Difference between revisions of "Energy conditions and the Raychaudhuri equation"

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[[Category:Dynamics of the Universe in the Big Bang Model|6]]
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[[Category:Dynamics of the Universe in the Big Bang Model|7]]
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==Energy conditions==
 
==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
+
<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>
conditions that limit the arbitrariness of $T^{\mu\nu}$." ''</p>
+
  
 
<p style="text-align: left;">The energy conditions are formulated in coordinate-independent way, but in the context of cosmology they are most useful in application to the energy-momentum tensor of a perfect fluid$^*$:
 
<p style="text-align: left;">The energy conditions are formulated in coordinate-independent way, but in the context of cosmology they are most useful in application to the energy-momentum tensor of a perfect fluid$^*$:
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\text{Name}&\text{Statement}&\text{For perfect fluid}\\
 
\text{Name}&\text{Statement}&\text{For perfect fluid}\\
 
\text{Weak}\phantom{\Big|}& T_{\mu\nu}v^{\mu}v^{\nu}\geq0&
 
\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&
 
\text{Null}& T_{\mu\nu}k^{\mu}k^{\nu}\geq 0&
 
\rho+p\geq 0;\\[0.2cm]
 
\rho+p\geq 0;\\[0.2cm]
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The conditions are assumed to hold for arbitrary timelike vectors $v^{\mu}$ and arbitrary null vectors $k^{\mu}$.</p>
 
The conditions are assumed to hold for arbitrary timelike vectors $v^{\mu}$ and arbitrary null vectors $k^{\mu}$.</p>
  
<p style="text-align: left;"> $^*$For more detailed discussion see textbooks: Carroll S. ''Spacetime and geometry: an introduction to General Relativity''. AW, 2003; ISBN 0805387323, 525p., and Poisson E. ''A relativist's toolkit''. CUP, 2004; ISBN 0521830915, 248p. (ch 2)</p>
+
<p style="text-align: left;"> $^*$For more detailed discussion see textbooks:
 +
 
 +
Carroll S. ''Spacetime and geometry: an introduction to General Relativity''. AW, 2003; ISBN 0805387323, 525p.,
  
 +
Poisson E. ''A relativist's toolkit''. CUP, 2004; ISBN 0521830915, 248p. (ch 2)</p>
  
  
 
<div id="EnCond1"></div>
 
<div id="EnCond1"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 1. ===
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=== Problem 1: energy conditions for perfect fluid ===
 
Derive the energy conditions for the perfect fluid, shown in the last column, from the coordinate-independent formulations.
 
Derive the energy conditions for the perfect fluid, shown in the last column, from the coordinate-independent formulations.
 
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<div class="NavFrame collapsed">
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=\gamma^{2}(\rho+v^{2}p)\geq0.\]
 
=\gamma^{2}(\rho+v^{2}p)\geq0.\]
 
Since this must hold for any $v\in[0,1)$, it follows that
 
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/>
 
<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
 
'''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
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Note that in the limit $v=1-0$ ($v$ tends to $1$ from below) the SEC is rewritten as
 
Note that in the limit $v=1-0$ ($v$ tends to $1$ from below) the SEC is rewritten as
 
\[\rho(1-0)+p(1+0)\geq 0,\]
 
\[\rho(1-0)+p(1+0)\geq 0,\]
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:
 
The strong energy condition is in the most simple way formulated for the Ricci tensor:
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=\gamma^{2}\big(\rho^{2}-v^{2}p^{2})\geq 0.\]
 
=\gamma^{2}\big(\rho^{2}-v^{2}p^{2})\geq 0.\]
 
Thus the DEC for a perfect fluid can be written as
 
Thus the DEC for a perfect fluid can be written as
\[|p|<\rho,\]
+
\[|p|\leq \rho,\]
 
which also implies $\rho\geq 0$.</p>
 
which also implies $\rho\geq 0$.</p>
 
   </div>
 
   </div>
</div>
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</div></div>
</div>
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+
  
  
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<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
  
=== Problem 2. ===
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=== Problem 2: weak or strong? ===
 
Does the weak energy condition follow from the strong one? Which of the energy conditions imply the others?
 
