Klein-Gordon equation

An ordinary mirror swaps left and right. Many laws of physics don't notice the swap: they are mirror-symmetric. But some phenomena, like the weak interaction, break this symmetry. Until now, it was thought that gravity and light each break mirror symmetry in their own independent way.

A stone drops into a pond, and ripples spread across the water. In the same way, massive bodies generate gravitational waves, except it's space itself that ripples. So far, these waves have been caught by giant detectors that barely flinch as a wave passes through. But what happens on the microscopic scale, inside a tiny piece of matter?

The neck of an hourglass is the narrowest point where grains of sand change sides. In quantum gravity, there is a similar object: an instanton (an instantaneous solution connecting two states of space). It acts like a tunnel through which the geometry of the universe flows from one form to another.

A black hole isn't just a cosmic vacuum cleaner. If it rotates, it can amplify waves around it, much like a mountain echo repeats sound. For ultralight dark matter (particles billions of times lighter than an electron), this effect, called superradiance, can create a huge cloud.

A black hole doesn't just spin: it's surrounded by bristles of an extra field (the Kalb–Ramond field). This paper explores how a massive field (waves with mass) behaves near such a black hole when it's almost spinning at its maximum rate. The Klein–Gordon–Fock equation for this field was separated into two independent parts — radial and angular.

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.

Light bends in a lens — everyone knows that. Gravitational waves passing a massive body also curve their path. But the authors of this work looked further: what happens to the wave's very "color" — its polarization, that is, the direction of oscillation? Gravitational lensing is the deflection of…

Two neutron stars whirl in a close dance, each feeling the tidal field of its partner. It's like the Moon raising oceans on Earth, but here it's spacetime itself and the star's matter that deform. Astrophysicists call this tidal deformation — a measure of how much an object 'bends' under the gravitational pull of its companion.

An eardrum at the very edge of a waterfall. Sound behaves strangely here: echoes don't multiply: they fade. That's exactly what gravitational waves look like at the horizon of a black hole — the boundary beyond which even light disappears.

When two black holes merge far away, we receive a tremor of space — gravitational waves. Usually, they are described by Einstein's equations, but what if gravity works differently at the tiniest distances? The authors took one such hypothesis — Hořava–Lifshitz gravity.

A taut trampoline sheet dips under a heavy ball. If the ball spins quickly, the sheet begins to twist around it. Spacetime near a rotating black hole behaves similarly: it is not just curved, but also dragged along by the rotation, like water swirling around a whirlpool.

Searching for the global minimum in the Sherrington–Kirkpatrick spin glass model resembles climbing a glacier, where every crevasse is a treacherous local optimum, and the true bottom lies buried beneath a chaos of ice blocks. A classical climber, trudging along the gradient, would almost certainly freeze in the nearest crevice.

An endless ocean where every wave carries information about what happens in the depths. That is roughly how our Universe works under the holographic principle: the physics inside a volume can be described by processes on its boundary.

A quantum wave is like a long-exposure photograph: everything’s blurred, but the blur itself reveals motion. Usually these frozen waves are treated as convenient abstractions. Yet if at each point in the image you estimate, by brightness, the moment on the particle’s internal clock it landed there, flat space suddenly acquires curves.

A black hole with a magnetic field resembles a whirlpool in a river, where water doesn't flow into the vortex but swirls in place. Usually, the hole's rotation picks up waves and amplifies them: this is called superradiance. But add a magnetic field, and everything changes.

At the centers of many galaxies lurk giant rotating black holes. Astrophysicists have long suspected that these objects are remarkably resilient: after swallowing a star or merging with another hole, they quickly settle down and forget the catastrophe.

The early universe was a restless place. Space trembled with gravitational waves — ripples in the very fabric of reality. In this churning cauldron, dark matter is thought to have been born. But the newborn particles had no mass: it had to come from somewhere. Usually, the Higgs field is responsible for mass.

Spacetime is not empty: it shivers. After the Big Bang, the universe was filled with a hum of gravitational waves, like the low rumble of a giant bell. In this ripple, as it turns out, particles of dark matter, the invisible substance that holds galaxies together, can be born. But the previous idea had a snag.

In the classical picture of the world, gravity is the smooth bending of space and time. But at the microscopic level, where quantum mechanics rules, these bends should turn into a grainy ripple. The particles of such a ripple are called gravitons. The trouble is that straightforward calculations with gravitons always run into infinities — the theory loses its meaning.

Deep inside a star compressed to its limit, particles behave like the strings of an unusual musical instrument. Their oscillations obey a complex equation that for nearly a century no one could solve exactly.

