On Wed, Feb 5, 2014 at 3:50 PM, Russell Standish <li...@hpcoders.com.au>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 everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
