# On a Geometrical Definition of Einstein's Stress-Energy Tensor (A translation of "Sur une définition géométrique du tenseur d'énergie d'Einstein")

Oussama Abdelghafour Belarbi, Ahmed Meziane

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Oussama Abdelghafour Belarbi, Ahmed Meziane. On a Geometrical Definition of Einstein's Stress-Energy Tensor (A translation of "Sur une définition géométrique du tenseur d'énergie d'Einstein"). 2023. hal-04185521

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# On a Geometrical Definition of Einstein's Stress-Energy Tensor†

(A translation of "Sur une définition géométrique du tenseur d'énergie d'Einstein")

Élie Cartan

(February 13, 1922)

We know that, in Einstein's generalized theory of relativity, the tensor that completely characterizes matter state in the neighbourhood of a point in the universe is identified with a tensor involving uniquely the *geometrical* properties of the universe in the neighbourhood of that point. The ten components of that tensor are given by more or less complicated calculations which are far from exhibiting the geometrical essence. It seemed to me that it would be useful to associate to this tensor a definition, which, while having the required precision, could be expressed in a purely geometrical language.

I. Before tackling the case of four-dimensional universe, we consider the much simpler case of three-dimensional *static* universe, namely a material continuous medium in equilibrium under the action of purely elastic forces. Such a medium is characterized physically by its tension state, namely in short, by a vector applied to an arbitrary oriented surface element of the medium and indicating the resultant of exerting actions through that element by the part of medium situated at the negative side of that element upon the part of medium situated at the positive side. Thus, the state of medium is defined with a vector tied to every oriented surface element of the three-dimensional space, a vector whose components are actually elements of surface integrals. The components satisfy, as we know, the symmetry law. Furthermore, the resultant of vectors attached to different surface elements bounding a given volume vanishes (conservation law).

We then consider a three-dimensional Riemannian space defined with a given $ds^2$, a quadratic differential form of three variables; the *curvature state* of that space, namely what distinguishes it more or less from Euclidean space, can be characterized exactly in the same manner as the tension state of an elastic medium in equilibrium. Indeed, if we describe an infinitesimal closed contour, the difference between the space under consideration and the Euclidean one is manifested upon this contour by the fact that a system with three rectangular directions issued from a point in the contour and transported *parallelly to themselves à la* Levi-Civita<fnref n="1" /> does not coincide with the same system when completely having covered

<footnote>†This note was originally published as *Comptes Rendus* **174** (1922) 437 under the original French title 'Sur une définition géométrique du tenseur d'énergie d'Einstein' and was reprinted in: Les (Euvres Complètes d'Élie Cartan, Partie III.1 (Gauthier-Villars: 1952) pp. 613-5. Translated from French and commented by O. A. Belarbi and A. Meziane (Laboratoire de Physique Théorique d'Oran (LPTO), Université d'Oran 1 Ahmed Ben Bella, 31100 Oran, Algeria. Emails: belarbi.oussama@edu.univ-oran1.dz and maptouk@yahoo.com). All footnotes and remarks in square brackets permeating the translated text are put by translators in order either to clarify probable misconceptions related to some words in the original text, or to refer the reader to a much richer bibliography. We also note that references at the end of the note are added by the translators.</footnote>

<footnote n="1">¹T. Levi-Civita, Nozione di parallelismo in qualsiasi varietà e conseguente specificazione geometrica della curvatura Riemanniana, *Rend. Circ. Mat. Palermo* **42** (1916) 173-204. [In this context, we should note that the contributions of Levi-Civita to the mathematical aspects of general relativity, especially the notion of parallel transport which is the cornerstone of Einstein's theory, are crystal clear and for his contributions to the other mathematical tools, namely tensor calculus (or following his appellation, *absolute differential calculus*) and the three-body problem in general relativity, the reader should consult the relatively recent work: P. Nastasi and R. Tazziali, Toward a scientific and personal biography of Tullio Levi-Civita (1873-</footnote>
the closed contour. In order to recover the initial system, one has to perform an infinitesimal rotation. The latter can be represented by a vector. The state of divergence between the given space and the Euclidean one can therefore be interpreted as a vector attached to each element of the oriented surface of the space. One can easily show that this vector, whose components are elements of surface integral, satisfies both of the symmetry and the conservation law (the vanishing of the geometric sum of vectors attached to surface elements defining an infinitesimal volume).

It follows from the above that we can explain the state of an elastic medium in equilibrium by assuming that the space which contains it [i.e. the medium] is deformed and that the tension state translates physically the geometrical deformation. A perfect fluid in equilibrium (and consequently with constant pressure) would correspond to a non Euclidean space with constant curvature, the vector that manifests the curvature of every surface element being normal [perpendicular] to that element.