A quantum system is like an island gradually eroded by the ocean. The waves, representing environmental influence, wash away the shores, stripping the island of its unique contours. The GKLS equation (Gorini–Kossakowski–Lindblad–Sudarshan) describes exactly how quickly this 'quantumness' fades. But every wave has an energy source.

Picture a pot of boiling soup, but every grain in it is not just a point but a tiny top spinning around its own axis. That's the world of spin-1 particles. For example, some short-lived mesons born in accelerator collisions.

A billiard game would be strange if the balls had no mass. They wouldn't collide but would pass through each other, ignoring obstacles. That's roughly what it looks like to imagine light as a stream of classical particles — tiny elastic bodies flying in straight lines. A new proof puts an end to that picture.

A neutron star is the leftover core of an exploded giant, compressed to the size of a city. A teaspoon of its matter weighs a billion tons. Inside such a star, particles are packed so tightly that quantum effects start to show. One of them is the intrinsic rotation of particles, or spin.

The light cone is a blueprint of causality, laser-cut onto the fabric of the universe. In classical physics, it is as sharp as a scalpel: events inside are linked by invisible threads of influence, while outside lies eternal isolation, sealed by the ultimate speed limit.

Two massive spheres slowly drift apart in empty space. From the standpoint of ordinary mechanics, nothing special is happening. But Einstein's equations say otherwise: accelerating masses stir spacetime itself, and from them spread gravitational waves — ripples that stretch and squeeze everything in their path.

Every quantum system (whether an atom, a crystal, or the entire Universe) has its own set of allowed energies, like the steps of a ladder. The hotter the system, the more readily particles climb to higher rungs.

A quantum particle traveling around a closed ring leaves two distinct 'records' along its path. The first is like a fingerprint on a doorknob: it remains unchanged no matter how many times you turn the knob, and it only tells you about the shape and material.

In the very beginning, moments after the Big Bang, the universe was ridiculously symmetric: particles and antiparticles in equal measure. But a tiny imbalance occurred, like one extra grain of sugar in a kilogram, and the entire familiar world, stars, planets, people, arose from that surplus. How nature created this imbalance remains a mystery.

Modern detectors of gravitational waves confirm that black holes are real. Spacetime can be compared to a giant trampoline: massive stars make it sag.

A guitar string decides for itself how tightly to stretch when a musician plays a chord in different octaves. That's roughly the picture physicists paint when they try to unite gravity, the weakest but most pervasive force, with the quirky laws of the microcosm.

In physics, there's an old ache. General relativity paints the beginning of the universe as a point — a singularity where equations break down and volume shrinks to zero. Quantum mechanics hints that at the most critical moment, something goes differently than classical physics predicts.

The riddle of quantum gravity has troubled minds for nearly a century. A classical field that curves spacetime refuses to fit into the probabilistic world of superpositions. Can gravitation, while remaining macroscopic, give rise to microscopic entanglement?

Sometimes an unwelcome guest sneaks into the equations, like a creaky floorboard that unsettles the whole house. In gravity theories, if you add curvature-squared terms to the equations (which you need to describe very strong fields), a ghost particle appears — heavy but with negative energy.

Every computer model of a black hole runs into the same inconvenience. A machine's memory is not infinite — at some point you have to cut off the calculation, as if placing a solid wall across space.

A tuning fork, once struck, makes neighboring strings vibrate. Much the same happens with a cloud of ultralight particles (also called axions) around a rapidly spinning black hole. The hole, like a motor, spins up the particles, and they begin to emit gravitational waves — a trembling of space itself.

The void can produce particles if a strong enough field is applied. Physicists call this the Dirac–Klein paradox. Previously, explaining it required the most complex calculations of quantum field theory. But we can make do with a visual image: a tightly stretched rubber band with a knot. The knot is a particle.

Two black holes spiral faster and faster until they merge into one. This dance shakes space itself: waves ripple outward from the collision site, like ripples from a stone thrown into a pond. Sometimes the merger also produces a flash of light.

A heavy spinning top launched near a black hole doesn't follow the curve that textbooks draw. Its own rotation and oblateness, like a pumpkin squashed at the poles, make the trajectory wobble. In a new study, this wobbling path has been calculated more precisely than ever before. Usually, a small body in a black hole's field is treated as a point — without size or spin.

The proton is not a solid ball but a seething cloud of quarks and gluons. Each particle carries its own bit of rotation, but if you add up all the bits, you don't get the full proton spin. Part of the rotation seems to hide in the cloud's slowest inhabitants: those carrying only a tiny share of the total motion.