docs/physics/classical-mechanics/newtonian-mechanics/newtonian-formalism.md

@@ -26,7 +26,7 @@ Implying that there is no fundamental limit to the precision of measurements of
 
 From the definition of a reference system and postulates 1, 2 and 3 the Galilean transformations may be posed, which may be used to transform between the coordinates of two reference systems. 
 
-> *Principle 1*: let $(\mathbf{x},t) \in \mathbb{R}^4$ be a general point in space and time. 
+> *Principle 1*: let $(\mathbf{x},t) \in \mathbb{R}^4$ be a general point in spacetime. 
 > 
 > A uniform motion with velocity $\mathbf{v}$ is given by
 >

docs/physics/electromagnetism/electromagnetic-dynamics.md

@@ -221,47 +221,6 @@ then it follows from Newton's second law that:
 
     Should be rewritten/reconsidered in Lagrangian/Hamiltonian formalism.
 
-### Waves
-
-We may decouple the posed equations in *Axiom 1*, obtaining second order equations describing the electric and magnetic field.
-
-> *Theorem 5*: In regions of space where $\rho = 0$ and $\mathbf{J} = \mathbf{0}$, the electric $\mathbf{E}: (\mathbf{r},t) \mapsto \mathbf{E}(\mathbf{r},t)$ and magnetic field $\mathbf{B}: (\mathbf{r},t) \mapsto \mathbf{B}(\mathbf{r},t)$ adhere to
->
-> $$
->   \begin{align*}
->       \nabla^2 \mathbf{E} &= \frac{1}{c^2} \partial_t^2 \mathbf{E},\\
->       \nabla^2 \mathbf{B} &= \frac{1}{c^2} \partial_t^2 \mathbf{B},
->   \end{align*}
-> $$
->
-> with a wave speed $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$.
-
-??? note "Proof:"
-    
-    For the electric field $\mathbf{E}$ in a region for which $\rho=0$ and $\mathbf{J}=\mathbf{0}$ we have
-
-    $$
-        \begin{align*}
-            \nabla^2 \mathbf{E} &= - \nabla \times (\nabla \times \mathbf{E}),\\
-            &= \nabla \times \partial_t \mathbf{B},\\
-            &= \partial_t (\nabla \times \mathbf{B}),\\
-            &= \mu_0 \epsilon_0 \partial^2_t \mathbf{E}.
-        \end{align*}
-    $$
-    
-    For the magnetic field $\mathbf{B}$ in a region for which $\rho=0$ and $\mathbf{J}=\mathbf{0}$ we have
-
-    $$
-        \begin{align*}
-            \nabla^2 \mathbf{B} &= - \nabla \times (\nabla \times \mathbf{B}),\\
-            &= -\nabla \times (\mu_0 \epsilon_0 \partial_t \mathbf{E}),\\
-            &= -\mu_0 \epsilon_0 \partial_t (\nabla \times \mathbf{E}),\\
-            &= \mu_0 \epsilon_0 \partial_t^2 \mathbf{B}.
-        \end{align*}
-    $$
-
-The second order equations posed in *Theorem 5* satisfy exactly the three-dimensional wave equation. Thus, vacuum supports the propagation of electromagnetic waves traveling at a constant speed $c$. In my opinion one of the most mesmerizing results of the theory of electromagnetism. 
-
 ## Electromagnetic dynamics in matter
 
 Let $\mathbf{J}_p = \partial_t \mathbf{P}$ denote the polarisation volume current density such that
@@ -277,7 +236,7 @@ $$
 
 complying to the conservation of charge.
 
