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\author{pod}
\title{\bf Relatividad}
\date{Primavera, 2002}


\begin{document}
\noindent
 $$ \frac{h\nu'}{h\nu} = \frac{m c^2}{mc^2 + m g h} \approx 1 - g h /c^2 $$
 $$ \Delta T_h \approx (1+g h /c^2) \Delta T_0 $$
 $$ \frac{1}{L^2} = \frac{G M}{r^3 c^2} $$

\sep
 $$ \vv = \deriv{\lambda} = \sum_i \left. \deriv[x^i]{\lambda} \right|_P \parcial{x^i}= \sum_i  v^i \parcial{x^i} $$
 $$ [\vX, \vY] = \sum_{i,j}^n \left(  x^i \parcial[y^j]{x^i} - y^j \right) \parcial{x^j} $$
 $$ x^i(\lambda_0+\epsilon) = \left. \expon{\epsilon \deriv{\lambda}} \right|_{\lambda_0} x^i = \left. \expon{\epsilon \vX} \right|_{\lambda_0} x^i $$
 $$ \tgrad{f} \left( \deriv{\lambda}  \right) := \left. \deriv[f]{\lambda} \right|_p \quad , \quad \tgrad f = \parcial[f]{x^i} \tgrad{x^i} $$
 $$\ve_{i'} = \mlambda{i}{i'} \ve_i \ ,\quad \tomega^{i'} = \mlambda{i'}{i} \tomega^i \ ,\quad v^{i'} = \mlambda{i'}{i} v^i \ ,\quad
  p_{i'} = \mlambda{i}{i'} p_i $$
 $$ g' = \Lambda^T g \Lambda $$
 Subvariedad $ \forall p\in S, \exists (\phi, U(p)) $ tq $\phi  :  U \cap \RR^{(n,m)} \to V \cap S $

\sep
 $\phi : \MM(x_i) \to \NN(y_i $ \\
 Push-fw $ \phi^* : T_x \to T_{\phi(x)} $, $ \vv^*(f) = \vv(f\circ\phi) $, $v^{*\alpha} = v^i \parcial[y^\alpha]{x^i}$ \\
 Pull-back $ \phi_* : T_{\phi(x)}x \to T_{x} $, $\omega^*(\vv) = \omega(\vv^*)$, $\omega^*_i = \omega_\alpha \parcial[y^\alpha]{x^i}$ \\
 Transporte Lie $g_{\Delta t} ( x^i(t) ) = x^i (t+\Delta t) $ \\
 Derivada Lie
 $$ \lie{\vv} f := \lim_{\Delta t \to 0} \frac{f^*_{\Delta t}(t_0)-f(t_0)}{\Delta t} = \vv(f) $$
 $$ \lie{\vu} \vv := \lim_{\Delta t} \frac{ (g^{-1}_{\Delta t})^* \vv(g_{\Delta t} \vx) - \vv(\vx)}{\Delta t} = [\vu,\vv] $$
 $$ \lie{\vu} (\tomega) := \lim_{\Delta t \to
   0} \frac{ (g^*_{\Delta t}) \tomega(g_{\Delta t} \vx) - \tomega(x^i)}{\Delta t} = ( u^k \partial_k \omega_j + \omega_k \partial_j u^k ) \tgrad{x^j} $$
 $$ \lie{\vu} \big( \tomega(\vv)\big) = \big( \lie{\vu} (\tomega) \big) \vv + \tomega \big( \lie{\vu} (\vv) \big) \quad,\quad \lie{[\vx,\vy]} =
  \big[\lie{\vx},\lie{\vy}\big]$$
 \begin{eqnarray*}
 \matriz{\left( \lie{\vu} T \right)}{i_1,\ldots,i_p}{j_1,\ldots,j_q} & = & u^k \parcial{x^k} \matriz{T}{i_1,\ldots,i_p}{j_1,\ldots,j_q} \\
  &&+\matriz{T}{i_1,\ldots,i_p}{k,j_2,\ldots,j_q} \parcial[u^k]{x^{j_1}} + \cdots + \matriz{T}{i_1,\ldots,i_p}{j_1,\ldots,j_{q-1}, k} \parcial[i^k]
   {x^{j_q}} \\
  &&-\matriz{T}{k,i_2,\ldots,i_p}{j_1,\ldots,j_q} \parcial[u^{u_1}]{x^k} - \matriz{T}{i_1,\ldots,i_{p-1},k}{j_1,\ldots,j_q} \parcial[u^{i_p}]{x^k}
 \end{eqnarray*}

