Tetradic Palatini action

The Einstein–Hilbert action for general relativity was first formulated purely in terms of the space-time metric. To take the metric and affine connection as independent variables in the action principle was first considered by Palatini.[1] It is called a first order formulation as the variables to vary over involve only up to first derivatives in the action and so doesn't overcomplicate the Euler–Lagrange equations with higher derivative terms. The tetradic Palatini action is another first-order formulation of the Einstein–Hilbert action in terms of a different pair of independent variables, known as frame fields and the spin connection. The use of frame fields and spin connections are essential in the formulation of a generally covariant fermionic action (see the article spin connection for more discussion of this) which couples fermions to gravity when added to the tetradic Palatini action.

Not only is this needed to couple fermions to gravity and makes the tetradic action somehow more fundamental to the metric version, the Palatini action is also a stepping stone to more interesting actions like the self-dual Palatini action which can be seen as the Lagrangian basis for Ashtekar's formulation of canonical gravity (see Ashtekar's variables) or the Holst action which is the basis of the real variables version of Ashtekar's theory. Another important action is the Plebanski action (see the entry on the Barrett–Crane model), and proving that it gives general relativity under certain conditions involves showing it reduces to the Palatini action under these conditions.

Here we present definitions and calculate Einstein's equations from the Palatini action in detail. These calculations can be easily modified for the self-dual Palatini action and the Holst action.

Some definitions


We first need to introduce the notion of tetrads. A tetrad is an orthonormal vector basis in terms of which the space-time metric looks locally flat,


where   is the Minkowski metric. The tetrads encode the information about the space-time metric and will be taken as one of the independent variables in the action principle.

Now if one is going to operate on objects that have internal indices one needs to introduce an appropriate derivative (covariant derivative). We introduce an arbitrary covariant derivative via


Where   is a spin (Lorentz) connection one-form (the derivative annihilates the Minkowski metric  ). We define a curvature via


We obtain


We introduce the covariant derivative which annihilates the tetrad,


The connection is completely determined by the tetrad. The action of this on the generalized tensor   is given by


We define a curvature   by


This is easily related to the usual curvature defined by


via substituting   into this expression (see below for details). One obtains,


for the Riemann tensor, Ricci tensor and Ricci scalar respectively.

The tetradic Palatini action


The Ricci scalar of this curvature can be expressed as   The action can be written


where   but now   is a function of the frame field.

We will derive the Einstein equations by varying this action with respect to the tetrad and spin connection as independent quantities.

As a shortcut to performing the calculation we introduce a connection compatible with the tetrad,  [2] The connection associated with this covariant derivative is completely determined by the tetrad. The difference between the two connections we have introduced is a field   defined by


We can compute the difference between the curvatures of these two covariant derivatives (see below for details),


The reason for this intermediate calculation is that it is easier to compute the variation by reexpressing the action in terms of   and   and noting that the variation with respect to   is the same as the variation with respect to   (when keeping the tetrad fixed). The action becomes


We first vary with respect to  . The first term does not depend on   so it does not contribute. The second term is a total derivative. The last term yields


We show below that this implies that   as the prefactor   is non-degenerate. This tells us that   coincides with   when acting on objects with only internal indices. Thus the connection   is completely determined by the tetrad and   coincides with  . To compute the variation with respect to the tetrad we need the variation of  . From the standard formula


we have  . Or upon using  , this becomes  . We compute the second equation by varying with respect to the tetrad,


One gets, after substituting   for   as given by the previous equation of motion,


which, after multiplication by   just tells us that the Einstein tensor   of the metric defined by the tetrads vanishes. We have therefore proved that the Palatini variation of the action in tetradic form yields the usual Einstein equations.

Generalizations of the Palatini action


We change the action by adding a term


This modifies the Palatini action to




This action given above is the Holst action, introduced by Holst[3] and   is the Barbero-Immirzi parameter whose role was recognized by Barbero[4] and Immirizi.[5] The self dual formulation corresponds to the choice  .

