Based on the postulate "That all inertial obeservers are equivalent" neither A nor B can establish which one is actual moving. This requires that the recorded moments when B receives A's flashes and when A receives B's flashes should be the same. That means that the dependence (ratio) a1 to B1 should be equal b1 to A1.
t = time ^ . B2 . . A2 a1 . .. b1 . . . . . . . . . . . . . . . . . . . B1 A1 . . . . . . . . . . . . . .. O................ >x Figure I(Angle a1,A1 with horizontal and Angle b1,B1 with horizontal should be 45 degrees)
B will receive flash A1 at point a1
coordinate of a1 = (x,t) = (2,4)
A will receive flash B1 at point b1
coordinate of B1 = (x,t) = (x,2x)
coordinate of b1 = (x,t) = (0,3x)
x = unknow, to be calculated
In order that A receives the first flash "simultaneous" with B receiving the first flash from A (in the sense that the time of A's clock is the same as B's clock) the following equation has to be true
Ob1 / OA1 = Oa1 / OB1 = ratio
or
3x / 2 = 2V5 / xV5 = 2 / x
x² = 4/3
x = V 4/3 V = sqrt
Coordinate of B1 = (V 4/3,2V 4/3)
coordinate of b1 = (0, 3V 4/3)
ratio = 3/2 V 4/3
A's clock flashes at the points A1, A2 and A3 etc
Coordinate A1 = (x,t) = (-a,4a)
Coordinate A2 = (x,t) = (-2a,8a)
B's clock flashes at the points B1,B2,B3
t = time ^ . . . . . . b1 . a1 .. . .. . . . . . . . . . . . . . . . . . . . . A2 . . . B2 . . . . . . . . . . . . . . . . . . A1 . B1 C . . . . . . . . . ... O................ >x Figure II(Angle a1,A1 with horizontal and Angle b1,B1 with horizontal should be 45 degrees)
B will receive flash A1 at point a1
coordinate of a1 = (x,t) = (x,4x)
A will receive flash B1 at point b1
coordinate of B1 = (x,t) = (a,4a)
coordinate of B2 = (x,t) = (2a,8a)
a and x = unknow
In order that A receives the first flash "simultaneous" with B receiving the first flash from A (in the sense that the time of A's clock is the same as B's clock) the following equation has to be true
Ob1 / OA1 = Oa1 / OB1 = ratio = x / a
B1C = b, a1C = 4b (b = temp variable)
A1C = a1C = 2a + b = 4b
or
2a = 3b
Also a + b = x or b = x - a
2a = 3b = 3(x-a)
3x = 5a
ratio = 5/3
compare this result with calculation 1
The mathematics I have used IMO is correct, that is not so much the problem
The question is: which of the above calculations is correct ? or are the both wrong ? or are they both right, each based on different assumptions ?
IMO only a real experiment can decide.