Stephen A. Lawrence wrote: > > > Harry Veeder wrote: >> Stephen A. Lawrence wrote: >> >>> This is not a paradox, and the "paradoxical" nature of the problem was >>> in fact resolved something on the order of a century ago. The traveling >>> twin accelerates; the stay-at-home twin does not; thus, the symmetry is >>> broken. >> >> That works in SR, but the solution is inconsistent with GR. > > Wrong. In fact the full solution can only be had using techniques > commonly considered to be part of GR. > > Acceleration is acceleration, in either SR or GR. In either case you > integrate the path length, using the pseudo-Riemannian metric of > Minkoski, in order to find the elapsed proper time of either twin, and > in either case the path which includes the acceleration comes out "shorter". > > A geodesic in GR is the longest distance between two points.
No, it is the shortest "distance" between two points on a spacetime manifold. > Acceleration pushes you off the geodesic, as a result of which you > follow a shorter path. If the two twins have worldlines which cross at > two points, and if one accelerates while the other follows a geodesic, > the one on the geodesic will "age" more. That's straight out of GR ... > or SR, take your pick. In GR, acceleration due to gravity is treated as indistinguishable from a manufactured acceleration. > If both accelerate, then neither follows a geodesic and you need to know > the details of the problem to determine who ages more (if either). > > GR and SR only really differ when you introduce gravity, which doesn't > enter into this problem. You can't ignore gravity. The raison d'être of GR is to explain gravity. Ignore gravity and you are back in the flat spacetime of SR. Harry