\sep Killing $\lie{\vx} g = 0$ , max $n(n+1)/2$
 $$ \nabla_{(\alpha} \xi_{\beta)} = 0 $$
 $$ \partial_j x^i +\partial_i x^j = 2 x_l \matriz{\Gamma}{l}{ji} $$

\sep
 $$ \tp \wedge \tq := \tp \otimes \tq - \tq \otimes \tp \quad,\quad \talpha = \frac{1}{p!}\alpha_{[i\ldots j]}\ \tomega^i \wedge \cdots \wedge \tomega^j $$
 $$ \tp \wedge \tq = \frac{1}{p! q!} p_{i\ldots j}\ q_{k\ldots l} \tomega^i \wedge \cdots \wedge \tomega^l = \frac{1}{(p+q)!} (\tp\wedge\tq)_{i\ldots
   l} \tomega^i \wedge \cdots \wedge \tomega^l $$
 $$\tp \wedge \tq = (-1)^{pq} \tq \wedge \tp  $$
 $$ (i_{\vx}) \tomega = \tomega( \vx ,\ , \ldots ,\ ) = \frac{1}{(p-1)!} x^i p_{ij\ldots k} \tomega^j \wedge \cdots \wedge \tomega^k $$

\sep
 $$ \det A =\epsilon_{il \ldots j} A^{1i} A^{2l} \cdots A^{nj} = \frac{1}{n!} \epsilon_{ab\ldots c} \epsilon_{il\ldots j} A^{a i} A^{bl}\cdots A^{cj}$$
 $$ \tomega = \frac{1}{n!} f \epsilon_{ij\ldots k} \tomega^i \wedge \tomega^j \wedge \cdots \wedge \tomega^k \quad,\quad f' = f \det\Lambda$$
 $$ \tomega = \tomega^1 \wedge \cdots \wedge \tomega^n = \sqrt{|g|} \tgrad{x}^1 \wedge \cdots \wedge \tgrad{x}^n $$
 $$ \textrm{dual: } (*\alpha)_{i\ldots j} = \frac{1}{q!} \omega_{k\ldots l i\ldots j} \alpha^{k\ldots l} $$
 $$*(\tomega^r \wedge \cdots \wedge \tomega^t) = \frac{1}{(n-p)!}\ g^{ri} \cdots g^{tk}  \ \omega_{i\ldots k l \ldots m} \tomega^l \wedge \cdots \wedge
  \tomega^m $$

\sep
 $$\tdd(\talpha + \tbeta) = \tdd \talpha + \tdd \tbeta \quad,\quad \tdd( \talpha \wedge \tbeta ) = \tdd\talpha \wedge \tbeta + (-1)^\alpha \talpha
  \wedge \tdd \tbeta $$
 $$\tdd\tdd \talpha = 0 \quad,\quad \tdd(f) = \tgrad{f}$$
 $$ \tdd \talpha = \frac{1}{\alpha!} \parcial[\alpha_{i\ldots j}]{x^m}\ \tgrad{x}^m \wedge \tgrad{x}^i \wedge \cdots \wedge \tgrad{x}^j $$
 $$ \mathrm{rot} \tp = * \tdd \tp \quad , \quad \mathrm{div}\tp = \tdd*\tp $$
 $$ \int_\Omega \tF = \int_\Omega \tF(\e{1},\ldots,\e{n}) \tgrad{x}^1 \wedge \cdots \wedge\ \tgrad{x}^k := $$ $$ :=\int_U \tF(\e{1},\ldots,\e{n}) \dx^1
  \cdots \dx^k $$
 $$ \textrm{Stokes}\  \int_\Omega \tdd \tF = \int_{\partial \Omega}\tF $$
 $$ \textrm{Gauss}\ \int_\Omega \dd (\tomega(\vA)) = \int_{\partial \Omega} \tomega(\vA) $$
 $$ \int_\Omega \frac{1}{\sqrt{g}} \parcial{x^\alpha}(A^\alpha \sqrt{g}) \tomega = \int_{\partial \Omega} A^\alpha n_\alpha \ts $$

