SR and GR aren't events in any spacetime-they're definitions of spacetimes: SR defines flat spacetime, GR is the generalization of SR and defines a more general spacetime.

Indeed spacetime can have not only curvature but, also, torsion. So saying that spacetime is curved isn't enough-it can have zero curvature, but non-zero torsion. It turns out that, if the spacetime geometry is defined only by the metric tensor, it is possible to set the torsion equal to zero. In extensions of general relativity (e.g. scalar-tensor theories or supergravity) this isn't the case.

1. Flat spacetime is one solution of Einstein's equations, that can have many others, depending on the boundary conditions.

2. Yes, if the spacetime geometry is defined only by the metric-as is the case in GR; no if other fields define the spacetime geometry, as in scalar-tensor theories, or supergravity.

3. Of course. The worldline of any physical object is a time-like curve.

All these topics are now understood and can be studied in textbooks, e.g. http://insti.physics.sunysb.edu/~siegel/errata.shtml

The space is based on the figure-ground relation, and all figure are closed by contours.

The example textbook does mention the "watt balance" on page 42 that has now come to pass. And “Euler’s theorem” on page 506 that has now opened the flood gates for applied differential topology. And it's free!

Dear Stam Nicolis

Thank you very much for bringing the most professional understanding. The well-trained is really enviable.

Thanks also for the link reference. I knew that Yang started the physics program at Stony Brook, but didn't know they had such a book (and of course, other don't know more).

There are a couple nagging questions that I need to keep asking.

First, you say that it is relativity that defines space-time. So there is a causality issue here. I'd like to know what you mean by "defines". Because I have always thought that some kind of "physics" defines "relativity", not the other way around, unless relativity is the most fundamental and correct "axiom". Obviously they don't seem to be, for example Einstein doesn't state a reason why matter changes spacetime, but only what matter changes spacetime according to.

Second, if spacetime can be bent, can an expression like ∂Ψ/∂x maintain its expression all the way from a flat spacetime into a bent (twisted) spacetime?

Also, when I say curved time, I don't mean curved world lines. I mean that the four-dimensional axes are orthogonal, and the definition of bending must be through the whole, or through the separate axes. Obviously bending must be expressed through the separate axes anyway. If time bending cannot be expressed, then neither can space bending.

Best Regards, Chian Fan

I agree with Chian Fan that we have to understand how the moving electron physically bends the surrounding space, but not how it is forced to move in a speculative space that satisfies the equations of SR or GR. (Although first it would be good to know what the surrounding space is.)

In particular, to understand how the “jitter” of the half-integer electron spin and its “reverse motion in time” with negative energy, resulting from the Dirac equation, helps the electron to move at relativistic speed, greatly to deform the surrounding space, but not to emit photons, since its movement is uniform and straight-line?

Dulin Mikhail.

It's been understood since the work of Emmy Noether in 1918, that the symmetries come first in the description of natural phenomena. So the symmetries do describe the physics.

So, while the relevance of special relativity was understood by noticing that the solutions of Maxwell's equations are solutions of the (in)homogeneous wave equation and the latter is invariant under global Lorentz transformations, it was then realized that Maxwell's equations are the only equations one can write, upon assuming global Lorentz invariance, gauge invariance and that the equations of motion don't have more than two derivatives.

An interesting issue is that two of the four Maxwell equations don't contain time derivatives-they impose constraints. These impose gauge invariance.

Similarly Einstein's equations seem to include four equations, that, also, impose constraints. That was what Noether found: They impose the constraints of invariance under diffeomorphisms. And, similarly as for electromagnetism, if one imposes invariance under diffeomorphisms, that the equations of motion don't contain more than two derivatives and that spacetime geometry is defined by the metric tensor only, one finds that only Einstein's equations are possible.

Matter affects the spacetime geometry, in general relativity, because the only way, that's consistent with the fact that the equations of motion be invariant under diffeomorphisms is that the properties of matter can be included in the equations of motion only through the energy-momentum tensor. If the spacetime geometry requires additional fields (e.g. in scalar-tensor theories or supergravity) then matter must affect the spacetime geometry through its fields, also, in a way consistent with invariance under diffeomorphisms.

Recall that the way matter enters into Maxwell's equations is, also, defined by global Lorentz invariance and gauge invariance, which describe the properties that the electric charge is invariant under global Lorentz transformations and that it is locally conserved-the variation in time of the charge density is due to the flow of a current and the charge density and current are components of a 4-vector.

