## Equations

### Einstein's Field Equations

R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} + \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^{4}} T_{\mu \nu}

The formula results in a set of 16 equations, while 10 of them are independent (non-symmetric). Vividly spoken $T_{\mu \nu}$ prescribes the appearance of $R_{\mu \nu}$ which in turn prescribes a body's trajectory along a worldline.

• $R_{\mu \nu}$: Ricci-Curvature-Tensor (Space curvature, gravitational field strength)
• $g_{\mu \nu}$: Metric Tensor (Potential of the gravity- and inertial field)
• $T_{\mu \nu}$: Stress-Energy-Tensor (Energy-, momentum- and stress density of all fields and bodys)
• $\Lambda$: Cosmological constant (Ideal liquid with negative pressure)
• $R$: Scalar curvature
• $G$: Gravitational constant
• $\mu, \nu$: Indexes [0,1,2,3]
• ### Relativistic Line Element

ds^{2} = \eta_{\mu \nu} dx^{\mu} dx^{\nu} = dx^{2}_1 + dx^{2}_2 + dx^{2}_3 - c^{2} dt^{2}
\eta_{\mu \nu} = diag(+, +, +, -) = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}
• $ds^2$: Invariant arc length of a worldline under Lorentz-Transformation. The worldline prescribes a bodys trajectory in space-time.
• $dx^{\mu \nu}$: Space-time coordinates
• ### Description for non-rotating Black Holes (Schwarzschild solution)

ds^{2} = - \left( 1-\dfrac{R_S}{r} \right) c^2 dt^2 + \dfrac{1}{1- \dfrac{R_S}{r}} dr^2 + r^2 d \theta^2 + r^2 sin^2 \theta \, d \phi^2
F_g \sim \dfrac{1}{r^3}

Observer: When a traveler is approaching the black hole's SSR $(r \rightarrow R_S)$ time elapses more and more slowly, while space bends and expands. When reaching $r = R_S$ time stands still and space becomes infinite. The traveler doesn't move anymore and appears infinitely red-shifted.
Traveler: When passing through the event horizon $(r < R_S)$ the algebraic signs of the first two terms inverse, thus causality and the dimensions of space $(3D \rightarrow 1D)$ and time $(1D \rightarrow 3D)$ turn. One has all the freedom moving through time, but is limited in space by only being able to move towards the singularity. Simultaneously, due to gravity, light falling into the black hole from the outside appears infinitely blue-shifted which results in a X-ray or Gamma-ray burst. The tidal forces become huge.

• $R_S$: Schwarzschild radius (SSR)
• $r$: Distance towards the singularity
• $\theta, \phi$: Angles for identification of direction
• $F_g$: Tidal force of the black hole
• ### Kerr-Newman metric

g_{11} = \frac{r^2 + a^2 cos^2 \theta}{r^2 + a^2 + Q^2 - 2mr}
g_{22} = r^2 + a^2 cos^2 \theta
g_{33} = \left( r^2 +a^2 - \frac{a^2 \left( Q^2 - 2mr \right) sin^2 \theta}{r^2 + a^2 cos^2 \theta} \right) sin^2 \theta
g_{34} = g_{43} = \frac{a^2 \left( Q^2 - 2mr \right) sin^2 \theta}{r^2 + a^2 cos^2 \theta}
g_{44} = - \left( 1+ \frac{Q^2 -2mr}{r^2 + a^2 cos^2 \theta} \right)

### Description for rotating Black Holes (Kerr solution)

Kr = R_{\kappa \lambda \mu \nu} R^{\kappa \lambda \mu \nu} = 48 \frac{m^2}{r^6} \frac{1- \left( \dfrac{a}{r} \right)^2 cos^2 \theta}{ \left[ 1+ \left( \dfrac{a}{r} \right)^2 cos^2 \theta \right]^6} \left[ 1-14 \left( \dfrac{a}{r} \right)^2 cos^2 \theta + \left( \dfrac{a}{r} \right)^4 cos^4 \theta \right]

