Equations of General Relativity - Revision history

Igor: /* Problem 14: energy conservation */

2012-10-11T17:18:14Z

‎Problem 14: energy conservation

Igor: /* Problem 13: Einstein equations */

2012-10-11T17:16:16Z

‎Problem 13: Einstein equations

Igor: /* Problem 13: Einstein equations */

2012-10-11T17:13:46Z

‎Problem 13: Einstein equations

Igor: /* Problem 6: principle of least action */

2012-10-11T17:10:59Z

‎Problem 6: principle of least action

Igor: /* Problem 5: geodesics */

2012-10-11T17:08:32Z

‎Problem 5: geodesics

Igor at 17:07, 11 October 2012

2012-10-11T17:07:30Z

Cosmo All at 11:57, 1 August 2012

2012-08-01T11:57:57Z

Cosmo All at 11:57, 1 August 2012

2012-08-01T11:57:18Z

Igor at 20:19, 24 July 2012

2012-07-24T20:19:40Z

Igor at 20:15, 24 July 2012

2012-07-24T20:15:05Z

@@ Line 628: / Line 628: @@
 	\left(T_{\mu\nu}- \frac 1 2 g_{\mu\nu}T\right).\]
-On the other hand, acting with $\nabla^{\mu}$ on the equations (\ref{EinsteinEq}), one gets zero in the left hand side due to Bianchi equality  (see [[#equ_oto6a|this problem]]), and therefore the equation for $T_{\mu\nu}$:
+On the other hand, acting with $\nabla^{\mu}$ on the equations (\ref{EinsteinEq}), one gets zero in the left hand side due to [[#equ_oto6a|the Bianchi identity]], and therefore the equation for $T_{\mu\nu}$:
 \[\nabla^{\mu}T_{\mu\nu}=0.\]
 In the flat space-time with Cartesian coordinates we have $\nabla_{\mu}=\partial_{\mu}$, so the equation reduces to differential conservation laws for the energy and momentum of matter

@@ Line 583: / Line 583: @@
 	\delta g^{\mu\nu}
 	\left( R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}\right).\]
-Variation of the action for matter, by the definition of the energy momentum tensor ([[#equ_oto6b|see problem]]) is
+Variation of the action for matter, by the [[#equ_oto6b|definition of the energy momentum tensor]] is
 \[\delta S_m = \frac{1}{2c}\int d^{4}x\sqrt{-g}
 	\delta g^{\mu\nu} T_{\mu \nu }.\]

@@ Line 526: / Line 526: @@
 \[S=S_{g}+S_{m};\qquad\text{where}\quad
 S_{g}=-\frac{c^{3}}{16\pi G}\int d^{4}x\sqrt{-g}\;R,\]
-$R=R^{i}_{i}=g^{ik}R_{ik}$ is scalar curvature, $G$ is the gravitational constant. Starting from the variational principle, derive the Einstein-Hilbert equations\footnote{Also called Einstein field equations, or just Einstein equation (either in singular or plural); the action $S_{g}$ is referred to as Hilbert or Einstein-Hilbert action.} for the
+$R=R^{i}_{i}=g^{ik}R_{ik}$ is scalar curvature, $G$ is the gravitational constant. Starting from the variational principle, derive the Einstein-Hilbert equations$^*$ for the
 gravitational field
 \begin{equation}\label{EinsteinHilbetEq}
@@ Line 597: / Line 597: @@
 	 R_{\mu \nu } - \frac{1}{2}Rg_{\mu \nu }
 	= \frac{8\pi G}{c^4} T_{\mu \nu }.
-\end{equation}</p>
+\end{equation}
    </div>
 </div></div>
@@ Line 603: / Line 605: @@
 <div id="equ_oto8"></div><div style="border: 1px solid #AAA; padding:5px;">
 === Problem 14: energy conservation ===
 Show that the Einstein's equation can be presented in the following form

