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Hyperbolic waves collide on the Poincaré disk. Bright blooms are constructive peaks, dark bands are cancellations. White tracers ride the phase-gradient, threading filaments. SU(1,1)...the set of Möbius moves that preserve the disk’s geometry...steers the beacons. Multiple tempos modulate the spatial frequency.

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Hyperbolic waves collide on the Poincaré disk. Bright blooms are constructive peaks, dark bands are cancellations. White tracers ride the phase-gradient, threading filaments. SU(1,1)...the set of Möbius moves that preserve the disk’s geometry...steers the beacons. Multiple tempos modulate the spatial frequency.

Hyperbolic waves collide on the Poincaré disk. Bright blooms are constructive peaks, dark bands are cancellations. White tracers ride the phase-gradient, threading filaments. SU(1,1)...the set of Möbius moves that preserve the disk’s geometry...steers the beacons. Multiple tempos modulate the spatial frequency.

Mathelirium

14,591 Aufrufe • vor 5 Monaten

A Blaschke-Quotient Flow on the Poincaré Disk Inside the unit disk, zeros and poles perform a Möbius ballet, twisting under SU(1,1) symmetries. Each point follows the gradient of an inner function, its path tracing folds of velvety hyperbolic fabric. Phase interference becomes light bending like thought itself. 🤯🤯🤩🤩🤗🫠

A Blaschke-Quotient Flow on the Poincaré Disk Inside the unit disk, zeros and poles perform a Möbius ballet, twisting under SU(1,1) symmetries. Each point follows the gradient of an inner function, its path tracing folds of velvety hyperbolic fabric. Phase interference becomes light bending like thought itself. 🤯🤯🤩🤩🤗🫠

Mathelirium

26,382 Aufrufe • vor 9 Monaten

Mathematics is beautiful🤩❤️ My favourite from 2025. Merry Christmas guys! 🎄 Zeros and poles of a Blaschke-Quotient spinning under SU(1,1) Möbius symmetries on the Poincaré disk with the phase shaded by the Fubini-Study metric.

Mathematics is beautiful🤩❤️ My favourite from 2025. Merry Christmas guys! 🎄 Zeros and poles of a Blaschke-Quotient spinning under SU(1,1) Möbius symmetries on the Poincaré disk with the phase shaded by the Fubini-Study metric.

Mathelirium

20,051 Aufrufe • vor 7 Monaten

Much better visually. We now visualize the 3D version of our stunning Blaschke-Quotient Flow from our previous post.🤗🤩 What follows is a mesmerizing break of dawn in the some realm of complex space with zeros and poles still spinning under SU(1,1) Möbius symmetries on the Poincaré disk…but phase is now shaded by the Fubini-Study metric. 🤯🤯🤯

Much better visually. We now visualize the 3D version of our stunning Blaschke-Quotient Flow from our previous post.🤗🤩 What follows is a mesmerizing break of dawn in the some realm of complex space with zeros and poles still spinning under SU(1,1) Möbius symmetries on the Poincaré disk…but phase is now shaded by the Fubini-Study metric. 🤯🤯🤯

Mathelirium

15,826 Aufrufe • vor 9 Monaten

You're watching a quantum fluid pull on itself. Bright clouds = high-density clumps ("proto-stars"); dark ripples = low-density filaments. The white dots are tracer particles...their motion draws the fine threads. It's a Schrödinger-Poisson self-gravity, in real time. 🤩🤗🫠

You're watching a quantum fluid pull on itself. Bright clouds = high-density clumps ("proto-stars"); dark ripples = low-density filaments. The white dots are tracer particles...their motion draws the fine threads. It's a Schrödinger-Poisson self-gravity, in real time. 🤩🤗🫠

Mathelirium

109,155 Aufrufe • vor 10 Monaten

What you're watching here is a torus cycling through its eigenfunctions...the surface's pure standing-wave modes. An eigenfunction is a mode that, when the surface Laplacian operator acts on it, reproduces itself scaled by a constant...that constant fixes the natural frequency. Each mode is indexed by two counts: how many oscillations wrap the small tube and how many wrap the big ring. Dark rings are nodes (no motion) and the bright bands between them are antinodes. As the counts rise, nodes crowd together, ridges sharpen, and the torus takes on a pine-cone look. #Eigenfunctions #SpectralGeometry #MathArt #Math

