6: Combination Rule/Regraduation Function- transforms speeds into quantities that are additive in the same way that slopes are not additive, but angles are. -Should satisfy: identity (0), annihilator (c), assosciativity, 4) d(u@v)/du and d(u@v)/dv exist and cnt in u and v 5) d(u@v)/du>0 and d(u@v)/dv>0 provided u,v!=0,c 8: If you assume that the combination rule must be symmetrical and linear, it must be of the form: l+m(u+v+w)+n(vw+wu+uv)+puvw=0 (u,v=input speeds k=resultant speed, w=-k) -When u,v=0, w must be 0 too, so l=0 -More generally, if v=0, then w=-u so l+n(-u^2)=0 therfore, since l=0, it follows that n=0 We now have: m(u+v+w)+puvw=0 -From the universal property if v=c, w=-c so: mu-puc^2=0 => m=pc^2 We now have: pc^2(u+v+w)+puvw=0 c^2(u+v+w)+uvw=0 w(c^2+uv)+(c^2)u+(c^2)v=0 k(c^2+uv)=c^2(u+v) k=c^2(u+v)/(c^2+uv) k=u+v/(1+uv/c^2) 9: This is similar to the hyperbloic tangent function: tanh(u+v)=tanh(u)+tanh(v)/(1+tanh(u)tanh(v)) So this suggests regraduating using: u=c*tanh(phi) Which makes phi=arctanh(u/c) Then cosh(phi)=1/sqrt(1-v^2/c^2)=gamma (originally Beta by Einstein) 12: Hyperbolic trig functions differ from ordinary trig functions only in their exponents differing by a factor of i. ---------End of Chapter 1--------- 51: F=dP/dt -equivalent way of expressing force, more compatible with QM 76: Twins Paradox- It seems like there is a paradox because both see the other going through time slower than themselves, but the spaceship twin leaves out a big period of time in the middle that corresponds to him turning around to come back to earth, which is the result of shifted lines of simultaneity. The earth twin has an uninterrupted view, so his calculations are correct and the spaceship twin ages less. 192: It is often said that the Special Theory implies that mass increases by the proportion of gamma, but this is only one of the two equivalent ways of interpreting it. Einstein didn't use this one, he used the convention that mass stays constant. 208: Orientable- doesn't allow continuous transformation of left-handed figures into right-handed figures. 214: Handed-ness has a fundamental asymmetry in some processes(favorative of one hand) but it appears that this is just the manifestation of a higher order symmetry (three factors combined are conserved, handedness alone isn't) 236: Einstein asked: Why is the inertial mass of a body the same as its gravitational mass? xxx: Covariant- varying together so as to preserve mathematical relations 248: Bradwardine- Galileo studied his formulation that acceleration is the derivative of velocity with respect to time rather than with respect to distance. 285: Huyghen's Principle- in 3D space, the wavefront remains sharp so that a sharply localized initial state is observed later as an effect that is equally sharply delimited. -Uses example of ripples in the surface of a pond to show that it doesn't work for 2D, but this is questionable because it seems like the only reason multiple ripples occur is because of the extra dimension of freedom that height allows for. It then claims that different scientists proved that different dimensions also didn't work until eventually only 3 was left. 293: Arago Spot- used initially to falsify the wave theory of light because it wasn't found, until Arago went looking for it and found it, hence falsifying the falsification 293: After metioning Rutherford's foil experiment: "Equally unexpected was the inversion of the tip-top." Then later: "The tip-top phenomenon, on the other hand, turned out to be deducible from classical mechanics." -I could not find further information on this!