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Claude Berthier

Mladen Horvatić

Marc-Henri Julien

Hadrien Mayaffre

Steffen Krämer

Nuclear magnetic resonance in high magnetic field: Application to condensed matter physics

Alain Aspect

C. R. Physique 18 (2017) 498–503Contents lists available at ScienceDirect
Comptes Rendus Physique
www.sciencedirect.com
The Comptes rendus de l’Académie des sciences throughout history / Les Comptes rendus de l’Académie des 
sciences à travers l’histoire
From Huygens’ waves to Einstein’s photons: Weird light ✩
Des ondes de Huygens aux photons d’Einstein : une étrange lumière
Alain Aspect
Institut d’optique Graduate School, Université Paris-Saclay, 2, avenue Augustin-Fresnel, 91120 Palaiseau, France
a r t i c l e i n f o a b s t r a c t
Article history:
Available online 26 November 2017
Keywords:
Light
Wave
Photon
Comptes rendus
French Academy of Sciences
Mots-clés :
Lumière
Onde
Photon
Comptes rendus
Académie des sciences
From Huygens to Kastler, the members of the Académie des sciences have played an 
important role in the development and/or the thorough understanding of the models of 
light. This has paved the way to modern Quantum Optics.
© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
r é s u m é
De Huygens à Kastler, les membres de l’Académie des sciences ont joué un rôle important 
dans le développement et/ou la compréhension profonde des modèles de la lumière. Ils 
ont ainsi permis l’émergence de l’optique quantique moderne.
© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
1. Emergence of a wave model of light
In the year 1666, Colbert invited the greatest scientists of the time to join the newly created “Académie des sciences”. 
One of the most eminent of them, Christiaan Huygens, born in 1629 and grown up in the Netherlands, had studied the 
works of Descartes, Pascal, and Fermat, and already produced major results in mechanics, mathematics and astronomy. He 
accepted to come and settle in Paris, where he would dwell until Colbert’s death and the revocation of the “Édit de Nantes” 
– events that led him to return to his native land, to spend there the rest of his life.
In 1672, Huygens discovered Newton’s work on the corpuscular model of light. At the time, the major problem was 
to find a model that would justify the laws of reflection and refraction, independently formulated by Descartes and Snell 
(Fig. 1). It is easy to understand that a bouncing corpuscle is a good model to describe the fact that a ray of light is reflected 
in a symmetrical way with respect to the normal, but the challenge is to understand why the refracted ray comes closer 
✩ This paper is based on a text that was published in the Lettre de l’Académie des sciences, as a record of a speech pronounced for the 350th anniversary 
of the foundation of the Academy, in 2016. It emphasizes the role of the members of the “Académie” in the evolution of the models of light, and extends 
over a period much broader than the existence of the Comptes rendus. Before the creation of the latter, the Mémoires de l’Académie des sciences, which can be 
found on the site Gallica of the “Bibliothèque nationale de France” (French National Library), are an invaluable source of major documents, such as Fresnel’s 
Mémoire sur la diffraction de la lumière, published in 1819. The long time it took for an article to be published, however, was a motivation to create the 
Comptes rendus, whose first role was to publish as fast as possible the discoveries presented in the weekly sessions of the “Académie”.
E-mail address: alain.aspect@iota.u-psud.fr.https://doi.org/10.1016/j.crhy.2017.11.005
1631-0705/© 2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
A. Aspect / C. R. Physique 18 (2017) 498–503 499Fig. 1. Christiaan Huygens soon considered the question of light beam reflection and refraction. (a) Reflection: θ2 = −θ1. (b) Refraction: n2 sin θ2 = n1 sin θ1. 
Adapted from Josell7 – Own work.
Fig. 2. Huygens’ wave model, expressed in 1678 and fully described in 1690, accounts for refraction provided that the speed of light is slower in the denser 
medium than in air, in contrast to Newton’s corpuscular theory.
to the normal when it passes from air to a denser medium, such as water or glass. Newton explained it by invoking the 
attraction of the denser medium: the particles of light are accelerated perpendicularly to the interface, and the trajectory 
is therefore closer to normal. But Huygens was looking for a model that would account for all known phenomena, and in 
particular the phenomenon of double refraction that is observed with some crystals, such as calcite. He discovered in 1678 
that a wave model of light would meet this requirement, while Newton’s corpuscular model did not. His model could render 
an account of refraction (Fig. 2), provided that the velocity of light is smaller in the denser medium, i.e., the contrary of 
Newton’s model.