Does the weak energy condition follow from the strong one? Which of the energy conditions imply the others?
 
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<p style="text-align: left;">All other variants of matter, obeying one condition but violating another, are hypothetically possible.</p>
 
<p style="text-align: left;">All other variants of matter, obeying one condition but violating another, are hypothetically possible.</p>
  
<p style="text-align: left;">$^*$Strictly speaking, the weak energy condition allows $\rho+p=0$, which is prohibited by the null condition, but from here on we will not usually distinguish the strict and non-strict equalities. However, this makes a difference when considering the cosmological constant (see chapter on dark energy).</p>
+
<p style="text-align: left;">$^*$Strictly speaking, the weak energy condition allows $\rho+p=0$, which is prohibited by the null condition, but from here on we will not usually distinguish the strict and non-strict equalities. However, this makes a difference when considering the cosmological constant (see the chapter on dark energy).</p>
 
   </div>
 
   </div>
</div>
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</div></div>
</div>
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<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
  
=== Problem 3. ===
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=== Problem 3: energy conditions through geometry ===
 
Express the energy conditions in terms of scale factor and its derivatives.
 
Express the energy conditions in terms of scale factor and its derivatives.
 
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\end{array}\right..\]</p>
 
\end{array}\right..\]</p>
 
   </div>
 
   </div>
</div>
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</div></div>
</div>
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+
  
  
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<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
  
=== Problem 4. ===
+
=== Problem 4: and in terms of redshift ===
 
Express the null, weak and strong energy conditions in terms of the Hubble parameter and redshift.
 
Express the null, weak and strong energy conditions in terms of the Hubble parameter and redshift.
 
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<div class="NavFrame collapsed">
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\[\begin{array}{ll}
 
\[\begin{array}{ll}
 
\text{Null:}&\displaystyle
 
\text{Null:}&\displaystyle
\phantom{\text{NEC \emph{and}}}
+
\phantom{\text{NEC and}}
 
\frac{\partial H^2}{\partial z}
 
\frac{\partial H^2}{\partial z}
 
\ge -\frac{2k\left(1 + z\right)}{a_0^2};\\[0.4cm]
 
\ge -\frac{2k\left(1 + z\right)}{a_0^2};\\[0.4cm]
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\end{array}\]</p>
 
\end{array}\]</p>
 
   </div>
 
   </div>
</div>
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</div></div>
</div>
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+
  
  
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<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
  
=== Problem 5. ===
+
=== Problem 5: restrictions on deceleration parameter ===
 
Find the restrictions that the energy conditions impose on the deceleration parameter in  a flat Universe with $\rho>0$.
 
Find the restrictions that the energy conditions impose on the deceleration parameter in  a flat Universe with $\rho>0$.
 
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<div class="NavFrame collapsed">
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\Rightarrow\; -1\leq q \leq 2\end{eqnarray*}</p>
 
\Rightarrow\; -1\leq q \leq 2\end{eqnarray*}</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
</div>
+
  
 
==Raychaudhuri equation==
 
==Raychaudhuri equation==
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<div id="Ray1"></div>
 
<div id="Ray1"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 6. ===
+
=== Problem 6: projection operators ===
 
Consider a timelike curve $x^{\mu}(\tau)$ and find the projection operators on its tangent vector and on its orthogonal complement.
 
Consider a timelike curve $x^{\mu}(\tau)$ and find the projection operators on its tangent vector and on its orthogonal complement.
 
