# Einstein, Barcelona, Symmetry & Cosmology: The Birth of an Equation for the Universe

## Abstract

**:**

## 1. Introduction

## 2. Who Was Einstein? What Had He Achieved by 1923?

## 3. Essential Principles Conforming the Theories of Relativity

#### 3.1. Galilean Relativity

“Lock yourself up with a friend in the main cabin, under the deck of a rather large ship, and bring flies, butterflies, and other small flying animals. Hang a bottle so that it drains, drop by drop, into a large container below. Make the ship go at the speed you prefer, but always the same: a smooth motion without fluctuations in one direction or the other. The drops will fall into this container without being diverted aft, even if the ship has moved forward while the drops are still in the air. The butterflies and flies will continue their usual flight from side to side as if they never tire of following the ship’s course, however fast it may go, and it will never happen that they concentrate on the stern of it.”

#### 3.2. The Special Theory of Relativity

^{2}. Einstein took time to write it in this form; he did not do so in his already-mentioned work of 1905, in which he expressed it indirectly, already. In principle, the formula describes the values that the magnitudes take in a reference system at rest, but it also extends to the values of relativistic mass and energy for a system in motion. Einstein clearly stated that the laws of energy conservation and conservation of mass were “the same” [25].

^{2}, they realized that the basic process of fission “was indeed energetically possible”. As Frisch described it [26]:

“We walked up and down the snow, me on skis and Lise on foot...and little by little the idea took shape... based on Bohr’s conception of the nucleus as a drop of liquid; the drop could stretch out and split apart... We knew there were very strong forces that would oppose it,... like the surface tension. But nuclei are different from normal droplets. At this point, we both sat on a tree trunk and started calculating on scraps of paper... the uranium nucleus could become a very unstable blob, ready to split... But... when the two blobs separated, they would be further separated by electrical repulsion, the equivalent (in energy) of about 200 MeV. Fortunately, Lise remembered by heart how the masses of nuclei were calculated... and discovered that the two nuclei formed... would be lighter by about one-fifth the mass of a proton. Now, every time mass disappears, energy is created, according to Einstein’s formula E = mc^{2}, and... the loss of mass was equivalent to 200 MeV! Everything fit!”

#### 3.3. The General Theory of Relativity

_{g}) is the same as the one which appears in the formula F = m

_{i}a (called inertial mass, m

_{i}), which is inversely proportional to the acceleration that the body acquires when a mechanical force is applied to it. In short, m

_{g}= m

_{i}. All these formulations of the principle of equivalence are equally valid.

“We assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system”.

“We arrive at a very satisfactory interpretation of this law of experience if we assume that the systems K and K’ are, physically, completely equivalent; that is, if we admit that we can also consider the system K as a space free of gravitational fields, but at the same time as a uniformly accelerated system. This assumption of exact physical equivalence makes it impossible for us to talk about the absolute acceleration of the reference system, just as the theory of special relativity forbids us to talk about the absolute speed of a system. And this makes the equal fall of all bodies in a gravitational field seem then to be a most natural thing.”

_{μν}is the Ricci curvature tensor, which represents the curvature of spacetime caused by matter; R is the scalar curvature, a measure of the overall curvature of spacetime; g

_{μν}is the metric tensor, which describes the geometry of spacetime; Λ is the cosmological constant, which in its modern conception represents the energy density of the vacuum state of space; and T

_{μν}is the stress-energy tensor, which describes the distribution of matter and energy in spacetime. The Ricci curvature tensor is a geometric object obtained by contraction of the first and third indices of the Riemann curvature tensor, R

^{ρ}

_{σμν}= ∂

_{μ}Γ

^{ρ}

_{νσ}− ∂

_{ν}Γ

^{ρ}

_{μσ}+ Γ

^{ρ}

_{μλ}Γ

^{λ}

_{νσ}− Γ

^{ρ}

_{νλ}Γ

^{λ}

_{μσ}, namely R

_{μν}= R

^{ρ}

_{μρν}, where ∂

_{μ}stands for ∂/∂x

^{μ}, Γ are the Christoffel symbols of the Levi-Civita connection corresponding to the spacetime metric, to wit: Γ

^{ρ}

_{μν}= 1/2 g

^{ρσ}(∂

_{μ}g

^{σν}+ ∂

_{ν}g

^{σμ}− ∂

_{σ}g

_{μν}). One must observe that Einstein only introduced the cosmological constant term in his paper of 1917, “Kosmologische Betrachtungen …” [6], where he used his field equations for the first time to obtain a static model for the universe.