\clearpage \noindent
  $$ \nabla_{\vu} \vv = \lim_{\epsilon \to 0} \frac{\vv^*_{\lambda_0 +\epsilon}(\lambda_0)- \vv(\lambda_0)}{\epsilon} \quad,\quad \nabla_{\e{j}} \e{i} =
  \matriz{\Gamma}{k}{ij}\e{k} $$
 \begin{enumerate}
  \item $\nabla_{\vu}(f \vv) = (\nabla_{\vu} f) \vv + f (\nabla_{\vu} \vv)$
  \item $\nabla_{\vu}(f) = \vu(f)$
  \item $\nabla_{\vu}(\vA \cdot \vB) = (\nabla_{\vu} \vA) \cdot \vB + \vA \cdot (\nabla_{\vu} \vB) $
  \item $\nabla_{\vu}(\tsigma(\vA)) = (\nabla_{\vu} \tsigma) (\vA) + \tsigma (\nabla_{\vu} \vA)$
  \item $\nabla_{f\vu+g\vv} \vA = f \nabla_{\vu} \vA + g \nabla_{\vv} \vA $
 \end{enumerate}
 $$ \nabla_{\vu} \vv = u^i\Big(\e{i}(v^k) + v^j \matriz{\Gamma}{k}{ji} \Big) \e{k} = u^i (\nabla_i v^k) \e{k} = u^i v^k_{;i}\e{k}$$
 $$ \nabla_{\e{k}} \tomega^j = - \matriz{\Gamma}{j}{lk} \tomega^l \qquad , \qquad \nabla_i \sigma_j =
  \partial_i \sigma_j - \matriz{\Gamma}{k}{ij} \sigma_k $$
 \begin{eqnarray*}
  \nabla_m \matriz{T}{i\ldots j}{k \ldots l} &=& \partial_m \matriz{T}{i\ldots j}{k \ldots l} + \matriz{T}{a\ldots j}{k \ldots l} \matriz{\Gamma}{i}{am}
  + \cdots + \matriz{T}{i\ldots a}{k \ldots l} \matriz{\Gamma}{j}{am} \\
  &&- \matriz{T}{i\ldots j}{a \ldots l} \matriz{\Gamma}{a}{km} - \ldots -  \matriz{T}{i\ldots j}{k \ldots a} \matriz{\Gamma}{a}{lm}
 \end{eqnarray*}
 $$ \matriz{T}{k}{ij} := \nabla_{\e{j}}\e{i} - \nabla_{\e{i}}\e{j} - [\e{i},\e{j}] = 0 \Leftrightarrow \matriz\Gamma k{ij} =\matriz\Gamma k{ji}$$
 $$\left(\lie{\vu} \vv \right)^i = u^j \partial_j v^i - v^j \partial_j u^i = u^j \nabla_j v^i - v^j \nabla_j u^i \qquad $$
 $$\left(\lie{\vu}\tsigma \right)_i = u^j \partial_j \sigma_i + \sigma_k \partial_i u^k = u^j \nabla_j \sigma_i + \sigma_k \nabla_i u^k $$

\sep
 $$ \textrm{Geodésicas}\ \nabla_{\vu}(\vu) = 0 \quad , \quad \nabla_{\vu}(\vu) = f \vu $$
 $$ \parcial[^2 x^k]{\lambda^2} + \matriz{\Gamma}{k}{ji} \parcial[x^i]{\lambda} \parcial[x^j]{\lambda} = 0 $$
 $$ \textrm{trans. paral.}\ \nabla_{\vu} \vv = 0 \to u^i \left( \parcial[v^k]{x^i} + v^j \matriz\Gamma k {ij} \right) = 0 $$

