Gravitational waves and matter - Revision history

Igor at 13:46, 15 January 2013

2013-01-15T13:46:39Z

Cosmo All at 03:08, 8 January 2013

2013-01-08T03:08:02Z

Cosmo All: /* Problem 5: Metric far from a non-relativistic source */

2013-01-08T02:29:05Z

‎Problem 5: Metric far from a non-relativistic source

Cosmo All: /* Problem 1: Particle motion in TT gauge */

2013-01-08T02:22:34Z

‎Problem 1: Particle motion in TT gauge

Cosmo All: /* Problem 1: Particle motion in TT gauge */

2013-01-08T02:18:09Z

‎Problem 1: Particle motion in TT gauge

Cosmo All: /* Problem 1: Particle motion in TT gauge */

2013-01-08T02:15:56Z

‎Problem 1: Particle motion in TT gauge

Cosmo All at 02:15, 8 January 2013

2013-01-08T02:15:15Z

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Cosmo All: Created page with "6"

2012-12-26T12:06:40Z

Created page with "6"

New page

[[Category:Weak field limit and gravitational waves|6]]

@@ Line 32: / Line 32: @@
 	\sim h_{\alpha\gamma} v.
 \end{align*}
-For particles at rest $v=0$ and therefore $u^\mu =const$. Thus the TT frame (the frame in which the TT gauge conditions hold) appears to be comoving with the test particles.
+For particles at rest $v=0$ and therefore $u^\mu =const$. Thus the TT frame (the frame in which the TT gauge conditions hold) appears to be comoving with the test particles.</p>
-) Now consider the congruence of particles, which are at the initial moment at rest in the TT frame. The Raychaudhuri equation for the geodesic deviation $s^\mu$ is
+<p style="text-align: left;">2) Now consider the congruence of particles, which are at the initial moment at rest in the TT frame. The Raychaudhuri equation for the geodesic deviation $s^\mu$ is
 \[\frac{D^2 s^\mu}{D\tau^2}=
 	{R^{\mu}}_{\nu\rho\sigma}u^{\nu}u^{\rho}s^{\sigma}

← Older revision		Revision as of 03:08, 8 January 2013
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	[[Category:Weak field limit and gravitational waves\|6]]		[[Category:Weak field limit and gravitational waves\|6]]
		+
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		+	__TOC__
		+

@@ Line 166: / Line 166: @@
 In the Lorenz gauge we only need the spatial components of the metric perturbation, as the rest are related to them via the gauge conditions (raising and lowering of spacial indices changes the sign)
-\[\pa_\mu \bar{h}^{\mu\nu}=0
+\[\partial_\mu \bar{h}^{\mu\nu}=0
 \quad\Leftrightarrow\quad
-\pa_0 \bar{h}_{0\mu}
+\partial_0 \bar{h}_{0\mu}
-	=\pa_{\alpha}\bar{h}_{\alpha\mu},\]
+	=\partial_{\alpha}\bar{h}_{\alpha\mu},\]
 or in Fourier space
 \[h_{0\mu}
-	=\frac{1}{i\omega}\pa_{\alpha}h_{\alpha\mu}.\]
+	=\frac{1}{i\omega}\partial_{\alpha}h_{\alpha\mu}.\]
 Very similar to this, the spatial components of the energy-momentum tensor can be expressed through the $T^{00}$ components, thanks to the energy conservation law, which in the first order takes the familiar form with partial (instead of covariant) derivatives
-\[\pa_{\mu}T^{\mu\nu}=0,
+\[\partial_{\mu}T^{\mu\nu}=0,
 	\quad\Leftrightarrow\quad
-	\pa_{\alpha}T_{\alpha\mu}=\pa_0 T_{0\mu}.\]
+	\partial_{\alpha}T_{\alpha\mu}=\partial_0 T_{0\mu}.\]
 Then for an isolated source, for which the surface integral vanishes, integration by parts twice gives
 \begin{align*}
 &=\int d^3 y\;
-	\pa_\gamma (y_{\alpha} T_{\beta\gamma})
+	\partial_\gamma (y_{\alpha} T_{\beta\gamma})
 	=\int d^3 y\; T_{\alpha\beta}
-	+\int d^3 y\; y_\alpha \pa_\gamma T_{\beta\gamma}
+	+\int d^3 y\; y_\alpha \partial_\gamma T_{\beta\gamma}
 	=\int d^3 y\; T_{\alpha\beta}
-	+\int d^3 y\; y_\alpha \pa_0 T_{0\beta},\\
+	+\int d^3 y\; y_\alpha \partial_0 T_{0\beta},\\
 &\Rightarrow\quad
 	\int d^3 y\; T_{\alpha\beta}=
-	-\int d^3 y\; y_\alpha \pa_0 T_{0\beta}
+	-\int d^3 y\; y_\alpha \partial_0 T_{0\beta}
 	=-i\omega \int d^3 y\; y_{\alpha}T_{0\beta};\\
 &=\int d^3 y\;
-	\pa_\gamma (y_{\alpha}y_{\beta} T_{0\gamma})
+	\partial_\gamma (y_{\alpha}y_{\beta} T_{0\gamma})
 	=\int d^3 y\; (y_{\alpha}T_{0\beta}
 						+y_{\beta}T_{0\alpha})
 	+\int d^3 y\; y_\alpha y_{\beta}
-		\pa_\gamma T_{0\gamma}=;\\
+		\partial_\gamma T_{0\gamma}=;\\
 &=2\int d^y y_\alpha T_{0\beta}
-	+\int d^3 y\; y_\alpha y_\beta \pa_0 T_{00}
+	+\int d^3 y\; y_\alpha y_\beta \partial_0 T_{00}
 	=2\int d^y y_\alpha T_{0\beta}
 	+i\omega \int d^3 y\; y_\alpha y_\beta T_{00},\\

