Physics

Everyone who has taken a linear algebra or physics course at university remembers this strange dualism. We were taught that vectors have TWO types of products. The first, scalar, takes two vectors and outputs a number. Geometrically, it's about projections and angles. The second, vector, also takes two vectors and... suddenly spits out a third vector, perpendicular to the first two. And this trick only works in 3D and 7D.

It always seemed like some kind of mathematical 'crutch'.

Why is it so complicated? Why two different products for different tasks? Why does one depend on the cosine and the other on the sine?

What if I told you that they really are 'crutches'? That there is a single, universal, and elegant geometric product, which includes both of these cases (and much more), and which is based on a single, crystal-clear idea. An idea that changes the way we look at the very essence of mathematics.

This article is an invitation to the world of Geometric Algebra. We are going to reinvent multiplication.

We all know this formula. .

This is perhaps the only piece of knowledge from school geometry that stays with a person for their entire life, even if they work as a barista or a copywriter.

But have you ever asked yourself the question: why squares specifically?
Why not cubes? Why not just the sum of the absolute values ?

If you ask a teacher, they will draw little squares on the sides of the triangle. If you ask a university professor, they will write down the definition of a scalar product.
And both of them, in essence, will deceive you. Or, to put it more mildly, they won't tell you the whole truth.

Today, we will unpack this 'black box' and see that the Pythagorean theorem is not about triangles at all. And it should be proven in a completely different way than we were taught.

The school curriculum doesn't provide an answer. Moreover, the history of teaching the Pythagorean theorem is a story of how living, visual geometry was turned into dry, dead algebra. We were led further and further away from understanding the essence and towards abstraction.

Today, we will analyze this path of degradation and show a proof that will bring you back to reality. Spoiler: the Pythagorean theorem is not about triangles. It's about mirrors.

Get ready to have all your preconceptions shattered!

Abstract. This work examines the physical foundations of Ekahau Sidekick measurements and the device offset mechanism from the perspectives of antenna theory, receiver noise theory, statistical signal theory, and the IEEE 802.11 standard family. It is shown that the scalar received signal strength indicator (RSSI) offset constitutes a linear level shift and does not model the true signal-to-noise ratio (SNR) of the client device, the quadrature amplitude modulation (QAM) constellation structure, the rate adaptation algorithm, or roaming behaviour. In addition to five independent physical and systemic sources of inaccuracy, the paper addresses modeling assumptions in Ekahau with respect to multiple-input multiple-output (MIMO) gain, multipath propagation, airtime estimation, and SNR visualisation. Verified numerical error estimates for representative deployment scenarios and practical recommendations are provided.

Quantum mechanics is one of the most successful theories in the history of science.

It underlies atomic physics, semiconductors, lasers, and modern quantum technologies.

However, nearly a century after its development, a peculiar situation remains:

we can predict experimental results with remarkable precision, yet we still do not fully understand what quantum theory actually represents.

Does it describe physical reality “as it is,” or does it instead describe the structure of conditions under which observable facts become possible?

In this article, I propose the following hypothesis:

quantum theory does not describe reality itself, but the conditions and mechanisms through which observable reality emerges.

Wilhelm Röntgen discovered X-ray November 8th 1895, when he did experiments with cathode rays in a vacuum tube. To capture and save images of the shadows from the X-rays, he used ordinary photoplates. Fortunately, sensitive to visible light silver based photoemulsions turned out to be sensitive to the X-ray too. These photoplates became the first X-ray detectors.

More than 100 years of scientific progress led to the creation of a number of various detectors for recording X-ray images. Developments of the microelectronics and semiconductor manufacturing technologies are crucial for development of the modern X-ray detectors. These detectors can transform the energy of the X-ray photon directly to the electrical signal. They allow capturing detailed, digital, high-resolution X-ray images.

Digital images are easy to work with. For example one can merge multiple macro images into an image of the whole object and represent monochrome images in false colors like Simon Procz did with this X-ray image of a flower he did in 2012.

