Tetradic Palatini action explained

See main article: Frame fields in general relativity.

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,

g _{\alpha} = e ^{I} _{\alpha} e ^{J} _{\beta} η _{IJ}

where

η _{IJ} = diag ( -1 , 1 , 1 , 1 ) 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

l{D} _{\alpha} V _{I} = \partial _{\alpha} V _{I} + {\omega _{\alpha} }^J V_J.

Where

{\omega _{\alpha} }^J is a spin (Lorentz) connection one-form (the derivative annihilates the Minkowski metric η _{IJ} ). We define a curvature via{\Omega _{\alpha} }^J V_J = (\mathcal_\alpha \mathcal_\beta - \mathcal_\beta\mathcal_\alpha) V_I

We obtain

{\Omega _{\alpha} }^ = 2 \partial__N \delta^M_ ^N = 0.

We show below that this implies that

{C ^{IJ} _{\alpha}} = 0 as the prefactor e ^{[a} _{M} e ^{b]} _{N} \delta ^{M} _{[I} \delta ^{K} _{J]} is non-degenerate. This tells us that \nabla coincides with D when acting on objects with only internal indices. Thus the connection D is completely determined by the tetrad and \Omega coincides with R . To compute the variation with respect to the tetrad we need the variation of e = \det e ^{I} _{\alpha} . From the standard formula\delta \det (a) = \det (a) \left (a ^{-1} \right ) _{ji} \delta a _{ij}

we have

\delta e = e e ^{\alpha} _{I} \delta e ^{I} _{\alpha} . Or upon using \delta \left (e ^{I} _{\alpha} e ^{\alpha} _{I} \right ) = 0 , this becomes \delta e = - e e ^{I} _{\alpha} \delta e ^{\alpha} _{I} . We compute the second equation by varying with respect to the tetrad,\begin{align} \delta S _{H-P} & = \int d ^{4} x e \left ( \left (\delta e ^{\alpha} _{I} \right ) e ^{\beta} _{J} {\Omega _{\alpha} }^ + e^\alpha_I \left (\delta e^\beta_J \right) ^ - e_\gamma^K \left (\delta e_K^\gamma \right) e^\alpha_I e^\beta_J ^ \right) \\&= 2 \int d^4 x \; e \left (e^\beta_J ^ - e_M^\gamma e_N^\delta e_\alpha^I ^ \right) \left (\delta e_I^\alpha \right)\end

One gets, after substituting

{\Omega _{\alpha} }^ for {R _{\alpha} }^ as given by the previous equation of motion,e ^{\gamma} _{J} {R _{\alpha} }^ - ^ e_M^\gamma e_N^\delta e_\alpha^I = 0

which, after multiplication by

e _{I} just tells us that the Einstein tensor R _{\alpha\beta} - \tfrac{ 1 }{ 2 } R g _{\alpha} 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

See main article: Holst action, Self-dual Palatini action, Barbero-Immirzi parameter and Plebanski action. We change the action by adding a term

- { 1 \over 2 \gamma} e e ^{\alpha} _{I} e ^{\beta} _{J} {\Omega _{\alpha} }^ [\omega] _

This modifies the Palatini action to

S = \int d ^{4} x e e ^{\alpha} _{I} e ^{\beta} _{J} {P ^{IJ} }_ ^

where

{P ^{IJ} }_ = \delta_M^ - _.

This action given above is the Holst action, introduced by Holst^[2] and

\gamma is the Barbero-Immirzi parameter whose role was recognized by Barbero^[3] and Immirizi.^[4] The self dual formulation corresponds to the choice \gamma = - i .

It is easy to show these actions give the same equations. However, the case corresponding to

\gamma = \pm i must be done separately (see article self-dual Palatini action). Assume \gamma \not = \pm i , then {P ^{IJ} }_ has an inverse given by{(P ^{-1} ) _{IJ} }^ = \frac \left (\delta_I^ + \frac ^ \right).