@@ Line 20: / Line 20: @@
 \frac{du^0}{d\tau}&\sim
 	\Gamma^{0}_{\alpha\beta}u^{\alpha}u^\beta
-	\sim \pa_0 h_{\alpha\beta} u^\alpha u^\beta
+	\sim \partial_0 h_{\alpha\beta} u^\alpha u^\beta
 	\sim h_{\alpha\beta} v^2,\\
 \frac{du^\gamma}{d\tau}&\sim
 	\Gamma^{\gamma}_{\alpha\mu}u^{\alpha}u^{\mu}
-	\sim \pa_0 h_{\alpha\gamma} u^\alpha
+	\sim \partial_0 h_{\alpha\gamma} u^\alpha
 	\sim h_{\alpha\gamma} v.
 \end{align*}
@@ Line 36: / Line 36: @@
 \[\frac{D^2 s^\alpha}{D\tau^2}
 	=R_{0\alpha 0\beta}s^{\beta}
-	=-\tfrac{1}{2}\pa_0^2 h_{\alpha\beta}s^\beta,\]
+	=-\tfrac{1}{2}\partial_0^2 h_{\alpha\beta}s^\beta,\]
 where we used the expression for the curvature tensor components through the metric in the TT gauge.
-On the right hand side we have (dot denotes $\pa_0$)
+On the right hand side we have (dot denotes $\partial_0$)
 \begin{align*}
 \frac{Ds^\alpha}{D\tau}&=u^\nu \nabla_\nu s^\alpha

@@ Line 43: / Line 43: @@
 \frac{Ds^\alpha}{D\tau}&=u^\nu \nabla_\nu s^\alpha
 	=\nabla_0 s^\alpha
-	=\pa_0 s^\alpha +{\Gamma^{\alpha}}_{0\mu}s^{\nu}
+	=\partial_0 s^\alpha +{\Gamma^{\alpha}}_{0\mu}s^{\nu}
 	=\dot{s}^\alpha-\Gamma_{\alpha,0\beta}s^\beta
 	=\dot{s}^\alpha
@@ Line 49: / Line 49: @@
 \frac{D^2 s^\alpha}{D\tau^2}&
 	=\nabla_0	\Big(\frac{D s^\alpha}{D\tau}\Big)
-	=\pa_0 \Big(\frac{D s^\alpha}{D\tau}\Big)
+	=\partial_0 \Big(\frac{D s^\alpha}{D\tau}\Big)
 	+{\Gamma^\alpha}_{0\beta}\frac{D s^\beta}{D\tau}
-	\approx\pa_0 (\dot{s}^\alpha
+	\approx\partial_0 (\dot{s}^\alpha
 		-\tfrac12 \dot{h}_{\alpha\beta}s^\beta)
 	-\tfrac12 \dot{h}_{\alpha\beta}\dot{s}^\beta\\

@@ Line 6: / Line 6: @@
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 1: Particle motion in TT gauge===
-Consider a plane wave with "$+$" polarization.
+Consider a plane wave with "$+$" polarization.<br/>
-) Write down the equations of motion for a non-relativistic particle; what happens to particles at rest?
+) Write down the equations of motion for a non-relativistic particle; what happens to particles at rest?<br/>
 ) Find the geodesic deviation from the Raychaudhuri equation and show that the result is the same;
 <div class="NavFrame collapsed">
@@ Line 67: / Line 67: @@
 <div id="gwm2"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 2: Proper distance variation===
 Find the variation of proper distance between two particles at rest in the TT frame in the presence of a plane gravitational wave with the "$+$" polarization. Show that a ring of test particles placed initially at rest in the plane orthogonal to the wave vector will be distorted into an ellipse with its axes directed along the polarization tensor's main axes and  oscillating with the wave's frequency.