What you're watching here is a torus cycling through its eigenfunctions...the surface's pure standing-wave modes. An eigenfunction is a mode that, when the surface Laplacian operator acts on it, reproduces itself scaled by a constant...that constant fixes the natural frequency. Each mode is indexed by two counts: how many oscillations wrap the small tube and how many wrap the big ring. Dark rings are nodes (no motion) and the bright bands between them are antinodes. As the counts rise, nodes crowd together, ridges sharpen, and the torus takes on a pine-cone look. #Eigenfunctions #SpectralGeometry #MathArt #Math

Mathelirium

54,411 Aufrufe • vor 8 Monaten

We now visualize the complex phase of our earlier self-gravitating quantum fluid twisting into smooth helicoids. Think of phase as the wave's "angle" at each point...mapping it to color lets you see how wavefronts curl and shear. Brightness tracks density. The faint moving bands are phase ribbons...contours of equal phase sliding as the field evolves. The tiny white dots are vortices: spots where density dips and the phase wraps by one full turn (2pi). Same Schrödinger-Poisson physics, revealed in phase...live!🫠❤️🫠

We now visualize the complex phase of our earlier self-gravitating quantum fluid twisting into smooth helicoids. Think of phase as the wave's "angle" at each point...mapping it to color lets you see how wavefronts curl and shear. Brightness tracks density. The faint moving bands are phase ribbons...contours of equal phase sliding as the field evolves. The tiny white dots are vortices: spots where density dips and the phase wraps by one full turn (2pi). Same Schrödinger-Poisson physics, revealed in phase...live!🫠❤️🫠

Mathelirium

14,200 Aufrufe • vor 10 Monaten

A physical system, its phase-space state, and the energy function that governs the flow, all linked together. The point of the animation is that these are not three different objects. They are three views of the same dynamics. The pendulum on the left is the physical motion. The curve on the floor is the evolution of its state. The translucent surface shows the energy landscape that organizes that motion.

A physical system, its phase-space state, and the energy function that governs the flow, all linked together. The point of the animation is that these are not three different objects. They are three views of the same dynamics. The pendulum on the left is the physical motion. The curve on the floor is the evolution of its state. The translucent surface shows the energy landscape that organizes that motion.

Mathelirium

19,865 Aufrufe • vor 3 Monaten

In 1998, Japanese Mathematician, Engineer and Machine Learning pioneer Shun-ichi Amari published "Natural Gradient Works Efficiently in Learning (722 Cites in Papers and 9 Cites in Patents)" and made a point that still feels fresh even today If your loss is L(θ), then the usual update θ̇ = −∇ L(θ) quietly assumes your parameter space is flat. Amari asked what happens when the model itself has geometry, described by G(θ), the Fisher information. Then the more natural direction becomes θ̇ = −G(θ)⁻¹∇ L(θ). This is the same loss, but with a different notion of distance. As the animation shows, ordinary Gradient Descent follows the raw slope and gets dragged around by distortion, while the natural gradient moves in a way that respects the geometry of the statistical model itself.

In 1998, Japanese Mathematician, Engineer and Machine Learning pioneer Shun-ichi Amari published "Natural Gradient Works Efficiently in Learning (722 Cites in Papers and 9 Cites in Patents)" and made a point that still feels fresh even today If your loss is L(θ), then the usual update θ̇ = −∇ L(θ) quietly assumes your parameter space is flat. Amari asked what happens when the model itself has geometry, described by G(θ), the Fisher information. Then the more natural direction becomes θ̇ = −G(θ)⁻¹∇ L(θ). This is the same loss, but with a different notion of distance. As the animation shows, ordinary Gradient Descent follows the raw slope and gets dragged around by distortion, while the natural gradient moves in a way that respects the geometry of the statistical model itself.

Mathelirium

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Normalize Working in Phase Space Ordinary space shows where the particle is. Phase space shows its complete state, position and momentum together, revealing the hidden geometry of its motion.

Normalize Working in Phase Space Ordinary space shows where the particle is. Phase space shows its complete state, position and momentum together, revealing the hidden geometry of its motion.