-> *Theorem 6*: The electric field $\mathbf{E}: (\mathbf{r},t) \mapsto \mathbf{E}(\mathbf{r},t)$, electric displacement $\mathbf{D}: (\mathbf{r},t) \mapsto \mathbf{D}(\mathbf{r},t)$, magnetic field $\mathbf{B}: (\mathbf{r},t) \mapsto \mathbf{B}(\mathbf{r},t)$ and auxiliary field $\mathbf{H}: (\mathbf{r},t) \mapsto \mathbf{H}(\mathbf{r},t)$ in a medium are described by
+> *Theorem 5*: The electric field $\mathbf{E}: (\mathbf{r},t) \mapsto \mathbf{E}(\mathbf{r},t)$, electric displacement $\mathbf{D}: (\mathbf{r},t) \mapsto \mathbf{D}(\mathbf{r},t)$, magnetic field $\mathbf{B}: (\mathbf{r},t) \mapsto \mathbf{B}(\mathbf{r},t)$ and auxiliary field $\mathbf{H}: (\mathbf{r},t) \mapsto \mathbf{H}(\mathbf{r},t)$ in a medium are described by
 >
 > $$
 >   \begin{align*}
@@ -313,58 +272,3 @@ complying to the conservation of charge.
     or $\nabla \times \mathbf{H} = \mathbf{J}_f + \partial_t \mathbf{D}$.
 
 Which require constitutive relations for closure.
-
-### Waves
-
-The posed equations in *Theorem 6* may de decoupled as well, obtaining, as expected, three-dimensional wave equations describing the electric and magnetic field.
-
-> *Theorem 7*: In a linear, homogeneous, dielectric medium where $\rho_f=0$ and $\mathbf{J}_f=\mathbf{0}$, the electric displacement $\mathbf{D}: (\mathbf{r},t) \mapsto \mathbf{D}(\mathbf{r},t)$ and the auxiliary field $\mathbf{H}: (\mathbf{r},t) \mapsto \mathbf{H}(\mathbf{r},t)$ adhere to
->
-> $$
->   \begin{align*}
->       \nabla^2 \mathbf{D} &= \frac{1}{v^2} \partial_t^2 \mathbf{D},\\
->       \nabla^2 \mathbf{H} &= \frac{1}{v^2} \partial_t^2 \mathbf{H},
->   \end{align*}
-> $$
->
-> with a wave speed $v = \frac{1}{\sqrt{\mu \epsilon}}$. 
-
-??? note "Proof:"
-
-    For the electric displacement in a linear, homogeneous, dielectric medium where $\rho_f=0$ and $\mathbf{J}_f=\mathbf{0}$ we have
-
-    $$
-        \begin{align*}
-            \nabla^2 \mathbf{D} &= \epsilon \nabla^2 \mathbf{E},\\
-            &= - \epsilon \nabla \times (\nabla \times \mathbf{E}),\\
-            &= \epsilon \nabla \times \partial_t \mathbf{B},\\
-            &= \epsilon \partial_t (\nabla \times \mathbf{B}),\\
-            &= \mu \epsilon \partial_t (\nabla \times \mathbf{H},\\
-            &= \mu \epsilon \partial_t^2 \mathbf{D}.
-        \end{align*}
-    $$
-
-    For the auxiliary field in a linear, homogeneous, dielectric medium where $\rho_f=0$ and $\mathbf{J}_f=\mathbf{0}$ we have
-
-    $$
-        \begin{align*}
-            \nabla^2 \mathbf{H} &= \nabla^2 \mathbf{H},\\
-            &= - \nabla \times (\nabla \times \mathbf{H}),\\
-            &= - \nabla \times \partial_t \mathbf{D},\\
-            &= - \partial_t (\nabla \times \mathbf{D}),\\
-            &= - \epsilon \partial_t (\nabla \times \mathbf{E}),\\
-            &= \epsilon \partial_t^2 \mathbf{B},\\
-            &= \mu \epsilon \partial_t^2 \mathbf{H}.
-        \end{align*}
-    $$
-
-Equivalently we may write *Theorem 7* in terms of the electric and magnetic fields in the linear, homogeneous, dielectric medium where $\rho_f = 0$ and $\mathbf{J}_f = \mathbf{0}$
-
-$$
-    \begin{align*}
-        \nabla^2 \mathbf{E} = \frac{1}{v^2} \partial_t^2 \mathbf{E},\\
-        \nabla^2 \mathbf{B} = \frac{1}{v^2} \partial_t^2 \mathbf{B},
-    \end{align*}
-$$
-
-since $\mathbf{D} = \epsilon \mathbf{E}$ and $\mathbf{B} = \mu \mathbf{H}$. 

