Introduction to Mathematical Physics/Relativity/Exercises

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:

$ds^{2}=dr^{2}+r^{2}d\theta ^{2}+dz^{2}$

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

$\Gamma _{hk}^{i}={\frac {1}{2}}g^{ij}(\partial _{h}g_{kj}+\partial _{k}g_{hj}-\partial _{j}g_{hk})$

Exercice:

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

$ds^{2}=g_{ij}dq_{i}dq_{j}$

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 $V$) :

${\frac {D{\dot {q}}_{i}}{Dt}}=-{\frac {\partial V}{\partial q_{i}}}$