Derivatives aren't invariant under diffeomorphisms, that's why the answer to that question is, No.

Time can be bent, as can space. In flat spacetime global Lorentz invariance simply implies that the light cone is globally defined. In curved spacetime it's locally defined. It's not possible to have closed time-like curves in flat spacetime, however. For a discussion on this point, beyond flat spacetime, cf. https://s3.cern.ch/inspire-prod-files-9/9e243ddb6420681ac2362781c9754c04

“How the view of space-time is unified (3)-If GR's space-time is not curved, what should it be?”

- the answer to this question in mainstream physics is direct and clear: space and time, if the spacetime is not curved, i.e. is flat, are unified in the SR yet from 1908, when it was claimed that “…Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality ….”.

However that in the answer is fundamentally incorrect. Really, as that is rigorously scientifically proven in the philosophical 2007 Shevchenko-Tokarevsky’s “The Information as Absolute” conception, recent version of the basic paper see

https://www.researchgate.net/publication/363645560_The_Information_as_Absolute_-_2022_ed

- both, “space” and “time” are concrete actualizations of the absolutely fundamental “Logos” set elements “Space” and “Time” in concrete informational system “Matter”, the elements are completely different phenomena, and so any “space”, including Matter’s space, and time, fundamentally cannot be “unified”.

The space and time “unifyings” in the SR – and in the GR – are postulated as purely bare declarations without some scientific grounds/explanations, however the SR and the GR are standard theories till now; what happened – and happens till now, because of that in mainstream philosophy and sciences, including physics, and for the SR/GR authors, the fundamental phenomena/notions, first of all “Matter” “Consciousness”, “Space”, “Time”, “Energy”, “Information”, are/were fundamentally completely transcendent/uncertain/irrational, and the postulates in the theories are really some transcendent illusory claims;

- which, moreover, really fundamentally aren’t adequate to the reality. More about what are “Space/space”, “Time/time”, and all other fundamental phenomena/notions above see the link above;

- to read yesterday SS post in https://www.researchgate.net/post/NO7_Is_there_a_minimum_value_of_m_in_the_mass-energy_equation_Emc2#view=64fbc3721b17d1bd0e0d4bac/5/6/5/6/6/5/4/3/2/1/2/3/4/5/4/5/6/5/6/6/7/6/7/6/7/7 and SS post 4 days ago now in

https://www.researchgate.net/post/How_the_View_of_Space-Time_is_Unified_2-Where_Does_the_Length_Contraction_and_Time_Dilation_of_SR_occur/1 it is useful as well.

A few concrete comments else.

First of all: Gravity fundamentally is nothing else than some fundamental Nature force, which in a few traits is similar to the Electric Force, and even to Nuclear Force – Matter is rather simple logical system, and, as that is rigorously scientifically shown in the Shevchenko-Tokarevsky’s 2007 initial models of Gravity and Electric Forces in

https://www.researchgate.net/publication/365437307_The_informational_model_-_Gravity_and_Electric_Forces; and 2023 model of Nuclear Force,

https://www.researchgate.net/publication/369357747_The_informational_model_-Nuclear_Force- all at least these 3 fundamental Nature forces in Matter act by the same scheme.

More concrete comment how Gravity acts see, say, SS post 2 days ago now in https://www.researchgate.net/post/Can_kinetic_and_gravitational_potential_energy_could_both_be_zero_and_can_gravitational_potential_energy_be_equal_to_kinetic_energy

Correspondingly, say, that

“…Matter affects the spacetime geometry, in general relativity, because the only way, that's consistent with the fact that the equations of motion be invariant under diffeomorphisms is that the properties of matter can be included in the equations of motion only through the energy-momentum tensor….”

- looks as rather questionable claim. Say, the equations of motion of electrically charged bodies are well invariant under diffeomorphisms and Lorentz group at the electromagnetic tensor components actions, and at that no any Matter affecting the spacetime geometry happens; etc.

