaxies in the data set that had redshifts greater than 0.3, roughly the most distant half of the sample (71 galaxies). The electromagnetic radiation from these galaxies has traversed a large fraction of the universe, and would serve as an ideal sample for a check of whether the apparent anisotropy seen in Figure 6 is more or less prevalent when the effects of distant regions of space are given more emphasis. Surprisingly, we found an even stronger signal of anisotropy for this galaxy set. We see in Figure 7 a well-connected cluster of peaks in 1/P when s is in the region s = (declination, right ascension)_s = (0 21, 21 hours 2 hours). In this region, P has a value of 0.001 or less. P = 0.001 is the lowest value we can resolve with 1000 randomizations of the data set. Several of the s-directions displayed in Figure 7 had no Monte Carlo events with R_rand R_data in the 1000 randomizations. As a conservative estimate, we assigned P = 0.001 to those directions.

Figure 7. For the 71 most distant galaxies in the data set (those with redshift z 0.3), the plot of inverse P-value versus spatial direction s shows that P is of order 10^-3 or smaller for s in the region s = (declination, right ascension)_s = (0 20, 21 hours 2 hours), while all other directions of s yield P-values that are about 100 times or more larger. This is quite a strong indication of anisotropy.

As seen in Figures 6 and 7, analysis of the synchrotron radiation data pinpoints only approximately the orientation s which yields a signal of anisotropy in the data, as quantified by a very small associated P-value. We may call this s-direction the "direction of anisotropy." We may visualize the direction by an infinitely long line, or "anisotropy axis" that runs through the universe through the two points Earth and Sextans. We see that the data strongly indicate that the anisotropy direction lies within an "anisotropy cone" that has its vertex at Earth, and its central axis pointing from Earth to the constellation Aquila, which is in the direction (declination, right ascension) = (0, 21 hours). Its surface makes approximately a 20 angle with the central axis. The data provide no support for an anisotropy direction anywhere outside this cone. In the opposite direction, from Earth to the constellation Sextans, which has the approximate coordinates (declination, right ascension) = (0, 7 hours), the anisotropy axis is confined within a similar cone, so that the anisotropy cone is really a "double cone."

In Figure 8, the double anisotropy cone is shown in red, positioned with its vertex at Earth, at the center of the figure, and opening up toward the constellation Aquila in one direction, and toward the constellation Sextans in the opposite direction. Our data, consisting of 160 radio galaxies, are shown as yellow dots. The most distant galaxies in the data set are about 7 billion light years away.

For the s-direction with highest 1/P value of the full data set, the distribution of R_rand is a Gaussian, having a mean = 0.60 and standard deviation = 0.032, with R_data = 0.66 = + 1.88 . In contrast, in a typical direction away from the anisotropy direction, like (declination, right ascension) = (60, 12 hours) for example, the distribution is given by = 0.47 and = 0.04, with R_data = 0.48 = + 0.25 .

For the data set of distant galaxies with z 0.3, and for an s-direction yielding a high 1/P value, the distribution of R_rand is also Gaussian, with a typical mean value of = 0.76 and standard deviation = 0.027, with R_data = 0.86 = + 3.7 . Distributions of R_rand with long tails were not seen. As mentioned above, the spatial distribution of galaxies in the sample is non-uniform, so that the number of galaxies assigned to in Equation 4, and the and values for R_rand, depend on the trial s-direction. The P-values, displayed in Figures 4 and 5, are therefore much more meaningful than the values of the correlation coefficients R_data themselves.

The average best fit value for the proportionality constant in Equation 3 is = (1.1 0.08) 10²⁵ (h₀ / h) meters for an s-direction of s = (declination, right ascension)_s = (0 20, 21 hours 2 hours) for the data with z 0.3. Here h₀ = (2/3) (10^-10 years^-1), and h is the Hubble constant. For the full data set of all 160 data points, we find that = (0.89 0.12) 10²⁵ (h₀ / h) meters for s = (declination, right ascension)_s = (-10 20, 2 hours 2 hours). So the length scale is of order a billion light years.

We also employed a second statistical test on the data, a test that is conceptually disparate from the one described above. For each "random" data set (r_i cos _i, [_{i, rand}, _{i, rand}]), we varied s over the celestial sphere (410 directions) to maximize R_rand. Again, galaxy positions were not randomized, as explained above. This "largest-R_rand" value was then recorded. A new random set was then generated, producing another "largest-R_rand." This calculation was repeated more than 1000 times, to create a set of largest-R_rand's. This procedure was motivated by the fact that there is an increased probability in the first procedure of obtaining a fit of s to the data due to the two degrees of freedom of s.

The important test is for the far-half galaxies with redshift z 0.3, since that galaxy set exhibited a P-value of order 0.001 in the first statistical procedure. For the far-half sample with z 0.3, we found that the fraction of the largest-R_rand's that exceeded R_data when s = (declination, right ascension)_s = (0 21, 21 hours 2 hours) was less than 0.006. This is again an indication of anisotropy. For the closest half of the galaxies with z < 0.3, the fraction of the largest R_rand's exceeding R_data when s = (declination, right ascension)_s = (0 20, 21 hours 2 hours) was 0.86, indicating that the anisotropic effect seems to be present only for the most distant half of the galaxies. These results corroborate the conclusion of the first statistical procedure described above.

