Motivation and symmetries / Introduction - Revision history

Igor at 12:52, 15 January 2013

2013-01-15T12:52:00Z

Cosmo All: /* Problem 10: Upper bound on frequency */

2012-12-26T12:46:02Z

‎Problem 10: Upper bound on frequency

Cosmo All: /* Problem 7: Natural luminosity from dimensional analysis */

2012-12-26T12:45:35Z

‎Problem 7: Natural luminosity from dimensional analysis

Cosmo All: /* Problem 7: Natural luminosity from dimensional analysis */

2012-12-26T12:45:12Z

‎Problem 7: Natural luminosity from dimensional analysis

Cosmo All: /* Problem 4: No dipole gravitational radiation */

2012-12-26T12:44:25Z

‎Problem 4: No dipole gravitational radiation

Cosmo All: /* Problem 4: No dipole gravitational radiation */

2012-12-26T12:43:47Z

‎Problem 4: No dipole gravitational radiation

Cosmo All: /* Problem 3: No monopole radiation */

2012-12-26T12:43:18Z

‎Problem 3: No monopole radiation

Cosmo All: /* Problem 3: No monopole radiation */

2012-12-26T12:42:59Z

‎Problem 3: No monopole radiation

Cosmo All at 12:42, 26 December 2012

2012-12-26T12:42:33Z

Cosmo All at 12:42, 26 December 2012

2012-12-26T12:42:16Z

Show changes

@@ Line 71: / Line 71: @@
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <p style="text-align: left;">Let us consider isolated systems of charges or masses that generate electromagnetic or gravitational waves correspondingly. The momentum of an isolated system is conserved, but existence of positive and negative charges (or just bodies with different rations $q/m$) allows one to create alternating dipole moment $\sum q_{i}\mathbf{r}_{i}$ while conserving the momentum $\sum m_{i}\mathbf{r}_{i}=const$. For gravity, however, the charge ''is'' the mass, which is the principle of equivalence lying at the heart of General Relativity, so that is impossible. This is the reason that in electromagnetic radiation the dominating term is usually the first one allowed, i.e. the dipole one, while for gravitational radiation the first possibly non-zero radiation term is the quadrupole one.
+     <p style="text-align: left;">Let us consider isolated systems of charges or masses that generate electromagnetic or gravitational waves correspondingly. The momentum of an isolated system is conserved, but existence of positive and negative charges (or just bodies with different rations $q/m$) allows one to create alternating dipole moment $\sum q_{i}\mathbf{r}_{i}$ while conserving the momentum $\sum m_{i}\mathbf{r}_{i}=const$. For gravity, however, the charge ''is'' the mass, which is the principle of equivalence lying at the heart of General Relativity, so that is impossible. This is the reason that in electromagnetic radiation the dominating term is usually the first one allowed, i.e. the dipole one, while for gravitational radiation the first possibly non-zero radiation term is the quadrupole one.</p>
-<br/>
 <gallery widths=400px heights=200px>
 File:Dipole.png|
 </gallery>
 Einstein writes of this in a letter to K.Shwarzshild (Feb 19, 1916):
 "Thus there are no gravitational waves analogous to light waves.  This probably is also related to the one-sidedness of the sign of scalar $T$, incidentally. (Nonexistence of the "dipole".)"
 <br/>
@@ Line 94: / Line 95: @@
 \[\frac{dE}{dt}\sim
 	\frac{G}{c^5}\cdot
-	\dddot{Q}_{\alpha\beta}
+	{\dddot{Q}}_{\alpha\beta}
-	\dddot{Q}^{\alpha\beta},\]
+	{\dddot{Q}}^{\alpha\beta},\]
 where
 \[Q_{\alpha\beta}=\int d^3 x \;
@@ Line 106: / Line 107: @@
      <p style="text-align: left;">As seen above, from symmetry considerations the gravitational wave amplitude far from the source should be expressed in terms of time derivatives of the second mass (quadrupole) moment $h^{\mu\nu}\sim \frac{d^n}{dt^n}I^{\mu\nu}$, and we expect the energy flux to be quadratic in the wave amplitude. The luminosity $L_{GW}$  (total radiated power) in gravitational radiation should, therefore, also be quadratic in $I^{\mu\nu}$  and its time derivatives.
 <br/>
-In units $G=c=1$ luminosity  $L_{GW}$ is dimensionless. The third  time derivative of $I^{\mu\nu}$ is also dimensionless; therefore, $L_{GW}$ must be a quadratic combination of $\dddot{I}^{\mu\nu}$. $L_{GW}$  also behaves as a scalar under rotations in space and so must be a quadratic scalar combination of $\dddot{I}_{\mu\nu}\dddot{I}^{\mu\nu}$ or $(\dddot{I}^\mu_\mu)^2$ . The precise combination is fixed by the demand that there is no radiation from a spherically symmetric system. In this case $I^{\mu\nu}\sim \delta^{\mu\nu}$ and the combination (reduced quadrupole momentum tensor)
+In units $G=c=1$ luminosity  $L_{GW}$ is dimensionless. The third  time derivative of $I^{\mu\nu}$ is also dimensionless; therefore, $L_{GW}$ must be a quadratic combination of ${\dddot{I}}^{\mu\nu}$. $L_{GW}$  also behaves as a scalar under rotations in space and so must be a quadratic scalar combination of ${\dddot{I}}_{\mu\nu}{\dddot{I}}^{\mu\nu}$ or ${(\dddot{I}}^\mu_\mu)^2$ . The precise combination is fixed by the demand that there is no radiation from a spherically symmetric system. In this case $I^{\mu\nu}\sim \delta^{\mu\nu}$ and the combination (reduced quadrupole momentum tensor)
 \[Q^{\mu\nu}\equiv I^{\mu\nu}
 	-\frac{1}{3}\delta^{\mu\nu}I_{\lambda}^{\lambda}\]
 vanishes. Consequently,
 \[L_{GW}\sim
-	 \dddot{Q}^{\mu\nu}\dddot{Q}_{\mu\nu}.\]
+	 {\dddot{Q}}^{\mu\nu}{\dddot{Q}}_{\mu\nu}.\]
 The principle of equivalence states that locally gravitational field can always be turned to zero by an appropriate choice of reference frame. Therefore local energy density of a gravitational field will always depend on the coordinate frame (and gauge) and cannot be well-defined: it is not a tensor quantity. So in order to obtain some meaningful notion of $L_{GW}$, one has to average  the obtained expression over spacetime volumes of the scales of the wavelength (which should be much less than curvature radius).</p>