Despite its value, Huygens’ wave model, described in full detail in 1690 in his Treatise on Light, [1] was ignored by most 
scientists for more than a century. If they had adopted the corpuscular model, it was because of the tremendous prestige 
conferred upon Newton, who had managed to explain the motions of planets through his law of universal gravitation. It 
took more than one century before Thomas Young, in England, and Augustin Fresnel, in France, developed the wave model 
of light, as the only one able to account for the phenomena of interference and diffraction, that they carefully studied in 
remarkable experiments (Fig. 3).
2. Wave vs. corpuscular theories
Young and Fresnel had to fight to impose this model, as evidenced by the story of Poisson’s bright spot, which involved 
several members of the Académie.
500 A. Aspect / C. R. Physique 18 (2017) 498–503Fig. 3. In France, Augustin Fresnel (1788–1827) developed the wave model in the frame of a contest launched by the “Académie des sciences”. In parallel, 
Thomas Young (1773–1829) conducted his work in England.
Fig. 4. The Poisson’s bright spot: Fresnel’s theory predicts the existence of a bright spot in the middle of the shade. The observation of that counterintuitive 
feature played an important role in the success of Fresnel’s wave theory of light.
1819 was the year, and the Académie des sciences had sponsored a contest on the diffraction of light, in other words, on 
the fact that, in the zone of transition between the shadow of a screen and an illuminated zone, fringes are observed, that 
is, alternating shadows and light. Fresnel, a young and brilliant mind educated at the “École polytechnique”, and who was a 
student of the academician Arago, presented a memoir [2] describing his experiments and giving a complete mathematical 
treatment of the problem, based on a convincing wave model. Many members of the “Académie”, though, were eminent 
scientists in mechanics, whose works were based on Newton’s mechanics, and they would not admit that Fresnel questioned 
their hero, even in a field other than mechanics. Siméon Denis Poisson, one of the die-hard Newtonians and a sophisticated 
mathematician, discovered a surprising consequence of Fresnel’s equations: if a light source is placed on the axis of a 
circular obstacle that blocks light, the theory predicts that there will be a bright spot behind the screen, at a place where 
common sense tells us that utter darkness reigns (Fig. 4). Does it not show that Fresnel’s theory is absurd? This is when 
the events took a surprising turn that the history of physics still remembers: Arago decided that an experiment should 
be conducted, and, to the amazement of Poisson and the Newtonians, the bright spot was indeed observed. The story has 
it that the Académie then shifted in favour of the wave theory, awarding Fresnel the prize of the competition and soon 
admitting him as a new member [3].
Some historians of science claim that the story has been embellished, and that the conversion to wave theory was far 
from being so massive. As a matter of fact, as, for example, Dominique Pestre explains it [4], Poisson was dazzled by the 
elegance and mathematical consistency of Fresnel’s theory, and was waiting only to welcome him at the “Académie”, while 
being at the same time not really convinced that Newton’s optics should be rejected. One cannot fail to notice, indeed, that 
the followers of the corpuscular theory remained active in the subsequent years until a new episode would clear the debate. 
Arago – he again – once more played a key role in 1838, when he proposed a crucial experiment to settle the argument 
between Newton’s corpuscular theory, then called the “system of emission”, and Fresnel’s wave theory, called the “system 
of undulations”. All it takes, he explained, is a comparison between the speed of light in air and in a refractive medium 
such as water. One remembers, indeed, that, in Newton’s theory, light accelerates when it penetrates a denser medium; on 
the contrary, according to the wave theory of Huygens and Fresnel, light goes slower in the denser medium (Fig. 2).
In 1849, Fizeau had measured the speed of light between Montmartre and Suresnes, over a distance of 8633 m, and found 
a value in agreement with the astronomy-based measurements [5]. The website of the “Observatoire de Paris” presents 
a remarkable description of this experiment, which was repeated in 2005 with a laser beam (Fig. 5). In 1850, Foucault 
improved the experimental set-up and was able to realize the measurement over a distance of only one meter. He could 
then directly compare the speed of light in air and water [6]. The thesis (Fig. 6) of the future member of the Academy is 
only 35 pages long and it plainly concludes: “always, light is delayed as it passes through the most refractive medium. The 
A. Aspect / C. R. Physique 18 (2017) 498–503 501Fig. 5. Measurement of the speed of light between the “Observatoire de Paris” and Montmartre, in 2005 (http://expositions.obspm.fr/lumiere2005/tir_
pratique.html).
Fig. 6. Sur les vitesses relatives de la lumière dans l’air et dans l’eau. PhD thesis by Léon Foucault, presented to the Paris Faculty of Sciences in 1853 (https://www.
bibnum.education.fr/sites/default/files/foucault-texte.pdf).
final conclusion of this work thus consists in declaring the system of emission incompatible with the reality of the facts.” 
