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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So without an atmosphere on earth, and assuming somehow muons are still produced at 10km above earth wrt an observer on earth,will the muon not see the distance between itself and the earth to be contracted as the physicl object in this case i.e. the atmosphere is no longer present? – Tim Crosby Oct 28 '23 at 19:11
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@TimCrosby : You can think of the distance from the earth to the muon source as the length of an imaginary rigid rod that stretches from the earth to the muon source. The length of that rod will be shorter in the muon frame than it is in the earth frame. By contrast, the distance from you to a basketball that you see moving toward you is not the length of any rigid rod, imaginary or otherwise. This is an entirely different scenario. – WillO Oct 28 '23 at 19:24
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Why is this a different scenario?The only difference is that muon was created with a velocity 'v' and the ball in my scenario has accelerated to v instantaneously.How does this fact change everything? – Tim Crosby Oct 28 '23 at 19:28
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@WillO The increased travel range of Muon experiment is typically explained by time dilation, i.e. that in Muon reference frame time ticks more slowly than compared to a lab frame, so Muon can cover greater distances, thereby increasing their flux to the lab. No, need for "imaginary rod" here as in any other scenario, because like I say "imaginary rods" does not have proper length. Also check relativistic Range – Agnius Vasiliauskas Oct 28 '23 at 19:34
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2@Tim Crosby : No, that's not the key difference. The distance from you to the muon source is fixed (in any given frame) while the distance from you to the ball is changing. Draw the spacetime diagrams for both scenarios and they will look very different. – WillO Oct 28 '23 at 19:35
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@WillO Actually our teacher skipped spacetimediagrams as this was an introductory course so It would be helpful if you would take the trouble to include that in your answer – Tim Crosby Oct 29 '23 at 08:47
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1Your teacher skipped the useful stuff to concentrate on the confusing and pointless "woo-woo" stuff. You have been ripped off. – m4r35n357 Oct 29 '23 at 09:17
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Regradless the muon and the earth observer are completely symmetric in their respective frames wrt their velocitites.E.G THE muon in its rest frame sees the earth moving at 0.9c and the earth sees the mun moving at 0.9c in its rest frame.Then why does then the earth measure the distance to be something and the muon the distance to be something else. – Tim Crosby Oct 29 '23 at 09:28
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@TimCrosby : "Then why does then the earth measure the distance to be something and the muon the distance to be something else." Because that is what relativity requires. – WillO Oct 29 '23 at 13:58
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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So is it correct if I say that ball observr also measures the distance b/w the observer to be not contracted ?Because both observers are symmetric in each other's frame. – Tim Crosby Oct 29 '23 at 08:46
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.