Etc., though what is else well more important is that in the GR the energy conservation law is violated, and so the GR equations have too transcendent solutions; which, however, are obtained as quite legitimate in innumerous mainstream physics publications – a number of transcendent/mysterious “holes”, “channels to some “multiverses””, etc. in spacetime, warped spaces, so on;

- while really Matters spacetime, as that is rigorously scientifically shown in the SS&VT models, is the fundamentally absolute, fundamentally continuous, fundamentally flat, and fundamentally “Cartesian”, (at least) [4+4+1]4D spacetime with metrics (at least) (cτ,X,Y,Z, g,w,e,s,ct); where all dimensions are fundamentally mutually independent, and the spacetime fundamentally cannot be impacted, including “contracted/dilated/curved” by anything in Matter, and fundamentally cannot impact, including “contract/dilate/force to move” anything in Matter;

More see in the links above.

Cheers

Dear Stam Nicolis.

Now it’s my turn to thank you for such an impressive and wonderful answer. It clearly shows that we have no choice in describing electromagnetism and gravity other than Maxwell's and Einstein's equations. If only we want to retain some obvious requirements, including symmetry for the former and diffeomorphism for the latter.

But then we remain at the level of the classical description of the phenomena under consideration. How then can we describe photons and particles with mass? How to describe their own structure, at least topologically, and the features of interaction with each other so as to come to the corresponding classical limit in the form of the indicated equations? Shouldn’t we describe photon in the form of wave packet, and particles with mass as point particles?! This description has long been unsatisfactory for many scientists.

I understand that there is no good answer to these questions, but I also don’t want to remain in the “golden cage” of Maxwell's and Einstein’s equations.

Dulin Mikhail.

Dear Mikhail N Dulin and Stam Nicolis

Thank you for the very professional discussion. For me personally, the amount of information is a bit overwhelming, and some of the math concepts need to take time to learn and comprehend yet. Nonetheless, I can't help but try to make some comments first and ask some questions that have been bothering me and are still immature. Criticisms are welcome.

I. Symmetry Dominates Interaction

1) Whether symmetry dominates conservation, or conservation dominates symmetry. Or maybe the two are indistinguishable. This is an important mathematical problem, a physical problem, and even an important physical-philosophical problem. It concerns the fundamental operating laws of nature. My understanding is that conservation comes first, and symmetry is a product of conservation. Conservatism determines more and is a manifestation of continuity, a larger set. The state of symmetry is not always continuous and is a smaller set relative to conservatism. We can use conservation indiscriminately in any situation, symmetry needs to be discovered and expressed in specific groups. But symmetry can reflect more specific physical structure and behavior.

2) Is Noether's theorem valid in non-uniform spacetime, curved spacetime, discrete spacetime?

According to the definition of symmetry, H(ξ+dξ)=H(ξ), infinitesimal space translation, infinitesimal time translation, is the most basic concept. Energy and momentum correspond to time and space, respectively, and represent the most fundamental things, and we can regard other conserved quantities as extensions of energy and momentum, such as conservation of angular momentum and conservation of charge ----. The implicit meaning here is that they must all be embodiments of the fundamental energy-momentum, another form of energy-momentum (this is a point that will be debated). The derivation of Noether's theorem does not take into account the nature of spacetime. In a flat spacetime, it certainly holds. But when spacetime is inhomogeneous, when spacetime is curved, when spacetime is discrete‡, is Noether's theorem still valid? If it is valid, how is it guaranteed; if not, does it mean that conservation follows?

3) According to Noether's theorem, does a constraint relation on the fundamental energy-momentum unit arise.

We examine the details of the behavior of translations. In classical mechanics, objects are usually treated as point particles* and are space-time homogeneous, so any shifts do not overlap. At non-point-particle states, H should be regarded as distributed and can never be a rigid body† or a solid, only a field. Assuming that dξ

Dear Mikhail N Dulin,

The quantum description of electrodynamics is QED. Indeed, in this case, the property that the group of gauge transformations, U(1), is compact, implies that it is possible to provide a consistent description of the quantum dynamics using the so-called lattice regularization. One example of how this works can be found here: Preprint The layered phase of anisotropic gauge theories: A model for...

How to take into account the quantum fluctuations explicitly is sketched here: Preprint Noisy SUSY

The quantum description of gravity is much more subtle, because the group of gauge transformations, the group of diffeomorphisms, is non-compact, so the approach, that works for electrodynamics, doesn't work here. One way to imagine realizing the corresponding description is presented here: Article Modular discretization of the AdS2/CFT1 Holography

Preprint The continuum limit of the modular discretization of AdS$_2

This is, however, still, far from the complete description of quantum gravity.

