Introduction to relativistic theory of small perturbations

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In the following section we consider only the linear in perturbations theory.

Perturbations on flat background

Problem 1

problem id: per15

Consider the stability of stationary Universe, filled with matter $(p=0)$ and by substance with $p = w\rho$.


Problem 2

problem id: per12

Determine the dependence of density fluctuations on scale factor in the flat Universe when
a) radiation,
b) matter
is dominating.


Problem 3

problem id: per13

Determine the dependence of density fluctuations on time in the closed Universe with $k=1$.


Problem 4

problem id: per14

Determine the dependence of density fluctuations on time in the open Universe with $k=-1$.


Metrics perturbations, coordinates transforms and perturbed energy--momentum tensor

Problem 5

problem id: per18

The inhomogeneities in matter distribution in the Universe generate perturbations of different kinds. In linear approximation these perturbations do not interact with each other and evolve independently. Construct the classifications of perturbations.


Problem 6

problem id: per19

Determine the number of independent functions required for description of perturbations, considered in the previous problem.


Problem 7

problem id: per20ntenzor

Assuming, that in the Universe with metrics \begin{equation}\label{met_ten_per} ds^2=dt^2-a^2(t)(\delta_{ik}-h_{ik})dx^idx^k\;(i,k=1,2,3), \end{equation} the four--velocity has the form $$u^0=\frac{1}{a}(1+\delta u^0),~u^i=\frac{1}{a} v^i,$$ determine the connection between components of four-velocity and metrics in linear approximation in perturbations. Consider $u^0$ and $v^i$ as first-order terms.


Problem 8

problem id: per21ntenzor

Demonstrate, that spatial velocity $v^i$ from previous problem is physical.


Problem 9

problem id: per22ntenzor

Using the results of two previous problems, calculate the components of energy--momentum tensor perturbation $\delta T^\mu_\nu$, which is induced by the perturbations of density and pressure $$\widetilde{\rho}=\rho+\delta \rho, ~ \widetilde{p}=p+\delta p,$$ where $\rho,p$ are their background values.


Problem 10

problem id: per23ntenzor

Obtain the equations of covariant convervation in linear approximation in components of tensor perturbations $h_{\mu\nu}$. Use the gauge $h_{0i}=0$ for simplicity.


Problem 11

problem id: per20

In the first order of $h_{ik}$ calculate the components of energy--momentum tensor for the Universe. filled with ideal fluid with equation of state$p=w\rho$ and metrics \eqref{met_ten_per}


Problem 12

problem id: per21

In linear approximation determine the transformation of metrics $g_{\mu\nu}$, which is generated by the space transformation $x^\alpha\rightarrow \tilde{x}^\alpha=x^\alpha+\xi^\alpha$, where $\xi^\alpha$ is infinitesimal scalar function.


Problem 13

problem id: per22

Using the results of previous problem determine the transformation of metrics, generated by the transformation $x^\alpha\rightarrow \tilde{x}^\alpha=x^\alpha+\xi^\alpha$. Here the four--vector $\xi^\alpha=(\xi^0,\xi^i)$ satisfies the condition $\xi^i=\xi^i_\bot+\zeta^i$, $\xi^i_\bot$ is a three--vector with zero divergence ($\xi^i_{\bot,i}=0$) and $\zeta^i$ is a scalar function.


Problem 14

problem id: per22_1

Demonstrate, that non--uniform flat Friedman metrics

 $$
  ds^2= \left(1-\frac{2}{\sqrt{\lambda}}\dot{f}(t,\vec{r})\right)dt^2-a^2(\delta_{ij}-2\mathcal{B}_{,ij})dx^idx^j,
  $$
    with arbitrary small function $f(t,\vec{r})$, can be trnsformed to the uniform one.