In What Feel Greene's Causality Hovering Slinky Explanation Is Right

I woke up, read some comments, too understood how to read Greene's explanation of the slinky behavior inwards the previous spider web log ship service too then that it isn't self-evidently wrong. In fact, it's strictly correct given some natural agreement too parameterization.

An effective partial differential equation describing surely variables inwards the falling slinky does resemble a moving ridge equation amongst a real depression "speed of signals" which is why I shout out back it's correct to apologize for the overreaction. Sorry, Brian, your comment may endure read too then that it conveys a truthful statement.

Every shout out for of the slinky is indeed hovering inwards mid-air upwardly to some shout out for too this declaration is exact inwards a expert plenty approximation of the problem. How does it work?




Before the slinky is released, each of its shout out for is inwards equilibrium. It agency that the sum forcefulness acting on that shout out for is zero. The downward gravitational forcefulness cancels the upward forcefulness trying to shrink the slinky, \[

F_{total}(z,t) = 0\quad {\rm for}\quad t=0.

\] The lower portions of the slinky are to a greater extent than compressed – less stretched – than the upper portions because there's a smaller forcefulness acting on them underneath. We may parameterize this shape either yesteryear a constituent \(z(\sigma)\) of an auxiliary parameter \(\sigma\) measurement the length along the slinky's circumference), or yesteryear a density constituent \(\rho(z)\) that tells us how much of the fabric is at a given tiptop \(z\). These 2 functions are some beingness "inverse to each other".




The enquiry is what happens subsequently the slinky is released. Let's pull the motion inwards price of the constituent \(\Delta x(\sigma,t)\) that describes the seat of the \(\sigma\)-th radian along the slinky circumference at fourth dimension \(t\), minus the equilibrium value at \(t=0\). For all \(t\leq 0\), nosotros receive got \(\Delta x(\sigma,t)=0\).

Because nosotros subtracted the equilibrium value, the gravitational forcefulness acting on that slice of the slinky cancels against the forcefulness from the tension within the slinky that existed upwardly to \(t=0\). So the exclusively forcefulness that contributes is the departure betwixt the forcefulness from tension at fourth dimension \(t\) minus the same forcefulness at \(t=0\).



The forcefulness from the tension is proportional to\[

\frac{\partial^2 \Delta x}{\partial \sigma^2}

\] Note that the stretched fountain drags from both sides too yous acquire some cyberspace forcefulness or acceleration of a given shout out for of the fountain if i side is to a greater extent than compressed than the other. The \(\sigma\)-parameterization has the payoff that nosotros receive got a constant majority density per unit of measurement modify of \(\sigma\) – every loop of the slinky has the same weight. For this reason, I create shout out back that inwards these variables \(\sigma,t\), the variable \(\Delta x\) obeys the exact moving ridge equation\[

\frac{1}{c_{slinky}^2}\frac{\partial^2 \Delta x}{\partial t^2} - \frac{\partial^2 \Delta x}{\partial \sigma^2} = 0

\] Here, the "speed" \(c_{slinky}\) is calculated from the fountain constant too majority density of the spring. Because \(\sigma\) is the length along the circumference of the slinky, its values or changes are much greater than the typical changes of the tiptop \(z\), peculiarly when the slinky is compressed, too that's why a relatively high value of the speed \(c_{slinky}\) which measures the propagation of signals along the circumference \(\sigma\) translates to a much lower speed inwards the actual spatial \(z\)-direction.

So I believe that \(\Delta x\) obeys the exact moving ridge equation inwards variables \(\sigma,t\), amongst a fixed value of the speed of signals. This moving ridge equation is studied at many places of physics. The initial perturbation at i point, such every bit the release of the transcend of the spring, indeed propagate strictly yesteryear the speed that is bounded, too then every lower shout out for of the slinky has to expect earlier it starts to displace at all, too the motion of the fountain is strictly nil earlier the upper constituent of the slinky arrives, indeed.

Again, the speed of the waves moving along the slinky is basically constant every bit a constituent of the "circumference length" – the same expose of loops is reached each second. This gets translated to higher speeds when the slinky is stretched, or a depression speed when the slinky is compressed or flaccid. For the "released slinky" experiment, the speed of the waves inwards the existent infinite is just depression plenty to endure plenty for the bottom of the slinky to expect for the freely falling upper portions of the fountain to arrive.

This causal behaviour is just analogous to that inwards relativity. If yous exclusively let the slinky compression waves to ship the information, Greene's comment is just right.

I would nonetheless non limited it inwards the same words. Each shout out for of the slinky is affected yesteryear both forces at all times too signals tin propagate much to a greater extent than speedily inwards general. But when nosotros exclusively bound the sending of the data to the slinky waves, it takes fourth dimension for whatever shout out for to acquire that the province of affairs has changed relatively to \(t=0\).

Note that I form out of needed to utilization the parameter \(\sigma\) too non the actual spatial coordinate \(z\) to acquire a stable plenty speed of the signals. If yous translated all the behaviour of \(\Delta x\) too waves on it into the variables \((z,t)\), yous would honour out that the speed of the slinky waves is 1) subject on how much the slinky is compressed at a given moment, every bit nosotros mentioned, too also 2) the speed of the signals inwards the \((z,t)\) infinite is increased or decreased yesteryear the actual speed of the corresponding shout out for of the spring. So of course, if yous through the whole slinky against someone, yous tin acquire the data to her much to a greater extent than speedily than the speed of the slinky waves relatively to the slinky.

Also, I needed to utilization the departure \(\Delta x\) too non only \(x\) to acquire a elementary moving ridge equation too eliminate gravity. The corresponding equation for \(x(\sigma,t)\) would incorporate the extra gravitational price whose irrelevance for the causal declaration wouldn't endure forthwith clear. This gravitational term would endure basically a constant term on the correct mitt side of the moving ridge equation. Because this term has no \(\sigma\)- or \(t\)-derivatives, it doesn't modify the bound on the speed of signals. (That's only similar the fact that the massive Klein-Gordon or Dirac equation preserves the speed of signals only similar their massless counterparts.)

H5N1 ground why I reacted intensely was the similarity of Greene's comment to a hilarious comment yesteryear a Czech philosopher virtually the "relativity of sound". The philosopher has argued that it's a discrimination to utilization the speed of lite inwards special relativity too this selection exclusively reflects our considering vision to endure the most of import sense. Blind people are governed yesteryear relativity amongst the speed of audio playing the role of the speed of light, he argued. ;-) Needless to say, I too a friend of mine asked him whether blind people are allowed to wing amongst supersonic airplanes.

The shout out for of this even out is that at that spot is exclusively i genuinely universal limitation on the propagation of signals, too it's given yesteryear the speed of lite too relativity. Mathematically analogous equations may look elsewhere – amongst the speed of audio which is 1 1000000 times lower or the speed of slinky waves that is some 100 times slower than the audio inwards this setup – but claims virtually the impossibility to propagate data speedily inwards these situations must endure acknowledged to depend on approximations too effective descriptions.