(note this diverges for

\gamma = \pm i ). As this inverse exists the generalization of the prefactor e ^{[a} _{M} e ^{b]} _{N} \delta ^{M} _{[I} \delta ^{K} _{J]} will also be non-degenerate and as such equivalent conditions are obtained from variation with respect to the connection. We again obtain {C ^{IJ} _{\alpha}} = 0 . 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

{R _{\alpha} }^ is defined by{R _{\alpha} }^ V_\delta = \left (\nabla_\alpha \nabla_\beta - \nabla_\beta \nabla_\alpha \right) V_\gamma.

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

V _{\gamma} = e ^{I} _{\gamma} V _{I} \begin{align} {R _{\alpha} }^ V_\delta &= \left (\nabla_\alpha \nabla_\beta - \nabla_\beta \nabla_\alpha \right) V_\gamma \\&= \left (\nabla_\alpha \nabla_\beta - \nabla_\beta \nabla_\alpha \right) \left (e_\gamma^I V_I \right) \\&= e_\gamma^I \left (\nabla_\alpha \nabla_\beta - \nabla_\beta \nabla_\alpha \right) V_I \\&= e_\gamma^I ^J e_J^\delta V_\delta\end

where we have used

\nabla _{\alpha} e ^{I} _{\beta} = 0 . Since this is true for all V _{\delta} we obtain{R _{\alpha} }^ = e_\gamma^I ^J e_J^\delta.

Using this expression we find

R _{\alpha} = {R _{\alpha} }^ = ^J e_\beta^I e_J^\gamma.

Contracting over

\alpha and \beta allows us write the Ricci scalarR = {R _{\alpha} }^ e_I^\alpha e_J^\beta.

Difference between curvatures

The derivative defined by

D _{\alpha} V _{I} 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 l{D} _{a} twice on V _{I} ,l{D} _{\alpha} l{D} _{\beta} V _{I} = l{D} _{\alpha} (\nabla _{\beta} V _{I} + {C _{\beta} }^J V_J) = \nabla_\alpha \left (\nabla_\beta V_I + ^J V_J \right) + ^K \left (\nabla_b V_K + ^J V_J \right) + \overline_^\gamma \left (\nabla_\gamma V_I + ^J V_J \right)

where

\overline{\Gamma} ^{\gamma} _{\alpha \beta} is unimportant, we need only note that it is symmetric in \alpha and \beta as it is torsion-free. Then\begin{align} {\Omega _{\alpha} }^J V_J &= \left (\mathcal_\alpha \mathcal_\beta - \mathcal_\beta \mathcal_\alpha \right) V_I \\&= \left (\nabla_\alpha \nabla_\beta - \nabla_\beta \nabla_\alpha \right) V_I + \nabla_\alpha \left (^J V_J \right) - \nabla_\beta \left (^J V_J \right) +^K \nabla_\beta V_K - ^K \nabla_\alpha V_K + ^K ^J V_J - ^K ^J V_J \\&= ^J V_J + \left (\nabla_\alpha ^J - \nabla_\beta ^J + ^K ^J - ^K ^J \right) V_J \end

Hence:

{\Omega _{ab} }^ - ^ = 2 \nabla_ ^ ^N \\ &= \delta \int d^4 x \; e \; e^_N ^K ^N \\ &= \int d^4 x \; e e^_N \left (\delta_\gamma^\alpha \delta^I_M \delta^K_J ^N + ^K \delta^\alpha_\beta \delta^I_K \delta^N_J \right) \delta ^J \\ &= \int d^4 x \; e \left (e^_N ^N + e^_J ^I \right) \delta ^J\end

or

e ^{[\alpha} _{I} e ^{\beta]} _{K} {C _{\beta} }^K + e^_J C_ = 0

or

{C _{\beta} }^K e^_J + ^K e^_K = 0.

where we have used

C _{\beta} = - C _{\beta} . This can be written more compactly ase ^{[\alpha} _{M} e ^{\beta]} _{N} \delta ^{M} _{[I} \delta ^{K} _{J]} {C _{\beta} }^N = 0.