docs/physics/spacetime/general-formalism.md

@@ -1,140 +0,0 @@
-# Relativistic formalism of mechanics
-
-From now on, we refer to space and time as spacetime. 
-
-## Fundamental assumptions
-
-> *Postulate 1*: spacetime is continuous.
-
-Implying that there is no fundamental limit to the precision of measurements of spatial positions, velocities and time intervals. 
-
-> *Postulate 2*: there exists a [neighbourhood]() in spacetime in which the axioms of [Euclidean]() geometry hold.
-
-A reformulation of the postulate in the Newtonian formalism compatible with the new formulation. 
-
-> *Postulate 3*: all physical axioms have the same form in all inertial frames.
-
-This principle is dependent on the definition of an inertial frame, which in my view is not optimal. It will have to be improved.
-
-> *Principle 1*: spacetime is not instantaneous.
-
-Implying that there exists a maximum speed with which information can travel. 
-
-> *Axiom 1*: spacetime is represented by a torsion-free pseudo Riemannian manifold $M$ with 3 spacial dimensions and 1 time dimension.
-
-Torsion-free means that $\mathbf{T} = \mathbf{0}$, the [torsion tensor]() is always zero.
-
-## Lorentz transformations
-
-Will be added later.
-
-## Results from the fundamental assumptions
-
-> *Theorem 1*: let $\bm{g} \in \Gamma(\mathrm{TM})$ be the pseudo Riemannian inner product on $\mathrm{TM}$, then it follows that from [Hamilton's principle]() that the covariant derivative is equal to zero:
->
-> $$
->   \forall i \in \{1, 2, 3, 4\}: D_i \bm{g} = \mathbf{0},
-> $$
->
-> which is called *metric compatibility*.
-
-??? note "*Proof*:"
-
-    Will be added later.
-
-A linear connection $\nabla$ on a torsion-free pseudo Riemannian manifold with metric compatibility is called the **Levi-Civita connection** with its linear connection symbols denoted as the **Christoffel symbols**.
-
-> *Theorem 2*: the Christoffel symbols $\Gamma_{ij}^k$ (of a Levi-Civita connection) are covariantly symmetric 
->
-> $$
->   \Gamma_{ij}^k = \Gamma_{ji}^k,
-> $$
->
-> for all $(i,j,k) \in \{1,2,3,4\}^3$, and may be given by
->
-> $$
->   \Gamma_{ij}^k = \frac{1}{2} g^{kl} (\partial_i g_{ij} + \partial_j g_{il} - \partial_l g_{ij}),
-> $$
->
-> for all $\bm{g} = g_{ij} dx^i \otimes dx^j \in \Gamma(\mathrm{TM})$. 
-
-??? note "*Proof*:"
-
-    Will be added later.
-
-Similarly, we have the following.
-
-> *Proposition 1*: let $\mathbf{R}: \Gamma(\mathrm{T^*M}) \times \Gamma(\mathrm{TM})^3 \to F$ be the Riemann curvature tensor on a manifold $M$ over a field $F$, defined under the Levi-Civita connection. Then it may be decomposed by
->
-> $$
->   \mathbf{R} = \frac{1}{8} R^i_{jkl} (\partial_i \wedge dx^j) \vee (dx^k \wedge dx^l).
-> $$
-
-??? note "*Proof*:"
-
-    Will be added later.
-
-Such that $R^i_{jkl}$ has a dimension of 
-
-$$
-    \frac{4^2 (4^2 - 1)}{12} = 20.
-$$
-
-## Curvature
-
