Jesse, Yes, that is correct. That's what I'm referring to. Epstein diagrams, which Brent highly recommends, are also based in this single unifying concept which I refer to as the STc Principle. I also use it in my book...
Edgar On Wednesday, February 5, 2014 4:21:47 PM UTC-5, jessem wrote: > > > > On Wed, Feb 5, 2014 at 3:50 PM, Russell Standish > <[email protected]<javascript:> > > wrote: > >> On Wed, Feb 05, 2014 at 07:53:16AM -0800, Edgar L. Owen wrote: >> > >> > In fact relativity itself conclusively falsifies block time as it >> requires >> > everything to be at one and only one point in clock time due to the fact >> > that everything always travels at the speed of light through spacetime. >> I >> > find it baffling that so many can't grasp this simple fact. >> > >> >> That is only true for clocks travelling along null geodesics. For all >> other geodesics, proper time increases at a rate of 1 second per >> second, or c metres per second, where c is the speed of light >> expressed in metres per second. >> > > I believe Edgar is referring to the fact that the magnitude (norm) of the > 4-velocity vector is always c, regardless of what worldline you're looking > at (Brian Greene explains that this is what he means by "speed through > spacetime is always c", and gives a short derivation, on p. 392 of The > Elegant Universe). > > But as I pointed out, this isn't a great revelation. We define 4-velocity > in natural units (c=1) as (dt/dtau, dx/dtau, dy/dtau, dz/dtau), where > t,x,y,z are coordinates of some inertial frame and tau is proper time. > Analogously, in a purely spatial scenario involving a curved wire in 3D > space we can define a similar vector V=(dx/dL, dy/dL, dz/dL) at every point > on the wire, where x,y,z are position coordinates and L is a parameter > giving distance along the wire (how far an ant would have to walk to get > from the end of the wire to any given point). Then the magnitude of this > vector V is given by the square root of (dx/dL)^2 + (dy/dL)^2 + (dz/dL)^2 > which can be rewritten as (dx^2 + dy^2 + dz^2)/dL^2, but dL^2 is just equal > to (dx^2 + dy^2 + dz^2)--if an ant moves a small distance dL along the wire > we can treat its path as a straight line segment, and by the Pythagorean > theorem the length of any line segment is just the square of the coordinate > differences between its endpoints--so the magnitude of V is always 1 at > every point on the wire, regardless of the shape or orientation of the > wire. Greene's derivation of the fact that the magnitude of the 4-velocity > is always c is pretty much the same, except the analogue of the pythagorean > theorem in SR spacetime is that dtau^2 = dt^2 - dx^2 - dy^2 - dz^2, and you > also have to keep track of minus vs. plus signs when you multiply a > 4-vector by itself to calculate its norm. > > Jesse > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
