Noether's theorem

A thrown stone arcs through the air, but a planet around the Sun follows a closed curve. Why an ellipse, and not a circle or a spiral? Kepler's first law provides the answer, but its usual proof requires heavy computation. This work offers a shorter path — through a single simple ratio of velocities.

The Kerr-NUT black hole is not just a cosmic vacuum cleaner. It's a solution to Einstein's equations where the hole has mass, rotation, and a strange NUT parameter. This parameter behaves like a magnetic charge for gravity: seemingly invisible, yet it alters the properties of surrounding space.

A pendulum in a noisy room is constantly jostled by people passing by. Its motion is a mix of regular swings and random jerks. Something similar happens with weak gravitational waves against the backdrop of an expanding universe. In Einstein's general relativity, these waves are tiny ripples in spacetime.

A pot of boiling water is covered with two lids — an inner and an outer one. The thermometer under each shows its own temperature, but their harmonic mean (a special averaging method where you add the reciprocals) always remains unchanged. Heat seems to obey an invisible rule.

Quantum calculations often resemble assembling a complex mechanism: a physicist knows a part will fit, but can't always explain why. One favorite trick is to insert a 'unity' into the middle of a formula — not an ordinary one, but one built from an infinite set of all possible states of the system. This turns a cumbersome expression into a chain of simple integrals.

10⁻¹²⁰. That's not just a small number: it's a chasm. That's the ratio of dark energy density to the Planck density, the natural scale of quantum gravity. The colossal difference of 120 orders of magnitude has long seemed an unsolvable puzzle. But the answer might lie in a drama played out at the dawn of time.

A stable system behaves like a ball in a bowl: give it a push, and it rolls back to the bottom. The spread of possible trajectories, uncertainty, shrinks over time. But the speed depends on the slope of the bowl's walls. In complex systems, this 'bowl' has sections of varying steepness.

Imagine a green beam that, passing through an ingenious optical setup, twists into a three-dimensional funnel of light — an optical vortex. At its heart, like in the eye of a hurricane, the intensity drops to zero. It is there that photons disappear.

An orchestra of billions of cesium atoms struck a single note, and it sounded for 17 hours without fading. That’s no exaggeration: Chinese physicists created a continuous time crystal with a record-breaking lifespan. A time crystal is a system that breaks time symmetry, spontaneously pulsating in an unchanging rhythm.

Inside planets and stars, in rotating oceans of plasma, turbulent flows, Coriolis forces, and magnetic tension collide. The central question is how the energy of this chaos crystallizes into a stable magnetic framework that can last billions of years.

Imagine a ball in an unfamiliar hall: two guests start from opposite corners. Their movements are not a smooth waltz but chaotic improvisation in search of rare treasures. Contrary to intuition, the outcome hinges not on tempo but on the geometry of the floor.

In a crystal that for centuries was called 'fool's gold,' physicists have spotted genuine treasure. The compound MnTe2 with a pyrite structure has become the stage for a quantum ballet: spin waves here twist into left- and right-handed vortices, with the twist direction firmly tied to momentum.

Eight hundred participants, firms and workers, enter the market. Each holds a strict preference list in mind. The challenge: pair up so that no one wants a do-over. In mathematics, this is called a game theory with a ban on repeat alliances, and its roots trace back to the work of John von Neumann and Oskar Morgenstern.

7 278 601 292 779 396 715. That’s exactly how many unlabeled chordal graphs, graphs with no induced cycles of length four or more, can be built on twenty vertices. This number isn’t a puzzle for a supercomputer; it’s the exact output of a new algorithm that treats symmetry as a tool, not an obstacle.

The explosion of a distant supernova told us something hard to believe: the universe is expanding faster and faster. The engine of this acceleration is dark energy — nearly 70% of everything that exists.

In the world of magnetism, a simple dichotomy long reigned: ferromagnets, where spins point in one direction, and antiferromagnets, where they line up in a checkerboard pattern, canceling each other out. But a recent 'third player' — altermagnets — forced a rewrite of the rules.