\sep
 $$ \textrm{Curvatura}\ \RR(\vu, \vv) = [ \nabla_{\vu} , \nabla_{\vv} ] - \nabla_{[\vu, \vv]} $$
 $$ \matriz{R}{m}{kij} = \partial_i \matriz{\Gamma}{m}{kj} - \partial_j \matriz{\Gamma}{m}{ki} + \matriz{\Gamma}{l}{kj}
  \matriz{\Gamma}{m}{li} - \matriz{\Gamma}{l}{ki} \matriz{\Gamma}{m}{lj} - \matriz{C}{l}{ij} \matriz{\Gamma}{m}{kl}$$
 $$ \matriz{\Gamma}{i}{jk} = \frac{1}{2} g^{il} ( g_{lj,k} + g_{lk,j} - g_{jk,l})$$
 \begin{itemize}
  \item $\nabla_i \nabla_j V^l - \nabla_j \nabla_i V^l = \matriz{R}{l}{kij} V^k $
  \item $\matriz{R}{m}{kij} = -\matriz{R}{m}{kji} \Leftrightarrow \matriz{R}{m}{k(ij)} = 0 $
  \item $\matriz{R}{m}{[kij]} = 0 $
  \item $\# = n^2 (n^2-1)/3$
  \item Bianchi $\nabla_{[m} \matriz{R}{l}{|k|ij]} = \matriz{R}{l}{k[ij;m]} = 0 $
  \item $ R_{ijkl} = - R_{ijlk} $
  \item $R_{ijkl} = R_{klij} $
  \item $R_{[ijkl]} = 0$
  \item $\# = n^2(n^2-1)/12$
 \end{itemize}
 $$ \commut u v = 0 \Leftrightarrow u^j u^k \nabla_j \nabla_k v^l =\matriz{R}{i}{jkl} u^j u^k v^l $$
 $$ \nabla_i v^i = \dd \tomega(\vv) \Rightarrow \matriz\Gamma i{ji} = \partial_j \ln \sqrt{g} $$
 $$ S = \int g_{ij} \dot{x}^i \dot{x}^j \dd\lambda \quad,\quad \delta S=0 \Leftrightarrow \ddot x^k + \matriz\Gamma k{ij}\dot x^i \dot x^j = 0 $$

\hrule\sep
 $T^{\alpha\beta} = $ flux $p^\alpha$ per $x^\beta = \cte$ \\
Polvo $n = N/\Delta V$, $n' = \gamma n$
 $$ \vN = n \vu = (\gamma nc, \gamma n v ) \quad, \quad T^{00} = \rho = n m $$
 $$ \mathbb{T} = \vN \otimes \vp \quad, \quad T^{\mu\nu} = n m u^\mu u^\nu $$
Fluido ideal
 $$ T^{\mu\nu} = (\rho + p) u^\mu u^\nu + p g^{\mu\nu} $$
 $$ a^i = u^\beta \nabla_\beta u^i = - \frac{\partial^i p}{\rho + p} $$
n partículas
 $$ T^{\alpha\beta} = \sum_N \int\dd\tau p_N^\alpha \deriv[x^i_N]{\tau} \delta^4\big( \vx - \vx_N(t) \big) $$
 $$ p = \frac13 \sum \frac{\vp_N^2}{E_N} \delta^3(\vx-\vx_N(t)) $$
 $$ \rho = \frac13 \sum {E_N} \delta^3(\vx-\vx_N(t)) $$
 $$ \textrm{(no rel)}\ \rho = n m + \frac32p \quad ,\quad \textrm{(molt rel)}\ \rho = 3p $$

\hrule\sep
 $$ G^{\alpha\beta} = R^{\alpha\beta} - \frac12 g^{\alpha\beta} R = \kappa T^{\alpha\beta} \quad , \quad \kappa = \frac{8\pi G}{c^4} $$
 Límite Newton $T_{i\alpha} \to 0$ , $\partial_0 \to 0 $
 $$ h_{00} = \frac{2\phi}{c^2} \quad,\quad R_{00} = - \frac12 \nabla^2 h_{00} $$

