Energy conditions and the Raychaudhuri equation - Revision history

Igor: /* Energy conditions */

2013-12-09T18:28:02Z

‎Energy conditions

Igor: /* Problem 1: energy conditions for perfect fluid */

2013-12-09T18:27:27Z

‎Problem 1: energy conditions for perfect fluid

Cosmo All at 22:03, 12 November 2012

2012-11-12T22:03:52Z

Igor at 12:44, 14 October 2012

2012-10-14T12:44:45Z

Show changes

Cosmo All: /* Problem 13. */

2012-07-23T19:23:06Z

‎Problem 13.

Cosmo All: /* Problem 13. */

2012-07-23T19:22:15Z

‎Problem 13.

Cosmo All: /* Problem 10. */

2012-07-23T19:20:34Z

‎Problem 10.

Cosmo All at 18:31, 23 July 2012

2012-07-23T18:31:39Z

Igor: /* Energy conditions */

2012-07-23T18:30:42Z

‎Energy conditions

Igor at 18:28, 23 July 2012

2012-07-23T18:28:23Z

@@ Line 9: / Line 9: @@
 \text{Name}&\text{Statement}&\text{For perfect fluid}\\
 \text{Weak}\phantom{\Big|}& T_{\mu\nu}v^{\mu}v^{\nu}\geq0&
-	\rho\geq 0,\quad \rho+p>0;\\[0.2cm]
+	\rho\geq 0,\quad \rho+p\geq 0;\\[0.2cm]
 \text{Null}& T_{\mu\nu}k^{\mu}k^{\nu}\geq 0&
 	\rho+p\geq 0;\\[0.2cm]

@@ Line 50: / Line 50: @@
 	=\gamma^{2}(\rho+v^{2}p)\geq0.\]
 Since this must hold for any $v\in[0,1)$, it follows that
-\[\rho\geq0,\quad \rho+p> 0.\]
+\[\rho\geq0,\quad \rho+p\geq 0.\]
 <br/>
 '''2)''' ''Null energy condition (NEC).'' <br/>In the same way using $k^{\mu}$ from (\ref{EnCond_VKparametrization})  with $k_{\mu}k^{\mu}=0$, we arrive to
@@ Line 71: / Line 71: @@
 Note that in the limit $v=1-0$ ($v$ tends to $1$ from below) the SEC is rewritten as
 \[\rho(1-0)+p(1+0)\geq 0,\]
-so, as opposed to the WEC, the non-strict inequality sign remains in $\rho+p\geq 0$.
+so the non-strict inequality sign remains in $\rho+p\geq 0$.
 The strong energy condition is in the most simple way formulated for the Ricci tensor:
@@ Line 83: / Line 83: @@
 	=\gamma^{2}\big(\rho^{2}-v^{2}p^{2})\geq 0.\]
 Thus the DEC for a perfect fluid can be written as
-\[|p|<\rho,\]
+\[|p|\leq \rho,\]
 which also implies $\rho\geq 0$.</p>
    </div>

← Older revision		Revision as of 22:03, 12 November 2012
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−	[[Category:Dynamics of the Universe in the Big Bang Model\|6]]	+	[[Category:Dynamics of the Universe in the Big Bang Model\|7]]
	__TOC__		__TOC__

@@ Line 486: / Line 486: @@
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 13. ===
-Let us consider the possibility of \textit{sudden future singularities}. The ``suddenness'' implies that they occur at some time in the future, while both the scale factor and the Hubble constant remain bounded and separated from zero:
+Let us consider the possibility of ''sudden future singularities''. The "suddenness" implies that they occur at some time in the future, while both the scale factor and the Hubble constant remain bounded and separated from zero:
 \[a\to a_{s}\neq 0,\infty,
 	\qquad	H\to H_{s}\neq 0,\infty.\]

@@ Line 493: / Line 493: @@
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <p style="text-align: left;">The first Friedman equation implies that under the imposed conditions of\; ``un\-ex\-pect\-ed\-ness'' density must be finite. However, $p$ can diverge along with $\ddot{a}$ and $\dot{\rho}$:
+     <p style="text-align: left;">The first Friedman equation implies that under the imposed conditions of "unexpected-ness" density must be finite. However, $p$ can diverge along with $\ddot{a}$ and $\dot{\rho}$:
 \[\frac{\ddot{a}}{a}\sim 4\pi k\,p,
 	\qquad \dot{\rho}\sim -3Hp.\]</p>
@@ Line 504: / Line 504: @@
 <div id="sing2"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 14. ===
 Consider a solution of Friedman equations of the form