II. I turn now to the Einsteinian universe. The generalization will be achievable if we define the physical state of the universe with a vector (with four components) attached to every three-dimensional volume element of the universe, satisfying the symmetry law: the 16 coefficients, which are reduced to 10 because of the symmetry law, can be, as we usually do, viewed as the coefficients of quadratic form with four variables, just as the 9 coefficients $p_{xx}, p_{xy}$ etc. which are reduced to 6, of the vector defining the tension of elastic medium, can be viewed as the coefficients of a quadratic form with three variables.

Let's start then from a given $ds^2$, a quadratic differential form with four variables. The difference between the universe defined by means of $ds^2$ and the Euclidean one appears upon every closed infinitesimal contour by the fact that a system with four directions, all issuing from a single point in the contour and *parallelly* transported to itself, does not coincide with the same system when traversing the entire closed contour: one has to, in order to achieve the initial system, perform a *rotation* (calling by that name an overall displacement of the Euclidean universe around a fixed point). To every closed infinitesimal contour, or to every oriented surface element, we thus associate a rotation.

Having posed this, we consider a volume element of the universe, and let's take it, for the sake of simplicity, in the form of an elementary parallelepiped; let O be one of its vertices, $OA_1, OA_2, OA_3$ be the three edges issuing from the vertex O. To the first side [which is actually a parallelogram] going through O, or rather to its covering contour in the sense $A_3OA_2$, one associates an infinitesimal rotation. Under the effect of that rotation, every point of the three-dimensional space which is perpendicular to $OA_1$ undergoes a displacement; if we only consider the component of this displacement that can be found in the three-dimensional space itself, it [refers to the component] could be obtained by the effect of a certain rotation of this space upon itself [refers to the space], rotation that we could call the *projection* of the primitive rotation upon this space. That projection can be geometrically represented in this space (perpendicular to $OA_1$) by a vector $\vec{R}_1$.

Ultimately, to every parallelogram of the elementary parallelepiped one can match three vectors $\vec{R}_1, \vec{R}_2, \vec{R}_3$ representing the projections, upon the perpendicular spaces to $OA_1, OA_2, OA_3$, of the three rotations associated with the three faces [parallelograms]. The geometric sum of these three vectors, multiplied by the lengths (or intervals) $OA_1, OA_2, OA_3$

<footnote n="1941">1941), *Historia Mathematica* **32** (2005) 203-236.]</footnote>
respectively, defines the required vector: Einstein's energy tensor<fnref n="2" />. This tensor vanishes in vacuum (any region devoid of matter).

The geometrical definition of that vector can be put in another form by starting with the notion of *complementary rotations* of four-dimensional Euclidean space with a fixed [preferred] point. [The end of Cartan's text]

References

[1] E. Cartan, Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion, *Comptes Rendus** **174** (1922) 593-5. English translation with the title 'On a generalization of the notion of Riemann curvature and spaces with torsion', in *Proc. 6th Course of International School on Cosmology and Gravitation: Spin, Torsion, Rotation, and Supergravity* (Eds.) P.G. Bergmann and V. De Sabbata (Plenum Press, New York: 1980) pp. 489-91

[2] M. Lazar and F. W. Hehl, Cartan's Spiral Staircase in Physics and, in Particular, in the Gauge Theory of Dislocations, *Found. Phys.* **40** (2010) 1298-325

[3] E. and F. Cosserat, Théorie des Corps Déformables (Hermann, Paris: 1909). Translated into English by D. Delphenich with the title 'Theory of Deformable Bodies' (2007)

<footnote n="2">²Provided that a geometrical interpretation of the material content of our universe is possible by means of Cartan's geometrization procedure of stress-energy tensor, an extension of the geometrical picture can lead to the introduction of Cartan's torsion notion. This could be done by enlarging our conceptual visualization of the geometry describing the universe. In the meantime, that idea inspired Cartan in a subsequent seminal paper [1] to generalize Einstein's theory of gravity in order to incorporate torsion. Subsequently, the existence of both Riemann curvature (which is related to local rotation) and torsion (related to local translation) into a unified arena constitutes the so-called Riemann-Cartan geometry for Einstein-Cartan theory of gravity. The coalescence of the two ingredients viz, local rotation and translation which are actually responsible for distorion of 3D Euclidean space was made manifest in an elegant picture of three-dimensional helical (spiral) staircase [2]. We should also notice that a resemblance between a geometrical Cartan's medium and the well-known material Cosserat's medium [3] was discussed in [2].</footnote>