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((Short description|Construction in relativity))
((More citations needed|date=February 2015))
[[General relativity]] and [[supergravity]] in all dimensions meet each other at a common assumption:

:''Any [[Configuration space (physics)|configuration space]] can be coordinatized by [[gauge field]]s <math>A^i_a</math>, where the index <math>i</math> is a [[Lie algebra]] index and <math>a</math> is a [[Three-dimensional space|spatial]] [[manifold]] index.''

Using these assumptions one can construct an [[effective field theory]] in low energies for both. In this form the action of general relativity can be written in the form of the '''Plebanski action''' which can be constructed using the [[Palatini action]] to derive [[Einstein's field equations]] of [[general relativity]].

The form of the action introduced by [[Jerzy Plebański|Plebanski]] is:

:<math>S_\mathrm{Plebanski} = \int_{\Sigma \times R} \epsilon_{ijkl} B^{ij} \wedge F^{kl} (A^i_a) + \phi_{ijkl} B^{ij} \wedge B^{kl} </math>

where

<math>i, j, l, k</math>

are internal indices,<math>F</math> is a curvature on the orthogonal group <math>SO(3, 1)</math> and the [[Connection (mathematics)|connection]] variables (the gauge fields) are denoted by <math>A^i_a</math>. The symbol <math>\phi_{ijkl}</math> is the  [[Lagrangian multiplier]] and <math>\epsilon_{ijkl}</math> is the [[antisymmetric symbol]] valued over <math>SO(3, 1)</math>.

The specific definition

:<math>B^{ij} = e^i \wedge e^j</math>

formally satisfies the [[Einstein's field equation]] of [[general relativity]].

Application is to the [[Barrett–Crane model]].<ref name=Barrett1998>((citation
 | author = Barrett, John W.
 | author2 = Louis Crane
 | title = Relativistic spin networks and quantum gravity
 | journal = J. Math. Phys. 
 | volume = 39
 | issue = 6
 | pages = 3296–3302
 | doi = 10.1063/1.532254
 | date = 1998
|arxiv = gr-qc/9709028 |bibcode = 1998JMP....39.3296B | s2cid = 1998581
 ))</ref><ref name=Barrett>((citation
 | author = Barrett, John W.
 | author2 = Louis, Crane
 | title = A Lorentzian signature model for quantum general relativity
 | journal = Classical and Quantum Gravity
 | volume = 17
 | issue = 16
 | pages = 3101–3118
 | doi = 10.1088/0264-9381/17/16/302
|arxiv = gr-qc/9904025 |bibcode = 2000CQGra..17.3101B | year = 2000
 | s2cid = 250906675
 ))</ref>

==See also==

* [[Tetradic Palatini action]]
* [[Barrett–Crane model]]
* [[BF model]]

==References==
((reflist))

* ((cite journal  |last1=Celada |first1=Mariano |last2=Gonzalez|first2=Diego |last3=Montesinos |first3=Merced |year=2016  |title=BF gravity |journal=Classical and Quantum Gravity |volume=33 |issue=21 |pages=213001 |arxiv=1610.02020  |bibcode=2016CQGra..33u3001C |doi=10.1088/0264-9381/33/21/213001|s2cid=119605468 ))

((DEFAULTSORT:Plebanski Action))
[[Category:Variational formalism of general relativity]]


((relativity-stub))
((math-physics-stub))

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