The last followers of the corpuscular theory then surrendered or passed – is it not a saying that new theories do prevail 
only when their detractors have passed?
In these times of debate and sometimes opposition between fundamental research and applications, it should be pointed 
out that the theoretician Augustin Fresnel was also the father of modern lighthouses, which he contributed to conceive as 
a “Ponts et Chaussées” civil engineer, and which are equipped with the marvellous Fresnel lenses (Fig. 7). As for his theory 
of wave optics, it is so perfect that we may continue to use it without making any change to it. But in 1818, Fresnel still 
lacked the answer to this question: what is the nature of this quantity that vibrates as it propagates?
It was the Scottish genius James Clerk Maxwell who gave the answer, in 1864. He had written equations enabling one 
to describe the whole set of facts that were known at the time in the fields of electricity and magnetism. In any genuine 
502 A. Aspect / C. R. Physique 18 (2017) 498–503Fig. 7. A huge Fresnel lens in a lighthouse.
Fig. 8. With Albert Einstein (1879–1955), 1905 marks the revival of the theory of particles, allowing him to make predictions on the photo-electric effect 
confirmed by experiments of Robert Millikan (1868–1953) in 1915.
physical theory, equations imply more than what has been used to establish them; they have new solutions whose traces 
may be looked for through experimental observation. In the case of Maxwell’s equations, the novelty was impressive: some 
solutions of the equations indicated waves propagating at a speed that Maxwell calculated. And he concluded: “This speed is 
so close to that of light that it seems we have good reason to conclude that light itself [...] is an electromagnetic disturbance 
in the form of waves propagating through the electromagnetic field according to the laws of electromagnetism.”
The case thus seems settled: light is now known to be an electromagnetic wave; in other words, electric and magnetic 
fields vibrating in harmony, perpendicularly to the direction of propagation. That is to say they are transversal waves, 
as Fresnel’s waves, and we can describe these waves with a comprehensive theory to which no phenomenon seems to 
be able to escape. Is it the end of theoretical research in physics, and is there nothing for physicists but to refine their 
measurements, as the American experimentalist Michelson reportedly put it? Surely not. In 1900, the great Kelvin caught 
sight of two “clouds above the dynamical theory of heat and light” [7]. Such two clouds, which Kelvin described as “very 
dense”, would lead Einstein to lay the foundations of the two revolutions of the 20th-century physics: relativity, as is well 
known, but also quantum physics, in the emergence of which he would play a key role [8].
3. The wave–corpuscle duality
Einstein’s first article on quantum physics was published in 1905. It propounds a drastic hypothesis: light is formed of 
grains, Lichtquanten, with well-defined energy and momentum, depending on the constant introduced by Planck in 1900. 
Today, these quanta of light are called photons. Drawing on this model, Einstein interpreted the photoelectric effect – in 
other words the ejection of electrons from matter under the effect of light – as a photon–electron collision, and thus 
deduced what was the energy of the ejected electrons. Among the works that Einstein published in 1905, a golden year, 
this article was certainly the least appreciated of all, as can be seen from a negative comment in the report that would 
however lead to his election at the Prussian Academy of Sciences in 1911. Yet it was this article that caught the attention of 
the Nobel Committee, awarding him the Prize in 1922 after Millikan’s experiments confirmed Einstein’s predictions on the 
photoelectric effect – a confirmation “contrary to all my expectations”, said Millikan in his memoirs [9] (Fig. 8).
A. Aspect / C. R. Physique 18 (2017) 498–503 503Fig. 9. Fruitful questioning: “Object of this course: to try to [. . .] answer the following question: could we do without the concept of photon, at least in the 
field of optics?”. Claude Cohen-Tannoudji, course at the “Collège de France”, 1979.
The corpuscular model was back on track. But then, how could one account for interferences, diffraction or double 
refraction, as Young and Fresnel showed they could only be consistently explained through the wave model? How could 
one make this corpuscular model compatible with Maxwell’s description of light as an electromagnetic wave? Einstein, 
an admirer of Maxwell, could not ignore the question and provided a masterful answer to it on his first appearance at a 
scientific conference, the conference of German physics that was held in Salzburg in 1909. Einstein developed there a series 
of arguments and concluded: “I only wanted to briefly illustrate the fact that the two structural properties (wave structure 
and quantum structure) [. . .] should not be seen as incompatible.” At the end of each argument, Einstein’s conclusion 
was the same: light is both wave and corpuscle. It would take fourteen years before Louis de Broglie would express this 
wave–corpuscle duality, this time not for light, but for material particles [10].
