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Tetrads in General Relativity

XIII. ECKS Theory

Hehl-Datta Equation

   Torsion enters ECD gravity through the coderivative (13.13) of the Dirac spinor in the Dirac Lagrangian (13.12). By separating off the torsion-free spin connection as in (13.30), one may decompose the Lagrangian into a part with the torsion-free (Levi-Civita) coderivative and a remainder which is a contraction between the contorsion tensor (13.33) and the spin density (13.27):

13.35

\[{\mathcal{L}_{\text{D}}}  = {{\mathop {\mathcal{L}}\limits^ \circ }_{\text{D}}}{\text{ + }}\frac{1}{4}i{C_{abc}}\bar \psi \left\{ {{\gamma ^a},{\sigma ^{bc}}} \right\}\psi ={{\mathop {\mathcal{L}}\limits^ \circ} _{\text{D}}} {\text{ + }}\frac{1}{2}{C_{abc}}{\sigma ^{abc}}\]

   The spin tensor of the Dirac field is completely antisymmetric. This implies that the torsion tensor, algebraically related to the spin density through the Cartan equations (13.5), and the contorsion tensor, are both completely anti-symmetric as well. Thus, in the case of the Dirac field, one has the simple relationship $2{C^{abc}} = {T^{abc}} = \kappa {\sigma ^{abc}}$. It concisely summarizes that in ECKS gravity the torsional degrees of freedom solely originate from the presence of fermions.

   When the linear relationship is inserted into equation (13.35), it induces an effective spin-spin interaction in the Lagrangian:

13.36

\[{{\mathcal{L}}_{\text{D}}} = {{\mathop {\mathcal{L}}\limits^ \circ} _{\text{D}}}{\text{ + }}\frac{1}{4}\kappa {\sigma _{abc}}{\sigma ^{abc}} = {{\mathop {\mathcal{L}}\limits^ \circ} _{\text{D}}} - \frac{3}{8}\kappa \left( {\bar \psi {\gamma _d}{\gamma ^5}\psi } \right)\bar \psi {\gamma ^d}{\gamma ^5}\psi \]

In the last term the spin density tensor has been expressed in terms of the Dirac axial spin vector, see (D.3):

13.37

\[{\sigma ^{abc}} = {\varepsilon ^{abcd}}{j_d} \quad {j_d} := \frac{1}{2}\bar \psi {\gamma _d}{\gamma ^5}\psi \]

The torsion due to the presence of fermions is now effectively contained in the matter sector in the form of an 4-spinor interaction term embedded in a Riemannian geometry, i.e. standard GRT.

   Variation of the corresponding action with respect to the spinor field $\bar \psi $ yields the nonlinear Hehl-Datta equation:

13.38

\[i\hbar {\gamma ^a}{{\mathop {\mathcal{D}}\limits^ \circ }_a} \psi = m \psi + \frac{{3\kappa {\hbar ^2}}}{8}(\bar \psi {\gamma _d}{\gamma ^5}\psi ){\gamma ^d}{\gamma ^5}\psi \]

The Planck constant $\hbar $ is made explicit to show that the coupling constant in this effective equation is proportional to square of the Planck length ${l_{\text{P}}}: = \sqrt{\hbar G / c^3} $. This indicates that deviations of GRT are only likely to occur in extreme density regimes. [Wikipedia: Nonlinear Dirac Equation]