Energy conditions and the Raychaudhuri equation

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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?