Physics Notes

When swimming you can notice the effect over the width of a small lake:

But if you want to see a noticeable effect over just two meters then it depends on what you mean by “noticeable” (eg it would be a trivial high school exercise to calculate the largest radius at which a two meter tangent line ends at least one mm from the corresponding arc).

Source: (1001) Alan Cooper’s answer to How close to the Earth’s core would you need to be to see an ‘uneven’ water level?Not regarding temperature etc making it impossible, how deep would you have to dig to see a noticeable curvature in, lets say a 2m length of water? – Quora

A writer I often admire, previously on Quora and more recently on Substack, is the New Zealand physician who goes by the pen name DrJo. His take-downs of anti-scientific nonsense (and sometimes actual fraud) are welcome and appear to be based on a good understanding of the relevant science and statistics, but getting closer to my own area of expertise, his take on quantum physics leaves me more doubtful.

Let’s first jump to the chase and look the concluding question in DrJo’s recent post on the subject:

“How can a space be both Hausdorff and non-local? What does ‘non-local’ even mean here? Can you explain the maths of this ‘non-local’ property, specifically the topology, without resorting to hand-waving?”

The main point is that DrJo appears here to be confusing the separability of points in space-time, just in the sense of how we can tell them apart (corresponding to his abuse of the word “Hausdorff”) and locality of a theory based on that space-time (which refers to some limit on how things happening at one point in space-time can affect those at another). Locality of a theory is a property that places some limit on how quickly actions at one place can affect the situation far away, but no such limit is required in order to tell events apart in either space or time. For example, I could still compute the distance from my home to the grocery store on the basis of how long it takes me to walk there – even if I had some way of instantly checking on the availability of a product and having them set it aside for me to pick up whenever I eventually arrive.

There is nothing wrong with feeling somewhat disoriented by the “spooky action at a distance” aspects of quantum theory – or with writing about that disorientation without having a technical background in the subject. But I do think that DrJo’s use of technical language for that purpose in his latest article is inappropriate – both because it is not correctly applied and because its effect is to create an aura of greater authority than is warranted.

Why do I say that? Well here goes.

DrJo’s post starts with what I consider a perfectly reasonable brief outline of the story behind the 2022 Nobel Prize in Physics (which was awarded to Alain Aspect, John F Clauser, and Anton Zeilinger for work on verifying the prediction of quantum mechanics that physical observations cannot be described by a theory in which the measurements correspond to permanent properties of classical particles which cannot somehow have instantaneous effects on one another). He follows this with a brief discussion of some aspects of the mathematical subject called topology which, while not wrong, strikes me as just introducing and throwing around technical terms for no useful purpose. And then comes the part about which I am actually “grumpy” but I hope not “inarticulate”:

Now here’s my problem. In terms of our current theories, spacetime is a Hausdorff differentiable manifold. Note the word ‘Hausdorff’ here. This assumption underpins both General Relativity and Quantum Mechanics.

But … if one of a pair of detectors interacts with one of a pair of entangled particles, and the other particle that is far removed instantaneously assumes a correlated state, how can this space possibly be Hausdorff? As has been pointed out ad nauseam by physicists like Einstein and Bell and Aspect and Clauser and Zeilinger, this is a non-local effect.

The points in the region of the second detector at an arbitrary distance from the first surely can’t be considered ‘Hausdorff’ (or ‘housed off’) if that distant interaction can influence them.

I guess the questions here are “How can a space be both Hausdorff and non-local? What does ‘non-local’ even mean here? Can you explain the maths of this ‘non-local’ property, specifically the topology, without resorting to hand-waving?”

Now, would a real physicist step up and explain.8 Please

OK. Let’s start with that first paragraph.

In terms of our current theories, spacetime is a Hausdorff differentiable manifold. Note the word ‘Hausdorff’ here. This assumption underpins both General Relativity and Quantum Mechanics.

The word ‘Hausdorff’ is not necessary here (as every differentiable manifold is Hausdorff) and appears to be just a way to appear impressive to the uninitiated by citing an esoteric-sounding, but not necessarily relevant, generalization when the actual spacetime frameworks of the physical theories have much stronger restrictions.

Next we have:

But … if one of a pair of detectors interacts with one of a pair of entangled particles, and the other particle that is far removed instantaneously assumes a correlated state, how can this space possibly be Hausdorff? As has been pointed out ad nauseam by physicists like Einstein and Bell and Aspect and Clauser and Zeilinger, this is a non-local effect.