Vanishing of

{C ^{IJ} _{\alpha}}

We will show following the reference "Geometrodynamics vs. Connection Dynamics"^[5] that

{C _{\beta} }^K e^_J + ^K e^_K = 0 \quad Eq. 1

implies

{C _{\alpha} }^J = 0. First we define the spacetime tensor field byS _{\alpha} : = C _{\alpha} e ^{I} _{\beta} e ^{J} _{\gamma.}

Then the condition

C _{\alpha} = C _{\alpha} is equivalent to S _{\alpha} = S _{\alpha} . Contracting Eq. 1 with e ^{I} _{\alpha} e ^{J} _{\gamma} one calculates that{C _{\beta} }^I e_\gamma^J e_I^\beta = 0.

As

{S _{\alpha} }^ = ^J e_\beta^I e_J^\gamma, we have {S _{\beta} }^ = 0. We write it as({C _{\beta} }^J e_J^\beta) e_\gamma^I = 0,

and as

e ^{I} _{\alpha} are invertible this implies{C _{\beta} }^J e_J^\beta = 0.

Thus the terms

{C _{\beta} }^K e^\beta_K e^\alpha_J, and {C _{\beta} }^K e^\alpha_I e^\beta_K of Eq. 1 both vanish and Eq. 1 reduces to{C _{\beta} }^K e^\alpha_K e^\beta_J - ^K e^\beta_I e^\alpha_K = 0.

If we now contract this with

e ^{I} _{\gamma} e ^{J} _{\delta} , we get\begin{align} 0 & = \left ( {C _{\beta} }^K e^\alpha_K e^\beta_J - ^K e^\beta_I e^\alpha_K \right) e^I_\gamma e^J_\delta \\&= ^K e^\alpha_K e^I_\gamma \delta_\delta^\beta - ^K \delta_\gamma^\beta e^\alpha_K e^J_\delta \\&= ^K e^I_\gamma e^\alpha_K - ^K e^J_\delta e^\alpha_K \end

or

{S _{\gamma} }^ = ^.

Since we have

S _{\alpha} = S _{\alpha} and S _{\alpha} = S _{(\alpha} , we can successively interchange the first two and then last two indices with appropriate sign change each time to obtain,S _{\alpha} = S _{\beta} = - S _{\beta} = - S _{\gamma} = S _{\gamma} = S _{\alpha} = - S _{\alpha}

Implying

S _{\alpha} = 0 ,

or

C _{\alpha} e ^{I} _{\beta} e ^{J} _{\gamma} = 0 ,

and since the

e ^{I} _{\alpha} are invertible, we get C _{\alpha} = 0 . This is the desired result.

See also

• Lanczos tensor
• Weyl tensor

Notes and References

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 [[Peter Bergmann|P.G. Bergmann]] and V. De Sabbata (eds.) Cosmology and Gravitation, Plenum Press, New York (1980)]
2. Holst . Sören . Barbero's Hamiltonian derived from a generalized Hilbert-Palatini action . Physical Review D . 53 . 10 . 1996-05-15 . 0556-2821 . 10.1103/physrevd.53.5966 . 5966–5969. 10019884 . gr-qc/9511026. 1996PhRvD..53.5966H . 15959938 .
3. Barbero G. . J. Fernando . Real Ashtekar variables for Lorentzian signature space-times . Physical Review D . 51 . 10 . 1995-05-15 . 0556-2821 . 10.1103/physrevd.51.5507 . 5507–5510. 10018309 . gr-qc/9410014. 1995PhRvD..51.5507B . 16314220 .
4. Immirzi . Giorgio . Real and complex connections for canonical gravity . Classical and Quantum Gravity . IOP Publishing . 14 . 10 . 1997-10-01 . 0264-9381 . 10.1088/0264-9381/14/10/002 . L177–L181. gr-qc/9612030. 1997CQGra..14L.177I . 5795181 .
5. Romano . Joseph D. . Geometrodynamics vs. connection dynamics . General Relativity and Gravitation . 25 . 8 . 1993 . 0001-7701 . 10.1007/bf00758384 . 759–854. gr-qc/9303032. 1993GReGr..25..759R . 119359223 .