\clearpage\noindent
 Gauge $ x^{\alpha'} = x^\alpha + t \xi^\alpha(x)$ , $t\ll1$ \\
 $$ \hat{h}_{\alpha\beta} = h_{\alpha\beta} - \big( \partial_\beta \xi_\alpha + \partial_\alpha \xi_\beta \big) = h_{\alpha\beta} - \lie{\vxi}
  (\eta_{\alpha\beta}) $$
 $$ R_{\alpha\beta\mu\nu} = \frac{1}{2} \Big( \partial_\beta \partial_\mu h_{\alpha\nu} + \partial_\alpha \partial_\nu h_{\beta\mu} - \partial_\beta
  \partial_\nu h_{\alpha\mu} - \partial_\alpha \partial_\mu h_{\beta\nu} \Big) $$
 $$ R_{\mu\nu}= \frac{1}{2} \Big( \partial_\mu \partial_\alpha \matriz{h}{\alpha}{\nu} + \partial_\alpha \partial_\nu \matriz{h}{\alpha}{\mu} -
  \partial_\mu \partial_\nu \matriz{h}{\alpha}{\alpha} - \partial_\alpha \partial^\alpha h_{\mu\nu} \Big) $$
 $$ R = \partial_\beta \partial_\alpha h^{\alpha\beta} - \partial_\beta \partial^\beta \matriz{h}{\alpha}{\alpha} $$
 $$ \textrm{pot}\ \bar h_{\mu\nu} = h_{\mu\nu} - \frac{1}{2} \eta_{\mu\nu} \matriz h \alpha \alpha $$
 $$ \textrm{g.Lorentz}\ \partial^\mu \bar h_{\mu\nu} = 0 \quad,\quad \Box^2 \xi_\nu = \partial^\mu h_{\mu\nu} $$
 $$ \hat{\bar h}_{\mu\nu} = \bar h_{\mu\nu} - \big( \partial_\mu \xi_\nu + \partial_\nu \xi_\mu \big) + \eta_{\mu\nu} \partial_\alpha \xi^\alpha $$
 $$ \Box^2 \bar h_{\mu\nu} = - 2\kappa T_{\mu\nu} $$
 $$ l.N\ \bar h_{00} = - 4 \phi \ ,\ \ds^2 = - (1+2\phi) \dt^2 + (1-2\phi) ( \dd x^2 + \dd y^2 + \dd z^2 ) $$
 $$ \textrm{Landau-Lifshitz}\ \partial_\mu \Big[ (-g) \big(T^{\mu\nu} + t^{\mu\nu} \big) \Big] = 0 $$
 $$ t^{\mu\nu} = \frac1{2 \kappa (-g)} \partial_\alpha \partial_\beta \left[ (-g) \left( g^{\mu\nu} g^{\alpha\beta} - g^{\mu\alpha} g^{\nu\beta} \right)
  \right] - \frac1\kappa G^{\mu\nu} $$
 $$ (-g) \big( G^{\mu\nu} + \kappa t^{\mu\nu} \big) = \partial_\alpha u^{\mu\nu\alpha} $$
 $$ \partial_\beta u^{\mu\nu\alpha\beta} = u^{\mu\nu\alpha} \longrightarrow (-g) \big( G^{\mu\nu} + \kappa t^{\mu\nu} \big) = \partial_\alpha
  \partial_\beta u^{\mu\nu\alpha\beta} $$
 $$ P^\mu := \int_\Sigma\!\!\! \dd^3 x\ \frac{\partial_\alpha u^{\mu0\alpha}}\kappa = \int_{\partial\Sigma}\!\!\!\! \dd^2 S_i\
  \partial_\beta \frac{ (-g) \left( g^{\mu0} g^{i\beta} - g^{\mu i} g^{0\beta} \right) }{2\kappa} $$

