Spacetime analogies

The elastic sheet model is one of the most used model for telling general relativity. It allows to show as smaller balls "orbit" around the larger ball in the center of the sheet in a way similar to the planets, at least until the balls lose energy due to friction and finally fall into the "gravitational wall".
This way to see the general relativity is directly connected with the embedded diagrams and Flamm paraboloids, the mathematical way to see the spacetime deformations. But this analogy has some problems not only because is inaccurate like all analogies, but also becuase it could be confusing about distorced space and spacetime especially among students. So we can ask: why is the sheet deformed? Because of the weight? This fact implies the use of a circluar argument: usinf gravity to explain gravity! But if the ball isn't in spacetime, where is it?⁽¹⁾

Come back on the road

Due to some family problems, 2017 was a difficult year and all my online activities, particularly blogs, had a considerable decrease in content. As for Doc Madhattan the effects were seen especially in the last months of 2017 and in 2018, when the last post came out in august.
In june of this year, almost a year later, I resumed posting also on this blog with the death of Murray Gell-Mann. However, I still had no idea if and how to regularly resume publications. Then the case wanted that, thanks to the first photograph of a quantum entaglement, I am selected among the contents of the week of Science Seeker, and this gives me the incentive to resume curating the contents also on this blog!
The program, which I hope to maintain, is to publish a couple of articles a month: in this way I should be able to keep up the pace, placing it between work and articles for my other italian blogs. As for the contents, I would like first of all to recover some of the notes that have remained in draft in all these months of silence, so try to write new contents from scratch. Probably I will also decrease the amount of mathematics present here on Doc Madhattan: the idea is also to take up my column on Mathematics in Europe, so probably the mathematical posts here will be an extract with the link to the complete article.
ON the other hand, physics will have all the space needed here on Doc Madhattan.
I hope that taking this road again may be even longer and more enjoyable than the one that was interrupted a year ago.

The great question about the Hubble constant

The Hubble-Lemaitre law is the mathematical formula bout the expanding universe. One of the collateral results of Einstein's theory of relativity was an expanding and non-static universe, a result that, in a first time, Einstein himself had disavowed. Yet various observations made in the second half of the 20s of the twentieth century instead confirmed the hypothesis of cosmic expansion^{(1, 2)}. \[z = H_0 \frac{D}{c}\] where $c$ is the speed of light, $H_0$ is the Hubble constant, while $z$ and $D$ are the light's redshift and the distance of the galaxy from the observer. The redshift, in particular, is due to the Doppler effect applied to electromagnetic waves. For example, when you hear the siren of an ambulance, it will seem to you stronger or weaker if approaching or moving away from your position. An electromagnetic wave, like light, instead will be closer to blue or red depending on whether it is closer to or away from the observer.
So, it has a certain importance to measure the redshift of the galaxies around us: evidently a null or little redshift was a clue to a static universe, otherwise we live in a dynamic universe, as you can see from the image present in the historical Hubble article⁽²⁾:

Imaging quantum entanglement

The main object of a paper published a couple of days ago on Science is to find an answer to the following question:

what kind of imaging process could reveal a Bell inequality?

The experimental set-up used a $\beta$-Barium Borate crystal pumped by a (quasi-cotinuous) laser. The pairs of entagled photons generated are subsequently separated on a beam splitter and propagate into two distinct optical systems like LIGO interferometer.
The results is the production of some images that shots the Bell inequality violation, like the following image:

Moreover, our demonstration shows that one can detect the signature of a Bell-type behavior within a single image acquired by an imaging setup. By demonstrating that quantum imaging can generate high-dimensional images illustrating the presence of Bell-type entanglement, we benchmark quantum imaging techniques against the most fundamental test of quantum mechanics.

Moreau, P. A., Toninelli, E., Gregory, T., Aspden, R. S., Morris, P. A., & Padgett, M. J. (2019). Imaging Bell-type nonlocal behavior. Science Advances, 5(7), eaaw2563. doi:10.1126/sciadv.aaw2563