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With [[Brian Green]] from YouTube#

The following is a summary with equations only from the series with [[Brian Green]] streamed on YouTube during the [[COVID]] time. The original playlist could be found here. Episode numbers is in squared brackets.

Einstein Special Relativity (1905)#

Einstein's famous equation [01]#

\[ E = m.c^2 \]

Time Dilation [02]#

\[ \Delta_{stationary} = \gamma . \Delta_{Moving} \]
\[ \gamma = \frac {1} { \sqrt{1 - {v^2}/{c^2} }} \]

Lorentz Contraction [03]#

\[ L_{outside} = \frac {L_{moving}} {\gamma} \]

Relativity of Simultaneity [04]#

Relativistic Mass [06]#

\[ m_{Rel} = m_{o@rest}.\gamma \]
\[ E = m_{Rel}.c^2 \]

Relativistic Velocity Combination [07]#

\[ \Delta V = \frac {V_1 - V_2} {1 - \frac{V_1.V_2}{c^2} } \]
\[ if \ V_1 = c \to \Delta V = c \]

Math#

Is \(1=0.999999...\) [05]
$$ 3* (1/3 = 0.333333...) \to 1 = 0.999999... $$

or

\[ \begin{align} 10x &= 9.999999... \\ -\phantom{1}x &= 0.999999... \\ \hline 9x &= 9 \to x = 1 \end{align} \]

Fourier Series [16]#

Order of operations [22]#

\[ 8 - 2 / 2 x 3 + 4 = ? \]

Noether's Theorem [25]#

Symmetry and Conservation

[[Euler]] Identity [11]#

Euler's number: \(e = \lim\limits_{n\to\infty} (1+\frac{1}{n})^n\)
$$ e^{i\pi} + 1 = 0 $$

Quantum Mechanics#

  • \(h\): Planck constant
  • \(p\): momentum
  • \(\lambda\): wave length
  • \(\nu\): frequency

Photoelectric Effect [08]#

\[ E_{proton} = h.\nu \]

De Broglie Wavelength [09]#

\[ \lambda = \frac {h} {p} \]

Quantum Physics and Probability [10]#

Schrödinger's Equation [12]#

  • Non-relativistic, for 1 particle, 1 dimension,
    $$ i\hbar \frac {\partial\psi(x,t)} {\partial t} = - \frac {\hbar^2}{2m} \frac {\partial^2\psi(x,t)}{\partial x^2} + V(x)\psi(x,t) $$

  • Non-relativistic, Generalized [13], leads to a high dimensional spaces, for the wave function (\(\psi\)) to live in.

Quantum Entanglement [14]#

Planck Length, Mass, Time [15]#

notion of: Natural Units, length, speed, and time.

\(c^{\alpha}.\hbar^{\beta}.G^{\gamma}\)

\[ Planck Length = \sqrt{ \frac{\hbar G}{c^3} } \simeq 10^{-33} cm \]
\[ Planck Time = \sqrt{ \frac{\hbar G}{c^5} } \simeq 10^{-44} s \]
\[ Planck Mass = \sqrt{ \frac{\hbar C}{G} } \simeq 2x10^{-8} kg \]

Heisenberg's Uncertainty Principle [18]#

\[ (\Delta X)(\Delta P) \geq \frac {\hbar}2{} \]

Bell's Theorem and the Non-locality of the Universe [21]#

Quantum Entanglement

Ehrenfest's Theorem [23]#

Classical from Quantum

$$ m \frac {d^2 < x >} {dt^2} = <-\frac {\partial V}{\partial x}> $$
where \(<x>\) is average values

Newtonian Gravity#

Planetary Motion [17]#

Kepler, Newton, and Gravity

\[ F_G = \frac {GMm} {r^2} \]
\[ F = ma \]

Euler-Lagrange Equations [19]#

The Principle of Least Action

\[ \frac{d}{dt} \frac {\partial L} {\partial \dot{x_i}} - \frac {\partial L} {\partial x_i} = 0 \]

1,000,000,001 - 1000,000,000 = 1 [20]#

Einstein General Relativity (1907)#

General Theory of Relativity [26]#

The Essential Idea

$$
R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R =
\frac {8 \pi G} {c^4} T_{\mu \nu}
$$
* \(R_{\mu \nu}\): Ricci tensor
* \(R\): Scalar curvature
* \(T_{\mu \nu}\): Energy-Momentum Tensor

Curvature [27]#

Riemann curvature tensor

\[ R^\rho{}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho{}_{\nu\sigma} - \partial_\nu\Gamma^\rho{}_{\mu\sigma} + \Gamma^\rho{}_{\mu\lambda}\Gamma^\lambda{}_{\nu\sigma} - \Gamma^\rho{}_{\nu\lambda}\Gamma^\lambda{}_{\mu\sigma} \]

The Big Bang, and the Expansion of the Universe [28]#

  • Space-time: Homogeneous / Isotropic
    $$ \frac {(da/dt)^2} {a_{(t)}^2 } = \frac {8 \pi G \rho}{3} - \frac {k}{a^2} $$
    where \(\rho\) is energy density

Repulsive Gravity [29]#

Dark Energy, and Accelerated Expansion
\(\Lambda\)

What Banged? [30]#

Cosmic Inflation
Scale factor: \(a(t)\)

[[Black Holes]] [31]#

Schwarzschild metric (Schwarzschild solution)
$$ g = (1- \frac {2GM}{c^2 r})c^2 dt^2 - (1- \frac {2GM}{c^2 r})^{-1}dr^2 - r^2 d\Omega $$
* Newtonian gravitational potential
* Special values of \(r\)
- \(r = 0\) singularity
- \(r = r_s = 2GM/c^2\)
* \(r_s\): Schwarzschild radius (event horizon)
* \(r_s = 3 km\) for the Sun

Entropy and the Arrow of Time [32]#

  • Entropy is Measure of disorder
  • Boltzmann's entropy formula:
    $$ S = k \log{W} $$
  • The second law of thermodynamic, is a tendency.
  • Entropy could go down, but it is unlikely to happen