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<div class="NavFrame collapsed">
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=g_{\mu\nu}e^{\mu}e^{\nu}.\]</p>
 
=g_{\mu\nu}e^{\mu}e^{\nu}.\]</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
</div>
+
 
+
  
  
 
<div id="Ray2"></div>
 
<div id="Ray2"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 7. ===
+
=== Problem 7: geodesic deviation ===
 
A ''congruence'' is a set of curves having the property that each point in a given region belongs to one and only one curve of the set. Consider a congruence of timelike geodesics. Let us mark two infinitely close geodesics and look at their relative evolution along their length. Let $\xi^{\mu}$ be the infinitesimal $4$-vector that is directed normal to one of the curves towards the other. Show that
 
A ''congruence'' is a set of curves having the property that each point in a given region belongs to one and only one curve of the set. Consider a congruence of timelike geodesics. Let us mark two infinitely close geodesics and look at their relative evolution along their length. Let $\xi^{\mu}$ be the infinitesimal $4$-vector that is directed normal to one of the curves towards the other. Show that
 
\[\frac{d\xi_{\nu}}{d\tau}
 
\[\frac{d\xi_{\nu}}{d\tau}
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so in particular we obtain $B_{\mu\nu}B^{\mu\nu}=B_{ij}B^{ij}\geq 0$.</p>
 
so in particular we obtain $B_{\mu\nu}B^{\mu\nu}=B_{ij}B^{ij}\geq 0$.</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
</div>
+
 
+
  
  
 
<div id="Ray3"></div>
 
<div id="Ray3"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 8. ===
+
=== Problem 8: tensor decomposition ===
 
Show that any tensor field of second rank defined on an $n-$dimensional Riemannian manifold with positive definite metric $g_{\mu\nu}$ can be uniquely decomposed into
 
Show that any tensor field of second rank defined on an $n-$dimensional Riemannian manifold with positive definite metric $g_{\mu\nu}$ can be uniquely decomposed into
 
\begin{equation}\label{TensorDecomposition}
 
\begin{equation}\label{TensorDecomposition}
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The contraction is $\Theta=B^{\mu}_{\mu}=\alpha \delta^{\mu}_{\mu}=\alpha n$, so we obtain  $\alpha=\Theta/n$ and thus derive the representation of $B_{\mu\nu}$ we had sought for.</p>
 
The contraction is $\Theta=B^{\mu}_{\mu}=\alpha \delta^{\mu}_{\mu}=\alpha n$, so we obtain  $\alpha=\Theta/n$ and thus derive the representation of $B_{\mu\nu}$ we had sought for.</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
</div>
+
  
  
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<div id="Ray4"></div>
 
<div id="Ray4"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 9. ===
+
=== Problem 9: geometrical meaning in 3D ===
 
Let there be a congruence in a three-dimensional Riemannian manifold. What is the geometric meaning of $\Theta$, $\sigma$ and $\omega$ for $B_{\mu\nu}=u_{\nu;\mu}$?
 
Let there be a congruence in a three-dimensional Riemannian manifold. What is the geometric meaning of $\Theta$, $\sigma$ and $\omega$ for $B_{\mu\nu}=u_{\nu;\mu}$?
 
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<div class="NavFrame collapsed">
 
   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
     <p style="text-align: left;">Let the vector $\xi^{\mu}$ be a two-dimensional vector that lies in the space $S$ orthogonal to the tangent vector, and let us consider the pipe of curves from the congruence in the neighborhood of the given curve and the evolution of its form with the parameter $\tau$. Each of the generating lines of the pipe is parametrized by the corresponding vector $\xi^{\mu}$.
+
     <p style="text-align: left;">Let the vector $\xi^{\mu}$ be a two-dimensional vector that lies in the space $S$ orthogonal to the tangent vector, and let us consider the pipe of curves from the congruence in the neighborhood of the given curve and the evolution of its form with the parameter $\tau$. Each of the generating lines of the pipe is parametrized by the corresponding vector $\xi^{\mu}$.</p>
  
a) Let $B_{\mu\nu}=\frac{\Theta}{2}g_{\mu\nu}$. Then
+
<p style="text-align: left;">a) Let $B_{\mu\nu}=\frac{\Theta}{2}g_{\mu\nu}$. Then
 
\[B^{\mu}_{\nu}=\frac{\Theta}{2}
 
\[B^{\mu}_{\nu}=\frac{\Theta}{2}
 
\begin{pmatrix} 1&0\\0&1\end{pmatrix}
 
\begin{pmatrix} 1&0\\0&1\end{pmatrix}
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The first term describes expansion in one direction in $(1+\sigma_{+})$ times, and contraction by the same factor in the perpendicular direction. The second term is the superposition of the same deformation without change of volume and a rotation. So the general deformation is a shear in arbitrary direction, without change of section area.</p>
 