Dear Chian Fan,

It is, indeed, quite a lot to understand... But, hopefully, it's not impossible to do so.

It's not ``symmetry dominates interaction'', but, rather, ``symmetry defines interaction''.

Noether's theorem is valid, by direct calculation, in curved spacetime and on the lattice. Care must be taken upon drawing conclusions, however.

The thread question is scientifically answered in the SS post on page1, so more see the post and papers that are linked in the post, here only a few points:

- again, Gravity is fundamentally nothing else than a fundamental Nature force, and so, as that fundamentally is at other – i.e. Weak, Electric, and Strong/Nuclear, Forces - ultimately gravitational interactions are quantized. Though Gravity critically differs from other Forces in that is extremely weak Force, and so “classical” QM systems, say “gravitational atoms” really cannot exist, since necessary in this case to form some stable QM system gravitational masses are too large, that would be only principally “classical scale” macro bodies, with corresponding huge other Forces background.

So really gravitational QM systems don’t exist in Matter, besides possible exotic cases, say, in central compact objects of “black bodies”.

However the quantum nature of Gravity is well observable – see proposed in 2007 experiments with photons in https://www.researchgate.net/publication/215526868_The_informational_model_-_possible_tests ; section 2.1.2. “Monochromatic photon beam distortion”.

Returning to “spacetime curvature” in concrete “gravitational time dilation”: really any “time dilation” fundamentally cannot, and so doesn’t, exist. However in gravitationally coupled systems of bodies the intrinsic processes in the bodies evidently are slowed down – that follows from well observed gravitational mass defect, and so in this case the GR really is adequate to a certain extent to the reality,

- nonetheless this slowing, at least at statics, is well probably two time lesser than that the GR predicts – i.e. as that the SS&VT model of Gravity, link see the SS post on page 1 predicts - and that can be quite easily now tested in proposed in 2013-2015 [though note, that this experiment was proposed by Clifford Will yet in 1986] experiments, see https://www.researchgate.net/publication/277710038_The_informational_model_-_gravity_a_next_experiment

As to

“…We examine the details of the behavior of translations…. H should be regarded as distributed and can never be a rigid body† [† Any rigid body violates the theory of relativity.] or a solid, only a field.….”

- really in Matter indeed fundamentally cannot exist only “absolutely rigid bodies”, however, as that is rigorously scientifically shown in the SS&VT model [see https://www.researchgate.net/publication/354418793_The_Informational_Conception_and_the_Base_of_Physics] the Galileo-Poincaré relativity principle, and so Lorentz transformations, are completely valid just only if a body or a system of such bodies, are rigid. Including that happens if in a system the distant bodies interact strongly enough, say, practically any system of bodies on Earth and nearly around compose rigid systems because of Earth gravity.

If a system is a free bodies system, in this case action of the relativity principle is limited, and so it is possible to observe absolute motion in the absolute Matter’s 3D space and to measure the absolute velocity of the bodies, see proposed in 2013-2016 experiments in https://www.researchgate.net/publication/259463954_Measurement_of_the_absolute_speed_is_possible

“…A number of people think that spacetime is discrete, and this is an issue that could be given more thought.…”

- as that is rigorously scientifically proven in the SS&VT “The Information as Absolute” conception [see the linked in the SS post on page 1 paper], dimensions in Matter’s spacetime [and in any other informational system, though] are fundamentally continuous. That is another thing, that Matter’s ultimate base is the (at least) [4+4+1]4D dense lattice of primary elementary logical structures – (at least) [4+4+1]4D binary reversible fundamental logical elements [FLE], which is placed in the corresponding Matter’s space [see SS post on page 1], and so everything on Planck scale exists and happens indeed discretely, more see the SS posts and links in the posts.

Cheers

Dear Chian Fan.

I have no doubt that it is conservation laws that work in the surrounding world, and symmetry or something else is a consequence of the manifestation of these laws.

For example, the so-called parity violation (CP symmetry). It is believed that in “weak interactions” the mirror symmetry P and charge conjugation C are not preserved at the same time. But in quantum chromodynamics it is preserved - paradox! (All that is due to the nuclear particles, as I already wrote to you, are representatives of another space.))

