Physics

Physical systems tend toward stable equilibrium configurations that minimize free energy or equivalently maximize total entropy under the relevant constraints.

Equations

Einstein’s special relativity: describes how measurements of space and time depend on an observer’s relative motion, while the laws of physics and the speed of light remain the same for all inertial observers; it also relates mass to energy: $E_0=m{c}^2$

Intuitively, the motion of an object with mass through space-time forms a four-velocity vector (composed of t,x,y,z) whose spacetime magnitude stays always c (speed of light).

Einstein’s general relativity: extends relativity to accelerating observers and describes gravity as spacetime curvature caused by mass-energy and momentum.

$G_{\mu \nu }+\mathrm{\Lambda }{g}_{\mu \nu }=\frac{8\pi G}{c^4}{T}_{\mu \nu }$

Schrödinger’s equation: governs the time evolution of a nonrelativistic quantum wavefunction.

$i\hslash \frac{\mathit{\partial }}{\mathit{\partial }t}\left|\psi \right(t)⟩=\hat{H}|\psi \left(t\right)⟩$

Heisenberg’s uncertainty principle: limits how precisely certain pairs of quantum properties can be simultaneously defined and measured.

$\mathrm{\Delta }x·\mathrm{\Delta }p\ge \frac{\hslash }{2}$

Forces

Gravity
Gravity is the interaction of mass-energy with spacetime. In general relativity, mass-energy curves spacetime, and objects follow paths through that curved spacetime. (attraction and repulsion -> only in universe expansion)

Electromagnetism
Electromagnetism acts between electrically charged particles. It causes both: attraction between opposite charges, repulsion between like charges.
Electrons have negative point charge and spin causing magnetic moment (aligned spins make a material magnetic). (attraction and repulsion)

Strong interaction
The strong interaction binds quarks together inside protons, neutrons and other hadrons.
A leftover effect called the residual strong force binds protons and neutrons together inside atomic nuclei, overcoming the electrical repulsion between positively charged protons at short distances. (attraction and repulsion)

Weak interaction
The weak interaction acts over extremely short distances and allows elementary particles to change type. For example, during beta decay, a neutron can transform into a proton while emitting an electron and an antineutrino.

Quantum Physics

In quantum field theory, particles are understood as localized, quantized excitations of underlying quantum fields (electron, quark, photon, gluon, Higgs fields, etc.).

A quantum system is described by a quantum state, often represented by a wavefunction. The wavefunction is not itself a probability distribution; it contains probability amplitudes. Its squared magnitude, $|\psi {|}^2$ gives the probability distribution for possible measurement outcomes.

For a nonrelativistic isolated system, the wavefunction evolves over time according to the Schrödinger equation.

When a measurement is made, an exact outcome is observed from the range of possible outcomes (superposition). In the traditional Copenhagen interpretation, this is described as wavefunction collapse. However, whether collapse is a literal physical process depends on the interpretation of quantum mechanics.

Symmetry and conservation laws

Continuous symmetries correspond to conservation laws.

By Noether’s theorem:
time-translation symmetry → conservation of energy
spatial-translation symmetry → conservation of momentum
rotational symmetry → conservation of angular momentum
gauge symmetries → interaction structures and conserved charges

Pauli exclusion principle

No two identical fermions can occupy the same quantum state at the same time.

Electrons are fermions, so in an atom no two electrons can have the same complete set of quantum numbers. That is why a single atomic orbital can hold at most two electrons, and those two must have opposite spin.

Important for:
atomic electron shells
chemistry
the size and rigidity of matter
electron degeneracy pressure
white dwarfs
neutron degeneracy pressure in neutron stars

Cosmology

The study of big structures like stars, planets and black holes.

Black Holes

How is it made? Just mass squashed into a tiny space? Is every black hole capable to grow infinitely? Hawking radiation?

Schwarzschild radius

Schwarzschild radius (what a coincidence that name is!) defined as the radius matter or light can not escape the pull of a blackhole.

$r_s=\frac{2GM}{c^2}$

If you compress a mass M inside that radius, an event horizon forms. For the Sun, $r_s$ is about 3 km. For Earth, it’s only about 9 mm. So if Earth’s entire mass were somehow compressed into roughly a marble-sized region, it would become a black hole.

Hawking radiation

Quantum effects imply that black holes are not perfectly black: they emit approximately thermal Hawking radiation with temperature

$T_H=\frac{\hslash c^3}{8\pi GM{k}_B}$

so, most importantly,

$T_H\propto \frac{1}{M}$.

Thus smaller black holes are hotter and evaporate faster, while larger black holes are colder and radiate extremely slowly. A solar-mass black hole has a temperature of only about

$6\times {10}^{-8},\mathrm{K}$,

far below the cosmic microwave background temperature of about $2.7,\mathrm{K}$, so present-day astrophysical black holes generally absorb more energy than they lose.

As a black hole loses mass, it gets hotter, causing faster radiation and accelerating evaporation. A stellar-mass black hole would take roughly ${10}^{67}\mathrm{y}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{s}$ to evaporate.

Hawking radiation is a strong theoretical prediction, but it has not yet been directly observed from a real gravitational black hole. The familiar “particle-antiparticle pair” explanation is only a simplified analogy, not the full quantum-field-theory derivation.

Because of this mechanism, we expect tiny (hypothetical) black holes to evaporate extremely quickly, potentially disappearing almost instantly after forming.

Questions

Why do electrons not repel but build bonds (in molecules)? Because positive attraction from nuclei protons dominates and the total quantum-mechanical energy of the combined system is lower.

Why do protons build atomic nuclei and not repel? Strong interaction binds them on small distances in the nuclei.

Open questions in physics itself

Gaps in current physics:

• No experimentally established quantum theory of gravity
• Unknown nature of dark matter
• Unknown nature of dark energy
• Matter–antimatter asymmetry
• Origin of neutrino masses
• Interpretation of quantum measurement
• Why fundamental constants have their observed values
• Whether the Standard Model is part of a deeper unified theory
• What happened at or before the earliest meaningful moments of the universe

#science