Tuesday, February 14, 2017

In what sense Greene's causality hovering slinky explanation is right

I woke up, read some comments, and understood how to read Greene's explanation of the slinky behavior in the previous blog post so that it isn't self-evidently wrong. In fact, it's strictly right given some natural understanding and parameterization.

An effective partial differential equation describing certain variables in the falling slinky does resemble a wave equation with a very low "speed of signals" which is why I think it's right to apologize for the overreaction. Sorry, Brian, your comment may be read so that it conveys a true statement.

Every point of the slinky is indeed hovering in mid-air up to some point and this statement is exact in a good enough approximation of the problem. How does it work?




Before the slinky is released, each of its point is in equilibrium. It means that the total force acting on that point is zero. The downward gravitational force cancels the upward force trying to shrink the slinky, \[

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

\] The lower portions of the slinky are more compressed – less stretched – than the upper portions because there's a smaller force acting on them underneath. We may parameterize this shape either by a function \(z(\sigma)\) of an auxiliary parameter \(\sigma\) measuring the length along the slinky's circumference), or by a density function \(\rho(z)\) that tells us how much of the material is at a given height \(z\). These two functions are close to being "inverse to each other".




The question is what happens after the slinky is released. Let's describe the motion in terms of the function \(\Delta x(\sigma,t)\) that describes the position of the \(\sigma\)-th radian along the slinky circumference at time \(t\), minus the equilibrium value at \(t=0\). For all \(t\leq 0\), we have \(\Delta x(\sigma,t)=0\).

Because we subtracted the equilibrium value, the gravitational force acting on that piece of the slinky cancels against the force from the tension inside the slinky that existed up to \(t=0\). So the only force that contributes is the difference between the force from tension at time \(t\) minus the same force at \(t=0\).



The force from the tension is proportional to\[

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

\] Note that the stretched spring drags from both sides and you get some net force or acceleration of a given point of the spring if one side is more compressed than the other. The \(\sigma\)-parameterization has the advantage that we have a constant mass density per unit change of \(\sigma\) – every loop of the slinky has the same weight. For this reason, I do think that in these variables \(\sigma,t\), the variable \(\Delta x\) obeys the exact wave 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 spring constant and mass 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 height \(z\), especially when the slinky is compressed, and 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 in the actual spatial \(z\)-direction.

So I believe that \(\Delta x\) obeys the exact wave equation in variables \(\sigma,t\), with a fixed value of the speed of signals. This wave equation is studied at many places of physics. The initial perturbation at one point, such as the release of the top of the spring, indeed propagate strictly by the speed that is bounded, so every lower point of the slinky has to wait before it starts to move at all, and the motion of the spring is strictly zero before the upper part of the slinky arrives, indeed.

Again, the speed of the waves moving along the slinky is basically constant as a function of the "circumference length" – the same number of loops is reached each second. This gets translated to higher speeds when the slinky is stretched, or a low speed when the slinky is compressed or flaccid. For the "released slinky" experiment, the speed of the waves in the real space is exactly low enough to be enough for the bottom of the slinky to wait for the freely falling upper portions of the spring to arrive.

This causal behavior is exactly analogous to that in relativity. If you only allow the slinky compression waves to send the information, Greene's comment is exactly right.

I would still not express it in the same words. Each point of the slinky is affected by both forces at all times and signals can propagate much more quickly in general. But when we only restrict the sending of the information to the slinky waves, it takes time for any point to learn that the situation has changed relatively to \(t=0\).

Note that I sort of needed to use the parameter \(\sigma\) and not the actual spatial coordinate \(z\) to get a stable enough speed of the signals. If you translated all the behavior of \(\Delta x\) and waves on it into the variables \((z,t)\), you would find out that the speed of the slinky waves is 1) dependent on how much the slinky is compressed at a given moment, as we mentioned, and also 2) the speed of the signals in the \((z,t)\) space is increased or decreased by the actual speed of the corresponding point of the spring. So of course, if you through the whole slinky against someone, you can get the information to her much more quickly than the speed of the slinky waves relatively to the slinky.

Also, I needed to use the difference \(\Delta x\) and not just \(x\) to get a simple wave equation and eliminate gravity. The corresponding equation for \(x(\sigma,t)\) would contain the extra gravitational terms whose irrelevance for the causal argument wouldn't be immediately clear. This gravitational term would be basically a constant term on the right hand side of the wave equation. Because this term has no \(\sigma\)- or \(t\)-derivatives, it doesn't change the limit on the speed of signals. (That's just like the fact that the massive Klein-Gordon or Dirac equation preserves the speed of signals just like their massless counterparts.)

A reason why I reacted intensely was the similarity of Greene's comment to a hilarious comment by a Czech philosopher about the "relativity of sound". The philosopher has argued that it's a discrimination to use the speed of light in special relativity and this choice only reflects our considering vision to be the most important sense. Blind people are governed by relativity with the speed of sound playing the role of the speed of light, he argued. ;-) Needless to say, I and a friend of mine asked him whether blind people are allowed to fly with supersonic airplanes.

The point of this story is that there is only one truly universal limitation on the propagation of signals, and it's given by the speed of light and relativity. Mathematically analogous equations may appear elsewhere – with the speed of sound which is 1 million times lower or the speed of slinky waves that is some 100 times slower than the sound in this setup – but claims about the impossibility to propagate information quickly in these situations must be acknowledged to depend on approximations and effective descriptions.

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