With wave–corpuscle duality, modern quantum physics was born. Optics would continue progressing all over the 20th 
century, notably with the invention of the laser, and the French scientists, perpetuating the tradition of their predecessors, 
would take an eminent stand there, supplying the Académie with important battalions. But the wave–corpuscle duality of 
light still remained quite mysterious, as is apparent from the questions of the future Nobel laureate Alfred Kastler, then a 
high-school professor in Bordeaux in 1932: “If this synthesis satisfies the mathematician, he wrote, it continues to worry 
the physicist, who shall not be content with abstract formulas. To him, the duality between the wave aspects and the 
corpuscle aspects of light remains an unresolved mystery...” [11]. Such questions led him to discover optical pumping with 
Jean Brossel [12].
Thirty-two years later, as a student in upper sixth at the “Lycée” of Agen, I asked a similar question to my physics teacher 
who had a hard time to answer it, despite his outstanding capacities. It would take me fifteen more years to find, as many 
of my contemporaries, illuminating answers in Claude Cohen-Tannoudji’s course at the “Collège de France”, as he clarified 
such questions and thus enabled the French School of Quantum Optics to exert a radiant influence throughout the world 
(Fig. 9).
References
[1] C. Huyghens, Traité de la lumière, reprinted and available at http://gallica.bnf.fr/ark:/12148/bpt6k5659616j/f4.image, 1690.
[2] A. Fresnel, Mémoire sur la diffraction de la lumière, available at www.academie-sciences.fr/.../Fresnel/Fresnel... /Mem1818_p339.pdf, 1818.
[3] The story is recounted, e.g., in the Éloge historique de Léon Foucault par J. Bertrand, published in the volume 42 of the Mémoires, see note 1.
[4] D. Pestre, La “tache de Poisson” fit triompher Fresnel, La Recherche, No. 436, December 2009.
[5] H. Fizeau, Sur une expérience relative à la vitesse de propagation de la lumière, C. r. hebd. séances Acad. sci. Paris 90 (1849) 132.
[6] L. Foucault, Méthode générale pour mesurer la vitesse de la lumière dans l’air et les milieux transparents. Vitesses relatives de la lumière dans l’air et 
dans l’eau. Projet d’expérience sur la vitesse de propagation du calorique rayonnant, C. r. hebd. séances Acad. sci. Paris XXX (1850) 551.
[7] L. Kelvin, Nineteenth century clouds over the dynamical theory of heat and light, Philos. Mag. Ser. 6 2 (1901) 1–40.
[8] See for instance the excellent book of A.D. Stone, Einstein and the Quantum, Princeton University Press, 2014.
[9] R.A. Millikan, Albert Einstein on his seventieth birthday, Rev. Mod. Phys. 21 (1949) 343.
[10] A. Aspect, J. Villain, The birth of wave mechanics (1923–1926), C. R. Physique 18 (2017) 583–585, in this issue.
[11] A. Kasler, Les propriétés corpusculaires de la lumière, in: Procès-verbaux des séances de la Société des sciences physiques et naturelles de Bordeaux, 
1931–1932, pp. 32–58.
[12] A. Kastler, J. Brossel, La détection de la résonance magnétique des niveaux excités – l’effet de dépolarisation des radiations de résonance optique et de 
fluorescence, C. r. hebd. séances Acad. sci. Paris 229 (23) (1949) 1213–1215.


From Huygens' waves to Einstein's photons: Weird light

Mehran Taki

Abolfazl Mirjalili

C. R. Physique 18 (2017) 66–71Contents lists available at ScienceDirect
Comptes Rendus Physique
www.sciencedirect.com
Original article / Article original
The Riemann tensor and the Bianchi identity in 5D 
space–time
Mehran Taki ∗, Abolfazl Mirjalili
Physics Department, Yazd University, 89195-741, Yazd, Iran
a r t i c l e i n f o a b s t r a c t
Article history:
Received 25 August 2015
Accepted 3 June 2016
Available online 5 August 2016
Keywords:
Riemannian geometry
Five-dimensional space–time
Bianchi identity
Einstein’s equation
The initial assumption of theories with extra dimension is based on the efforts to yield 
a geometrical interpretation of the gravitation field. In this paper, using an infinitesimal 
parallel transportation of a vector, we generalize the obtained results in four dimensions 
to five-dimensional space–time. For this purpose, we first consider the effect of the 
geometrical structure of 4D space–time on a vector in a round trip of a closed path, which 
is basically quoted from chapter three of Ref. [5]. If the vector field is a gravitational field, 
then the required round trip will lead us to an equation which is dynamically governed 
by the Riemann tensor. We extend this idea to five-dimensional space–time and derive 
an improved version of Bianchi’s identity. By doing tensor contraction on this identity, we 
obtain field equations in 5D space–time that are compatible with Einstein’s field equations 
in 4D space–time. As an interesting result, we find that when one generalizes the results to 
5D space–time, the new field equations imply a constraint on Ricci scalar equations, which 
might be containing a new physical insight.