The first term describes expansion in one direction in $(1+\sigma_{+})$ times, and contraction by the same factor in the perpendicular direction. The second term is the superposition of the same deformation without change of volume and a rotation. So the general deformation is a shear in arbitrary direction, without change of section area.</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
</div>
+
 
+
  
  
 
<div id="Ray5"></div>
 
<div id="Ray5"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 10. ===
+
 
 +
=== Problem 10: Landau-Raychaudhuri equation ===
 
Derive the Raychaudhuri equation for a congruence of timelike geodesics in spacetime
 
Derive the Raychaudhuri equation for a congruence of timelike geodesics in spacetime
 
\begin{equation}\label{Raychaudhuri}
 
\begin{equation}\label{Raychaudhuri}
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<div id="Ray6"></div>
 
<div id="Ray6"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<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 $-\infty$.
+
=== Problem 11: congruence orthogonal to a hypersurface ===
 +
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|energy conditions]] and the next problem), 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 $-\infty$.
 
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   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
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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.
 
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>
 +
 
 +
<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>
</div>
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</div></div>
</div>
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<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
  
=== Problem 12. ===
+
=== Problem 12: Raychaudhuri equation for FLRW ===
 
Write out the Raychaudhuri equation for the geodesics of comoving matter in the FLRW Universe and show that it is reduced to the second Friedman equation.
 
Write out the Raychaudhuri equation for the geodesics of comoving matter in the FLRW Universe and show that it is reduced to the second Friedman equation.
 
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<div class="NavFrame collapsed">
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=\frac{8\pi G}{c^4}\big(T_{00}-\tfrac{1}{2}Tg_{00})
 
=\frac{8\pi G}{c^4}\big(T_{00}-\tfrac{1}{2}Tg_{00})
 
=\frac{4\pi G}{c^4}\big(\varepsilon +3p\big),\]
 
=\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;">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$^{*}$</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>
</div>
+
</div></div>
</div>
+
 
+
  
  
 
==Sudden Future Singularities==
 
==Sudden Future Singularities==
The following problems are composed in the spirit of ([http://arxiv.org/abs/gr-qc/0403084 John D. Barrow, Sudden Future Singularities, arXiv:0403084v3]).
+
The following problems are composed in the spirit of [http://arxiv.org/abs/gr-qc/0403084 John D. Barrow, Sudden Future Singularities, arXiv:0403084v3].
  
 
<div id="sing1"></div>
 
<div id="sing1"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 13. ===
+
=== Problem 13: a sudden singularity ===
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,
 
\[a\to a_{s}\neq 0,\infty,
 
\qquad H\to H_{s}\neq 0,\infty.\]
 
\qquad H\to H_{s}\neq 0,\infty.\]
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   <div class="NavHead">solution</div>
 
   <div class="NavHead">solution</div>
 
   <div style="width:100%;" class="NavContent">
 
   <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,
 
\[\frac{\ddot{a}}{a}\sim 4\pi k\,p,
 
\qquad \dot{\rho}\sim -3Hp.\]</p>
 
\qquad \dot{\rho}\sim -3Hp.\]</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
</div>
+
 
+
  
  
 
<div id="sing2"></div>
 
<div id="sing2"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 14. ===
+
 
 +
=== Problem 14: asymptotics ===
 
Consider a solution of Friedman equations of the form
 
Consider a solution of Friedman equations of the form
 
\[a(t)=A+Bt^{q}+C(t_{s}-t)^{n},\]
 
\[a(t)=A+Bt^{q}+C(t_{s}-t)^{n},\]
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Note that $n>1$ is needed for $\dot{a}$ to stay finite and $n<2$ for $\ddot{a}$ to diverge. For $2<n<3$ the values of $\ddot{a}$ and $p$ would remain bounded, while $\dddot{a}$ and $\dot{p}$ diverge.</p>
 