-> *Definition 1*: let $\mathbf{W}: \Gamma(\mathrm{TM}) \times \Gamma(\mathrm{TM}) \to F$ denote the **Ricci tensor** which is defined as 
->
-> $$
-> \begin{align*} 
->     \mathbf{W} &= \frac{1}{2} R_{ijk}^k dx^i \vee dx^j,\\
->                 &= \frac{1}{2} W_{ij} dx^i \vee dx^j,
-> \end{align*}
-> $$
->
-> with $R_{ijk}^k$ the contracted Riemann holor and let $W$ be the **Ricci scalar** be defined as $W = W_{ij} g^{ij}$ with $g^{ij}$ the dual metric holor.
-
-The Ricci tensor and scalar are normally denoted by the symbol $R$ but this would impose confusion with the curvature tensor, therefore it has been chosen to assign symbol $W$ to the Ricci tensor and scalar.
-
-The **Ricci tensor** is a contraction (simplification) of the Riemann curvature tensor. It provides a way to summarize the curvature of a manifold by focusing on how volumes change in the presence of curvature. The **Ricci scalar** summarizes the curvature information contained in the **Ricci tensor**.
-
-> *Definition 2*: let $\mathbf{G}: \Gamma(\mathrm{TM}) \times \Gamma(\mathrm{TM}) \to F$ denote the **Einstein tensor** which is defined as
->
-> $$
->   \mathbf{G} = \mathbf{W} - \frac{1}{2} W \bm{g},
-> $$
->
-> with $\mathbf{W}$ the Ricci tensor, $\bm{g}$ the metric tensor and $W$ the Ricci scalar.
-
-The **Einstein tensor** encapsulates the curvature of the manifold while satisfying the posed conditions (Lovelock's theorem). Such as the following proposition.
-
-> *Proposition 2*: the Einstein tensor $\mathbf{G}: \Gamma(\mathrm{TM}) \times \Gamma(\mathrm{TM}) \to F$ has the following properties
->
-> 1. $\mathbf{G} = G_{|ij|} dx^i \vee dx^j$,
-> 2. $D_i \mathbf{G} = 0$.
-
-??? note "*Proof*:"
-
-    Will be added later.
-
-## Energy and momentum
-
-> *Definition 3*: let $\mathbf{T}: \Gamma(\mathrm{T^*M}) \times \Gamma(\mathrm{T^*M}) \to F$ denote the **energy momentum tensor** which is defined by the following properties,
->
-> 1. $\mathbf{T} = T^{|ij|} \partial_i \vee \partial_j \in \bigvee^2(\mathrm{TM})$,
-> 2. $D_i \mathbf{T} = 0$.
-
-Property 1. is a result of the zero torsion axiom and property 2. is the demand of conservation of energy and momentum.
-
-The **energy momentum tensor** describes the matter distribution at each event in spacetime. It acts as a *source* term.
-
-## Einstein field equations
-
-> *Axiom 2*: the Einstein tensor $\mathbf{G}: \Gamma(\mathrm{TM}) \times \Gamma(\mathrm{TM}) \to F$ relates to the energy momentum tensor $\mathbf{T}: \Gamma(\mathrm{T^*M}) \times \Gamma(\mathrm{T^*M}) \to F$ by
->
-> $$
->   \mathbf{G} + \Lambda \bm{g} = \kappa \mathbf{T},
-> $$
->
-> with $\kappa = \frac{8 \pi G}{c^4}$ and $\Lambda, G$ the cosmological and gravitational constants respectively.
-
-This equation (these equations) relate the geometry of spacetime to the distribution of matter within it. For a given $\mathbf{T}$ the system of equations can solve for $\bm{g}$ and vice versa.

