# General relativity in terms of differential forms

Is there a formulation of general relativity in terms of differential forms instead of tensors with indexs and subindexs? If yes, where can I find it and what are the advantages of each method? If not, why is it not possible?

• staff.science.uva.nl/~jpschaar/report/node52.html – Jerry Schirmer Dec 31 '13 at 0:34
• Differential forms are tensors. And tensors don't have indices. – MBN Dec 31 '13 at 8:47
• Just to add a comment. The formulation you need is the tetrad/veibein/frame one. The main objects are one-form tetrad $e^a_\mu dx^\mu$ and one-form spin-connection $\omega^{a,b}_\mu dx^\mu$. The Riemann two-form $F^{a,b}=d\omega^{a,b}+\omega^{a,}{}_c\wedge \omega^{c,b}$ can be used to rewrite the Einstein action as $\int F^{a,b}\wedge e^c\wedge...\wedge e^u \epsilon_{abc...u}$. It is the only formulation where one make fermions, i.e. matter fields, feel gravitational field, so $g_{\mu\nu}$ has to be abandoned anyway. I would recommend "Gravity and Superstrings" by T.Ortin – John Dec 31 '13 at 11:47
• @MBN I know differential forms are tensors, but they are tensors with special properties. As for the index, I mean the usual $g_{ab}$ formulation, with the Einstein convention and index raising. – nabla Dec 31 '13 at 13:44

Short answer. Example : The Palatini action , where the action is a functional of a tetrad/vierbien $e$ and a spin connection $\omega$.

$S(e, \omega) = \int \epsilon_{abcd} e^a \wedge e^b \wedge \Omega^{cd}$

where $\Omega$ is the curvature associated to the connection form $\omega$ :

$\Omega = D\omega = d\omega + \omega \wedge \omega$

Other examples listed here (with references) :

Plebansky action

Samuel-Jacobson-Smolin action

Goldberg action:

differential forms. It's not 100% clear how to read the question so I'll fire in all directions.

In general relativity, the field of interest is the symmetric metric tensor $g$ which you see written $g_{\mu\nu}$ or e.g. $g_{ab}$. The latter often implies abstract index notation.

The base vectors on the cotangential space are $\mathrm dx^i$ so there you have forms, but it's not like you want to write the symmetric tensors like the energy-momentum tensor $T$ as image of $\mathrm d$. You can drop the indices from objects like $g_{\mu\nu}$ any time you want. E.g. You can write down Einsteins field equations as $G=\kappa\ T$ and then it's in your face covariant per definition.

In this answer someone rants about why you wouldn't want to dismiss index notation anyway. (But the person has governed zero reputation on this site yet, so I would take his opinions with a grain of salt.)

• Ron has/had more rep than 1, but for whatever reason was given a year-long ban. His posts are usually pretty good. – Kyle Kanos Dec 31 '13 at 1:37
• @KyleKanos, "pretty good" is a pretty good understatement. – Alfred Centauri Dec 31 '13 at 1:55
• @AlfredCentauri: The link answer is in fact quite bad. Only the first sentence can be considered as an answer to the question. – MBN Dec 31 '13 at 8:45
• I also have to say that many people say his answers are "pretty good", but I am yet to see a good answer by him. – MBN Dec 31 '13 at 8:46
• @NickKidman when you say covariant per definition you mean "invariant"/"independent" of the coordinates? The "manifest covariance" and diverse meanings of covariance always give me headaches. – nabla Dec 31 '13 at 16:28

It was Cartan who developed General Relativity in his book "ON MANIFOLDS WITH AN AFFINE CONNECTION AND THE THEORY OF GENERAL RELATIVITY " relying only on "Affine Connections", it is not clear to me what to be called a "formulation of General relativity in terms of differential forms", but I take it granted from the question that one is trying to develop a theory using index free notation and keeping in mind "Christoffel Symbols" are fundamental building blocks in deriving Field equations. Actually "bundle of linear forms" - what Cartan mentioned, gives rise to a variant of Christoffel symbols (hence in some loose sense FORMS), and torsion of the Geometrical space considered (actually Cartan gives more, he actually predicts "spin" like systems that are absent in Einstein's formulation of GTR as Einstein considered Manifolds with ZERO torsion and torsion is a FORM also, made precise in the above mentioned text), this is also the birth place of modern day "Fiber Bundle" theory ( see his book "Riemannian Geometry in an Orthogonal Frame") this fiber bundle theory has the power to accommodate Yang-Mills theory into solid Geometrical Ground and brings GTR and YM having a common mathematical background, both of this book contains enough material to satisfy the approach of the seeker. Now this theory is called Cartan-Einstein theory, and Einstein's theory is contained in this Cartan-Einstein theory as sub-theory.

One can consult the following letters (Elie Cartan - Albert Einstein Letters on Absolute Parallelism 1929-1932 ) between Einstein and Cartan, to have a taste of how far this theory can be treated as "physical"

It is not sensible to write any theory - GR included - in terms of differential forms. Differential forms are just completely antisymmetric tensors. The antisymmetric tensors are just one kind of irreducible representation of the general linear group GL(m,C); the completely symmetric tensors are another irrep and so are all the irreps that are labelled by Young's diagrams . Since physical quantities are irreps of groups, it is not sensible to set up any physical theory in a restrictive mathematical arena which only supports one irrep of GL(m,C).

• Could you give me any reference on physical quantities being represented by irreducible representations? – nabla Dec 31 '13 at 16:00
• Uh, what about electromagnetism? Let $F=\mathrm{d}A$, then it's $\mathrm{d}F=0$ and $*\mathrm{d}*F=J$. See, e.g., these rudimentary notes. A similar formalism holds for Yang-Mills theory. So what do you mean "It's not sensible to write any theory...in terms of differential forms"? – Alex Nelson Dec 31 '13 at 17:48
• @Alex Nelson : The notes you linked to don't mention the energy-momentum tensor of the electromagnetic field: it's been excluded because it's not an antisymmetric tensor. – Stephen Blake Dec 31 '13 at 18:16
• @gerardpc : The argument that a physical quantity must carry a group representation and hence that the simplest physical quantities carry irreducible representations is made in the section entitled "Weyl's argument" in chapert 5 of the book "Eddington's search for a fundamental theory" by C.W. Kilmister. For groups theory I like "Group Theory in Physics" by Wu-Ki Tung, especially chapters 5 and 13 for the things we are talking about here. – Stephen Blake Dec 31 '13 at 18:32
• @StephenBlake But what about, e.g., arXiv:1201.3504v3, or Ch. 5 of Pietro Giuseppe Frè's Gravity, a Geometrical Course, or Rovelli's Quantum Gravity when he reviews the classical theory using differential forms, or...? – Alex Nelson Dec 31 '13 at 19:20