On Mathematical Beauty in Physics

> the Penrose singularity theorems in General relativity were impossible until Einstein's theory was formulated correctly with rigorous mathematics

I wonder if you would please justify that statement, specifically and only with respect to one of the things called the [Hawking-]Penrose singularity theorem? In particular, what was sufficiently unrigorous or alternatively missing from classical General Relativity that blocked a reasonable choice -- especially the early 1960s ones (example and commentary below) -- of such a theorem?

[Penrose 1965] Gravitational Collapse and Space-Time Singularities https://doi.org/10.1103/PhysRevLett.14.57 (2.5 pages)

(You can also find a copy in the usual place)

(I submit the reason it took until 1964 for a Penrose singularity theorem is explained by the first two lines of [Penrose 1965]: the surprising discovery that QSO 3C 273's highly extragalactic redshift z ~ 0.16 having been published in 1963 and QSO 3C 147's z ~ 0.55 following in 1964 strongly suggested SMBH activity. In other words the underlined statement in the fourth paragraph did not follow any sort of mathematical development (ADM, for instance) but rather was motivated by one of the most provocative observations of nature I can think of off the top of my head, up there with the (also 1964) discovery of the CMB.)

Mostly serious comment: I'm not sure why the form e^{i\pi}=-1 isn't better. Only it doesn't have 0, but is it worse, less beautiful? (It doesn't seem to have the same "relationship between 5 fundamental constants", although it adds the negative number realm, to the imaginary and transcendental–neat.) Would E-mc^2=0 be similarly be better than E=mc^2, because it has an additional "fundamental constant"?

Just in time I started reading ``Drawing Physics'' by Don S Lemons

Just a note on the author's comment under Goya's painting. She advances that "unlike painting", in physics beauty can only be appreciated by those who understand their meaning. I disagree there's a difference in these disciplines, case in point, I knew Goya's painting long before becoming interested in grecoroman mythology, and I plainly didn't like it. Today I can appreciate the brutal tone of the painting knowing the story behind, and that has turned my view of the opus.

Another shocking example is El Expolio by El Greco, shocking technique but doubtful setting... until the author's intention is explained (or understood without further word), at that point it just becomes beautiful.

And in the same way that it's not necessary to know the story behind in art (you could like a painting for technique, looks alone), a layperson could like a mathematical formula for purely aesthetic reasons.

So all in all, I don't think mathematics and physics stand out in this regard.

To me, General Relativity has some pleasing qualities: it's mathematically complete, and it's very flexible.

As to the Einstein Field Equations (EFEs) themselves, the aesthetically pleasing quality is the terse notation brought about by the Einstein summation convention, abstraction into the Einstein tensor G_{\mu\nu}, the stress-energy-momentum tensor T_{\mu\nu} and other objects, and so forth (it's even shorter with geometrized units c = G = 1, ignoring the cosmological constant (say if one relies on thin shells and junctions[1]), \pi = 1, and/or 0 on the RHS).

This terseness can hide several pages of partial derivatives that look like http://4.bp.blogspot.com/-0e2Zl5QrRiA/UZfL29xVTZI/AAAAAAAAAF... (from NCSA's (offline) numerical relativity mathmine1.html originally) -- there will be sixteen of those for the Ricci tensor, plus some more pages for the other parts of the left-hand-side, although by introducing lots of symmetries and simplifications we can cut down by quite a bit.

One can get a feel for how these are generated with the example at https://www.maplesoft.com/support/help/Maple/view.aspx?path=... (eqn 12 for the metric at eqn 6)

Hartle's (anathe)Mathematica examples https://web.physics.ucsb.edu/~gravitybook/math/curvature.pdf are nice but don't take you to an interesting Ricci tensor.

Instead, perhaps see the answer https://mathematica.stackexchange.com/a/8908

Frankly, the mechanics of working with known solutions of the EFEs can be a pain. Perturbing against those can be even more of a pain. Junction conditions[1] between different exact solutions can be a super pain. Modern symbolic computing systems help a lot: https://en.wikipedia.org/wiki/Tensor_software (sigh, that needs updating, e.g. GRTensorII->GRTensorIII https://github.com/grtensor/grtensor/wiki )

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[1] https://arxiv.org/abs/gr-qc/9510052v3 since absorbed into GRTensorIII