Linear algebra (Osnabrück 2024-2025)/Part II/Exercise sheet 40

Exercises

Exercise

Compute

$\left\langle {\begin{pmatrix}-8\\3\\-3\\5\end{pmatrix}},{\begin{pmatrix}-1\\0\\7\\-4\end{pmatrix}}\right\rangle$

in a four-dimensional standard-Minkowski space.

Exercise

Let ${}V$ be a Minkowski space.

a) Show that a scalar multiple of a timelike (spacelike, lightlike) vector is again timelike (spacelike, lightlike).

b) Show that the sum of two timelike (spacelike, lightlike) vectors is in general not timelike (spacelike, lightlike) again.

Exercise

Is the restriction of a Minkowski form in ${}\mathbb {R} ^{n}$ to an ${}n-1$-dimensional linear subspace again a Minkowski form?

Exercise

Let ${}V$ be a Minkowski space with the Minkowski form ${}\left\langle -,-\right\rangle$. Show that for every observer vector ${}v\in V$, there exists a direct sum decomposition

${}V=\mathbb {R} v\oplus (\mathbb {R} v)^{\perp }\,,$

where the restriction of the Minkowski form to ${}\mathbb {R} v$ is negative definite, and the restriction of the Minkowski form to ${}(\mathbb {R} v)^{\perp }$ is positive definite.

Exercise

We equip ${}\mathbb {R} ^{2}$ with the standard-Minkowski form. Show that ${}{\begin{pmatrix}{\frac {25}{24}}\\{\frac {7}{24}}\end{pmatrix}}$ is the velocity vector of an observer. Determine the space component of this vector.

Exercise

We equip ${}\mathbb {R} ^{2}$ with the standard-Minkowski form. Show that ${}{\begin{pmatrix}{\frac {\sqrt {5}}{2}}\\{\frac {3}{2}}\end{pmatrix}}$ is an observer vector, and determine its space component.

Exercise

We equip ${}\mathbb {R} ^{2}$ with the standard-Minkowski form. Show that for every oserver vector ${}{\begin{pmatrix}a\\b\end{pmatrix}}$, the space component of the observer is the reflection of the time component at the diagonal.

The hyperbolic functions are introduced in Analysis 1.

Exercise

We equip ${}\mathbb {R} ^{2}$ with the standard-Minkowski form. Show that for ${}\alpha \in \mathbb {R}$, the vector ${}{\begin{pmatrix}\sinh \alpha \\\cosh \alpha \end{pmatrix}}$ is the velocity vector of an observer. Determine the space component of this vector.

Exercise

We equip ${}\mathbb {R} ^{2}$ with the standard-Minkowski form. Show that for ${}z\in \mathbb {R}$, ${}z\neq 0$, the vectors

${\frac {1}{2}}{\begin{pmatrix}z-{\frac {1}{z}}\\z+{\frac {1}{z}}\end{pmatrix}}\,\,{\text{ and }}\,\,{\frac {1}{2}}{\begin{pmatrix}-z+{\frac {1}{z}}\\z+{\frac {1}{z}}\end{pmatrix}}$

are velocity vectors of an observer. Show that every observer vector has this form.

Exercise

Let ${}V$ be a Minkowski space with the Minkowski form ${}\left\langle -,-\right\rangle$, and let ${}v,w$ be observer vectors from the same half cone. Show

${}\left\langle v,w\right\rangle <0\,.$

Exercise

Let ${}V$ be a Minkowski space. Show that the set of all observer vectors has two path-connected components. Show that two observer vectors ${}v,w$ belong to the same component if and only if

${}\left\langle v,w\right\rangle <0\,$

holds.

Exercise

Let ${}V$ be a Minkowski space with the Minkowski form ${}\left\langle -,-\right\rangle$, and let ${}v,w$ denote timelike vectors. Show the estimate

${}\left\langle v,w\right\rangle ^{2}\geq \left\langle v,v\right\rangle \cdot \left\langle w,w\right\rangle \,.$

Exercise

Suppose that in a four-dimensional Minkowski space, a certain event has the coordinates ${}{\begin{pmatrix}2\\-3\\1\\4\end{pmatrix}}$ with respect to a Minkowski basis. Determine the decomposition in the space component and the time component of this event with respect to the observer vector ${}{\begin{pmatrix}{\frac {3}{4}}\\0\\0\\{\frac {5}{4}}\end{pmatrix}}$.