In 1957, an experiment to study the beta decay of polarized 60Co nuclei was carried out by Chien Shiung Wu. It turned out that electrons fly mainly in the direction opposite to the direction of the spin of the nucleus. It was due to the fact that for the three particles formed after the decay (60Ni + e + antineutrino), the law of conservation of momentum and total spin must be satisfied.

Or change in symmetry during the phase transition - melting. Symmetry is broken - because the rotational motion is added to the translational motion of atoms or molecules, which I talked about in the report Preprint Thermal expansion of solids or how the space in crystals is ...

Dulin Mikhail.

Discrete symmetries (charge conugation, parity, time reversal, along with the structutaral symmetries of crystals) are a-slightly-different story, since Noether's theorem doesn't apply to them. Their role is different from the symmetries that correspond to continuous transformations-they impose so-called ``selection rules''.

Just some remarks about C and P in the weak interactions: It's a fact-not a belief-that the weak interaction-that describes radioactivity-does not preserve parity: The electrons that are emitted during beta decay all are found to have only one helicity (the projection of their spin along their momentum), not both. That was found experimentally in 1957.

In 1964 it was found that the weak interaction doesn't preserve the combination of charge conjugation and parity: Neutral K-mesons, when prepared in a state of definite charge conjugation AND parity, could be found in a state that was a linear combination thereof. That this transformation is due to the weak, not the strong, interaction was a non-trivial result at a time when the Standard Model wasn't known.

That the strong interaction (now known to be described by quantum chromodynamics)preserves-as does electrodynamics-the combination of charge conjugation and parity, while the weak interaction doesn't, isn't a paradox, however. The coefficient of the interaction that would be responsible for describing the breaking of CP (the so-called ``theta-term'') is zero, that's all. And the selection rule implies that fluctuations can't make it contribute.

There's no mystery about the thermal expansion of solids, since the contact with a thermal bath means that the solid isn't a closed system.

Space-time consists of two layers: chord (tonality) space-time and non-chord (atonality) space-time; The former is based on chords (quantum spectrum, strings), often used in music, painting, etc.; The latter masks chords, is given quantification by an external reference frame (clock, ruler, etc.) and is often used in classical physics, etc. Both layers involve figure-ground and motion.

The constant speed of light principle belongs to the quantum space-time category.

However it must be stressed that there is a subtle interplay between the coefficient of the so-called ``theta-- term'' of quantum chromodynamics and the ``mass matrix'' of the quarks and the consequences aren't fully understood-but there's not a paradox, but a realization that the description is incomplete. How to complete it isn't, for the moment, fully understood.

Cf. here: https://indico.nikhef.nl/event/534/contributions/8504/attachments/3873/4534/Nikhef_TopicalLectures_EDMs_2.pdf for a nice presentation.

The reason CP violation is of practical interest is because we observe mainly matter and antimatter is hard to find. CP violation does imply that matter would be more frequently encountered than antimatter; the problem is that the terms in the Standard Model (now known as the Standard Theory) that describe the effects of CP violation don't lead, quantitatively, to the level of CP violation that's necessary to describe the matter/antimatter asymmetry that we do observe. So there must be further mechanisms of CP violations to be discovered.

“…The reason CP violation is of practical interest is because we observe mainly matter and antimatter is hard to find. CP violation does imply that matter would be more frequently encountered than antimatter; the problem is that the terms in the Standard Model (now known as the Standard Theory) that describe the effects of CP violation don't lead, quantitatively, to the level of CP violation that's necessary to describe the matter/antimatter asymmetry that we do observe. So there must be further mechanisms of CP violations to be discovered.…..”

- really at high energy physics experiments now seems practically all “particles”

[i.e., as that is rigorously scientifically shown in the SS&VT modelhttps://www.researchgate.net/publication/354418793_The_Informational_Conception_and_the_Base_of_Physics,

brief comment see SS post August 28 page 7 in https://www.researchgate.net/post/No9_Is_the_spin_of_an_electron_really_spin/7, “particles” are specific disturbances in the Matter’s ultimate base - the (at least) [4+4+1]4D dense FLE lattice, as are close-loop algorithms that constantly run, basing on FLE “hardware”, as sequential FLE-by-FLE flipping; at that the algorithms sequences some FLE sections have marks of fundamental Nature forces – essentially the Forces charges - that radiate in the lattice the specific disturbances “Forces’ mediators”, which aren’t particles. If algorithms are stable – that are stable particles, if algorithms are damaged, on some tick it can be broken, when the particle decays]

- in a few hundreds ones zoo; and their reactions and decays channels are already well discovered and studied. Further in energy with a well non-zero probability is a huge desert gap between “particles” scale and Planck scale, where some only interacting by fundamental Nature Gravity force particles can exist; which so can have very short algorithms – up to [and seems rather probably really existent] Planck mass particles, algorithms of which contains 6 FLEs and so tick with frequency 1/6Planck times, and so the particles masses are ~19GeV.