\sep
 Gauge TT $ h_{\alpha 0} = 0 $ , $ \matriz h \alpha \alpha = 0 $ , $ \partial^i h_{ik} = 0 $ \\
 $$ \bar h_{\mu\nu} = \mathrm{Re} \left( A_{\mu\nu}\ \expon{i k_\alpha x^\alpha} \right) \quad,\quad k_\alpha k^\alpha = k^\alpha A_{\mu\alpha} =
   \matriz A \mu \mu = 0 $$
 $$ R_{j0k0} = R_{0j0k} = - \frac 1 2 \partial_0 \partial_0 h^{TT}_{jk} $$
 $$ x_A^{\hat \imath} = 0 \quad,\quad x_B^{\hat \imath}=x_B^j \left(\matriz\delta {\hat\imath} j + \frac 1 2 \matriz h {\hat \imath} j (t, x^i=0)\right)$$
 $$ {e}_+ =  \begin{pmatrix}
     1 & 0 \\
     0 & -1 \\
   \end{pmatrix} \quad , \quad
   {e}_X =  \begin{pmatrix}
     0 & 1 \\
     1 & 0 \\
   \end{pmatrix}  \quad,\quad k^\alpha = \omega(1,0,0,1) $$
  $$ h^{TT}_{ij} = \mathrm{Re} \left\{ A_+ e_+ \expon{-i\omega(t-z)} + A_X e_X \expon{-i\omega(t-z)} \right\} $$
  $$ \matriz h{TT}{ij} = P_{il} h_{lm} P_{mj} - \frac12 P_{ij} P_{ij} ( P_{lm} h_{lm} ) \quad,\quad P_{lm} = \delta_{lm} - k_l k_m $$

\sep
 $ \la h\ h_{\mu\nu|\alpha\beta} \ra = \la h\ h_{\alpha\beta|\mu\nu} \ra $ ,
 $ \la \left( h_{|\alpha} \ h_{\mu\nu} \right)_{|\beta} \ra = 0 $ \\
 $ \la h\ h_{\mu\nu|\alpha\beta} \ra =-\la h_{|\beta}\ h_{\mu\nu|\alpha} \ra $
 $$ \matriz T {OG} {\mu\nu} = \frac 1 {4\kappa} \la h_{\alpha\beta|\mu} {h^{\alpha\beta}}_{|\nu} \ra \quad , \qquad {\matriz T \mu \mu}^{OG} = 0 $$
 $$ R_{\mu\nu} = \mpar R F \mu \nu + \underbrace{\mpar R 1 \mu\nu (\pare h1)}_{o(A/\lambda^2)} + \underbrace{\mpar R 1 \mu\nu (\pare h2) +
    \mpar R 1 \mu\nu (\pare h2)}_{o(A^2 / \lambda^2)} + o(A^3/\lambda^2) $$
 \begin{eqnarray*}
  \mpar R 1 \mu \nu (h) &=& \frac12 \left( {h_{\alpha\mu|\nu}}^\alpha + {h_{\alpha\nu|\mu}}^\alpha - \matriz h \alpha {\alpha|\mu\nu} -
  {h_{\mu\nu|\alpha}}^\alpha \right) \\
  \mpar R 2 \mu \nu (h) &=& \frac12 \left\{ \frac 12 h_{\alpha\beta|\mu} {h^{\alpha\beta}}_{|\nu} + h^{\alpha\beta} \left( h_{\alpha\beta|\mu\nu} +
  h_{\mu\nu|\alpha\beta} \right. \right. \\
  & & \left. \left. - h_{\alpha\mu|\nu\beta} - h_{\alpha\nu|\mu\beta} \right) +{h_\nu}^{\alpha|\beta} \left( h_{\alpha\mu|\beta} - h_{\beta \mu|\alpha}
  \right)\right. \\
  & & \left.  - \left( {h^{\alpha\beta}}_{|\beta} - \frac12
  h^{|\alpha} \right) \left( h_{\alpha\mu|\nu} + h_{\alpha\nu|\mu}-h_{\mu\nu|\alpha} \right) \right\}
 \end{eqnarray*}

