In the book "The classical theory of fields" by Landau and Lifshitz,after defining interval showed that "if ds = 0 in one inertial system, then ds' = 0 in any other system. On the other hand, ds and ds' are infinitesimals of the same order.From these two conditions it follows that ds2 and ds'2 must be proportional to each other: ds2 = ads'2 where the coefficient a can depend only on the absolute value of the relative velocity of the two inertial systems."It is further proved that a=1. There could be higher order terms connecting ds and ds' like ds'^2 = ads^2+ bds^4 + c*ds^6 ... Those higher order terms may be neglected since they are extremely small.But still, won't that be an approximation to say that the interval is exactly equal?
$\begingroup$
$\endgroup$
1
-
$\begingroup$ Related : (1) Invariance of spacetime interval directly from postulate. (2) Schutz's geometrical proof that spacetime interval is invariant. $\endgroup$– FrobeniusJan 28 at 3:45
Add a comment
|