A billiard ball strikes another, and the sum of their velocities before and after the collision obeys strict rules. The same happens in the microscopic world: energy, momentum, and charge cannot simply vanish.

Imagine: superfluid helium-3 in a thin film receives a sudden heat shock. In that instant, a whole orchestra of collective oscillations awakens within it — like striking an invisible drum produces a multi-voiced chord. The Kibble–Zurek mechanism, familiar from the cracking of ice during rapid freezing, here acquires a quantum elegance.

In the late 1990s, the night sky delivered a surprise: Saul Perlmutter and Adam Riess found that the expansion of the Universe is accelerating. Ever since, the greatest riddle has been dark energy — a substance making up nearly 70% of the cosmos, yet evading laboratory study.

A quantum spin chain is like a string of beads, where each bead is a microscopic magnet pointing up or down. In the XXZ model, these little magnets interact differently along the three axes, and at special parameter values, roots of unity, the system awakens with a hidden memory.

Ordinary neural networks learn from examples, like a person memorizing tunes by ear. But networks with physical laws embedded from the start resemble a musician who knows notes and harmony: they can play a new piece from just a few chords. Instead of mindless memorization, they use equations like energy conservation or motion.

When a grain of sand falls on a table, its impact is predictable. But if you squeezed it down to the size of an atomic nucleus, the familiar laws of gravity would break down — the force of gravity would grow without bound.

A taut sheet wrinkles. If you tug a corner, a smooth wave runs across; if you sharply bunch the fabric, a sharp crease forms. Space and time in Einstein's equations behave much the same way. Gravitational waves resemble the gentle ripples of the sheet, while shock fronts are like stiff folds where matter's properties change abruptly.

A drop of honey spreads slowly across a table, while water spreads quickly. The resistance to flow, viscosity, differs from one fluid to another. But what if the viscosity changes on the fly, as the substance compresses or expands?

Since time immemorial, physicists have dreamed of tunnels piercing the cosmos — wormholes, capable of connecting far-flung corners of the Universe in an instant.

Throw a stone into a pond, and ripples spread across the water. Throw two stones nearby, and the ripples intersect, creating a pattern of reinforced and canceled waves. That's interference.

A stone falls into a pond — ripples spread across the water. In physics, a similar conservation law works for energy and momentum: what goes in comes out. But in curved spacetime, where gravity rules, conserved quantities are harder to define.

A hot spoon in a cup of tea — heat runs along the metal from the hot end to the cold one. Textbooks describe this process as atoms jiggling and passing on their jolts to each other. But atoms don't just jiggle: they also spin, like tiny tops. Until now, physicists had no common language to describe both motions at once. The authors of the work built such a language.

If you pour sand into a glass, it settles in a flat layer. But instead of sand, take the tiniest particles of gas: in a gravitational field they don't just fall: each particle races along its own trajectory, collides with neighbors, and all this commotion creates pressure.

Cells on the surface of an embryo resemble grains of rice that can align in one direction. This order is called nematic. It arises when cells start acting in concert, like a school of fish swimming in the same direction.

A neutron star is the ultra-dense remnant of a supernova explosion, whose existence was predicted by Fritz Zwicky back in the 1930s. Later, Jocelyn Bell Burnell detected them as pulsars — rotating sources of strictly periodic radio signals.

A quantum particle, an inhabitant of the elementary particle world, resembles a coin spinning in the air: neither heads nor tails, but a blend of possibilities. Yet when it lands on the table, the outcome becomes definite. What serves as the table in a measurement?

When entropy reaches its maximum, the play is over. The curtain falls, the actors leave, the scenery freezes. But the stage remains. So it is with time: the heat death of the universe erases the direction 'past—future', but not time itself.

Two translators listen to the same symphony. One hears only the fundamental tone: a clean, predictable melody. The other catches the overtones that color every note.

In an accelerator, two Higgs particles collide. For a fleeting moment, their fates intertwine, then separate, carrying away information about the encounter.