I've mentioned several times my admiration for Richard Feynman and tonight while roaming freely, I chanced across this little gem. http://www.mathpages.com/home/kmath320/kmath320.htm <http://www.mathpages.com/home/kmath320/kmath320.htm> Yes, Mathpages is one of my favorite comedy sites. The section on stellar aberration attempts to validate mathematically that aberration proves the earth is moving, using the MS paradigm of special relativity. http://www.mathpages.com/rr/s2-05/2-05.htm <http://www.mathpages.com/rr/s2-05/2-05.htm> From within this section: ??. For example, consider a binary star system in which one large central star is roughly stationary (relative to our Sun), and a smaller companion star is orbiting around the central star with a large angular velocity in a plane normal to the direction to our Sun, as illustrated below. It might seem that the periodic variations in the velocity of the smaller star relative to our Sun would result in significantly different amounts of aberration as viewed from the Earth, causing the two components of the binary star system to appear in separate locations in the sky - which of course is not what is observed. Fortunately, it's easy to show that the correct application of the principles of special relativity, accounting for the non-uniform variations in the orbiting star's velocity, leads to prediction that agree perfectly with observation of binary star systems. At any moment of observation on Earth we can consider ourselves to be at rest at the point P0 in the momentarily co-moving inertial frame, with respect to which our coordinates are Notice anything wrong? The first error: This is a GC system, when it should be HC, to be MainStream. The Earth should be in polar coordinates centered on the Sun, representing the earth in orbit while light is in transit from the binary stars. As set up, the earth never moves; at any time t, it?s at the origin! Suppose the large central star of a binary pair is at point P1 at a distance L from the Earth with the coordinates V is the speed of stellar rotation as seen from earth - again, the GC view. The fundamental assertion of special relativity is that light travels along null paths, so if a pulse of light is emitted from the star at time t = T and arrives at Earth at time t = 0, we have This is derived from the null space-time interval for light: r2 ?(ct)2 = 0 . Units are being used in which c = 1, so r2 ? t2 = 0 So r02 ? t02 = 0 and r12 ? t12 = 0 and thus r02 ? t02 = r12 ? t12 Plug in the initial values to obtain the equation above. Now the second error: Light is emitted from the star at a time T AFTER it arrives at Earth at time zero!. Light is traveling backwards in time, received before emitted. The MS trademark ? contradictions. The correction: set t = 0 in the star system #1 and t= T on earth ? system #0. Inserting these values into the last equation, 0 ? T2 = L2 ?0 ; ? T2 = L2 Since the left side is always negative and the right side always positive, we have found another MS trademark contradiction! Error 3, and counting. But why the error, if GC was used (albeit under disguised pretenses)? Suppose r02 ? t02 = k and r12 ? t12 = k, where k is constant and non-zero. Then it would also be true that r02 ? t02 = r12 ? t12 , but they are not each equal to zero !! The equation is degenerate, folks, because the value of the constant k has been removed??. Error # 4. This site is the reference of many other MS sites for explanations of relativity math?. God help us all. There are more errors below, but if you get the point - what?s the point ?? Robert and so from which it follows that x1/z1 at time T is . Thus, for the central star we have the aberration angle Now, what about the aberration of the other star in the binary pair, the one that is assumed to be much smaller and revolving at a radius R and angular speed w around the larger star in a plane perpendicular to the Earth? The coordinates of that revolving star at point P2 are where q = wt is the angular position of the smaller star in its orbit. The fundamental principle of special relativity is that light travels along null paths, so a pulse of light arriving on Earth at time t = 0 was emitted at time t = T satisfying the relation Solving this quadratic for T (and noting that the phase q depends entirely on the arbitrary initial conditions of the orbit) gives If the radius R of the binary star's orbit is extremely small in comparison with the distance L from those stars to the Earth, and assuming v is not very close to the speed of light, then the quantity inside the square root is essentially equal to 1. Therefore, the tangents of the angles of incidence in the x and y directions are The leading terms in these tangents are obviously just the inherent "static" angular separation between the two stars viewed from the Earth, and the first term in the x tangent is completely negligible (assuming R/L and v are both small compared with 1), so the aberration angle is essentially which of course is the same as the aberration of the central star. Indeed, binary stars have been carefully studied for over a century, and the aberrations of the components are consistent with the relativistic predictions for reasonable Keplerian orbits. (Incidentally, recall that Bradley's original formula for aberration was tan(a) = v, whereas the corresponding relativistic equation is sin(a) = v. The actual aberration angles for stars seen from Earth are small enough that the sine and tangent are virtually indistinguishable.)