I am reading Landau's Volume 2 of the course of theoretical physics. I have a doubt after reading the first few pages of it which I explain below.
Landau first defines intervals and on pages 5 and 6 shows that two events having time like interval between them can never occur simultaneously in any reference system. Then he goes on to construct a 2D space-time graph (for visualization) with an event O occurring at (0,0,0,0). Then he considers any event which occurs in future in that frame and is time-like w.r.t. origin and says on page no. 7,
But two events which are separated by a time-like interval cannot occur simultaneously in any reference system. Consequently, it is impossible to find a reference frame in which any of the events in region aOc occurred "before" the event O, i.e. at time t<0.
The argument above just proves that because interval square should be positive, i.e. the events can't be simultaneous. But, if I replace the difference in time in the original frame with its negative in my proposed frame and let the space distance between them to be same in both frames, then I get an in my proposed frame an interval which is time like but in it the order of events is changed. Am I making some gross error or Landau has missed some argument?