“Aber eines ist sicher, dass ich mich im Leben noch nicht annähend so geplagt habe und dass ich große Hochachtung vor der Mathematik eingeflößt bekommen habe, die ich bis jetzt in ihren subtileren Teilen in meiner Einfalt für puren Luxus gehalten habe!”

“But one thing is certain, that I have never nearly so toiled in life and have instilled in me such a high regard for mathematics, which I had considered until recently, in my naivety, as for its subtler parts, as a simple luxury!”

^{2}= (1 − 2GM/rc

^{2}) c

^{2}dt

^{2}− dr

^{2}/(1 − 2GM/rc

^{2}) − r

^{2}(dθ

^{2}+ sin

^{2}θ dφ

^{2}).

#### 3.4. Two Important Observations

#### 3.5. Summary of the Relativity Theories

- The principle of relativity or general covariance
- Galileo: it makes sense to formulate laws of physics (inertial systems)
- Laws for inertial or accelerated systems. Form of eqs changes (Galilean, Lorentz, Poincaré transf.)
- No total relativity (Mach’s principle). Truncated to 1st-/2nd-order eqs.

- The speed of light in a vacuum is constant, c, in all inertial systems
- Together with Galileo’s relative principle (inertial system) → Special Theory of Relativity

- The principle of equivalence
- Gravity is like all other forces. Equiv. of inertial mass and gravitational mass: m
_{i}= m_{g} - Spacetime is a mathematical manifold, locally Minkowskian

- 4.
- Zero torsion hypothesis (∇
_{X}Y − ∇_{Y}X = [X,Y])- Christoffel symbols are symmetrical. You can relax it (Einstein-Cartan, string theory)

- 5.
- Reduction to Newton’s laws (at small speeds as compared to c)
- To define the universal constants of the new theory

- ➢
- Are the 1st and 3rd principles independent?
- ▪
- The answer is tricky: yes and no
- ▪
- They are in their presentation
- ▪
- But the approximations made in the math formulation of the GTR (cut to 2nd order) render the equivalence principle also approximate
- ▪
- Higher-order differentials and gradients do differ
- ▪
- This will become noticeable in extremely high-energy processes.

- ➢
- Einstein’s theory is not the final one (AE dixit)
- ▪
- Mach’s principle of general relativity is not fulfilled
- ▪
- Einstein was the first to admit his theory was approximate and was convinced that someone would soon perfect it
- ▪
- We are trying to do this now, a century later: S-T and f(R) theories, QG? etc.
- ▪
- The symmetry-breaking paradigm could be useful

## 4. Events of 1923 and Einstein’s Six-Month Journey

#### 4.1. Notable Events in the World, in 1923

#### 4.2. Einstein’s Long Journey

Travel diary for Japan, Palestine, and Spain [6 October 1922–12 March 1923].

6 October Night trip in overfilled train after reunion with Besso and Chavan. Lost wife at the border.

7 October Sunrise shortly before arrival in Marseille. Silhouettes of austere flat houses surrounded by pines. Marseille, narrow alleyways. Voluptuous women. Vegetative living. We were taken in tow by seemingly honest youth and dropped off at a ghastly inn by the railway station. Bugs in morning coffee. Made our way to the shipping company and the old harbor near the old city quarter. At the ship …

“The Japanese people attract me... even more than all the peoples I have met so far: quiet, modest, intelligent, appreciative of art and considerate. Nothing is about appearances, but everything is about substance...”

“...countless small temples, dedicated to natural deities. Stone figures are often delightful. Of steps cut in granite rocks (altitude around 700 m). Memorial to the Japanese love of nature and all kinds of endearing superstitions.”

^{2}, had led Lise Meitner, Otto Hahn, and other scientists [35], as discussed before, to investigate whether it could make real sense. And later, this led to advances in creating a chain reaction of uranium and plutonium and its use in the Second World War. Although this had never been his intention, the moment Einstein heard the tragic news, he exclaimed: “Woe is me!”.

#### 4.3. Einstein in Spain

Doc. 379. Travel diary [March 1923], p. 325–326

“17, 18, 19 February. Indigestion from bad food. High seas and rain. 19 in the morning, Stromboli well in sight. Afternoon, 6 o’clock, Naples. Vesuvius with gray clouds cloudy sky. So cold and unpleasant that one is glad to stay on the boat. An Englishman from Australia turns out to be from Mecklenburg. News of a rail strike in France and more and more retaliation in the Ruhr, how will things go? In Toulon, friendly people. In Marseille, dangerous to speak German. The manager of the freight depot refuses to send our baggage to Berlin or even to Zurich.