In particular, the rate of rotation of the polarization plane caused by the new effect depends on the angle (denoted above) between the direction of travel of the polarized wave and a fixed direction in space (denoted s above), pointing approximately toward the constellation Aquila from Earth. The more parallel the direction of straight-line travel of the wave is with this fixed direction, the greater the rotation of the polarization plane of the wave (as given by Equation 3 above). The amount of polarization rotation is also proportional to the distance of travel of the wave. These are the only two dependencies of the rotation.

The anisotropic effect is illustrated in Figure 9. In this diagram, Earth is at the center, and the direction toward Sextans is represented by an infinitely long "anisotropy axis" (red). The axis extends from Earth toward Sextans in one direction, and toward the constellation Aquila in the opposite direction. A plane-polarized radio wave emitted by Galaxy A (green) travels in a straight line toward Earth in a direction almost parallel to the anisotropy axis (red). On the other hand, a plane-polarized radio wave emitted by Galaxy B (blue) approaches Earth in a direction almost perpendicular to the anisotropy axis.

As the two waves propagate along straight lines through space, their planes of polarization rotate around those lines, as represented by the green and blue helices. The distances of travel are the same for both waves, but the wave traveling nearly parallel to the anisotropy direction (green wave) has its polarization plane rotated more than the wave traveling in a direction nearly perpendicular to the anisotropy direction (blue wave). In general, we find that the polarization rotation increases systematically as a wave's direction of travel approaches that of the fixed anisotropy direction (red line). For illustrative purposes, the rotation effect in this diagram is exaggerated. The actual effect is extremely tiny: we find that, on the average, one full revolution of the polarization plane is completed after the wave has voyaged for about ten billion years (as found from Equation 3 and the constant above).

It is important to note that the infinite anisotropy axis running through Aquila, Earth and Sextans, as shown in Figures 7 and 8, only represents a direction, or, in the vernacular of Mathematics, a vector, in space. Any other axis - possibly vastly remote from Earth, Sextans and Aquila - parallel to the anisotropy axis shown here, will suffice in defining the anisotropy vector. No particular location in space, like the location of Earth for example, is relevant - only directions are relevant.

We have of course considered the possibility that a local effect of the galaxy, via some unanticipated conventional physics, might account for our correlation. However, the fact that the correlation is seen for z 0.3, but not for z < 0.3, rules out a local effect. Strong magnetic fields at a galaxy might generate unexpected initial polarization orientations, or upset the Faraday-based fits, and this could plausibly depend on redshift. But since the correlation is observed over the sky angle , any such explanation requires an unnatural, if not impossible, conspiracy between distant galaxies at widely separated sky angles.

One is left, then, with the option of contemplating new physics. In the language of the Quantum Field Theory branch of Physics, we show that the anisotropic polarization rotation , illustrated in Figure 9, can be generated by a coupling of the electromagnetic field of the wave, represented by its so-called "electromagnetic field tensor" F and "electromagnetic four-potential" A , to a new, four-dimensional vacuum field s, whose "spatial part" s is the anisotropy vector we discovered [Nodland 1995; Nodland and Ralston, 1997]. The so-called "Lagrangian density" L for an extended theory of electromagnetism that incorporates this coupling is, to first order in ^-1, given by

L = -(1/4) F F + (1/4) ^-1 A s.

The second term in this equation represents an anisotropic extension to electrodynamics. is here a scale of dimension length, and is the four-dimensional "Levi-Civita tensor." From the so-called "Euler-Lagrange equations," this Lagrangian density yields a modified set of Maxwell equations for the electromagnetic field. From these equations, one obtains the so-called "dispersion relation" between the wave's wavenumber k and frequency given by

k = (1/2) ^-1 cos

We see from Equation 8 that the wave has two propagation modes, one given by k₊, and one by k_-. A rotation of the plane of polarization of the total wave arises from the difference in phase speeds between the two modes, and is given by

= (1/2) r (k₊ - k_-).

Substitution of Equation 8 into Equation 9 finally yields the anisotropic expression in Equation 3 for the polarization rotation . When subjected to coordinate transformations such as "time reversal" and "space inversion," the new field s behaves in the same manner that the intrinsic spin of an atom or elementary particle does, when the atom or particle is subjected to such transformations. One may therefore affix some sort of "spin" to s.

Other proposed theories and explanations for the anisotropic polarization rotation have recently appeared after our article in Physical Review Letters [Nodland and Ralston, 1997] was published [Bracewell and Eshleman, 1997; Dobado and Maroto, 1997; Kühne, 1997; Obukhov, Korotky and Hehl, 1997; Moffat, 1997; Mansouri and Nozari, 1997; Sachs, 1997].

Over the centuries, we have gradually learned more about the world we live in. We once thought the Earth was flat, then realized it is a sphere. We thought the sun revolved around the Earth, then realized the Earth revolves around the sun. And now most people believe the universe is isotropic, or directionless - maybe this is not so either.

In one sense, the anisotropic polarization twist that seems to take place does not really matter in our daily lives. However, part of being human is having an innate curiosity about the world. Who are we, and why are we here? Millions of people around the world ask these questions. A similar yearning drives physicists, who ask the same questions on a more cosmic scale: What is this universe, and how - and why - did it come into existence? I hope that our findings can contribute in some small way to answering these questions, and satisfying the curiosity we all share.

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