docs/physics/spacetime/special-formalism.md

@@ -1,320 +0,0 @@
-# Special formalism of spacetime
-
-The assumption had been that the three-dimensional geometry of the universe was distinct from time. However, with the uncovering of the dynamics of electromagnetism, light has been found to have a constant propagation speed $c$ in vacuum. This finding, together with the principle of relativity, refutes the absoluteness and independence of space and time. We, thus, require to fuse space and time into four-dimensional **spacetime**. 
-
-In the special formalism of spacetime we only consider spacetime in its non-perturbed (flat) state. 
-
-## Geometric postulates
-
-> *Special postulate*: Flat spacetime is described by a *differential manifold* $(\mathrm{M},\bm{\eta})$ with a metric 
-> 
-> $$
->   \bm{\eta} = c^2 \mathbf{d}t \otimes \mathbf{d} t - \sum_i \mathbf{d}x^i \otimes \mathbf{d}x^i
-> $$ 
->
-> and $\dim \mathrm{M} = 4$ called **Minkowski spacetime** and contains points called **events**. 
-
-We may now state that the evolution of a body in **Minkowski spacetime** is a sequence of events constituting a smooth curve in $\mathrm{M}$ that we call the **world-line** of that body. 
-
-Such a world-line $\gamma: \mathscr{D}(\gamma) \to \mathrm{M}: \lambda \mapsto \gamma(\lambda)$ is thus a smooth curve on the manifold parameterized by an open interval $\mathscr{D}(\gamma) \subset \mathbb{R}$. If $\gamma$ is a geodesic on $\mathrm{M}$ then the attached reference system to $\gamma$ may be denoted as an **inertial reference system**. 
-
-> *Princple of relativity*: All physical axioms are of identical form in all inertial reference systems.
-
-Generally there is a second statement considering the constancy of the speed of light $c$ in all inertial reference systems. But this statement is already implicitly included in the principle of relativity. 
-
-## Lorentz transformation
-
-> *Theorem 1*: Let $X$ and $Y$ be two inertial reference systems. The transformation $\psi: X \to Y$ obeys the *principle of relativity* and the *special postulate* if $\psi \in \mathrm{O}(1,3)$, i.e. the transformation is in the **Lorentz group**. 
-
-??? note "Proof:"
-
-    I am not sure if there exists a formal proof for this notion, hopefully there is...
-
-The transformations in the (non-abelian) Lorentz group are called **Lorentz transformations** and they enable us to relate the inertial reference systems to each other. The Lorentz group consists of rotations, reflections and boosts. For translations we need to extend the Lorentz group to the Poincaré group.
-
-### Rotation
-
-Let $\psi_\theta$ denote a rotation in the $x,y$ plane with respect to an angle $\theta$. The indices of $\psi_\theta$ may then be given by
-
-$$
-    (\psi_\theta)_\nu^\mu = \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & \cos \theta & \sin \theta & 0\\ 0 & -\sin\theta & \cos\theta & 0\\ 0 & 0 & 0 & 1\end{pmatrix}.
-$$
-
-### Reflection
-
-Let $\psi$ denote a time reversal, the indices of $\psi$ may then be given by
-
-$$
-    \psi_\nu^\mu = \begin{pmatrix} -1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{pmatrix}.
-$$
-
-### Boost
-
-Let $\psi_\xi$ denote a boost in the $x$ direction of speed $v = \tanh\xi$. The indices of $\psi_\xi$ may then be given by
-
-$$
-    (\psi_\xi)^\mu_\nu = \begin{pmatrix} \cosh\xi & -\sinh\xi & 0 & 0\\ -\sinh\xi & \cosh\xi & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{pmatrix}.
-$$
-
-## Poincaré transformation
-
-> *Theorem 2*: Let $X$ and $Y$ be two inertial reference systems. The transformation $\psi: X \to Y$ obeys the *principle of relativity* and the *special postulate* if $\psi \in \mathbb{R}^{1,3} \times \mathrm{O}(1,3)$, i.e. the transformation is in the **Poincaré group**.
-
-??? note "Proof:"
-
-    Proof follows from Lorentz transformation, this is just a semidirect product with the spacetime translations group $\mathbb{R}^{1,3}$. 
-