\sep
 $$ \tilde{\bar h}_{ij} = - \frac{k}{4\pi} \omega^2 \frac{\expon{i\omega r}}{r} \tilde D_{ij} \quad \textrm{on} \quad \tilde D_{ij} = \int
 \tilde T^{00} x^{i'} x^{j'} \dd^3 x' $$
 $$ \bar h_{ij} = \frac{k}{4\pi r} \deriv[^2]{t^2} D_{ij}(t-r) \quad \textrm{on} \quad D_{ij} = \int T^{00} x^{i'} x^{j'} \dd^3 x'$$
 $$ \matriz{\bar h}{TT}{ij} = \frac{c^4}{4\pi r}\deriv[^2]{t^2} \matriz I {TT}{ij} = \frac{2G}{r}\deriv[^2]{t^2} \matriz I {TT}{ij} $$
 $$ \deriv[P]{\Omega} = r^2 \cdot n_i T^{0i} = - \frac{n^i}{4\pi} r^2 \la \matriz h{TT}{jk,0} \matriz h{TT\ jk}{,i} \ra$$
 $$ I_{ij} = D_{ij} - \frac13 \delta_{ij} \matriz Dkk = \int \rho \left( x^{i'} x^{j'} - \frac13 \delta^{ij} {r'}^2 \right) \dd^3x' $$
 $$ \deriv[P]{\Omega} = \frac \kappa {64 \pi^2 c} \la \dddI_{jk} \dddI_{jk} \ra =\frac G {8 \pi c^5} \la
    \dddI_{jk} \dddI_{jk} \ra $$
 $$ P = \deriv[E]{t} = \frac \kappa {40 \pi c} \la \dddI_{jk} \dddI_{jk} \ra =\frac G {5 c^5} \la
    \dddI_{jk} \dddI_{jk} \ra $$

\clearpage\noindent
 $$ \ds^2 = - \expon{2\phi(t,r)} \dt^2 + \expon{2\Lambda(t,r)} \dr^2 + ( \dd \theta^2 +\sin^2\theta \dd \varphi^2 ) $$
 $$\textrm{Schwarchild}\ \ds^2 = - \left( 1 - \frac{2M}{r} \right) \dt^2 + \frac{\dr^2}{1 - \frac{2M}{r}} + r^2 \dd^2 \Omega $$
 $$ \begin{array}{cclccclcccl}
   \chist 111&=&\Lambda'                    &,\ &\chist133&=&-r\sin^2\theta\expon{-2\Lambda}&,\ &\chist323&=&\cotg\theta                         \\
   \chist 110&=&\dot \Lambda                &,\ &\chist212&=&1/r                            &,\ &\chist011&=&\dot\Lambda \expon{2(\Lambda-\phi)} \\
   \chist 122&=&-r \expon{-2\Lambda}        &,\ &\chist233&=&-\sin\theta \cos\theta         &,\ &\chist010&=&\phi'                               \\
   \chist 100&=&\expon{2(\phi-\Lambda)}\phi'&,\ &\chist313&=&1/r                            &,\ &\chist000&=&\dot \phi                           \\
 \end{array} $$
 \begin{eqnarray*}
   R_{00} & = & \expon{2(\phi-\Lambda)} \left( \phi'' + {\phi'}^2 - \phi'\Lambda' + \frac{2\phi'}{r} \right) - \ddot\Lambda - \dot\Lambda^2 +
                \dot\Lambda \dot\phi \\
   R_{11} & = & -\phi'' - {\phi'}^2 + \phi'\Lambda' + \frac{2\Lambda'}{r} + \expon{2(\Lambda-\phi)}\left(\ddot\Lambda + \dot\Lambda^2 -
                \dot\Lambda \dot \phi \right) \\
   R_{22} & = & - \expon{-2\Lambda} \left[ 1 + r(\phi'-\Lambda') \right] + 1 \\
   R_{33} & = & - \sin^2 \theta \left\{ \expon{-2\Lambda} \left[ 1 + r(\phi'-\Lambda') \right] - 1 \right\}  \\
   R_{10} & = & \frac{2 \dot \Lambda}{r}
 \end{eqnarray*}