Quantum computers have gradually migrated from “natural” quantum objects like ions and quantum dots to “artificial” objects like transmons, where natural frequencies of LC circuit mimic energy levels. Coaxial cables attached to such circuits excite the system, read their state, and adjust their properties. They are still quantum objects and operate at low temperatures.

I wondered if a similar system can be built with large “classical” mechanical objects. Here I demonstrate 2 -qubit quantum computer built of coupled pendulums.

The director Denis Villeneuve and cinematographer Greig Fraser in their Dune: Part Two movie made a curious decision to film the scenes on the surface of the Giedi Prime planet in the infrared spectrum. It turned out to have interesting aesthetics and there are some interesting related physics to discuss and speculate about how realistic the look of it is.

3.What's an ugly smiling face?

It's the cat from curved space.

V. Komen, I. Tikhonenkov

Here we are discussing in general the majority of metrics for a stationary gravitation fields in one dimension. The only accepted approach so far to apply the equations of field (A. Einstein):

V. Komen, I. Tikhonenkov

In the previous post we've considered a model example of a motion of a free particle within a uniform gravitation field where a coupling to the field is defined by an observed inertion mass (see eq. (2) in https://habr.com/en/articles/739714/). The equation of motion was:

Uniform gravity, can it exist?

1. The motion of a free particle-like cat

V. Komen, I. Tikhonenkov

Good morning! You've woke up. Having prepared coffee and toasts of bread you are drifting from a kitchen to a table before a large wall TV. The left hand keeps a small plate with toasts and the right one controls coffee level within your beloved mug. The life is plotted out for ten seconds to come. But Ervin is already worried that you, as usual, made conspiracy and decided not to share your breakfast with him. So the damned cat thrusts himself across you pass, hits your legs. By the next moment the plate, toasts and the mug are falling. And – yes! All those precious things reached a floor level by the same time. Physics...It's how it shows up, unexpected. And we used to identify ourselves as physicists. The unexpected thing is that we still do. It's whence our motivation originates. We cannot pass by any falling objects quietly.

I explain experimental results of Bell’s Theorem by superdeterminism. I follow with insights into how such a universe may arise and be compatible with the subjective experience of free will.

In this post there are a lot of references to the previous one – it is essential that you read it before getting down to this.

In my previous posts (see the list below below) I tried to express my doubts whether there is a real physical substrate to the Lorentz transforms. The assumptions about the constancy of the speed of light, the homogeneity of space-time, and the principle of relativity do not and cannot lead to the deduction of the Lorentz transforms – Einstein himself, for one, gets quite different transforms, and from those he goes over directly to the Lorentz transforms obviously missing a logical link (see Einstein p. 7, and also Part 1 of this discussion). As for the light-like interval being equal to zero, we saw that it can be attached to such assumptions only in error and cannot in itself be a foundation of a theory. I have to conclude that all that fine, intricately latticed construction of scientifictitious, physics-like arguments with the air of being profound is nothing but a smokescreen creating the appearance of a physical foundation while there is none.

What is then the real foundation of the Lorentz transforms? Let’s start from the rear end, the Minkowski mathematics. Historically, this appeared later than special relativity as a non-contradictory model of the Lorentz mathematical world; previously mentioned Varićak was among those who took part in its creation. Notwithstanding its coming later in history, it can be used as the starting point for derivation of the Lorentz transforms.

I thought my previous post rather funny, and was surprised seeing it initially receive so few views. I thought the entertainment flopped, but fortunately I was wrong. I therefore feel it my duty before my readers to address the subject of the Landau & Lifschitz proof of the invariance of the interval.

You can find the summary of it in Wikipedia. Making their starting point the light-like interval always being equal to zero, Landau & Lifschitz seem to make a great fuss about it. The Wikipedia article even says: ‘This is the immediate mathematical consequence of the invariance of the speed of light.’ No, it is not.

