Geometry and fixed external functions

Displayed formulas in this passage

• XII-C@1#eq:curvature · line 437

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

• unlabeled · line 390

  $$M={\mathbb{R}}^4,\eta =\mathrm{diag}(1,-1,-1,-1),\square ={{\partial }_t}^2-{\Delta }_{\mathbf{x}},P_m=\square +m^2,$$

• unlabeled · line 396

  $$M_T=[0,T]\times {\mathbb{R}}^3,{M_T}^{\circ }=(0,T)\times {\mathbb{R}}^3,0<\epsilon <T.$$

• unlabeled · line 402

  $${\sigma }_{\epsilon }(t)=\chi (t/\epsilon ),\chi (s)=\{\begin{array}{ll}0,& s\le 0,\\ \frac{e^{-1/s}}{e^{-1/s}+e^{-1/(1-s)}},& 0<s<1,\\ 1,& s\ge 1.\end{array}$$

• unlabeled · line 413

  $$g\in {C_c}^{\infty }({M_T}^{\circ };[0,1]),P_g=\mathrm{int}\{g=1\}\ne \varnothing ,$$

• unlabeled · line 418

  $$\overline{U}\subset P_g\cap \{\epsilon <t<T\},{\mathfrak{g}}_U={C_c}^{\infty }(U;TM).$$

• unlabeled · line 426

  $$q_{\mu \nu }={\eta }_{\mu \nu }+\kappa {\gamma }_{\mu \nu },\mathcal{P}r=r-\frac{1}{2}\eta {\mathrm{tr}}_{\eta }r,$$

• unlabeled · line 430

  $$D^{\nu }\gamma ={\partial }_{\mu }(\mathcal{P}\gamma {)}^{\mu \nu },(K\xi {)}_{\mu \nu }={\partial }_{\mu }{\xi }_{\nu }+{\partial }_{\nu }{\xi }_{\mu }.$$