@@ Line 212: / Line 212: @@
 	\frac{1}{4\sqrt{2}\pi }\frac{c^{3}}{GM}
 	\simeq 10^{4}\Big(\frac{M_{\odot}}{M}\Big)Hz.\]
-[\href{http://arxiv.org/abs/astro-ph/0210481 arXiv:astro-ph/0210481]</p>
+[http://arxiv.org/abs/astro-ph/0210481 arXiv:astro-ph/0210481]</p>
    </div>
 </div></div>

@@ Line 150: / Line 150: @@
 === Problem 7: Natural luminosity from dimensional analysis===
 Construct the limiting luminosity from dimensional analysis
-<--! <div class="NavFrame collapsed">
+<!-- <div class="NavFrame collapsed">
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">

@@ Line 155: / Line 155: @@
      <p style="text-align: left;"></p>
    </div>
-</div>--></div>
+</div> --> </div>
@@ Line 162: / Line 162: @@
 <div id="gw8"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 8: Quantum luminosity limit===
 Why the upper limit on luminosity, obtained in General Relativity, should not change in the future quantum theory of gravitation?

@@ Line 72: / Line 72: @@
    <div style="width:100%;" class="NavContent">
      <p style="text-align: left;">Let us consider isolated systems of charges or masses that generate electromagnetic or gravitational waves correspondingly. The momentum of an isolated system is conserved, but existence of positive and negative charges (or just bodies with different rations $q/m$) allows one to create alternating dipole moment $\sum q_{i}\mathbf{r}_{i}$ while conserving the momentum $\sum m_{i}\mathbf{r}_{i}=const$. For gravity, however, the charge ''is'' the mass, which is the principle of equivalence lying at the heart of General Relativity, so that is impossible. This is the reason that in electromagnetic radiation the dominating term is usually the first one allowed, i.e. the dipole one, while for gravitational radiation the first possibly non-zero radiation term is the quadrupole one.
-</br>
+<br/>
 <gallery widths=400px heights=200px>
 File:Dipole.png|

@@ Line 73: / Line 73: @@
      <p style="text-align: left;">Let us consider isolated systems of charges or masses that generate electromagnetic or gravitational waves correspondingly. The momentum of an isolated system is conserved, but existence of positive and negative charges (or just bodies with different rations $q/m$) allows one to create alternating dipole moment $\sum q_{i}\mathbf{r}_{i}$ while conserving the momentum $\sum m_{i}\mathbf{r}_{i}=const$. For gravity, however, the charge ''is'' the mass, which is the principle of equivalence lying at the heart of General Relativity, so that is impossible. This is the reason that in electromagnetic radiation the dominating term is usually the first one allowed, i.e. the dipole one, while for gravitational radiation the first possibly non-zero radiation term is the quadrupole one.
 </br>
-<gallery widths=600px heights=500px>
+<gallery widths=400px heights=200px>
 File:Dipole.png|
 </gallery>
@@ Line 89: / Line 89: @@
 <div id="gw5"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 5: Quadrupole formula from dimensional analysis===
 Obtain, using dimensional analysis, the quadrupole formula for the energy loss by a system due to emission of gravitational waves

@@ Line 55: / Line 55: @@
 <br/>
 Monopole radiation can only be caused by a change in the source which is perfectly spherically symmetric. But purely radial oscillation as shown does not change the inverse-square law fields such as electromagnetism and gravity. A source simply disappearing would produce monopole radiation, but the conservation laws of charge and mass forbid this.
-<gallery widths=600px heights=100px>
+<gallery widths=150px heights=100px>
 File:Monopole2.png|
 </gallery></p>

@@ Line 178: / Line 178: @@
 <div id="gw9"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1: Gravitationally bound systems===
+=== Problem 9: Gravitationally bound systems===
 Estimate the gravitational luminosity for gravitationally bound systems
 <div class="NavFrame collapsed">
@@ Line 198: / Line 198: @@
 <div id="gw10"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
-=== Problem 1: Upper bound on frequency===
+=== Problem 10: Upper bound on frequency===
 Estimate the upper bound of the gravitational wave frequency generated by a compact source with size $R$ and mass $M$.
 <div class="NavFrame collapsed">