Mathelirium

102,300 Aufrufe • vor 13 Tagen

The Elusive Concept of Time. Your instincts treat time like a background meter the whole universe shares. Relativity does not take time away, it forces you to earn it operationally. Events are points. Motion is a curve through them. A clock is not a metaphor, it is a worldline with a number attached to it. Two observers can disagree on which distant events are simultaneous, and nothing contradictory happens, because causal structure is still pinned down by light cones and invariants. Even in weak gravity the rule shows up. A clock deeper in a gravitational potential ticks more slowly relative to one far away. Near a compact object the difference becomes hard to ignore. Add rotation and spacetime itself picks up a twist. That twist is frame dragging, not a new force, just geometry telling you that time and angle are coupled in a rotating spacetime. In the animation You are watching a geometry lesson disguised as a black hole scene. The fabric is a visualization of the field shaping clock-rates and the paths light can take. The ripples are driven by local proper time, so their phase visibly slows as you approach the horizon. The accretion disk is lensed through Kerr ray tracing, and its brightness is pushed by redshift and beaming so the approaching side can flare while the receding side dims. Beacon points at different radii pulse at different rates, so you can see time dilation without any labels. The bead ring is a redshift tracer, with intensity scaled by a g³ proxy so deeper emission arrives weaker and shifted. The math breakdown Start with what a clock actually measures. Proper time τ is the accumulated time along an observer’s worldline. In special relativity, the invariant interval is ds² = c² dt² − dx² − dy² − dz² Along a timelike path, dτ = (1/c) √(ds²) = √( dt² − (1/c²)(dx²+dy²+dz²) ) If the observer moves with speed v, so dx²+dy²+dy²+dz² = v² dt², then dτ = dt √(1 − v²/c²) That is time dilation as geometry. The moving clock accumulates less τ between the same pair of events. Now add gravity. General relativity replaces the flat interval with a metric gᵤᵥ that depends on position: ds² = gᵤᵥ dxᵘ dxᵛ For a stationary clock in Schwarzschild geometry (mass M), the time component is g_tt = −(1 − 2GM/(rc²)) If the clock sits at fixed r (no spatial motion), ds² = g_tt c² dt², so dτ = dt √(1 − 2GM/(rc²)) Closer to the mass means a smaller factor, so the clock ticks more slowly relative to a clock far away. That is the rule used to drive the fabric phase in the animation. Now connect time to light. A gravitational field shifts photon frequency. Between an emitter at rₑ and an observer at rₒ, f_obs / f_emit = √( (1 − 2GM/(rₑ c²)) / (1 − 2GM/(rₒ c²)) ) For a far-away observer rₒ → ∞, f_obs / f_emit = √(1 − 2GM/(rₑ c²)) Deeper emission arrives redshifted. Lower frequency. Lower energy per photon. In the render, the disk intensity uses a Kerr-derived redshift factor g (clipped for stability). The bead ring uses a simple radiative proxy I_obs ∝ g³ I_emit to make that effect visible. Finally, why rotation looks like a twist. A rotating black hole is Kerr geometry. The key structural change is a nonzero g_tφ term, which couples time to angle. That coupling is frame dragging in equations. Near the hole, being stationary is not the same notion everywhere, because the local inertial frames are being pulled around the spin axis. So the moral stays clean. Time is not a universal substance flowing everywhere at one rate. It is what clocks accumulate along worldlines. Light cones constrain what can influence what. Invariants are what everyone agrees on. The rest is operational detail that only feels universal because our daily corner of the universe is slow and mild. #GeneralRelativity #Gravity #FrameDragging #BlackHoles #Spacetime