-The Poincaré group extends the Lorentz group with translations on $\mathrm{M}$, denoted with the semidirect product of the spacetime translations group $\mathbb{R}^{1,3}$ and the Lorentz group $\mathrm{O}(1,3)$. The transformations in the Poincaré group are called **Poincaré transformations**.
-
-### Translation
-
-Let $\psi$ denote a translation of $\mathbf{a} \in X$, then
-
-$$
-    \psi(\mathbf{x}) = \mathbf{x} + \mathbf{a},
-$$
-
-for all $\mathbf{x} \in X$. 
-
-## Causal structure 
-
-Consider a light cone in the origin of an inertial reference system $X$. Then the subsystem $A \subset X$ inside the cone has the property that 
-
-$$
-    \forall \mathbf{a} \in A: \|\mathbf{a}\|^2 > 0,
-$$
-
-such that all events in $A$ are **timelike**. The world-lines of massive bodies that intersect the origin are always in $A$.
-
-The subsystem $B \subset X$ on the surface of the cone has the property that
-
-$$
-    \forall \mathbf{b} \in B: \|\mathbf{b}\|^2 = 0,
-$$
-
-such that all events in $B$ are **lightlike**. The world-lines of massless bodies that intersect the origin are always in $B$.
-
-Finally, the subsystem $C \subset X$ outside the cone has the property that
-
-$$
-    \forall \mathbf{c} \in C: \|\mathbf{c}\|^2 < 0,
-$$
-
-such that all event in $C$ are **spacelike**. 
-
-A trivial remark is that $A \cup B \cup C = X$ and $A \cap B \cap C = \empty$. 
-
-For timelike events we may define a proper time $\tau$. The proper time between two timelike events $\mathscr{A}$ and $\mathscr{B}$ measures the time as observed by an inertial observer on the geodesic connecting $\mathscr{A}$ and $\mathscr{B}$ such that
-
-$$
-    \tau_{\mathscr{B}\mathscr{A}} = \|\mathscr{B} - \mathscr{A}\|.
-$$
-
-## Mass, energy and momentum
-
-In the consideration of the world-lines of massive bodies it is convenient to use the proper time $\tau$ as the parameter. By choosing a suitable $\lambda$ we may invert and express $\lambda$ in terms of the proper time $\lambda: \tau \mapsto \lambda(\tau)$, such that our world-line $\gamma$ can be expressed in terms of $\tau$, i.e. $\gamma: \tau \mapsto \gamma(\tau)$. 
-
-The vector tangent to this world-line is known as the **four-velocity**
-
-$$
-    \mathbf{u} = d_\tau \gamma \overset{\mathrm{def}}{=} \mathbf{d}_\tau,
-$$
-
-with 
-
-$$
-    \|\mathbf{u}\| = c.
-$$
-
-This absolute normalization is a reflection of the fact that the four-velocity is a velocity in spacetime, through which one always travels at the same rate.
-
-A related concept is the **four-momentum** which in terms of the action $S$ may be defined as
-
-$$
-    \mathbf{p} = \mathbf{d}S,
-$$
-
-with 
-
-$$
-    \|\mathbf{p}\|^2 = \Bigg(\frac{E}{c}\Bigg)^2 - \sum_i p_i p_i.
-$$
-
-The **four-momentum** thus extends the notion of momentum to spacetime by the consideration of an inherent energy $E$ tied to the constant rate at which one travels through spacetime. A second definition thus follows (linear approximation) in terms of the four-velocity
-
-$$
-    \mathbf{p} = m \mathbf{u},
-$$
-
-with $m$ the (rest) mass of the body. 
-
-> *Theorem 3*: The mass $m$, energy $E$ and momentum $p_i$ of a massive body is related by
->
-> $$
->   E^2 = \big(mc^2\big)^2 + \sum_i p_i p_i.
-> $$
-
-??? note "Proof:"
-
-    We have from the definition of the four-momentum in terms of the mass and the four-velocity that
-
-    $$
-        \begin{align*}
-            \|\mathbf{p}\| &= \|m\mathbf{u}\|,\\
-            &= mc,
-        \end{align*}
-    $$
-
-    and from the definition of the four-momentum in terms of the action (energy and momentum) that
-
-    $$
-        \|\mathbf{p}\|^2 = \Bigg(\frac{E}{c}\Bigg)^2 - \sum_i p_i p_i,
-    $$
-
-    thus
-
-    $$
-        \Big(mc\Big)^2 = \Bigg(\frac{E}{c}\Bigg)^2 - \sum_i p_i p_i,
-    $$
-
-    imposes
-
-    $$
-        E^2 = \big(mc^2\big)^2 + \sum_i p_i p_i.
-    $$
-
-In the particular case that the momentum of a body is zero we obtain the (reasonably famous) mass energy equivalence
-
-$$
-    E = mc^2.
-$$
-