Exercise

Suppose that in a four-dimensional Minkowski space, two observers ${}B$ and ${}C$ are given, with the corresponding space components ${}V_{B}$ and ${}V_{C}$. What can we say about ${}V_{B}\cap V_{C}$?

Exercise

Let ${}V$ be a two-dimensional Minkowski space.

a) Show that there exists a basis of ${}V$ with the property that all diagonal entries in the Gram matrix with respect to this basis equal ${}1$.

b) Show that there exists a basis of ${}V$ with the property that all diagonal entries in the Gram matrix with respect to this basis equal ${}-1$.

c) Show that there exists a basis of ${}V$ with the property that all diagonal entries in the Gram matrix with respect to this basis equal ${}0$.

Exercise

Determine the velocity vector of an observer ${}B$ in a Minkowski space relative to itself, and the relative velocity.

Exercise

Let ${}B$ and ${}C$ be observers with the four-velocities

${}v_{B}={\begin{pmatrix}2\\0\\0\\{\sqrt {5}}\end{pmatrix}}\,$

and

${}v_{C}={\begin{pmatrix}0\\0\\-1\\{\sqrt {2}}\end{pmatrix}}\,.$

a) Determine the velocity vector of ${}C$ relative to ${}B$.

b) Determine den velocity vector of ${}B$ relative to ${}C$.

c) Determine the relative velocity of the two observers.

Exercise

Show that the relative velocity of two observers in a Minkowski space is between ${}0$ and ${}1$. Is ${}1$ possible? What is the significance of this statement for physics?

Hand-in-exercises

Exercise (1 mark)

Compute

$\left\langle {\begin{pmatrix}5\\8\\-11\\-13\end{pmatrix}},{\begin{pmatrix}-4\\-9\\17\\6\end{pmatrix}}\right\rangle$

in a four-dimensional standard-Minkowski space.

Exercise (4 marks)

Let ${}\mathbb {R} ^{3}$ be given the standard-Minkowski form. Show that ${}{\begin{pmatrix}{\frac {\sqrt {6}}{3}}\\{\frac {1}{3}}\\{\frac {4}{3}}\end{pmatrix}}$ is an observer vector, and determine an orthonormal basis of its space component.

Exercise (4 marks)

Suppose that in a four-dimensional Minkowski space, a certain event has the coordinates ${}{\begin{pmatrix}-1\\5\\2\\-3\end{pmatrix}}$ with respect to a Minkowski basis. Determine the decomposition in the space component and the time component of this event with respect to the observer vector ${}{\begin{pmatrix}0\\{\frac {5}{12}}\\0\\{\frac {13}{12}}\end{pmatrix}}$.

Exercise (6 (2+2+2) marks)

We equip ${}\mathbb {R} ^{3}$ with the standard-Minkowski form.

a) Give a basis of ${}\mathbb {R} ^{3}$ with the property that all diagonal entries in the Gram matrix with respect to this basis equal ${}1$.

b) Give a basis of ${}\mathbb {R} ^{3}$ with the property that all diagonal entries in the Gram matrix with respect to this basis equal ${}-1$.

c) Give a basis of ${}\mathbb {R} ^{3}$ with the property that all diagonal entries in the Gram matrix with respect to this basis equal ${}0$.

Exercise (3 (1+1+1) marks)

Let ${}B$ and ${}C$ be observers with the four-velocities

${}v_{B}={\begin{pmatrix}3\\-2\\5\\{\sqrt {39}}\end{pmatrix}}\,$

and

${}v_{C}={\begin{pmatrix}1\\1\\1\\2\end{pmatrix}}\,.$

a) Determine the velocity vector of ${}C$ relative to ${}B$.

b) Determine den velocity vector of ${}B$ relative to ${}C$.

c) Determine the relative velocity of the two observers.

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