# Motivation and symmetries / Introduction

### Problem 1: Decoupling time

Show that relic neutrinos can provide us with information on the Universe at temperatures less that 1 MeV (i.e. from the age of the order of seconds). What is the corresponding limit for gravitons?

### Problem 2: Gravitational waves on curved background

The basic theory of gravitational waves, discussed below, deals mostly with small perturbations on flat Minkowski background. Can this approximation be useful for studying waves on a non-trivial non-flat background, and if yes then in which cases?

### Problem 3: No monopole radiation

Show that generation of either electromagnetic or gravitational monopole radiation is impossible.

### Problem 4: No dipole gravitational radiation

Show that dipole gravitational radiation is prohibited by the momentum conservation law.

### Problem 5: Quadrupole formula from dimensional analysis

Obtain, using dimensional analysis, the quadrupole formula for the energy loss by a system due to emission of gravitational waves $\frac{dE}{dt}\sim \frac{G}{c^5}\cdot {\dddot{Q}}_{\alpha\beta} {\dddot{Q}}^{\alpha\beta},$ where $Q_{\alpha\beta}=\int d^3 x \; ( x_\alpha x_\beta -\tfrac13 r^2 \delta_{\alpha\beta}) \rho(\mathbf{x}).$ is the reduced quadrupole moment of the system.

### Problem 6: Natural luminosity

Using the quadrupole formula, find the upper limit for gravitational luminosity of a source.

### Problem 7: Natural luminosity from dimensional analysis

Construct the limiting luminosity from dimensional analysis

### Problem 8: Quantum luminosity limit

Why the upper limit on luminosity, obtained in General Relativity, should not change in the future quantum theory of gravitation?

### Problem 9: Gravitationally bound systems

Estimate the gravitational luminosity for gravitationally bound systems

### Problem 10: Upper bound on frequency

Estimate the upper bound of the gravitational wave frequency generated by a compact source with size $R$ and mass $M$.