© 2016 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
1. Introduction
The idea of the existence of higher-dimensional space–times has a long history. It has been founded by Kaluza [1] and 
Klein [2,3] in the 1920s, as they were trying to unify the gravitational and electromagnetic interactions by assuming a 
five-dimensional space–time (5D). A new version of the idea of Kaluza and Klein appeared later, around the 1970s, which 
is still considered by the majority of the physics community to be the best hope for a completed unified theory of all 
fundamental interactions [4], despite growing opposition against extra-dimensional theories.
All methods about unification are based on the fact that the Riemannian geometry in general relativity can be generalized 
to higher dimensions without implying any constraint on the gravitational field. In other word, as we will discuss in this 
paper, the antisymmetric nature of Riemann’s tensor and the Bianchi identity as a constraint on Riemann’s tensor are 
implied when we consider two and three spatial dimensions of space–time. But no one is concerned by any constraint on 
gravitational field equations when four or more spatial dimensions of space–time are considered!
In this paper, by following the geometrical approach described in Ref. [5] and using the concept of parallel transport [6]
in five dimensional space–time, we get a new second-order differential identity for the Riemann tensor. As we will see 
by doing contractions on the new identity, we can define the gravitational field equations in 5D space–time, which is 
* Corresponding author.
E-mail addresses: metaki@stu.yazd.ac.ir (M. Taki), a.mirjalili@yazd.ac.ir (A. Mirjalili).http://dx.doi.org/10.1016/j.crhy.2016.06.003
1631-0705/© 2016 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
M. Taki, A. Mirjalili / C. R. Physique 18 (2017) 66–71 67compatible with Einstein’s field equations in 4D space–time. Also we will prove that in order to convert the new 5D field 
equations to Einsteins’s field equations that are supposed to be correct in all extra-dimensional spaces, the d’Alembertian of 
5D Ricci scalar R should be zero (R = 0).
The organization of this paper is as follows. In section 2, the geometry of curved space is reviewed. Bianchi identities 
in five dimensional space are obtained in Section 3. In section 4, we will generalize the parallel transportation of a vector 
in a four-dimensional space–time to a five-dimensional one. We will see that this leads us to a new identity. By using this 
identity, the new 5D gravitational equation is derived. We give finally our conclusion in section 5.
2. An overview of the geometrical features of a curved space
Riemannian geometry rests essentially on the consequences derived from the displacement element ds in which the 
related metric tensor is appeared as it follows
ds2 = gμνdxμdxν, μ,ν = 0,1,2,3 (1)
Riemannian geometry requires a symmetric linear connection coefficient, , in which the infinitesimal parallel transport 
of a vector always preserves the length of the vector. These coefficients have been called Christoffel symbols and defined 
by [7]:
λμν =
1
2
gλσ
(
∂μgνσ + ∂νgσμ − ∂σ gμν
)
(2)
The Christoffel symbols are used to determine the parallel transport of vector Aμ:
Aμ → Aμ + λμνdxλ Aν (3)
where dx is related to the vector displacement of Aμ . The Christoffel symbols measure the curvature of the coordinate axes. 
The covariant derivative is then defined as:
Dμ A
ν = ∂μ Aν + νμλ Aλ (4)
This derivative results from the comparison of a vector at coordinate x with its parallel transformed one at x + dx. The 
equations that are written in terms of the covariant derivatives preserve their physical properties under gauge transforma-
tions.
The curvature of a manifold manifests itself when a vector is transported parallel around a closed path. By displacement 
around an infinitesimal square loop with elements dx and dy which is plotted in Fig. 1, the difference between the initial 
gauge field vector Aμ and the transported vector A′ μ , using the Taylor expansion, is given by [5,8]:
Aμ = A′ μ − Aμ = − [Dσ , Dλ]μρ Aρdxσ dyλ = Rμρσλ AρSσλ (5)
where Sσλ represents the area of the performed rectangle, and the curvature tensor Rμρσλ is given by
Rμρσλ = ∂σ μλρ − ∂λμσρ + μσννλρ − μλννσρ (6)
So Aμ is non-zero if and only if the space is intrinsically curved. Now if A′ μ moves in the opposite direction along the 
path to reach the point where Aμ exists, we are expecting the difference between Aμ and A′′ μ to be zero, i.e.(
Rμρσλ + Rμρλσ
)
AρSσλ = 0 (7)
This leads to an anti-symmetric nature of the curvature tensor, which is confirmed by Eq. (6) [7]:
Rμρσλ = −Rμρλσ (8)
This is the first property of the curvature tensor. This tensor is called the Riemann tensor.