But… despite lots of badly written pop-sci nonsense, there is nothing in quantum theory which suggests that “the other particle that is far removed instantaneously assumes a correlated state”

And perhaps more importantly, wtf does he mean by “this space”? The discussion of experiments in which particles are separated usually requires our theory to involve observables whose possible values correspond to space-time coordinates relative to an observer, and the set of all possible such coordinate values can be assigned metrics in many ways – with the corresponding topologies all satisfying (among other things) the technical condition used to define what is called a Hausdorff space. So the question of whether or not there are surprising correlations between the spins of separated particles says nothing at all about whether or not the set of possible values of their positions can or cannot be endowed with a Hausdorff topology.

In fact no real physicist has ever claimed that the Bell-defying correlation of spins “is a non-local effect” but rather just that it would have to be a non-local effect IF the underlying physics was classical (with the spin or polarization values being permanent classical properties of the particles or photons in question).

Next we have:

The points in the region of the second detector at an arbitrary distance from the first surely can’t be considered ‘Hausdorff’ (or ‘housed off’) if that distant interaction can influence them.

But, leaving aside the fact that it’s spaces not points that are Hausdorff, there is absolutely no reason to think that the possibility of some instantaneous interaction causing influence from one point to another makes it impossible to distinguish those points in some other way.

Then at last we come to the final question – which I believe I have addressed above.

Source: Cracked up about Quantum Physics – Dr Jo

The fact that a sharp horizon exists at a distance which increases noticeably over a fairly small increase in altitude should be convincing evidence of the Earth’s curvature for anyone who thinks about it for a moment or two. And Domenico Barillari is right to emphasize that the only evidence for the Earth’s curvature that is visible in a static picture is the angle of depression of the horizon – which is NOT noticeable by eye (ie without using some scientific instrument) at any altitude that most of us will ever experience. But the only “optical physics principle” involved in either of those cases is the assumption that light travels in straight lines.

However, although we can deduce the Earth’s curvature, we do not actually see it in the form of a curved horizon. Pictures showing a horizon that curves down at the edges arise only when the camera’s line of sight is angled downwards and would look the same if the visible part of the Earth was a perfectly flat disc. And again, the only “optical physics principle” involved is the assumption that light travels in straight lines.

Source: (1001) Alan Cooper’s answer to When you see the Earth’s curvature from high altitude, what optical physics principle is at play that makes this phenomenon observable? – Quora

If two electron beams are travelling side-by-side, then, from the point of view of an observer relative to whom they are moving, the force between them has two components, one electrostatic and the other magnetic. But from the point of view of an observer who is travelling with them, the electrons in both beams are stationary so there is no current and the only force is electrostatic. The Lorentz transformation explains the difference in terms of a change of the distances between the charges in their direction of motion (and so a change of the charge densities in the beams); but regardless of that, what is seen as a sum of “electric” and “magnetic” forces in one frame is seen as just an “electric” field effect in the other, and so there is no inherent observer-independent distinction between the electric and magnetic components. The formulation in terms of a second order tensor allows us to express both the apparent electric and magnetic fields seen by an observer as aspects of a single Lorentz invariant tensor field; but while it describes how they are related as aspects of a single tensor field, I wouldn’t really say that it shows they are so related.

Source: (1000) Alan Cooper’s answer to How does the relativistic transformation of electric and magnetic fields show they are two aspects of a single electromagnetic field? – Quora

The claim of “mass increase with velocity” is based on using the word “mass” for something that is not a property of the object itself but rather depends also on the relative velocity of the observer.

Although this usage is not “wrong” and was fairly common in the early days, most physicists soon chose to use the word “mass” just for the invariant “rest mass” and followed the title of Einstein’s initial paper on the effect by just using the word “inertia” for the velocity-dependent quantity that used to be called the “inertial (or relativistic) mass”.

In order to prove that the observed “inertia” depends on the relative speed (and direction) of motion of the observer it is necessary first to define it.

One way is to define it as the ratio of applied force to observed acceleration, but this in turn requires a clear definition of force. For example you could perhaps look at the dynamics of a collision with an object of known mass at rest in the observer’s frame.

Or perhaps you could just read Einstein’s original paper on the subject.

Source: (915) Alan Cooper’s answer to Why does mass increase with velocity as an object approaches the speed of light? How can I prove it mathematically? – Quora