-## Force
-
-Extending the consideration of the world-lines of massive bodies in spacetime we may define the **four-acceleration** in terms of the four-velocity as
-
-$$
-    \mathbf{a} = d_\tau \mathbf{u},
-$$
-
-such that
-
-$$
-    0 = d_\tau \langle \mathbf{u}, \mathbf{u}\rangle = 2 \langle \mathbf{u}, \mathbf{a}\rangle \implies \langle \mathbf{u}, \mathbf{a}\rangle = 0.
-$$
-
-The four-acceleration can only change the direction of the four-velocity, as the rate at which one travels through spacetime must always stay constant $(\|\mathbf{u}\| = c)$. 
-
-A related concept is the **four-force** which in terms of the four-momentum may be defined as
-
-$$
-    \mathbf{f} = d_\tau \mathbf{p} \overset{\mathrm{def}}{=} m d^2_\tau \mathbf{a}.
-$$
-
-In the case that a constant force $F$ in the $x$ direction acts on a body we have that the four-force on that body adheres to
-
-$$
-    \|\mathbf{f}\|^2 = -F^2,
-$$
-
-obtained by considering the instantaneous reference system of the body, and thus its four-acceleration should adhere to
-
-$$
-    \|\mathbf{a}\|^2 = -(F/m)^2.
-$$
-
-Also called the **proper acceleration** of the body.
-
-Using the properties of the four-velocity and four-acceleration we may obtain the evolution of the four-acceleration, four-velocity and the position of the accelerated body
-
-$$
-    \begin{align*}
-        \mathbf{a} &= \xi c\sinh(\xi\tau) \bm{\partial}_t + \xi c \sinh(\xi\tau) \bm{\partial}_x,\\
-        \mathbf{u} &= c \cosh(\xi\tau) \bm{\partial}_t + c \sinh(\xi\tau) \bm{\partial}_x,\\
-        \mathbf{x} &= \frac{c}{\xi} \sinh(\xi\tau) \bm{\partial}_t + \frac{c}{\xi} \cosh(\xi\tau) \bm{\partial}_x,
-    \end{align*}
-$$
-
-with respect to an inertial reference system intersecting at $\tau = 0$ and $\xi = \frac{F}{mc}$. 
-
-We thus obtain hyperbolic motion as one accelerates with a constant proper acceleration, an interesting property of this motion is that it defines a horizon (Rindler horizon), i.e. no timelike or light like geodesics between events behind this horizon and the accelerating body exist. 
-
-??? note "Proof:"
-    
-    We consider
-
-    $$
-        \begin{align*}
-            c^2 &= \|\mathbf{u}\|^2 &= (u^t)^2 - (u^x)^2,\\
-            0 &= \langle \mathbf{u}, \mathbf{a}\rangle &= u^t a^t - u^x a^x,\\
-            -(F/m)^2 &= \|\mathbf{a}\|^2 &= (a^t)^2 - (a^x)^2,
-        \end{align*}
-    $$
-
-    then
-
-    $$
-        \begin{align*}
-            u^t a^t &= u^x a^x,\\
-            (u^t a^t)^2 &= \big((u^t)^2 - c^2\big)\big((a^t)^2 + (F/m)^2\big),\\
-            &= (u^t a^t)^2 - c^2 (a^t)^2 + (u^t)^2 (F/m)^2 - c^2 (F/m)^2,
-        \end{align*}
-    $$
-
-    implies
-
-    $$
-        \begin{align*}
-            (a^t)^2 &= \xi^2 \big((u^t)^2 - c^2\big),\\
-            &= \xi^2 (u^x)^2,
-        \end{align*}
-    $$
-
-    with $\xi = \frac{F}{mc}$.
-
-    Obtains
-
-    $$
-        \begin{align*}
-            a^t &= \pm \xi u^x,\\
-            a^x &= \pm \xi u^t,
-        \end{align*}
-    $$
-
-    and since we are moving in the positive $x$ direction we only take $(+)$, now using the fact that $a^\mu = d_\tau u^\mu$ we have
-
-    $$
-        d^2_\tau u^x = \xi^2 u^x.
-    $$
-
-    With as solution
-
-    $$
-        u^x(\tau) = \chi_1 e^{\xi\tau} + \chi_2 e^{-\xi\tau},
-    $$
-
-    and using $u^t = \frac{1}{\xi} d_\tau u^x$ obtains
-
-    $$
-        u^t(\tau) = \chi_1 e^{\xi\tau} - \chi_2 e^{-\xi\tau}.
-    $$
-
-    Applying the boundary conditions obtains
-
-    $$
-        \begin{align*}
-            u^x(0) = 0 &\implies \chi_1 = - \chi_2,\\
-            \|\mathbf{u}\|^2 = c^2 &\implies \chi_1 = \frac{c}{2},
-        \end{align*}
-    $$
-
-    Such that
-
-    $$
-        \begin{cases}
-            u^t(\tau) &= \frac{c}{2} \Big(e^{\xi\tau} + e^{-\xi\tau}\Big) &= c \cosh\xi\tau,\\
-            u^x(\tau) &= \frac{c}{2} \Big(e^{\xi\tau} - e^{-\xi\tau}\Big) &= c \sinh\xi\tau,
-        \end{cases}
-    $$
-
-    and the rest follows.