We now consider a 3-dimensional closed path and transport the vector Aμ through the path. The path consists of circuits 
round the top and bottom face of the box with elements dx and dy, which are connected by another element of the cube 
with length dz (see the right panel of Fig. 1). By moving the vector from the starting point and transport it around the 
closed path, the vector will be changed by [8]:
Aμ = (Dρ Rμνσλ) AνV σλρ (9)
where V σλν = dxσ dyλdzν is the volume of the box. The concerned cube contains clearly another two similar circuits 
enclosing the two other pairs of faces. If we transport the vector along these two paths, the following result for the new 
transported vector is obtained [5]:
A′ μ =
(
Dρ R
μ
νσλ + Dσ Rμνλρ + DλRμνρσ
)
AνV σλρ (10)
68 M. Taki, A. Mirjalili / C. R. Physique 18 (2017) 66–71Fig. 1. Two- and three-dimensional closed paths. These two figures have been taken from Ref. [5].
Obviously, the total path traverses each side of the box as many times in one direction as the opposite one, so the path is 
equivalent to its own reverse, and consequently the vector Aμ would not be changed, and so A′μ = 0, and hence [5]
DλRρσμν + Dρ Rσλμν + Dσ Rλρμν = 0 (11)
This is known as the Bianchi identity. This constraint is related to the features of the Riemann tensor, which was obtained 
just by considering the geometrical feature of space–time.
3. Bianchi identity in five-dimensional space–time
We are now going to generalize the method employed in the previous section to five-dimensional (5D) space–time. For 
simplicity, we make the new extra-dimension non-compact, like the other three spatial dimensions. Since we are concerned 
about the very small hyper-volume of space, the result can be generalized to a space with higher dimensions. Therefore, the 
new metric tensor in 5D metric is given by:
ds2 = gμνdxμdxν, μ,ν = 0,1,2,3,4 (12)
The functional form of the covariant derivative will not be changed, and is:
Dμ A
ν = ∂μ Aν + νμλ Aλ, μ,ν,λ = 0,1,2,3,4 (13)
The four-dimensional analog of the cube in space coordinates is a tesseract, which consists of 8 cubical cells. So for our 
closed path in 4D, we consider two connected 3D cubical cells, whose extra-dimension is denoted by w . According to what 
we have already done with a three-dimensional cube, for the four-dimensional tesseract we should choose cube pair paths, 
while a transported vector is considered along them. By moving the vector field from the starting point and transport it 
parallel around the closed path, it will be changed into
A′ μA =
(
1 − dwλDλ
) (
1 − V ηνρ Dη Rμσνρ
) (
1 + dwη Dη
) (
1 − V ηνρ Dη Rμσνρ
)
AσA (14)
After doing some algebra manipulations, we will arrive at
A′ μA =
(
1 − T ληνρ DλDη Rμσνρ
)
AσA (15)
or
Aμ = (Dη Dρ Rμνσλ) AνT σλρη (16)
where T σλρη is the hyper-volume of the performed box in 5D space–time. A tesseract is comprised of four cube pairs 
paths, so the whole path leads to
Dη DλRρσμν + DλDρ Rσημν + Dρ Dσ Rηλμν + Dσ Dη Rλρμν = 0 (17)
This new identity can be considered as the improved Bianchi identity in 5D space–time. Substituting the Riemann tensor 
into the left-hand side of Eq. (17) automatically confirms its right-hand side (see Appendix A). This confirms the validity of 
the obtained Eq. (17). We try to employ this equation in the next section to achieve the gravitational field equations in 5D 
space–time.
4. Gravitational field equations in five-dimensional space–time
Einstein’s tensor as an outstanding equation in general relativity can be obtained by twice contracting the Bianchi iden-
tity, Eq. (11). So we can write:
giσ g jλ
(
DλRρσ i j + Dρ Rσλi j + Dσ Rλρi j
) = 0
⇒ Di Rρi − Dρ R + D j Rρ j = 0
⇒ Di
(
Rρi − 12 Rgiρ
)
= 0
(18)
M. Taki, A. Mirjalili / C. R. Physique 18 (2017) 66–71 69where σ , λ, ρ, i, j = 0, 1, 2, 3. The Einstein tensor is then defined by
Gij = Rij − 12 gi j R (19)
The Einstein tensor Gij , which is constructed from the Riemann metric and the Ricci tensor, does not have any divergence:
Di Gij = 0 (20)
By considering the conservation law of energy and momentum, i.e.