The Elusive Concept of Time. Your instincts treat time like a background meter the whole universe shares. Relativity does not take time away, it forces you to earn it operationally. Events are points. Motion is a curve through them. A clock is not a metaphor, it is a worldline with a number attached to it. Two observers can disagree on which distant events are simultaneous, and nothing contradictory happens, because causal structure is still pinned down by light cones and invariants. Even in weak gravity the rule shows up. A clock deeper in a gravitational potential ticks more slowly relative to one far away. Near a compact object the difference becomes hard to ignore. Add rotation and spacetime itself picks up a twist. That twist is frame dragging, not a new force, just geometry telling you that time and angle are coupled in a rotating spacetime. In the animation You are watching a geometry lesson disguised as a black hole scene. The fabric is a visualization of the field shaping clock-rates and the paths light can take. The ripples are driven by local proper time, so their phase visibly slows as you approach the horizon. The accretion disk is lensed through Kerr ray tracing, and its brightness is pushed by redshift and beaming so the approaching side can flare while the receding side dims. Beacon points at different radii pulse at different rates, so you can see time dilation without any labels. The bead ring is a redshift tracer, with intensity scaled by a g³ proxy so deeper emission arrives weaker and shifted. The math breakdown Start with what a clock actually measures. Proper time τ is the accumulated time along an observer’s worldline. In special relativity, the invariant interval is ds² = c² dt² − dx² − dy² − dz² Along a timelike path, dτ = (1/c) √(ds²) = √( dt² − (1/c²)(dx²+dy²+dz²) ) If the observer moves with speed v, so dx²+dy²+dy²+dz² = v² dt², then dτ = dt √(1 − v²/c²) That is time dilation as geometry. The moving clock accumulates less τ between the same pair of events. Now add gravity. General relativity replaces the flat interval with a metric gᵤᵥ that depends on position: ds² = gᵤᵥ dxᵘ dxᵛ For a stationary clock in Schwarzschild geometry (mass M), the time component is g_tt = −(1 − 2GM/(rc²)) If the clock sits at fixed r (no spatial motion), ds² = g_tt c² dt², so dτ = dt √(1 − 2GM/(rc²)) Closer to the mass means a smaller factor, so the clock ticks more slowly relative to a clock far away. That is the rule used to drive the fabric phase in the animation. Now connect time to light. A gravitational field shifts photon frequency. Between an emitter at rₑ and an observer at rₒ, f_obs / f_emit = √( (1 − 2GM/(rₑ c²)) / (1 − 2GM/(rₒ c²)) ) For a far-away observer rₒ → ∞, f_obs / f_emit = √(1 − 2GM/(rₑ c²)) Deeper emission arrives redshifted. Lower frequency. Lower energy per photon. In the render, the disk intensity uses a Kerr-derived redshift factor g (clipped for stability). The bead ring uses a simple radiative proxy I_obs ∝ g³ I_emit to make that effect visible. Finally, why rotation looks like a twist. A rotating black hole is Kerr geometry. The key structural change is a nonzero g_tφ term, which couples time to angle. That coupling is frame dragging in equations. Near the hole, being stationary is not the same notion everywhere, because the local inertial frames are being pulled around the spin axis. So the moral stays clean. Time is not a universal substance flowing everywhere at one rate. It is what clocks accumulate along worldlines. Light cones constrain what can influence what. Invariants are what everyone agrees on. The rest is operational detail that only feels universal because our daily corner of the universe is slow and mild. #GeneralRelativity #Gravity #FrameDragging #BlackHoles #Spacetime

Mathelirium

149,753 Aufrufe • vor 6 Monaten

The Torus Becomes the Canvas A Jacobi Theta Function lives naturally on a complex torus. Instead of drawing it on a flat plane, we wrapped the mathematics onto the torus. The surface is driven by θ₁(z|τ), with its moving zeros, phase winding, and logarithmic derivative shaping the colour, seams, and raised divisor points across the geometry. What you are see is a periodic quantum-like field painted onto the space where it actually belongs. #JacobiTheta #ComplexTorus #MathematicalArt #ComplexAnalysis #RiemannSurfaces #MathAnimation

The Torus Becomes the Canvas A Jacobi Theta Function lives naturally on a complex torus. Instead of drawing it on a flat plane, we wrapped the mathematics onto the torus. The surface is driven by θ₁(z|τ), with its moving zeros, phase winding, and logarithmic derivative shaping the colour, seams, and raised divisor points across the geometry. What you are see is a periodic quantum-like field painted onto the space where it actually belongs. #JacobiTheta #ComplexTorus #MathematicalArt #ComplexAnalysis #RiemannSurfaces #MathAnimation

Mathelirium

22,197 Aufrufe • vor 1 Monat

The Dance of Three Masses on a Warped Spacetime Fabric Newtonian Gravity says masses attract one another through force. General Relativity changes the picture by treating gravity as geometry itself, where matter shapes Spacetime, and motion follows that evolving curved structure. In a three-body system, the bodies are not just pulling on each other. They are moving through a geometric landscape that shifts as the mass arrangement changes.

The Dance of Three Masses on a Warped Spacetime Fabric Newtonian Gravity says masses attract one another through force. General Relativity changes the picture by treating gravity as geometry itself, where matter shapes Spacetime, and motion follows that evolving curved structure. In a three-body system, the bodies are not just pulling on each other. They are moving through a geometric landscape that shifts as the mass arrangement changes.

Mathelirium

10,806 Aufrufe • vor 3 Monaten

The Hidden Flow Inside Schrödinger’s Equation Once you define ρ = |ψ|² and differentiate it, Schrödinger equation rearranges into a continuity law, ∂ρ/∂t + ∇·j = 0, with j = (1/m)Im(ψ*∇ψ). Which means that probability is not just sitting there, it flows. Write ψ = r e^(iθ), and the current becomes j = (ρ/m)∇θ. So, the phase gradient is the steering field. When the phase winds by 2πn around a loop, the circulation is quantized, and the cores appear where ρ collapses so the phase can go undefined. In the animation, the ribbons trace that current, the color field shows phase, and the curls in the streamlines mark quantized vortex defects.