22–28 February. Stop in Barcelona. We are tired but friendly people (Terradas, Campalans, Lana, Tirpitz’s daughter). Popular songs, dances. Refectory. How beautiful it was! (Figure 13)

1 March Arrival in Madrid. Departure from Barcelona, farewell. Terradas, German consul with Tirpitz’s daughter, etc. (Figure 14)

3 March First lecture at the university.

4 March Car ride with Kocherthaler—answer to Cabrera. I wrote the academy speech. Academy session in the afternoon chaired by the king. Magnificent speech by the president of the acad. Afterwards, tea in the society of artists. Ladies. You felt at home but in a very Catholic atmosphere.

5th In the morning. Honorary member of the Mathematical Society. Debate on general relativity. Lunch at Kuno’s. Visit with Kuchal. A wonderful old thinker. Very sick, good conversation. Invitation to dinner in the afternoon from Mr. Vogel. He has a good heart, humorous pessimism.

6th Excursion to Toledo hidden through many lies. One of the best days of my life. Radiant skies Toledo is like a fairy tale. We were guided by an old enthusiast, who supposedly had written something important about Gra [El Greco]. Streets and market, city views, the Tagus with stone bridges, stone covered hills, lovely level cathedral, synagogue, sunset on the return trip with brilliant colors. Small gardens with views near the synagogue. Greco’s magnificent fresco in a small church (burial of a nobleman) is one of the most profound images I have ever seen. Wonderful day.

7th Audience at noon with the King and Queen Mother. The latter shows that she knows about science. You realize that no one tells her what they think. The king, simple and dignified, I admire him for his ways. In the afternoon, the third university conference, a devoted audience that probably couldn’t understand practically anything because the latest problems were being discussed. In the evening, a great reception at the home of the German envoy. The envoy and family are magnificent and modest people. Socializing is as heavy as ever.

8th Honorary doctorate. Spanish speeches with associated firecrackers. Long but with good content that of the envoy on German-Spanish relations but in genuine German. No rhetoric. Then, in the afternoon, visit with the technical students. Speeches and nothing more than speeches, but very meaningful. Talk in the evening. Then playing music at Kuno’s. A professional (director of the conservatory), Poras [Bordás] played the violin exquisitely.

9th Excursion to the mountain and El Escorial. Glorious day. An evening reception in the student residence with talks by Ortega and me.

10th Prado (mainly looking at paintings by Velazquez and El Greco). Farewell visits. Lunch at the home of the German envoy. A night with Lina and the Ullmanns in a small and primitive dance venue. Fun evening.

11th Prado (splendid masterpieces by Goya, Raphael, Fra Angelico).

12th Trip to Zaragoza.”

## 5. The Important Scientific Context Around Einstein at the Time of His Visit

“...I have been working on the axiomatics of the principle of relativity, starting from two propositions: a) uniform movement continues to be uniform for all observers; b) the speed of light is constant (the same for both a static and a moving observer). Moreover, I have obtained formulas for a Universe with only one spatial dimension, which are more general than the Lorentz transformations...”

“...I am sending you a short note on the shape of a possible Universe, more general than Einstein’s cylinders and De Sitter’s spheres. Apart from these two cases, a world also arises whose space has a radius of curvature that varies with time. I thought this question might be of interest to you. As soon as I can, I will send you a German translation of this note. And, if you think the matter is interesting, please be so kind as to endorse me with a view to its publication in a scientific journal...”

“...The case of a stationary universe includes only two possibilities, which have already been considered by Einstein and De Sitter. The case of a variable universe admits, on the other hand, many possible situations. In some cases, the radius of curvature of the universe increases steadily with time. And other situations correspond to a radius of curvature that changes periodically...”

“...As for the non-stationary universe, the results contained in the work seem suspicious to me. The solution given for this case turns out not to satisfy the field equations...”

“...Considering that the possible existence of a non-stationary universe is of interest, I would like to present the calculations I have made here so that you can verify and critically evaluate them. [He details all mathematical operations]. If you find the calculations, I present in this letter to be correct, please be so kind as to inform the editors of Zeitschrift für Physik about this conclusion. Perhaps in that case, you would like to publish a correction to the statement you have made, or at least allow me to publish the calculations part of this letter...”