Di Tij = 0 (21)
we can write
Gij = κ4DTij (22)
or
Rij − gi j 12 R = κ4DTij (23)
where κ4D, as the proportional coefficient, is Einstein’s gravitational constant in four-dimensional space–time. Eq. (23) is 
known as Einstein’s field equation, which is the basic equation in theory of general relativity [7].
The general form of Einstein’s field equation can be written as follows:
Rij − gi j 12 R + gi j = κ4DTij (24)
where  is the cosmological constant that is used to justify the expansion of the universe [7].
Let us now apply the contraction to the improved Bianchi identity, Eq. (17), which leads us to (see details of the calcu-
lation in Appendix B):
DμDν
(
Rμν − 1
3
Rgμν
)
= 0, μ,ν = 0,1,2,3,4 (25)
Now, according to the procedure that leads us to the Einstein equation above, we can choose a second-rank tensor that is 
proportional to the bracket term in Eq. (25). The second covariant derivative of this stress tensor should be zero, and we 
call it μν . Also, in a way similar to the one in Eq. (24), we can add this stress tensor to the concerned second-rank tensor 
in Eq. (25). Therefore, we can write the field equations derived from Eq. (25) as
Rμν − 1
3
Rgμν + μν = κ5DTμν (26)
where
Dν Dμμν = 0 (27)
and κ5D is proportional coefficient that suppose to be a constant. By choosing
μν = −1
6
Rgμν + gμν (28)
Eq. (26) will turn into Einstein’s equation in 5D, i.e.
Rμν − 1
2
Rgμν + gμν = κ5DTμν (29)
But, according to Eq. (27), considering the fact that the covariant derivative of the metric tensor is always zero, the Ricci 
scalar in five-dimensional space–time should now satisfy the following relation:
5D R = 0 (30)
where 5D stands for the 5D d’Alembertian operator that contains the second covariant derivative ( = DμDμ).
We now say that Eq. (30) can be considered as a constraint on the curvature of 5D space–time when we try to gen-
eralize the 4D Einstein equation to higher dimensions, i.e. Eq. (26). The obtained equation, Eq. (26), with the supposed 
tensor μν as in Eq. (28), is completely similar to the 4D Einstein field equation. Investigating the general form of the 
assumed second-rank tensor, whose second-order covariant derivative is zero and also satisfies Eq. (26), can be considered 
as an open research area.
In summary, using the improved Bianchi identity, we derive a gravitational field equation that contains the second 
covariant derivatives as in Eq. (25). This equation satisfies automatically the Einstein field equation in 4D space–time. But 
it additionally contains an extra feature, which is related to the constraint, imposed on Ricci’s scalar by Eq. (30). The Ricci 
scalar in 4D space–time is in fact a quantity that provides us with a specified curvature for the universe, but appears as a 
scalar field in 5D space–time, which can describe a universe with a changing curvature. Other features of Eq. (29), which 
obviously contain more information than the Einstein’s equation in 4D space–time, can be considered as a new research 
area in future.
70 M. Taki, A. Mirjalili / C. R. Physique 18 (2017) 66–715. Conclusion
A curved space–time may be defined by measuring the change of a vector when a parallel transport is occurring around 
an infinitesimal closed loop. We began with three-dimensional space–time, where, by transporting a vector around an 
infinitesimal square loop, we could extract the antisymmetric nature of the Riemann tensor. In 4D space–time, the same 
method led us to the Bianchi identity.
A second-order differential identity for the Riemann tensor was obtained as the improved Bianchi identity, when we 
applied parallel transportation to five-dimensional space–time. By doing contractions on the new identity, we could define 
the gravitational field equations in 5D space–time. As we saw, the new 5D field equations that arise from the geometrical 
feature of space–time imply a constraint on Ricci’s scalar. On the other hand, by a proper choice of the stress tensor, we 
arrived at a field equation in 5D space–time whose apparent form is similar to Einstein’s field equation 4D space–time. 
Investigating the features of the new gravitational field, Eq. (29), can be considered as a research task in future.
Appendix A
We intend here to evidence that the improved Bianchi identity, which was obtained by a geometrical interpretation of 
5D space–time, is satisfied by the Riemann tensor. For this purpose, by considering the second-order covariant derivative on 
the Riemann tensor, evaluated in locally inertial coordinates, we can write:
Dη DλRρσμν = ∂η∂λRρσμν = 1
2
∂η∂λ
(
∂μ∂σ gρν − ∂μ∂ρ gνσ − ∂ν∂σ gρμ + ∂ν∂ρ gμσ
)
(31)
One has to notice that using the inertial coordinates does not impose any limitation on our calculations because the 
terms that we are neglecting are all proportional to ∂σ gμν , and therefore automatically vanish [7].
