Worke it out!

Let $E$ be the space of all real polynomials on $\R$ of degree less than $n$. Suppose $x_1,\ldots,x_n\in\R$ are pairwise distinct. Show that the linear functionals $x_j^*(p)\colon=p(x_j)$ form a basis of $E^*$.

Let $u\in\Hom(\R^2)$ be the linear map with matrix (with respect to some basis) $$ \left(\begin{array}{cc} 0&1\\ 1&0 \end{array}\right) $$ 1. Verify that for any $t\in\R$ the matrix of $e^{tu}$ (with respect to the same basis) is given by $$ \left(\begin{array}{cc} \cosh t&\sinh t\\ \sinh t&\cosh t \end{array}\right) $$ 2. Find functions $a_0(t)$ and $a_1(t)$ such that $e^{tu}=a_0(t)+a_1(t)u$.

Let $E$ be the space of all real polynomials of degree less than $n$ and $u$ the mapping which sends a polynomial $p$ to its formal derivative $p^\prime$. Compute the trace of $u$.

The sum of Euclidean products is again a Euclidean product, but the sum of two Lorentz products need not be an inner product.

Verify that $(x_1,x_2),(y_1,y_2)\mapsto x_1y_2+x_2y_1-x_3y_3$ defines an inner product on $\R^3$. Find its index and determine an orthonormal basis.

Use the polarization formula to prove that an isomorphism $u\in\Hom(E)$ is an isometry iff for all $x\in E$: $\la u(x),u(x)\ra=\la x,x\ra$.

Describe all skew-symmetric linear operators $u:\R_1^2\rar\R_1^2$ and $u:\R_1^3\rar\R_1^3$.

If a subspace $F$ of $E$ contains a vector $x\neq0$ orthogonal to itself, then $F$ need not be degenerated. If $\dim F=1$ then $F$ is degenerated iff for some $x\in F\sm\{0\}$: $\la x,x\ra=0$.

Verify that the subspace $F$ of $\R_1^3$ generated by $(1,1,0)$ and $(1,1,1)$ is degenerated.

Suppose $F=\lhull{(1,1,1),(0,1,1)}$. Compute the orthogonal projection onto $F$ in $\R_1^3$.

Suppose $E$ is an inner product space and $u:E\rar E$ is self-adjoint.

If $F$ is an invariant subspace, i.e. $u(F)\sbe F$, the so is $F^\perp$.
If $x$ and $y$ are eigen vectors with different real eigen values, then $x\perp y$.

Suppose $e_1,\ldots,e_n$ is an orthonormal basis for $E$ and $\vol{}$ the associated volume form, i.e. $\vol{}(e_1,\ldots,e_n)=1$. Then for any orthonormal basis $b_1,\ldots,b_n$ for $E$: $\vol{}(b_1,\ldots,b_n)=\pm1$.

$\R\times S^{n-1}\colon=\{(x_0,x_1,\ldots,x_n)\in\R_1^{n+1}: x_1^2+\cdots+x_n^2=1\}$ is a Lorentz submanifold of $\R_1^{n+1}$.

$S^{n}\colon=\{(x_0,x_1,\ldots,x_n)\in\R_1^{n+1}: x_0^2+\cdots+x_n^2=1\}$ is not a Pseudo-Riemannian submanifold of $\R_1^{n+1}$.

On $S^{2n-1}(\sbe\R^{2n})$ there is a Lorentz metric.

A hyperplane $F$ in $E$ is a subspace of co-dimension one, i.e. $\dim F=\dim E-1$. Thus $F$ is the kernel of a linear functional $x^*\in E^*\sm\{0\}$. Prove that $F$ is space-like iff $x^{*\sharp}$ is time-like; $F$ is light-like iff $x^{*\sharp}\in F$; $F$ is time-like iff $x^{*\sharp}$ is space-like.

Suppose $F$ is the subspace of $\R_1^4$ generated by $b_1\colon=e_0+2e_3+e_1$, $b_2\colon=-e_0+e_1+e_2$ and $b_3\colon=-e_0+e_3$. Determine the causal character of $F$.