It is easy to show these actions give the same equations. However, the case corresponding to   must be done separately (see article self-dual Palatini action). Assume  , then   has an inverse given by


(note this diverges for  ). As this inverse exists the generalization of the prefactor   will also be non-degenerate and as such equivalent conditions are obtained from variation with respect to the connection. We again obtain  . While variation with respect to the tetrad yields Einstein's equation plus an additional term. However, this extra term vanishes by symmetries of the Riemann tensor.

Details of calculation


Relating usual curvature to the mixed index curvature


The usual Riemann curvature tensor   is defined by


To find the relation to the mixed index curvature tensor let us substitute  


where we have used  . Since this is true for all   we obtain


Using this expression we find


Contracting over   and   allows us write the Ricci scalar


Difference between curvatures


The derivative defined by   only knows how to act on internal indices. However, we find it convenient to consider a torsion-free extension to spacetime indices. All calculations will be independent of this choice of extension. Applying   twice on  ,


where   is unimportant, we need only note that it is symmetric in   and   as it is torsion-free. Then




Varying the action with respect to the field  


We would expect   to also annihilate the Minkowski metric  . If we also assume that the covariant derivative   annihilates the Minkowski metric (then said to be torsion-free) we have,




From the last term of the action we have from varying with respect to  






where we have used  . This can be written more compactly as


Vanishing of  


We will show following the reference "Geometrodynamics vs. Connection Dynamics"[6] that


implies   First we define the spacetime tensor field by


Then the condition   is equivalent to  . Contracting Eq. 1 with   one calculates that


As   we have   We write it as


and as   are invertible this implies


Thus the terms   and   of Eq. 1 both vanish and Eq. 1 reduces to


If we now contract this with  , we get




Since we have   and  , we can successively interchange the first two and then last two indices with appropriate sign change each time to obtain,






and since the   are invertible, we get  . This is the desired result.

See also



  1. ^ A. Palatini (1919) Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton, Rend. Circ. Mat. Palermo 43, 203-212 [English translation by R.Hojman and C. Mukku in P.G. Bergmann and V. De Sabbata (eds.) Cosmology and Gravitation, Plenum Press, New York (1980)]
  2. ^ A. Ashtekar "Lectures on non-perturbative canonical gravity" (with invited contributions), Bibliopolis, Naples 19988.
  3. ^ Holst, Sören (1996-05-15). "Barbero's Hamiltonian derived from a generalized Hilbert-Palatini action". Physical Review D. 53 (10): 5966–5969. arXiv:gr-qc/9511026. Bibcode:1996PhRvD..53.5966H. doi:10.1103/physrevd.53.5966. ISSN 0556-2821. PMID 10019884. S2CID 15959938.
  4. ^ Barbero G., J. Fernando (1995-05-15). "Real Ashtekar variables for Lorentzian signature space-times". Physical Review D. 51 (10): 5507–5510. arXiv:gr-qc/9410014. Bibcode:1995PhRvD..51.5507B. doi:10.1103/physrevd.51.5507. ISSN 0556-2821. PMID 10018309. S2CID 16314220.
  5. ^ Immirzi, Giorgio (1997-10-01). "Real and complex connections for canonical gravity". Classical and Quantum Gravity. 14 (10). IOP Publishing: L177–L181. arXiv:gr-qc/9612030. Bibcode:1997CQGra..14L.177I. doi:10.1088/0264-9381/14/10/002. ISSN 0264-9381. S2CID 5795181.
  6. ^ Romano, Joseph D. (1993). "Geometrodynamics vs. connection dynamics". General Relativity and Gravitation. 25 (8): 759–854. arXiv:gr-qc/9303032. Bibcode:1993GReGr..25..759R. doi:10.1007/bf00758384. ISSN 0001-7701. S2CID 119359223.