I'm calculating spacetime intervals in standard setup with frames A and B
where t = t' = 0 and x = x' = 0, and all the motion is going only in x direction.
Objects A and B move in relative velocity v = 0.99c, (giving us $\gamma= 7.089$) for duration of what A perceives to be 100 seconds.
I need to calculate spacetime interval undergone by the objects A and B, from each of the frames. Meaning ie sB is spacetime interval of B as calculated from reference frame A and s'A is spacetime interval of A calculated from B ( the apostrophe signals from which frame we are calculating)
Now spacetime interval undergone by B as seen from reference frame A should be
$sB^2= (ct)^2 - x^2 = (100c)^2 - (99c)^2 = 199c^2$
time is 100 seconds as seen from A, in that time with speed 0.99c the B moves 99 light seconds
Now we know that what A perceives to be distance of 99 light seconds will B perceive to be contracted by factor of $\gamma$, so in their proper time they will traverse this distance in shorter time $t' = t /\gamma$
$s'B^2= (100c / \gamma)^2 = (14.1c)^2 = 199c^2$
so far so good, even though im not sure this is correctly reasoned about
now for the spacetime interval of point A, first from the reference frame of A :
$sA^2 = (100c)^2 = 10 000c^2$
and them from reference frame of B:
$s'A = (100c/\gamma)^2 - (99c/\gamma)^2 = 4c^2$
completely different number!
Shouldnt they be equal, given that spacetime interval is invariant?
I must have made an error in my reasoning but I dont see where, can someone help me?