Suppose $u:\R_1^2\rar\R_1^2$ is a Lorentz transformation; let $A$ be the matrix representation of $u$ with respect to the canonical basis $e_0,e_1$ for $\R_1^2$. Prove that there is a unique number $v\in(-1,1)$ such that $$ A =\frac1{\sqrt{1-v^2}}\left(\begin{array}{cc} 1&v\\ v&1 \end{array}\right)~. $$ In coordinates: $u$ maps the point with coordinates $(x_0,x_1)$ to the point with coordinates $$ \Big(\frac{x_0+vx_1}{\sqrt{1-v^2}},\frac{vx_0+x_1}{\sqrt{1-v^2}}\Big)~. $$

Once we know that a Lorentz transformation $u\in\Hom(E)$ is a boost, we can determine $\vp$ quite easily: $\tr u=2(\cosh\vp-1)+\dim E$.

Let $X=\cosh(a)Z+\sinh(a)E$ and $Y=\cosh(b)Z+\sinh(b)E$. Then energy, momentum, velocity and speed of $Y$ with respect to $X$ are given by $$ \cosh(b-a),\quad \sinh(b-a)F,\quad \tanh(b-a)F,\quad |\tanh(b-a)| $$ where $F\colon=\sinh(a)Z+\cosh(a)E$ is a unit vector orthogonal to $X$.

For any $c\in\R$ the submanifolds $[x_0=c]$ is a rest space of the observer field $Z\colon=E^0$ in Minkowski space $\R_1^{n+1}$. What is the distance of two points in $[x_0=c]$ for $Z$?

Put $M\colon=\R^{n+1}$ and let $h:M\rar\R^+$ be any smooth function. At $x\in M$ we define $$ \la E^0,E^0\ra_x=-1, \quad\mbox{and for all $j\geq1$:}\quad \la E^j,E^j\ra_x=h(x)~. $$ Then $(M,\la.,.\ra)$ is a local Lorentz manifold - its called the Einstein-de Sitter spacetime - and $Z\colon=E^0$ defines a time orientation.

Compute the velocity of the world line $c(t)=(e^t-1,t+t^2/2)$, $t\geq0$, in $2$-dimensional Minkowski space $\R_1^2$ with respect to the observer field $Z_x\colon=\cosh(x_0)E_x^0+\sinh(x_0)E_x^1$ at each point $c(t)$, $t\geq0$.

Deduce from the rocket equation that $$ M\la\bnabla X,V\ra=M^\prime\norm V^2~. $$

Find all geodesics on the submanifold $$ H^n\colon=\{(x_0,\ldots,x_n):-x_0^2+x_1^2+\cdots+x_n^2=-1, x_0 > 0\} $$ of Minkowski space $\R_1^{n+1}$.

Let $T$ be an instantaneous observer and $X,Y$ unit vectors in $T^\perp$. The light-like vectors $X=T+F$ and $Y=T-F$ point into opposite directions for $T$ but not for $Z=\b(T+vE)$. Compute the angle $\a$ of $X$ any $Y$ for $Z$ in case $\la E,F\ra=0$.

Prove that a linear map $A:\R^n\rar\R^n$ is conformal iff there exists some $r\neq0$ such that $rA$ is an isometry.

Prove that the map $F:(-\pi,\pi)\times(-\pi/2,\pi/2)\rar S^2$, $$ F(\vp,\theta)\colon=(\cos\theta\cos\vp,\cos\theta\sin\vp,\sin\theta) $$ is not conformal - the set $(-\pi,\pi)\times(-\pi/2,\pi/2)$ carries the canonical Euclidean metric. Find a Riemannian metric on $(-\pi,\pi)\times(-\pi/2,\pi/2)$ such that $F$ is a local isometry.

The mapping $z\mapsto e^z$ from $\C$ onto $\C\sm\{0\}$ as a real mapping is given by $F:(x,y)\mapsto e^x(\cos y,\sin y)$. Verify by direct calculation that $$ DF(x,y)^*DF(x,y) =e^{2x}\left( \begin{array}{cc} 1&0\\ 0&1 \end{array}\right)~. $$ Thus $F$ is conformal with scaling function $h(x)=e^x$.

The Cayley transform $F(z)\colon=(z-i)/(z+i)$ maps the upper half plane $H^+\colon=[\Im z > 0]$ onto the unit disk $B^2=[|z| < 1]$.

← Index → Glossary

Last modified: Thu Sep 5 15:12:47 CEST 2024