### Kerr-Newman solution (with charge)

Kr = \frac{8}{\left( r^2 + a^2 cos^2 \theta \right)} \left[ 6m^2 \left( r^6 - 15a^2 r^4 cos^2 \theta + 15a^4 r^2 cos^4 \theta - a^6 cos^6 \theta \right) - 12m Q^2 r \left( r^4 - 10a^2 r^2 cos^2 \theta + 5a^4 cos^4 \theta \right) + Q^4 \left( 7r^4 - 34 a^2 r^2 cos^2 \theta + 7a^4 cos^4 \theta \right) \right]
• $Kr$: Kretschmann-Scalar
• $R_{\kappa \lambda \mu \nu}$: Riemann Tensor
• $Q$: Electric charge of the black hole (0...1)
• $r$: Normalized radial coordinate
• $\theta$: Arc length [rad] (0, $\pi$: poles, $\frac{\pi}{2}$: equator)
• Positive curvature in 3D-space: Volume increases
• Negative curvature in 3D-space: Volume decreases
• ### Mass-Energy Equivalence

E = \sqrt{m_{0}^2 c^{4} + p^2 c^2}
• $E = m_{0} c^{2}$ is only an approximation, derived from the linearization of the Lorentz factor, only valid for $v << c$.
• H. Lesch: "If and only if radiation is created that oscillates with a certain frequency $\nu$ equal to $\frac{m_{0} c^{2}}{h}$ where $m_{0}$ is the rest mass, a particle/antiparticle pair can emerge and so this equation is valid."
• The equation says: Mass can't be converted to energy, but mass is exactly the same as energy!
• You have two identical, relaxed springs. If one of them is being compressed, its mass increases and therefore it becomes heavier.
• A charged battery is heavier than a discharged.
• ### Relativistic Time Dilation

t' = \frac{t}{\sqrt{1 - \left( \frac{v}{c} \right)^{2}}} = t \sqrt{1 + \frac{2 \phi_{G}}{c^{2}}}

An observer, moving with a constant speed $\textbf{v}$, a time $\textbf{t}$ in the observed rest frame will be dilated to $\textbf{t'}$.
If both, the observer and the observed system remain in their inertial system, they must be treated equally and it's not possible to distinguish who is moving or to whom the effect shall be applied. If the observer changes his state of movement, he leaves his related inertial system and so this effect will be attributed to him - even retrospectively.

### Relativistic Length Contraction

l' = l \sqrt{1 - \left( \frac{v}{c} \right)^{2}}

An observer, moving with a constant speed $\textbf{v}$ along a path $\textbf{l}$ in the observed rest frame, will see this path being shortened to $\textbf{l'}$. (Details: see obove.)

### Equation of Continuity

\frac{\partial \rho}{\partial t} + \nabla \cdot \vec{j} = \sigma

The density $\boldsymbol{\rho}$ of some quantity decreases over time in a certain volume if the divergence $\boldsymbol{\nabla}$ of the flux $\textbf{j}$ of that quantity is positive.

• $\rho$: Density
• $j$: Flux
• $\sigma$: Production rate (= 0 if quantity is conserved)
• ### Schrödinger Equation

i \hbar \frac{\partial}{\partial t} \psi(t) = \hat{H} \psi(t)

While investigating a particle as is there is no need to describe particles as waves with relation to circular motion ( $2\pi$). Thus $\boldsymbol{\hbar}$ is used to describe particles' quantum of action, not only if angular frequencies are used!

### Einstein's Deviation of Light in the Gravitational Field

\theta = \dfrac{4GM}{Rc^2}
• $\theta$: Angle of deviation
• $G$: Gravitational constant
• $M$: Mass of the star
• $R$: Radius of the star
• $c$: Lightspeed
• ### Gaussian Law

\varepsilon_{0} \nabla \cdot \vec{E} = \rho

### Gaussian Law for Magnetism

\nabla \cdot \vec{B} = 0

\nabla \times \vec{E} = -\frac{d \vec{B}}{dt}

### Ampére's Circuital Law

\nabla \times \vec{B} = \mu_{0} \left( j + \varepsilon_{0} \frac{d \vec{E}}{dt} \right)

