Curvature and the fixed unintegrated density

Displayed formulas in this passage

• Em01-C@3#eq:finite-curvature · line 1966

  $$\begin{array}{rl}{{\Gamma }_n}^{\rho }{}_{\mu \nu }& =\frac{1}{2}{u_{n-1}}^{\rho \lambda }\left({\partial }_{\mu }{\gamma }_{\nu \lambda }+{\partial }_{\nu }{\gamma }_{\mu \lambda }-{\partial }_{\lambda }{\gamma }_{\mu \nu }\right),\\ R_{n,\mu \nu }& ={\partial }_{\rho }{{\Gamma }_n}^{\rho }{}_{\mu \nu }-{\partial }_{\nu }{{\Gamma }_n}^{\rho }{}_{\mu \rho }\\ & +\sum _{\begin{array}{c}a+b=n\\ a,b\ge 1\end{array}}\left({{\Gamma }_a}^{\rho }{}_{\rho \lambda }{{\Gamma }_b}^{\lambda }{}_{\mu \nu }-{{\Gamma }_a}^{\rho }{}_{\nu \lambda }{{\Gamma }_b}^{\lambda }{}_{\mu \rho }\right),\\ r_n& ={\sum _{i=0}}^{n-1}{u_i}^{\mu \nu }{R}_{n-i,\mu \nu }.\end{array}$$

• Em01-C@3#eq:ell3 · line 1990

  $${\ell }_3=-2(r_3+b_1{r}_2+b_2{r}_1).$$

• Em01-C@3#eq:J3 · line 2011

  $${J_3}^{\mu \nu }=2\left({V_2}^{\mu \nu }+b_1{V_1}^{\mu \nu }-{\gamma }^{\mu \alpha }{V}_{1,\alpha }{}^{\nu }-{\gamma }^{\nu \alpha }{V}_{1,\alpha }{}^{\mu }\right).$$

• unlabeled · line 1944

  $$R_{\mu \nu }={\partial }_{\rho }{{\Gamma }^{\rho }}_{\mu \nu }-{\partial }_{\nu }{{\Gamma }^{\rho }}_{\mu \rho }+{{\Gamma }^{\rho }}_{\rho \lambda }{{\Gamma }^{\lambda }}_{\mu \nu }-{{\Gamma }^{\rho }}_{\nu \lambda }{{\Gamma }^{\lambda }}_{\mu \rho }.$$

• unlabeled · line 1956

  $${u_0}^{\mu \nu }={\eta }^{\mu \nu },{u_1}^{\mu \nu }=-{\gamma }^{\mu \nu },{u_2}^{\mu \nu }={\gamma }^{\mu }{}_{\lambda }{\gamma }^{\lambda \nu },$$

• unlabeled · line 1961

  $$b_0=1,b_1=\frac{1}{2}\tau ,b_2=\frac{1}{8}{\tau }^2-\frac{1}{4}{\gamma }_{\mu \nu }{\gamma }^{\mu \nu }.$$

• unlabeled · line 1995

  $${\ell }_3=-2[s^3]\left(\sqrt{-\det (\eta +s\gamma )}R[\eta +s\gamma ]\right),$$

• unlabeled · line 2003

  $$V_{1,\mu \nu }=R_{1,\mu \nu }-\frac{1}{2}{\eta }_{\mu \nu }{r}_1,$$

• unlabeled · line 2006

  $$V_{2,\mu \nu }=R_{2,\mu \nu }-\frac{1}{2}{\eta }_{\mu \nu }{r}_2-\frac{1}{2}{\gamma }_{\mu \nu }{r}_1.$$