\sep
 $$ \expon{2\Lambda} = \frac1{1 - 2 \frac{m(r)}{r}} \quad , \quad m(r) = \frac \kappa 2 \int_0^r\!\!\!\! \dr\, r^2 \rho(r) $$
 $$ \deriv[\phi]{r} = \frac{m(r) + \frac{\kappa}{2} p r^3}{r(r-2m(r))} $$
 $$ \textrm{Oppenheimer-Volkov}\ \deriv[p]{r} = - \frac{(p+\rho) \left( m(r) + 4\pi r^3 \rho \right)}{r(r-2m(r))} $$
 Clàssica $ \deriv[p]{r} = - \rho \frac{m(r)}{r} $
 $$ M_\textrm{mat} = \int_0^{r_0} \left.\sqrt{-g}\right|_{t=\cte} \rho \dd^3 x = 4\pi \int_0^{r_0} \expon{\Lambda} \rho r^2 \sin\theta \dr $$
 $$ p(r_0) = 0 \quad , \quad r_0 > \frac94M \Rightarrow p(o) = \infty $$

\sep $m\ne 0 \rightarrow p_0 = - \hat E m $ , $p_\varphi = \hat L m $
 $$ \hat E = \left( 1 - \frac{2M}{r} \right) \deriv[t]{\tau} $$
 $$ \left( \deriv[r]{\tau} \right)^2 = {\hat E}^2 - \left( 1 - \frac{2M}{r} \right) \left( 1+\frac{{\hat L}^2}{r^2} \right) = {\hat E}^2 - {\hat
 V}^2(r)$$
 $$ r_0 = \frac{{\hat L}^2}{2M} \left( 1 \pm  \sqrt{1 - \frac{12M^2}{{\hat L}^2} } \right) $$
 $m=0 \rightarrow p_0 = -E$ , $ p_\varphi = L$ , $r_0 = 3 M$
 $$ \left( \deriv[r]{\lambda} \right)^2 = E^2 - \left( 1 - \frac{2M}{r} \right) \frac{L^2}{r^2} = E^2 - V^2(r) $$

\sep Precesión perihelio $u=1/r = y + M / \hat L^2$
 $$ \left( \deriv[r]{\varphi}\right)^2 = \frac{{\hat E}^2 - \left( 1-2M/r \right) \left( 1+{\hat L}^2/r^2 \right)}{{\hat L}^2 / r^4} $$
 $$ \left( \deriv[u]{\varphi} \right)^2 = \frac{{\hat E}^2}{{\hat L}^2} - \left( 1-2M u \right) \left( \frac1{{\hat L}^2} +u^2 \right) $$
 \begin{eqnarray*} \left( \deriv[y]{\varphi} \right)^2 &=& \frac{\hat E^2 + (M/\hat L)^2 + 2(M/\hat L)^4 -1}{\hat L^2} \\ &&+ \frac{6M^3}{\hat L^2}y +
  \left(\frac{6 M^2}{\hat L^2}-1\right) y^2 + (2 M y^3)
 \end{eqnarray*}
 $$ \left( \deriv[y]{\varphi} \right)^2 = A + B y - k^2 y^2 \ , \quad  k^2 = 1-\frac{6 M^2}{\hat L^2} $$
 $$ y = C \cos(k \varphi + \varphi_0) \quad , \quad $$
 $$ \Delta \varphi = \phi-2\pi = 2\pi \left( \frac1{1-\frac{6 M^2}{\hat L^2}} - 1 \right) \approx \frac{6\pi M^2}{\hat L^2} = \frac{6\pi M}{r} $$

\sep Deflexión lluz
 $$ \left( \deriv[u]{\varphi} \right)^2 + \left( 1 - {2M}{u} \right) u^2 = \frac{E^2}{L^2} = \frac1{b^2} $$
 $$ M=0 \Longleftrightarrow u_0(\varphi) = \frac1b \sin\varphi $$
 $$ \deriv[^2 u]{\varphi^2} + u = 3Mu^2 \approx \frac{3M}{b^2} \sin^2\varphi $$
 $$ u(\varphi) = \frac1b \sin\varphi + \frac{M}{b^2} (1+\cos\varphi)^2 $$
 $$ u(\varphi) = \frac1b \Delta\varphi + \frac{4M}{b^2} = 0 \longrightarrow  \Delta\varphi = -\frac{4M}{b}$$
\end{document}