### Light Speed in Vacuum

c_{0} = \frac{1}{\sqrt{\varepsilon_{0} \mu_{0}}}

### Entropy Change

dS = \frac{dE}{T}

### Navier-Stokes Equation

{\rho \left( \frac{\partial \vec{u}}{\partial t} + \vec{u} \left( \nabla \cdot \vec{u} \right) \right) = - \nabla p + \nabla \cdot \left( \mu \left[ \nabla \cdot \vec{u} + \left( \nabla \cdot \vec{u} \right)^{T} \right] - \frac{2}{3} \mu \left( \nabla \cdot \vec{u} \right) \vec{I} \right)+f}

Term 1: Intertial forces, Term 2: Pressure forces, Term 3: Viscous forces, Term 4: External forces

• $u$: Velocity field
• $p$: Fluid pressure
• $\rho$: Fluid density
• $\mu$: Fluid dynamic viscosity
• ### General Momentum Balance

\sum_{i} \vec{F}_{i} + \vec{F}_{g} + \sum_{j} \vec{I}_{p,j} = \dot{p}
\sum_{i} m \vec{a}_{i} + m \vec{g} + \sum_{j} \vec{v}_{j} I_{m,j} = m \dot{v}

### Heat Transfer Equation (convective, diffusive)

\rho c_{p} \left( \frac{\partial T}{\partial t} + \vec{v} \cdot \nabla T \right) = \nabla \cdot \left( k \nabla T \right) +Q \left[ \frac{W}{m^{3}} \right]

|S| = \sqrt{\frac{\varepsilon_{0}}{\mu_{0}}} |E|^2

\rho_{\omega} = \frac{h \omega^3}{\pi^{2} c^{3}} \frac{1}{e^{\hbar \omega / k_{B} T} -1} = \frac{A_{n'n}}{B_{n'n}} \cdot \frac{1}{\frac{B_{nn'}}{B_{n'n}} \cdot e^{\hbar \omega / k_{B} T} -1} = \frac{A_{n'n}}{B_{nn'} \cdot e^{\hbar \omega / k_{B} T} - B_{n'n}}
• $\rho_{\omega}$: Spectral radiation density
• ### Einstein Coefficents

\frac{A_{n'n}}{B_{n'n}} = \frac{h \omega^{3}}{\pi^{2} c^{3}} = \frac{8\pi h}{\lambda^{3}}
\tau = \frac{1}{A_{n'n}}
B_{nn'} = B_{n'n}

Because the spontaneous transitions dramatically increase with higher frequencies of light, it is cumbersome to construct a Short-Wavelength-LASER. The solution for this problem is to increase the intensity of the radiation field. In MRI-Technology the probability for induced effects is close to certainty.

• $A_{n'n}$: Coeff. of spontaneous emission
• $A_{nn'}$: Coeff. of spontaneous absorption
• $B_{n'n}$: Coeff. of induced emission
• $B_{nn'}$: Coeff. of induced absorption
• $\tau$: Lifetime of an excited state
• ### Coefficent Form PDE

e_{a} \frac{\partial^2 u}{\partial t^2} + d_{a} \frac{\partial u}{\partial t} + \nabla \cdot \left( -c \nabla u - \alpha u + \gamma \right) +\beta \cdot \nabla u + au = f
• $c$: Diffusion coeff.
• $a$: Absorption coeff.
• $d_a$: Damping coeff.
• $e_a$: Mass coeff.
• $\alpha$: Flux convection coeff.
• $\beta$: Convection coeff.
• $\gamma$: Flux source
• $f$: Source term
• ### Laplace Equation

\nabla \cdot \left( -\nabla u \right) = 0

### Poisson Equation

\nabla \cdot \left( -c\nabla u \right) = f

### Wave Equation

e_{a} \frac{\partial^2 u}{\partial t^2} + \nabla \cdot \left( -c \nabla u \right) = f

### Helmholtz Equation

\nabla \cdot \left( -c \nabla u \right) + au = f

### Heat Equation

d_{a} \frac{\partial u}{\partial t} + \nabla \cdot \left( -c \nabla u \right) = f

### Convection-Diffusion Equation

d_{a} \frac{\partial u}{\partial t} + \nabla \cdot \left( -c \nabla u \right) + \beta \cdot \nabla u = f

### Boundary Conditions

• Dirichlet:
u = u_{0}
• Neumann:
\vec n \cdot \nabla u = u_{0}
• Neumann with zero flux:
\vec n \cdot \nabla u = 0
• Robin:
au + b\vec n \cdot \nabla u = c