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.
4 Answers
I didnt understand why this partial treatment of two lengths ?.They are identical in properties with regard to lorentz transformations
No, not identical. Special relativistic effects, like length contraction applies to a physical objects which proper length can be measured. Distance wrt to different frames can be different, but this is due to a coordinate system choice and/or relativistic length contraction of a moving object. However, distance between something (or space so to say) does not undergo "length contraction", because proper length for space has no meaning. Unless you add effect of general relativity, in where space can be tampered by gravity.
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If all parts of the ball accelerate simultaneously in the observer's frame, then (obviously) the diameter of the ball does not change in the observer's frame.
Equally obviously, the distance between the ball and the observer (in the observer's frame) changes at the rate at which the ball travels toward the observer.
Any change in the diameter of the ball is due to different parts of the ball accelerating at different times. Any change in the distance to the (close end of) the ball is due to the motion of that end of the ball.
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I can see why you were confused by this. When you bring accelerations into SR, additional nuances arise that are easily overlooked.
The gap between the ball and the observer, as far as the observer is concerned, is measured in the observer's frame. Imagine if the observer was holding a long ruler that stretched to the initial position of the ball. Although the ball accelerates, the ruler remains at rest relative to the observer- that's why neither the ruler nor the measurement of the distance to the ball are length contracted.
Length contraction arises from the relativity of simultaneity. Effectively what happens is that in a frame in which it is moving, an object's two ends are being viewed at times that are not synchronous in the object's rest frame. The forward end of the object is viewed before the rear end, which gives the rear end time to move forward and thus makes the object shorter. When you accelerate an object, you have to distinguish between two possible scenarios. In one, every part of the object accelerates simultaneously in the object's instantaneous rest frame- that will cause the object to be length contracted in its original frame. What happens is that the acceleration of the front and the rear are out of synch in the original frame, with the rear of the object accelerating before the front, which causes the object to be shorter. In the other scenario, every part of the object accelerates instantaneously in its original frame, in which case the object remains the same length in the original frame. Length contraction is still at work in the second scenario- because what is happening is that the object actually stretches in its instantaneous rest frame, and the stretching is cancelled out by length contraction so the object remains the same length in the original frame. That effect is known as Bell's spaceship paradox.
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In this post I explained why you should stick with one reference frame for good.
Regarding OP's question, the answer is:
- The ball's geometry remains the same, i.e. no change in diameter according to ground observer. (Let's forget the relativistic stress issue when you try to uniformly make a body move. )
- Change in distance according to ground observer is very easy: change rate is just $v$ -- we learnt this when we were children.
- What being compared in 'length contraction' is not 'length before move' and 'length when moving' but length according to two inertial frames. The body considered is keeping its inertial motion forever. NO dynamical change!
- When you set the ball suddenly into motion, the ball comoving observer would find the diameter in the motion direction suddenly increased by $\frac{1}{\sqrt{1-v^2/c^2}}$. This can be easily seen in Minkowski spacetime diagram.
- Ball's frame in the above setting is not a inertial frame, rather some strange combination of two different inertial frame. Therefore there's no symmetry between ball and ground.
More on 4.: if you imagine the ball have been move with $\vec{v}$ w.r.t ground from very distant past (or think another inertial body instantaneously comoving with ball), than you recover the normal 'length contraction'.
To remove strangeness due to sudden change in velocity, you may learn how to deal with mild acceleration from Rindler coordinates wiki page and other resources.
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