<Dd> P (t + d t) = (m − d m) (v + d v) + d m (v − u) = m v + m d v − u d m = P (t) + m d v − u d m (\ displaystyle P (t + dt) = (m - dm) (v + dv) + dm (v-u) = mv + mdv - udm = P (t) + mdv - udm \ qquad \ mathrm ()) </Dd> <P> Newtonian physics assumes that absolute time and space exist outside of any observer; this gives rise to Galilean invariance . It also results in a prediction that the speed of light can vary from one reference frame to another . This is contrary to observation . In the special theory of relativity, Einstein keeps the postulate that the equations of motion do not depend on the reference frame, but assumes that the speed of light c is invariant . As a result, position and time in two reference frames are related by the Lorentz transformation instead of the Galilean transformation . </P> <P> Consider, for example, a reference frame moving relative to another at velocity v in the x direction . The Galilean transformation gives the coordinates of the moving frame as </P> <Dl> <Dd> t ′ = t x ′ = x − v t (\ displaystyle (\ begin (aligned) t' & = t \ \ x' & = x-vt \ end (aligned))) </Dd> </Dl>

Derivation of conservation of linear momentum from newton's second law