“...On Monday, May 7, I was with Einstein, reading Friedmann’s article of Zeitschrift für Physik in detail...”

“...I managed to defeat Einstein in the argument of Friedmann’s work. Petrograd’s honor is saved!”

“...In my previous note, I criticized Friedmann’s work on the curvature of space. However, a letter from Mr. Friedmann, which Mr. Krutkov handed me, has convinced me that my criticism was based on an error in my calculations. Now I consider that the results of Mr. Friedmann are correct and bring new light. It is shown that the field equations and the static solution also admit dynamic solutions (i.e., with a variable time-coordinate), with central symmetry for the spatial structure.”

^{−2}).

^{−2}, and c is the speed of light in vacuum. Moreover, ρ and p are the volumetric mass density and pressure, respectively; k, corresponding to the curvature, is constant throughout a particular solution but may vary from one solution to another.

## 6. Summary and Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Galileo Galilei (1564–1642). Oil portrait of an Italian painter believed to be from the 18th Century. Reproduced with permission from Wellcome Trust, UK. Fair use.

**Figure 2.**On the left, Hendrik Lorentz (1853–1928) and, on the right, Henri Poincaré (1854–1912)—Image: Wikimedia Commons. Public Domain.

**Figure 3.**Albert Einstein, around 1905, his “annus mirabilis”, in which he published four momentous articles. One of them: “Zur Elektrodynamik bewegter Körper”, the work in which he built the special theory of relativity. Public Domain.

**Figure 4.**On the (

**left**), a ball falls to the ground in a suitably accelerated rocket in the absence of gravity. On the (

**right**), the ball falls to the ground in the usual way. The effect is identical in both situations, completely indistinguishable in the chamber that isolates the observer from the outside world. Fair use.

**Figure 5.**Albert Einstein in 1916, in the house library by Paul Ehrenfest, in Leiden, where he stayed for a few days. Public Domain.

**Figure 7.**Albert Einstein giving the 11th Josiah Willard Gibbs lecture at the American Association for the Advancement of Science meeting on 28 December 1934—Public domain.

**Figure 8.**On 11 July 1923, Einstein spoke at the Congress Hall in Gothenburg, Sweden, at the Scandinavian Congress of Naturalists. Public domain.

**Figure 9.**Albert and Elsa Einstein in Japan, November–December 1922. Author unknown, courtesy of Meiji Seihanjo. Public domain.

**Figure 10.**Einstein at the YMCA in Moji, Japan, December 1922. Courtesy of Kenji Sugimoto. Einstein Archives Collection, Hebrew University of Jerusalem. Fair use.

**Figure 11.**The Einsteins at the Government House in Jerusalem, with the British High Commissioner. February 1923. Einstein Archives Collection, Hebrew University of Jerusalem. Fair use.

**Figure 12.**Albert Einstein in front of the Fonda Ibérica, in Espluga de Francolí, on 25 February 1923. What attracted the children’s interest was not Einstein’s charm but the magnificent automobile in which he had arrived, from Casa Elizalde, a type 29 torpedo, unmistakable (in 1922, its price was 33,000 pesetas)—public domain.

**Figure 13.**Einstein at the event held at the Industrial School of Barcelona on 28 February 1923, where he attended the performances of the Barcelona sardanist couple and the Penya de la Dansa of the New University Student Association. Public domain.

**Figure 14.**Einstein inside the train, at the France station in Barcelona, before leaving for Madrid on 1 March 1923. Public domain.

**Figure 15.**Retractation note by Albert Einstein, published in Zeitschrift für Physik on 31 May 1923—public domain.

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**MDPI and ACS Style**

Elizalde, E.
Einstein, Barcelona, Symmetry & Cosmology: The Birth of an Equation for the Universe. *Symmetry* **2023**, *15*, 1470.
https://doi.org/10.3390/sym15071470

**AMA Style**

Elizalde E.
Einstein, Barcelona, Symmetry & Cosmology: The Birth of an Equation for the Universe. *Symmetry*. 2023; 15(7):1470.
https://doi.org/10.3390/sym15071470

**Chicago/Turabian Style**

Elizalde, Emilio.
2023. "Einstein, Barcelona, Symmetry & Cosmology: The Birth of an Equation for the Universe" *Symmetry* 15, no. 7: 1470.
https://doi.org/10.3390/sym15071470