I beg everyone’s pardon, but the light-like interval always being equal to zero is nothing else but the following statement: ‘The length of a ray of light will always be equal to the length of this ray of light’. Sounds like a cool story, bros and sis, but I cannot see what further inferences can be drawn from it. The ‘proof’ of this truism cannot fail under any circumstances whatever – whether you keep the speed of light invariant, or keep or change the metric of space or time or both – or make both metric and speed of light change – the light-like interval will remain equal to zero. I am okay with anyone wanting to prove it if they feel like it, but you cannot make it an ‘immediate mathematical consequence of the invariance of the speed of light’. Neither is it possible to make the constancy of the speed of light a consequence of the invariance of the light-like interval for the reason already mentioned: this is a truism. It does not prove anything, nor can it be a consequence of anything. When Landau & Lifschitz insist that this is a consequence of the constancy of the speed of light, that is either an error or a downright subterfuge, a means employed to create a spectre of logical connection between two unconnected notions, and charge this ghostly connection with pretended significance. And, since the following proof of invariance of an arbitrary interval hangs on the invariance of the light-like interval, we can altogether dismiss it: the necessity of introduction of such a measure as interval cannot be derived from the statement that a length of something will be equal to itself in whatever frame of reference it is measured.

The previous discussion done, we have surmounted the difficult waters and are now sailing into something much more pleasure-like and hopefully even entertaining.

As I promised, we will be discussing the invariance of the interval, that is to say, the following relation:

Even as I am posting this, I can see that my previous post received a hundred and twenty plus views, but no comments yet. I am saying again that my pursuit is not to give an answer, but to ask a question. I only wonder if there is in fact no answer to the questions I am asking – but anyway, I will continue asking them. If you know how to deal with the problems I am setting – or happen to understand they are not problems at all, I will be most grateful for a constructive input in the comments section. I am sorry to say I was unable to make this post sound as light and unpretentious as the previous one. This one deals with harder questions, is a little wordy, and requires at least elementary knowledge of calculus to be read properly.

In my previous post we discussed the ‘Galilean’ velocity composition used for introduction or substantiation of relative simultaneity. It is not the only point where Einstein resorts to sums c + v or c – v: he does that actually to deduce the Lorentz transforms, notwithstanding the fact that a corollary of the Lorentz transforms is a different velocity composition which makes the above sums null and void. It looks like the conclusions of this deduction negate its premises – but this is not the only strange thing about Einstein’s deduction of the Lorentz transforms undertaken by him in his famous 1905 article.

In Paragraph 3 of that paper Einstein is considering the linear function τ (the time of the reference frame in motion) of the four variables x′ = x – vt, y, z, and t (the three spatial coordinates and time of the frame of reference at rest) and eventually derives a relation between the coefficients of this linear function.

There is one only too obvious problem with relativity of simultaneity in the way it is normally introduced, and I have never found an answer to it – what’s more, I never read or heard anyone formulate it. I will be grateful for an enlightening discussion.

The framework of the thought experiment introducing relativity of simultaneity is this. Two rays of light travel in opposite directions and reach their destination simultaneously in one frame of reference and at different moments in the other.

For example, in the Wikipedia article on the subject you can read:

‘A flash of light is given off at the center of the traincar just as the two observers pass each other. For the observer on board the train, the front and back of the traincar are at fixed distances from the light source and as such, according to this observer, the light will reach the front and back of the traincar at the same time.

‘For the observer standing on the platform, on the other hand, the rear of the traincar is moving (catching up) toward the point at which the flash was given off, and the front of the traincar is moving away from it. As the speed of light is finite and the same in all directions for all observers, the light headed for the back of the train will have less distance to cover than the light headed for the front. Thus, the flashes of light will strike the ends of the traincar at different times’.

I am always not a little surprised at the modesty displayed by the authors of such illustrations. If we grant the statement ‘the light headed for the back of the train will have less distance to cover than the light headed for the front’ to be true – how then do we evaluate the magnitude of the effect? Or, in other words, how much longer is one distance in comparison to the other?