Question

Imagine a ball with diameter a at rest and another observer at rest at a distance of 'l' w.r.t to the ball in this rest frame.Say the ball suddenly starts moving at velocity v towards the observer .(Assume it accelerates instantaneously).What does the observer measure the distance between them?What does the observer measure the diameter of the ball to be?If the diameter is contracted by gamma factor then why is not 'l' contracted by the same?Because acc to me by time dilation,as the time interval is zero wrt to the observer between the events (ball moving and ball at rest),so the distance moved is zero acc to observer and hence observed distance remains 'l'.I asked my professor,he also said that diameter would contract but distance of ball wrt observer wont , but I didnt understand why this partial treatment of two lengths ?.They are identical in properties with regard to lorentz transformations so if one contracts wrt observer the other should also.

Answer 1

Length contraction affects only the measurements of lengths parallel to the direction of motion, so while the ball's diameter contracts due to its perpendicular alignment to the motion, the distance 'l' does not contract as it is not a part of the object in motion but a position coordinate from the observer to the ball.

Step-by-step explanation:

The length contraction phenomenon in special relativity only applies to the dimensions of objects that are parallel to the direction of their relative motion. In the scenario described, the observer perceives the ball's diameter, which is perpendicular to the direction of motion and thus experiences length contraction due to its motion towards the observer.

However, the distance 'l' remains unchanged for the observer because the length contraction only affects lengths measured along the direction of relative motion between the observer and the observed object.

Proper length (Lo) is the term used to describe the distance between two points in the frame where they are at rest, while the contracted length (L) is the length observed by an observer in relative motion to the object. According to the Lorentz transformation, the perceived contraction only happens in the direction of motion.

Since the ball's diameter is aligned perpendicular to its velocity vector towards the observer, it will appear contracted. The distance 'l', however, is aligned along the ball's velocity vector. But because 'l' is the position coordinate of the ball itself which is moving towards the observer, in this context, 'l' does not represent the distance between two points both moving with the ball and thus does not contract.