mkdocs.yml

@@ -187,8 +187,13 @@ nav:
       - 'Hamiltonian mechanics': 
         - 'Hamiltonian formalism': physics/classical-mechanics/hamiltonian-mechanics/hamiltonian-formalism.md
         - 'Equations of Hamilton': physics/classical-mechanics/hamiltonian-mechanics/equations-of-hamilton.md
-
+    - 'Relativistic mechanics':
-
+      - 'Relativistic formalism': physics/relativistic-mechanics/relativistic-formalism.md
+      - 'Schwarzschild geometry': physics/relativistic-mechanics/schwarzschild-geometry.md
+      - 'Kerr geometry': physics/relativistic-mechanics/kerr-geometry.md
+      - 'Wave geometry': physics/relativistic-mechanics/wave-geometry.md
+#    - 'Quantum mechanics':
+#    - 'Statistical mechanics':
     - 'Electromagnetism': 
       - 'Electrostatics': physics/electromagnetism/electrostatics.md
       - 'Magnetostatics': physics/electromagnetism/magnetostatics.md
@@ -201,16 +206,6 @@ nav:
         - 'Interference': physics/electromagnetism/optics/interference.md
         - 'Diffraction': physics/electromagnetism/optics/diffraction.md
         - 'Polarisation': physics/electromagnetism/optics/polarisation.md
-#    - 'Quantum mechanics':
-#    - 'Statistical mechanics':
-    - 'Spacetime':
-      - 'Special formalism': physics/spacetime/special-formalism.md
-      - 'General formalism': physics/spacetime/general-formalism.md
-      - 'Schwarzschild geometry': physics/spacetime/schwarzschild-geometry.md
-      - 'Kerr geometry': physics/spacetime/kerr-geometry.md
-      - 'Isotropic geometry': physics/spacetime/isotropic-geometry.md
-#    - 'Relativistic mechanics':
-#      - 'Relativistic formalism': physics/relativistic-mechanics/relativistic-formalism.md
 #   - 'Thermodynamics':
 #     - 'Classical thermodynamics'
 #     - 'Statistical thermodynamics'