@@ Line 397: / Line 397: @@
 <div id="Ray6"></div>
 <div style="border: 1px solid #AAA; padding:5px;">
 === Problem 11. ===
 Show that for a congruence of geodesics orthogonal to a family of hypersurfaces $\omega_{\mu\nu}=0$. Prove further, that in case the strong energy condition $R_{\mu\nu}u^{\mu}u^{\nu}\geq0$ holds (see [[#EnCond1|problem]] and the following), then the following (the focusing theorem) also is true:  if $\Theta=\Theta_{0}<0$ at some initial moment, then in a finite period of proper time $\Theta$ diverges and tends to $-\infty$.

@@ Line 1: / Line 1: @@
 [[Category:Dynamics of the Universe in the Big Bang Model|6]]
 __NOTOC__
 ==Energy conditions==
 <p style="text-align: left;">S. Carroll writes$^{*}$: '' "Sometimes it is useful to think about Einstein's equation without specifying the theory of matter from which $T^{\mu\nu}$ is derived. This leaves us with a great deal of arbitrariness; consider for example the question, What metrics obey Einstein's equation? In the absence of some constraints on $T^{\mu\nu}$, the answer is any metric at all; simply take the metric of your choice, compute the Einstein tensor $G^{\mu\nu}$ for this metric, and then demand that $T^{\mu\nu}$ be equal to $G^{\mu\nu}$. It will automatically be conserved, by the Bianchi identity. Our real concern is with the existence of solutions to Einstein's equation in the presence of "realistic" sources of energy and momentum, whatever that means. One strategy is to consider specific kinds of sources, such as scalar fields, dust, or electromagnetic fields. However, we occasionally wish to understand properties of Einstein's equations that hold for a variety of different sources. In this circumstance it is convenient to impose energy conditions that limit the arbitrariness of $T^{\mu\nu}$." ''</p>

@@ Line 423: / Line 423: @@
 so if at some moment $\Theta_{0}<0$, then for $\Delta\tau\to |\Theta_{0}|/3$ we obtain from the inequality that $\Theta\to -\infty$, i.e. the geodesics of the congruence are focused into a point.
-The focusing theorem implies that the geodesics form a caustics, which in general does not necessarily mean a singularity. Some further elaboration is needed, making use of the focusing theorem and usually some energy conditions, in order to prove that a singularity actually takes place\footnote{Hawking, Ellis; Wald.}.</p>
+The focusing theorem implies that the geodesics form a caustics, which in general does not necessarily mean a singularity. Some further elaboration is needed, making use of the focusing theorem and usually some energy conditions, in order to prove that a singularity actually takes place$^{*}$</p>
    </div>
 </div>
@@ Line 466: / Line 468: @@
 	=\frac{8\pi G}{c^4}\big(T_{00}-\tfrac{1}{2}Tg_{00})
 	=\frac{4\pi G}{c^4}\big(\varepsilon +3p\big),\]
-we obtain the second Friedman equation.
+we obtain the second Friedman equation.</p>
-We see now that the focusing theorem in the cosmological context implies that the geodesics of comoving matter must converge at some time in the past, as is already known to be a major feature of the solutions of the Friedman equations. Though in general a caustic of geodesics does not necessarily mean a singularity, in this case the considered geodesics are actually the geodesics of all the comoving matter in the Universe, so their focusing actually implies the Big Band singularity. The theorem becomes inapplicable, however, at small times, when particles' interaction has to be taken into account. This is where the more general singularity theorems work\footnote{Hawking, Ellis; Wald.}.</p>
+<p style="text-align: left;">$^{*}$ For more on this better see textbooks Hawking S.W., Ellis G.F.R. ''The large scale structure of space-time'', CUP, 1973 (ISBN 0521099064) and Wald R.M, ''General relativity.'' U. Chicago, 1984, 505p (ISBN 0226870332).</p>
    </div>
 </div>