Note that $n>1$ is needed for $\dot{a}$ to stay finite and $n<2$ for $\ddot{a}$ to diverge. For $2<n<3$ the values of $\ddot{a}$ and $p$ would remain bounded, while $\dddot{a}$ and $\dot{p}$ diverge.</p>
 
   </div>
 
   </div>
</div>
+
</div></div>
</div>
+
 
+
  
  
 
<div id="sing3"></div>
 
<div id="sing3"></div>
 
<div style="border: 1px solid #AAA; padding:5px;">
 
<div style="border: 1px solid #AAA; padding:5px;">
=== Problem 15. ===
+
=== Problem 15: energy conditions ===
 
Is any energy condition violated by the solutions with the sudden future singularity? What physical constraint on matter can be introduced that would prevent it?
 
Is any energy condition violated by the solutions with the sudden future singularity? What physical constraint on matter can be introduced that would prevent it?
 
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     <p style="text-align: left;">Due to the second Friedman equation, we see that as (\ref{sing_eq}), both $\rho$ and $(\rho+3p)$ remain positive, so all common energy conditions are satisfied. The considered singularity is possible, however, only if pressure is allowed to be unbounded at finite values of density. If we demand that $p<C\rho$ for some $C>0$, for example, along with the common conditions $\rho>0$ and $\rho+3p>0$, then the sudden future singularity is eliminated.</p>
 
     <p style="text-align: left;">Due to the second Friedman equation, we see that as (\ref{sing_eq}), both $\rho$ and $(\rho+3p)$ remain positive, so all common energy conditions are satisfied. The considered singularity is possible, however, only if pressure is allowed to be unbounded at finite values of density. If we demand that $p<C\rho$ for some $C>0$, for example, along with the common conditions $\rho>0$ and $\rho+3p>0$, then the sudden future singularity is eliminated.</p>
 
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Latest revision as of 21:28, 9 December 2013

Energy conditions

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 $T^{\mu\nu}$."

The energy conditions are formulated in coordinate-independent way, but in the context of cosmology they are most useful in application to the energy-momentum tensor of a perfect fluid$^*$: \[\begin{array}{lcc} \text{Name}&\text{Statement}&\text{For perfect fluid}\\ \text{Weak}\phantom{\Big|}& T_{\mu\nu}v^{\mu}v^{\nu}\geq0& \rho\geq 0,\quad \rho+p\geq 0;\\[0.2cm] \text{Null}& T_{\mu\nu}k^{\mu}k^{\nu}\geq 0& \rho+p\geq 0;\\[0.2cm] \text{Strong}& (T_{\mu\nu}-\tfrac{1}{2}Tg_{\mu\nu}) v^{\nu}v^{\nu}\geq 0\quad& \quad\rho+p\geq 0,\quad \rho+3p\geq 0;\\[0.2cm] \text{Dominant}& \quad T^{\mu}_{\nu}v^{\nu}\; \mbox{is non-spacelike and future-directed}\quad & \rho\geq |p\,|. \end{array}\] The conditions are assumed to hold for arbitrary timelike vectors $v^{\mu}$ and arbitrary null vectors $k^{\mu}$.

$^*$For more detailed discussion see textbooks: Carroll S. Spacetime and geometry: an introduction to General Relativity. AW, 2003; ISBN 0805387323, 525p., Poisson E. A relativist's toolkit. CUP, 2004; ISBN 0521830915, 248p. (ch 2)


Problem 1: energy conditions for perfect fluid

Derive the energy conditions for the perfect fluid, shown in the last column, from the coordinate-independent formulations.


Problem 2: weak or strong?

Does the weak energy condition follow from the strong one? Which of the energy conditions imply the others?


