From: g.scholten@gmx.de   
      
   Luigi Fortunati wrote:   
      
   > Twin A is at the base of the tower where the watch  of twin B goes   
   > slower than that at the top.   
      
   You mean twin A is at the bottom of the tower and twin B is at the top   
   of the tower?   
      
      
   > Is the distance AB in the reference of the twin A greater, less or   
   > equal to the same distance measured in the reference of the twin B?   
      
   Obviously, you are referring to General Relativity (GR) and   
   gravitational time dilation. Right? However, this implies that the   
   concept of frames of reference is defined on short scales only, for   
   which the curvature of spacetime can be neglected. Near the location on   
   twin A, you can define a frame of reference for twin A, and near the   
   location of twin A, you can define a frame of reference for twin B, but   
   there is no frame of reference that covers a spacetime region that   
   contains both twins, A and B.   
      
   To describe such a spactime region, you can construct a coordinate   
   system, e.g. one in which both twins have fixed spatial positions. In   
   that coordinate system, there is a unique spatial distance AB between   
   both twins.   
      
   And before you ask: the light speed in that coordinate system is not   
   constant (unlike in the local frames of twin A and B that are defined on   
   local scales around twin A and B), but is lower at the bottom of the tower.   
      
      
   > Second question: if there is a light signal that goes from A to B and   
   > back reflecting on 2 mirrors placed in A and B, the round-trip time   
   > (from A to B and from B to A) measured by the clock of the twin A is   
   > greater, less or equal to the time measured by B for the path from B to   
   > A and from A to B?   
      
   With twin A at the bottom and twin B at the top of the tower, the round   
   trip time for the trip A -> B -> A measured by the clock of twin A is   
   less than the round trip time for the trip B -> A -> A measured by the   
   clock of twin B.   
      
   Assume you the coordinate system you constructed shows time translation   
   invariance, like Schwarzschild coordinates do. Then the half trip A -> B   
   always takes the same coordinate time Delta_t_AB, and the half trip B ->   
   A always takes the same coordinate time Delta_t_BA. So, the round trip   
   coordinate time for both trips A -> B -> A and B -> A -> B is the same:   
      
   Delta_t_ABA = Delta_t_BAB = Delta_t_AB + Delta_t_BA   
      
   Considered in Schwarzschild coordinates, the clock of twin A runs slower   
   than the clock of twin B, compared to coordinate time. So, the round   
   trip coordinate time Delta_t_ABA corresponds to a proper time interval   
   Delta_tau_A in the proper time of twin A that is shorter than the   
   corresponding proper time interval Delta_tau_B in the proper time of twin B.   
      
   That's why twin A sees signals from twin B blue-shifted and twin B sees   
   signals from twin A red-shifted.   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)