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

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(Energy conditions)
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     <p style="text-align: left;">No. Small negative $\rho$ with large positive $p$ obey the SEC and NEC but violate WEC and DEC.
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No. Small negative $\rho$ with large positive $p$ obey the SEC and NEC but violate WEC and DEC.</p>
  
Despite the naming, which is a bit confusing here, the ''null'' energy condition is the weakest, ''not'' the weak energy condition$^**$. The weak, strong and dominant conditions all imply NEC, but of all three only the dominant energy condition implies the weak one.
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<p style="text-align: left;">Despite the naming, which is a bit confusing here, the ''null'' energy condition is the weakest, ''not'' the weak energy condition$^**$. The weak, strong and dominant conditions all imply NEC, but of all three only the dominant energy condition implies the weak one.</p>
  
All other variants of matter, obeying one condition but violating another, are hypothetically possible.
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<p style="text-align: left;">All other variants of matter, obeying one condition but violating another, are hypothetically possible.</p>
  
 
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<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>
$^**$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 10).</p>
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=== Problem 3. ===
 
=== Problem 3. ===
 
Express the energy conditions in terms of scale factor and its derivatives.
 
Express the energy conditions in terms of scale factor and its derivatives.

Revision as of 18:10, 23 July 2012


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$^*$: \[\begin{array}{lcc} \text{Name}&\text{Statement}&\text{For perfect fluid}\\ \text{Weak}\phantom{\Big|}& T_{\mu\nu}v^{\mu}v^{\nu}\geq0& \rho\geq 0,\quad \rho+p>0;\\[0.2cm] \text{Null}& T_{\mu\nu}k^{\mu}k^{\nu}\geq 0& \rho+p\geq 0;\\[0.2cm] \text{Strong}& (T_{\mu\nu}-\tfrac{1}{2}Tg_{\mu\nu}) v^{\nu}v^{\nu}\geq 0\quad& \quad\rho+p\geq 0,\quad \rho+3p\geq 0;\\[0.2cm] \text{Dominant}& \quad T^{\mu}_{\nu}v^{\nu}\; \mbox{is non-spacelike and future-directed}\quad & \rho\geq |p\,|. \end{array}\] 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., and Poisson E. A relativist's toolkit. CUP, 2004; ISBN 0521830915, 248p. (ch 2)


Problem 1.

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


Problem 2.

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


Problem 3.

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


Problem 4.

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


Problem 5.

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

Raychaudhuri equation

Problem 6.

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


Problem 7.

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.

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.

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.

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.

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 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 12.

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.

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


Problem 14.

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.

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?