Problem 3: energy conditions through geometry

Express the energy conditions in terms of scale factor and its derivatives.


Problem 4: and in terms of redshift

Express the null, weak and strong energy conditions in terms of the Hubble parameter and redshift.


Problem 5: restrictions on deceleration parameter

Find the restrictions that the energy conditions impose on the deceleration parameter in a flat Universe with $\rho>0$.

Raychaudhuri equation

Problem 6: projection operators

Consider a timelike curve $x^{\mu}(\tau)$ and find the projection operators on its tangent vector and on its orthogonal complement.


Problem 7: geodesic deviation

A congruence is a set of curves having the property that each point in a given region belongs to one and only one curve of the set. Consider a congruence of timelike geodesics. Let us mark two infinitely close geodesics and look at their relative evolution along their length. Let $\xi^{\mu}$ be the infinitesimal $4$-vector that is directed normal to one of the curves towards the other. Show that \[\frac{d\xi_{\nu}}{d\tau} =B_{\nu\mu}\xi^{\mu},\] where \[B_{\nu\mu}=u_{\nu;\mu},\] is a three-dimensional spacelike tensor orthogonal to $u^{\mu}$, and $\tau$ is the parameter along the geodesic.


Problem 8: tensor decomposition

Show that any tensor field of second rank defined on an $n-$dimensional Riemannian manifold with positive definite metric $g_{\mu\nu}$ can be uniquely decomposed into \begin{equation}\label{TensorDecomposition} B_{\mu\nu}=\frac{1}{n}\Theta g_{\mu\nu} +\sigma_{\mu\nu}+\omega_{\mu\nu}, \end{equation} where $\Theta={B^{\mu}}_{\mu}$, $\sigma_{\mu\nu}$ is the symmetric traceless part of $B_{\mu\nu}$, and $\omega_{\mu\nu}$ is the antisymmetric part of $B_{\mu\nu}$.


Problem 9: geometrical meaning in 3D

Let there be a congruence in a three-dimensional Riemannian manifold. What is the geometric meaning of $\Theta$, $\sigma$ and $\omega$ for $B_{\mu\nu}=u_{\nu;\mu}$?


Problem 10: Landau-Raychaudhuri equation

Derive the Raychaudhuri equation for a congruence of timelike geodesics in spacetime \begin{equation}\label{Raychaudhuri} \frac{d\Theta}{d\tau}= -\frac{1}{3}\Theta^{2} -\sigma_{\mu\nu}\sigma^{\mu\nu} +\omega_{\mu\nu}\omega^{\mu\nu} -R_{\mu\nu}u^{\mu}u^{\mu}. \end{equation} Here $\Theta$, $\sigma_{\mu\nu}$ and $\omega_{\mu\nu}$ are the components (\ref{TensorDecomposition}) of decomposition of $B_{\mu\nu}=u_{\mu;\nu}$.


Problem 11: congruence orthogonal to a hypersurface

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 energy conditions and the next problem), 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 $-\infty$.


Problem 12: Raychaudhuri equation for FLRW

Write out the Raychaudhuri equation for the geodesics of comoving matter in the FLRW Universe and show that it is reduced to the second Friedman equation.


Sudden Future Singularities

The following problems are composed in the spirit of John D. Barrow, Sudden Future Singularities, arXiv:0403084v3.

Problem 13: a sudden singularity

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: \[a\to a_{s}\neq 0,\infty, \qquad H\to H_{s}\neq 0,\infty.\] What scalars can in principle become unbounded in this scenario?


Problem 14: asymptotics

Consider a solution of Friedman equations of the form \[a(t)=A+Bt^{q}+C(t_{s}-t)^{n},\] where $A,B,q,n>0$ and $C$ are some free constants. What values of $q$ and $n$ are compatible with the sudden singularity of the previous problem?


Problem 15: energy conditions

Is any energy condition violated by the solutions with the sudden future singularity? What physical constraint on matter can be introduced that would prevent it?