R7

1. Find the laws of transformation of light-cone coodinates under a boost and give the expression of proper time in terms of these coodinates.
2. Show that the functions that define the generalised Bogolyubov transform must have a wronskian equal to i to conserve the algebra of creation and annihilation operators.
3. What is the time dependence of conformal time in a radiation-dominated universe and in a matter-dominated universe? What is the value of conformal time today? At what value of the conformal time did the cosmic microwave background decouple ([math]\displaystyle{ z=1000 }[/math])?
4. Calculate (in K) the Hawking temperature of a 1-solar-mass black hole. How large should the acceleration of an observer be for the corresponding Unruh temperature to be 300 K?
5. Particles and antiparticles of given mass m (neutrinos, electrons and so on) can be emitted only if the mass M of the black hole is less than some critical mass. Estimate the critical mass of a black hole as a function of m. How small must a black hole be to emit electrons?