Introduction to Mathematical Physics/Relativity/Exercises

From Wikibooks, open books for an open world
< Introduction to Mathematical Physics‎ | Relativity
Jump to: navigation, search

Exercice:

Show that a rocket of mass m that ejects at speed u (with respect to itself) a part of its mass dm by time unit dt moves in the sense opposed to u. Give the movement law for a rocket with speed zero at time t=0, located in a earth gravitational field considered as constant.

Exercice:

Doppler effect. Consider a light source S moving at constant speed with respect to reference frame R. Using wave four-vector (k,\omega/c) give the relation between frequencies measured by an experimentator moving with S and another experimentater attached to R. What about sound waves?

Exercice:

For a cylindrical coordinates system, metrics g_{ij} of the space is:


ds^2=dr^2+r^2d\theta^2+dz^2

Calculate the Christoffel symbols \Gamma^{i}_{hk} defined by:


\Gamma^{i}_{hk}=\frac{1}{2}g^{ij} (\partial_hg_{kj}+ \partial_kg_{hj}-
\partial_jg_{hk})

Exercice:

Consider a unit mass in a three dimensional reference frame whose metrics is:


ds^2=g_{ij}dq_idq_j

Show that the kinetic energy of the system is:


E_c=\frac{1}{2}g_{ij}\dot q_i\dot q_j

Show that the fundamental equation of dynamics is written here (forces are assumed to derive from a potential V) :


\frac{D\dot q_i}{Dt}=-\frac{\partial V}{\partial q_i}