Physics · Gravitation and the universe · Grade 9

Scales of the universe

From quarks to the Universe: which laws of physics rule at each scale?

The situation

Imagine a person standing near the Tashkent TV Tower. If the camera moves away 10 times farther with each step, after 7 steps the whole Earth is visible, and after 21 steps the Milky Way galaxy. If the camera goes inwards, we see a cell, DNA, an atom, a nucleus and the quarks inside a proton. Over the whole journey the scale changes by about 10⁴² times. Are the laws of physics the same at every scale, or does each scale have its own “ruling” law?

Topic

Physical picture of the universe: scales. From a quark to a galaxy: a journey through the powers of ten

Key ideas

Powers of ten
Fundamental constants
Branches of physics

Formulas

l_{P} = \sqrt{\frac{ℏ G}{c^{3}}} \approx 1.6 \cdot 10^{- 35} m, t_{P} = \frac{l_{P}}{c} \approx 5.4 \cdot 10^{- 44} s

The Planck length and time — a scale built from three fundamental constants.

R_{s} = \frac{2 G M}{c^{2}}, λ_{C} = \frac{h}{m c}

Two limits: if a body is smaller than $R_{s}$ , it is a black hole; at scales smaller than $λ_{C}$ , quantum effects dominate.

{(\frac{v}{c})}^{2}, \frac{ℏ}{m v L}, \frac{G M}{r c^{2}}

Dimensionless numbers: if they are smaller than the required accuracy, classical (Newtonian) physics is enough.

\vec{r} = x \vec{i} + y \vec{j} + z \vec{k}, \vec{v} = \frac{d \vec{r}}{d t}, s = \int_{t_{1}}^{t_{2}} | \vec{v} | d t

Reference frame: the position vector, its derivative is the velocity, and the integral of the velocity is the distance travelled.

t = \frac{L}{c} : \frac{1.496 \cdot 10^{11} m}{3.00 \cdot 10^{8} m/s} \approx 499 s

The speed of light is finite: when we look far away, we see the past.

Theory

Powers of ten: space and time

The diameter of a proton is ≈ 1.7·10⁻¹⁵ m, and the diameter of the observable Universe is ≈ 8.8·10²⁶ m. Such numbers are convenient to write with powers of ten: each step is a factor of 10. Time is the same: light crosses a proton in ≈ 6·10⁻²⁴ s, while the age of the Universe is ≈ 4.4·10¹⁷ s (13.8 billion years). Units of length are chosen according to the scale: fm (nuclei), nm (molecules), m, km, AU (Earth–Sun), the light-year and the parsec (stars), Mpc (galaxies). Space and time are the stage of all phenomena; special relativity unites them into a single space-time.

Reference frame and position vector

To say where a body is, you need a reference frame: a reference body, coordinate axes and a clock. In the “Powers of ten” scene, at the scale of 10² m, the origin is at the foot of the TV tower: $x$ points east, $y$ up, $z$ south. The person's position is the position vector $\vec{r} = (- 120; 0; 80)$ m. The derivative of the position vector with respect to time is the velocity, and the integral of the velocity is the distance travelled. Velocity depends on the reference frame: a “stationary” person in Tashkent moves at ≈ 350 m/s with the Earth's rotation, at ≈ 30 km/s with the Earth around the Sun, and at ≈ 230 km/s with the Sun around the centre of the Galaxy.

The four interactions: a single picture of the world

All phenomena are explained by four interactions. The strong one holds quarks in protons and neutrons and nucleons in the nucleus; its range is ≈ 10⁻¹⁵ m. The weak one governs β-decay and the first stage of fusion in the Sun; ≈ 10⁻¹⁸ m. The electromagnetic one gives atoms, molecules, chemistry, biology, friction and elasticity; its range is infinite. Gravity is the weakest (≈ 10³⁶ times weaker than the electric repulsion between two protons), but because matter is usually electrically neutral, it dominates at large scales — planets, stars, galaxies. The history of physics is a history of unification: electricity and magnetism (Maxwell), electromagnetic and weak interactions (electroweak theory), the Standard Model; uniting gravity with quantum physics is still unsolved.

The cube of theories: the limits of classical physics

Each theory can be marked by three constants. In the limit $c \to \infty$ , $ℏ \to 0$ , $G \to 0$ — Newtonian mechanics. When $v$ approaches the speed of light — special relativity; when the action ( $m v L$ ) is close to $ℏ$ — quantum mechanics; both together — quantum field theory; strong gravity and $c$ — general relativity. Which theory is needed also depends on the accuracy: for GPS clocks even corrections of $10^{- 10}$ matter, while for a football 1 % accuracy is enough. The old theory remains the limiting case of the new one — this is the correspondence principle.

The microworld: milestones of atomism and particle physics

The size of an atom is ≈ 10⁻¹⁰ m (the Bohr radius is 5.29·10⁻¹¹ m), while the nucleus is ≈ 10⁻¹⁵ m — about 100 000 times smaller than the atom, yet it holds more than 99.9 % of the mass. The microworld is governed by fundamental constants: the elementary charge $e$ , the electron mass $m_{e}$ , Planck's constant $h$ , and the speed of light $c$ . The quantum (wave nature, spin, energy levels) and relativistic (antiparticles, $E = m c^{2}$ ) properties of particles appear together at this scale. Milestones: the electron (1897, J. J. Thomson), the nucleus (1911, E. Rutherford), the neutron and positron (1932), the quark model (1964), quarks “seen” in experiment (1968–1969, SLAC), W and Z bosons (1983, CERN), the top quark (1995), the Higgs boson (2012, LHC).

Physics, other sciences and society; the model

Physics studies the most general laws of nature through observation, experiment, models and theory; its language is mathematics. It is the foundation of chemistry (the structure of atoms and molecules), biology and medicine (biophysics, X-rays, MRI, PET), geology, astronomy and engineering. Electricity grids, the transistor and computers, lasers, GPS, nuclear power stations and solar panels are products of discoveries in physics; they have transformed society's productive forces. Model: sizes are approximate; the Moon and planets are enlarged to be visible; galaxies and the cosmic web are schematic; the positions of the TV tower and the person are arbitrary.

Did you know?

The Tashkent TV Tower is 375 m tall. It was put into operation on 15 January 1985 and from 1985 to 1991 was the fourth tallest tower in the world.
A “stationary” person beside the TV tower moves with the Earth's rotation at about 350 m/s — faster than the speed of sound.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

The Solar System — Grade 7
Newton's cannon: an artificial satellite — Grade 10
Special relativity — Grade 10
Law of universal gravitation — Grade 10
Relativistic dynamics — Grade 10
Kepler's laws — Grade 10
Physics of stars — Year 1
The expanding universe — Year 1
Black holes — Year 4