Finite density definition

Displayed formulas in this passage

• XVI-K@1#eq:Gamma · line 519

  $$\begin{array}{rl}{{\Gamma }_n}^{\rho }{}_{\mu \nu }& =\frac{1}{2}{u_{n-1}}^{\rho \lambda }({\partial }_{\mu }{\gamma }_{\nu \lambda }+{\partial }_{\nu }{\gamma }_{\mu \lambda }-{\partial }_{\lambda }{\gamma }_{\mu \nu }),\\ R_{n,\mu \nu }& ={\partial }_{\rho }{{\Gamma }_n}^{\rho }{}_{\mu \nu }-{\partial }_{\nu }{{\Gamma }_n}^{\rho }{}_{\mu \rho }\\ & +\sum _{\begin{array}{c}a+b=n\\ a,b\ge 1\end{array}}({{\Gamma }_a}^{\rho }{}_{\rho \lambda }{{\Gamma }_b}^{\lambda }{}_{\mu \nu }-{{\Gamma }_a}^{\rho }{}_{\nu \lambda }{{\Gamma }_b}^{\lambda }{}_{\mu \rho }),\\ r_n& ={\sum _{i=0}}^{n-1}{u_i}^{\mu \nu }{R}_{n-i,\mu \nu },\boxed{{\ell }_3=-2(r_3+b_1{r}_2+b_2{r}_1).}\end{array}$$

• XVI-K@1#eq:Ricci · line 519

  $$\begin{array}{rl}{{\Gamma }_n}^{\rho }{}_{\mu \nu }& =\frac{1}{2}{u_{n-1}}^{\rho \lambda }({\partial }_{\mu }{\gamma }_{\nu \lambda }+{\partial }_{\nu }{\gamma }_{\mu \lambda }-{\partial }_{\lambda }{\gamma }_{\mu \nu }),\\ R_{n,\mu \nu }& ={\partial }_{\rho }{{\Gamma }_n}^{\rho }{}_{\mu \nu }-{\partial }_{\nu }{{\Gamma }_n}^{\rho }{}_{\mu \rho }\\ & +\sum _{\begin{array}{c}a+b=n\\ a,b\ge 1\end{array}}({{\Gamma }_a}^{\rho }{}_{\rho \lambda }{{\Gamma }_b}^{\lambda }{}_{\mu \nu }-{{\Gamma }_a}^{\rho }{}_{\nu \lambda }{{\Gamma }_b}^{\lambda }{}_{\mu \rho }),\\ r_n& ={\sum _{i=0}}^{n-1}{u_i}^{\mu \nu }{R}_{n-i,\mu \nu },\boxed{{\ell }_3=-2(r_3+b_1{r}_2+b_2{r}_1).}\end{array}$$

• XVI-K@1#eq:ell3 · line 519

  $$\begin{array}{rl}{{\Gamma }_n}^{\rho }{}_{\mu \nu }& =\frac{1}{2}{u_{n-1}}^{\rho \lambda }({\partial }_{\mu }{\gamma }_{\nu \lambda }+{\partial }_{\nu }{\gamma }_{\mu \lambda }-{\partial }_{\lambda }{\gamma }_{\mu \nu }),\\ R_{n,\mu \nu }& ={\partial }_{\rho }{{\Gamma }_n}^{\rho }{}_{\mu \nu }-{\partial }_{\nu }{{\Gamma }_n}^{\rho }{}_{\mu \rho }\\ & +\sum _{\begin{array}{c}a+b=n\\ a,b\ge 1\end{array}}({{\Gamma }_a}^{\rho }{}_{\rho \lambda }{{\Gamma }_b}^{\lambda }{}_{\mu \nu }-{{\Gamma }_a}^{\rho }{}_{\nu \lambda }{{\Gamma }_b}^{\lambda }{}_{\mu \rho }),\\ r_n& ={\sum _{i=0}}^{n-1}{u_i}^{\mu \nu }{R}_{n-i,\mu \nu },\boxed{{\ell }_3=-2(r_3+b_1{r}_2+b_2{r}_1).}\end{array}$$

• unlabeled · line 508

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

• unlabeled · line 514

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

• unlabeled · line 551

  $$V_{\mathrm{I}}(h)={\int }_M{\chi }_{\mathrm{I}}(w){\ell }_3(h)(w){\mathrm{d}}^{4}w.$$