So the four terms in Eq. (17) appear as:
∂η∂λRρσμν = 12
(
∂η∂λ∂μ∂σ gρν − ∂η∂λ∂μ∂ρ gνσ − ∂η∂λ∂ν∂σ gρμ + ∂η∂λ∂ν∂ρ gμσ
)
∂λ∂ρ Rσημν = 12
(
∂λ∂ρ∂μ∂η gσν − ∂λ∂ρ∂μ∂σ gνη − ∂λ∂ρ∂ν∂η gσμ + ∂λ∂ρ∂ν∂σ gμη
)
∂ρ∂σ Rηλμν = 12
(
∂ρ∂σ ∂μ∂λgην − ∂ρ∂σ ∂μ∂η gνλ − ∂ρ∂σ ∂ν∂λgημ + ∂ρ∂σ ∂ν∂η gμλ
)
∂σ ∂η Rλρμν = 12
(
∂σ ∂η∂μ∂ρ gλν − ∂σ ∂η∂μ∂λgνρ − ∂σ ∂η∂ν∂ρ gλμ + ∂σ ∂η∂ν∂λgμρ
)
(32)
Therefore, the sum of cyclic permutations of the first four indices in the Riemann tensor lead us to:
Dη DλRρσμν + DλDρ Rσημν + Dρ Dσ Rηλμν + Dσ Dη Rλρμν =
1
2 (∂η∂λ∂μ∂σ gρν − ∂η∂λ∂μ∂ρ gνσ − ∂η∂λ∂ν∂σ gρμ + ∂η∂λ∂ν∂ρ gμσ+ ∂λ∂ρ∂μ∂η gσν − ∂λ∂ρ∂μ∂σ gνη − ∂λ∂ρ∂ν∂η gσμ + ∂λ∂ρ∂ν∂σ gμη
+ ∂ρ∂σ ∂μ∂λgην − ∂ρ∂σ ∂μ∂η gνλ − ∂ρ∂σ ∂ν∂λgημ + ∂ρ∂σ ∂ν∂η gμλ
+ ∂σ ∂η∂μ∂ρ gλν − ∂σ ∂η∂μ∂λgνρ − ∂σ ∂η∂ν∂ρ gλμ + ∂σ ∂η∂ν∂λgμρ
+ ∂σ ∂η∂μ∂ρ gλν − ∂σ ∂η∂μ∂λgνρ − ∂σ ∂η∂ν∂ρ gλμ + ∂σ ∂η∂ν∂λgμρ) = 0
(33)
This result, as we were expecting, confirms the validity of the improved Bianchi identity, i.e. Eq. (17).
Appendix B
Here we are going to evidence how we can extract the second derivative in Eq. (25). The improved Bianchi identity in 
5D space–time is:
Dη DλRρσμν + DλDρ Rσημν + Dρ Dσ Rηλμν + Dσ Dη Rλρμν = 0 (34)
Applying the contraction to this equation will lead us to:
gρμgσν
(
Dη DλRρσμν + DλDρ Rσημν + Dρ Dσ Rηλμν + Dσ Dη Rλρμν
) = 0
⇒ −Dη DλR + DλDμRημ + DμDν Rηλμν + Dν Dη Rλν = 0
⇒ gηλ (−Dη DλR + DλDμRημ + DμDν Rηλμν + Dν Dη Rλν) = 0
⇒ Dη DμRημ + DμDν Rμν + Dν DλRνλ − DλDλR = 0
(35)
By using the fact that all the indices are dummy, it is obvious that the first three terms in the last row of Eq. (35) are equal 
to each other; thus we arrive at:
DμDν
(
Rμν − 1
3
gμν R
)
= 0 (36)
M. Taki, A. Mirjalili / C. R. Physique 18 (2017) 66–71 71References
[1] T. Kaluza, Math. Phys. K1 (1921) 966.
[2] O. Klein, Z. Phys. 37 (1926) 895.
[3] O. Klein, Nature 118 (1926) 516.
[4] T. Appelquist, A. Chodos, P.G.O. Freund (Eds.), Modern Kaluza–Klein Theories, Addison-Wesley, Menlo Park, 1987.
[5] Lewis H. Ryder, Quantum Field Theory, Cambridge University Press, Cambridge, UK, 1996.
[6] B.G. Schmidt, Commun. Math. Phys. 29 (1973) 55.
[7] S.M. Carroll, Space–Time and Geometry, An Introduction to General Relativity, vol. 1, Addison Wesley, 2004.
[8] R. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982) 255.


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