@@ Line 272: / Line 272: @@
 	[\partial_{\nu}g_{\mu\lambda}-
 \partial_{\mu}g_{\lambda\nu}].\]
-Multiplying by $u^{\mu}=dx^{\mu}/ds$ and symmetrizing the bracket over $\mu\lambda$, we eventually obtain the geodesic equation
+Multiplying by $u^{\mu}=dx^{\mu}/ds\;$ and symmetrizing the bracket over $\mu\lambda$, we eventually obtain the geodesic equation
 \[\frac{du^{\sigma}}{ds}+u^{\mu}u^{\lambda}
 	\Gamma^{\sigma}_{\mu\lambda}=0,\]
@@ Line 282: / Line 282: @@
 \[p_{\mu}=-\frac{\delta S}{\delta x^{\mu}}
 	=mcu_{\mu}.\]
-Then in case the spacetime has the symmetry with respect to translations along $x^{\mu}$, then the $\mu$-component of momentum is conserved. If it is invariant under time translations, then the conserved quantity is energy $p_{0}=mcu_{0}$.
+Then in case the spacetime has the symmetry with respect to translations along $x^{\mu}$, the $\mu$-component of momentum is conserved. If it is invariant under time translations, then the conserved quantity is energy $p_{0}=mcu_{0}$.
-Likewise for massless particles, e.g. photons, the $4$-momentum and wave vector are proportional to the $4$-velocity $u^{\mu}=dx^{\mu}/d\lambda$. However, in this case the normalization of $u^{\mu}$ cannot be fixed since $u^{\mu}u_{\mu}=0$, so it is arbitrary. Therefore the quantity $dt/d\lambda$ along a zero geodesics coincides with the photon's energy up to a constant factor.</p>
+Likewise for massless particles, e.g. photons, the $4$-momentum and wave vector are proportional to the $4$-velocity $u^{\mu}=dx^{\mu}/d\lambda$. However, in this case the normalization of $u^{\mu}$ cannot be fixed since $u^{\mu}u_{\mu}=0$, so it is arbitrary. Therefore the quantity $dt/d\lambda$ along a null geodesics coincides with the photon's energy up to a constant factor.</p>
    </div>
 </div></div>
@@ Line 290: / Line 290: @@
 <div id="equ_oto-kill1"></div><div style="border: 1px solid #AAA; padding:5px;">
 === Problem 7: symmetries and Killing vectors ===
 A Killing vector field, or just Killing vector, is a vector field $K^{\mu}(x)$, such that infinitesimal coordinate transformation $x\to x'+\varepsilon K $ (where $\varepsilon\to 0$) leaves the metric invariant in the sense*

@@ Line 194: / Line 194: @@
 === Problem 5: geodesics ===
-Free falling particles' worldlines in General Relativity are \textit{geodesics} of the spacetime--- i.e the curves $x^{\mu}(\lambda)$ with tangent vector $u^{\mu}=dx^{\mu}/d\lambda$, such that covariant derivative of the latter along the curve equals to zero:
+Free falling particles' worldlines in General Relativity are ''geodesics'' of the spacetime, i.e the curves $x^{\mu}(\lambda)$ with tangent vector $u^{\mu}=dx^{\mu}/d\lambda$, such that covariant derivative of the latter along the curve equals to zero:
 \[u^{\mu}\nabla_{\mu}u^{\nu}=0.\]
@@ Line 231: / Line 231: @@
 <div id="equ_oto5"></div><div style="border: 1px solid #AAA; padding:5px;">
 === Problem 6: principle of least action ===
 Consider the action for a massive particle of the form

@@ Line 186: / Line 186: @@
 		-\tilde{\Gamma}^{i'}_{k'l'}=	0.\]
-Thus for any given point $X$ we can always bring the metric in the point to the diagonal form and then carry out the above coordinate transformation, turning to zero the connection coefficients. This frame, in which the metric tensor differs from $\eta_{\mu\nu}=diag(1,-1,-1,-1)$ only in second derivatives, is called the locally inertial frame, or locally geodesic frame. Its existence is very useful for calculations, as will be seen in [[#equ_oto6a|problem]]</p>
+Thus for any given point $X$ we can always bring the metric in the point to the diagonal form and then carry out the above coordinate transformation, turning to zero the connection coefficients. This frame, in which the metric tensor differs from $\eta_{\mu\nu}=diag(1,-1,-1,-1)$ only in second derivatives, is called the locally inertial frame, or locally geodesic frame. Its existence is very useful for calculations, as will be seen in [[#equ_oto6a|the problem on Bianchi identity]]</p>
    </div>
 </div></div>
@@ Line 623: / Line 623: @@
 	\left(T_{\mu\nu}- \frac 1 2 g_{\mu\nu}T\right).\]
-On the other hand, acting with $\nabla^{\mu}$ on the equations (\ref{EinsteinEq}), one gets zero in the left hand side due to Bianchi equality  ([[#equ_oto6a|see problem]]), and therefore the equation for $T_{\mu\nu}$:
+On the other hand, acting with $\nabla^{\mu}$ on the equations (\ref{EinsteinEq}), one gets zero in the left hand side due to Bianchi equality  (see [[#equ_oto6a|this problem]]), and therefore the equation for $T_{\mu\nu}$:
 \[\nabla^{\mu}T_{\mu\nu}=0.\]
 In the flat space-time with Cartesian coordinates we have $\nabla_{\mu}=\partial_{\mu}$, so the equation reduces to differential conservation laws for the energy and momentum of matter

@@ Line 1: / Line 1: @@
 [[Category:Dynamics of the Expanding Universe|2]]
+__TOC__
 <p align="right">''I have thought seriously about this question,''<br/>
@@ Line 10: / Line 10: @@
 ''The Road to Reality''</p>
-__TOC__
 <div id="equ_oto1"></div><div style="border: 1px solid #AAA; padding:5px;">

← Older revision		Revision as of 20:19, 24 July 2012
Line 10:		Line 10:
	''The Road to Reality''</p>		''The Road to Reality''</p>

		+	__TOC__

	<div id="equ_oto1"></div><div style="border: 1px solid #AAA; padding:5px;">		<div id="equ_oto1"></div><div style="border: 1px solid #AAA; padding:5px;">