So it seems as that a probability of discovering of some particles/events/effects where “the level of CP violation that's necessary to describe the matter/antimatter asymmetry would be sufficient in this case is practically for sure equal to zero.

At that particles and antiparticles differ first of all in that though they are identical algorithms, but which run in opposite commands orders, and so differ really because in the algorithms the Electric and Strong Forces [and Weak?] charges sequences aren’t symmetrical.

However Gravity Force is symmetric Force, and so algorithms of particles that have only Gravity charge are the same in both command orders, i.e. such particles are their antiparticles.

Correspondingly in the SS&VT initial cosmological model [more see https://www.researchgate.net/publication/355361749_The_informational_physical_model_and_fundamental_problems_in_physics, section “Cosmology”, it is proposed that at Beginning of Matter’s creation having only Gravity charge particles were created, which had rest masses, but which so logically could have momentums that are directed only in one – “positive” direction along cτ-axis of the 4D space with metrics (cτ,X,Y,Z), so Matter contained only matter yet at Beginning; further at interactions of primary particles because of momentum conservation law practically only “ordinary” particles were created, that are observed now, while essential part of the primary particles compose now “dark matter”]

- while particles and antiparticles that have charges of other Forces move in opposite directions in cτ-dimension [“Feynman–Stueckelberg interpretation” in QED]. Etc., more see the links above.

Cheers

Dear Stam Nicolis

Thank you very much for the math principles and the great resource links.

In addition to your reply here, I also noticed your discussion with @Wolfgang Konle about gravity*. You said "because their symmetries aren't appropriate - symmetries implying dynamics." "The entity that provides the ``energy'' needed to accelerate an object is the curvature of spacetime-it's the fact that time translations aren't globally defined that implies that there's a force. That's what the geodesic equation expresses."

I share his view that your answer is correct, but not the final one. It is true that "asymmetry" should be expressed as a force, but it doesn't answer why physical reality seeks to be "symmetric"‡; "geodesics" is the shortest path, but it doesn't answer why an object must move along the geodesic†, unless you assume that moving along the geodesic is "axiomatic" (which Einstein doesn't seem to have said) and provide a good reason for it; and "mass" causes spacetime to change, but doesn't answer why spacetime must be changed by mass.

I still maintain my view that the root of all this is the definition of energy-momentum, and the absolute requirement of conservation of energy-momentum, which is independent of occasion and is the largest set. And symmetries, whether external or internal, continuous or discrete, complete or approximate, symmetries in the steady state or symmetries in the breaking, all have specific conditions, and they are all supposed to be individual subsets under the conservation of energy-momentum.

The relationship between symmetry and conservation is an extremely important issue concerning the nature of physics. Therefore, we'd better start a new topic in the hope that it will attract more attention. https://www.researchgate.net/post/Symmetry_Invariance_and_Conservation_1-Who_is_the_Primary

BTW, "symmetry dominates interaction" is "Symmetry Dictates Interaction", not me, but C.N. Yang¶ , Gross also thinks so. I wrote it wrongly. Gross puts it another way, "Today we realize that symmetry principles are even more powerful-they dictate the form of the laws of nature. nature."

Best Regards, Chian Fan

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* Dr. Wolfgang Konle, https://www.researchgate.net/post/The_ultimate_reason_for_the_gravitational_force/1

‡ One of the examples is symmetry in crystallography. This geometric level of symmetry is a consequence rather than a cause, but it still describes many of the properties of crystals.

† Chian Fan, "How do light and particles know that they are choosing the shortest path?" https://www.researchgate.net/post/NO4_How_do_light_and_particles_know_that_they_are_choosing_the_shortest_path

¶ Yang, C. N. (1980). "Einstein's impact on theoretical physics." Physics Today 33(6): 42-49.

Gross, D. J. (1996). "The role of symmetry in fundamental physics." Proceedings of the National Academy of Sciences 93(25): 14256-14259.