The Hidden Flow Inside Schrödinger’s Equation Once you define ρ = |ψ|² and differentiate it, Schrödinger equation rearranges into a continuity law, ∂ρ/∂t + ∇·j = 0, with j = (1/m)Im(ψ*∇ψ). Which means that probability is not just sitting there, it flows. Write ψ = r e^(iθ), and the current becomes j = (ρ/m)∇θ. So, the phase gradient is the steering field. When the phase winds by 2πn around a loop, the circulation is quantized, and the cores appear where ρ collapses so the phase can go undefined. In the animation, the ribbons trace that current, the color field shows phase, and the curls in the streamlines mark quantized vortex defects.

Mathelirium

26,511 Aufrufe • vor 4 Monaten

The Branch-Point Loom Some equations do not draw curves but instead draw weave sheets. This scene comes from the algebraic curve y² = Δ(z,t), where Δ(z,t) = Πⱼ(z - βⱼ(t)). The moving points βⱼ are branch points. These are special places where two sheets of the curve touch and swap identities. In the animation, those branch points appear as glowing mineral spindles moving across a dark surface. The landscape is shaped by -log|Δ(z,t)|. Therefore, the branch points rise into sharp seams while the surrounding surface folds around them. The bright threads follow the horizontal trajectories of the quadratic differential Δ(z,t)dz². #Mathematics #ComplexAnalysis #AlgebraicGeometry #MathArt #MathematicalArt #Geometry #PhysicsVisualization #STEM #Art

The Branch-Point Loom Some equations do not draw curves but instead draw weave sheets. This scene comes from the algebraic curve y² = Δ(z,t), where Δ(z,t) = Πⱼ(z - βⱼ(t)). The moving points βⱼ are branch points. These are special places where two sheets of the curve touch and swap identities. In the animation, those branch points appear as glowing mineral spindles moving across a dark surface. The landscape is shaped by -log|Δ(z,t)|. Therefore, the branch points rise into sharp seams while the surrounding surface folds around them. The bright threads follow the horizontal trajectories of the quadratic differential Δ(z,t)dz². #Mathematics #ComplexAnalysis #AlgebraicGeometry #MathArt #MathematicalArt #Geometry #PhysicsVisualization #STEM #Art

Mathelirium

15,287 Aufrufe • vor 1 Monat

Speech Carries Its Own Hidden Geometry in Phase Space Speech is usually shown as a waveform of amplitude changing in time. Phase-space reconstruction gives a different view by pairing the signal with a delayed version of itself, so the voice is seen as a path rather than just a trace. On the left is the raw sound signal. On the right is the geometric structure formed by rhythm, syllables, and emphasis as the voice unfolds. * Real human speech data from the public GitHub repository voxserv/audio_quality_testing_samples, using the 16 kHz file test01_20s.wav.

Speech Carries Its Own Hidden Geometry in Phase Space Speech is usually shown as a waveform of amplitude changing in time. Phase-space reconstruction gives a different view by pairing the signal with a delayed version of itself, so the voice is seen as a path rather than just a trace. On the left is the raw sound signal. On the right is the geometric structure formed by rhythm, syllables, and emphasis as the voice unfolds. * Real human speech data from the public GitHub repository voxserv/audio_quality_testing_samples, using the 16 kHz file test01_20s.wav.

Mathelirium

18,725 Aufrufe • vor 4 Monaten

Ever Heard of a Metric in Differential Geometry? How About a Geodesic? These two ideas are central to how Mathematics and Physics describe Spacetime. A metric is how a surface measures distance and angle. A geodesic is the path that counts as "straight" after that measuring rule is chosen. Once the metric changes from point to point, straightness is no longer something you judge by eye. On a curved surface, the shortest paths are geodesics. They are chosen by the metric. Take a surface written as r(x,y) = (x, y, f(x,y)) Its metric is gᵢⱼ = ∂ᵢr · ∂ⱼr A geodesic q(t) = (x(t), y(t)) obeys q̈ᵏ + Γᵏᵢⱼ q̇ᶦq̇ʲ = 0 where Γᵏᵢⱼ are the Christoffel symbols of the surface metric. Launch many almost-parallel paths. At the start, they look like a clean beam until curvature takes over. Some paths focus, others separate and some slide around ridges. The surface is essentially the rulebook for motion. The shortest path is local. The geometry remembers everything. #DifferentialGeometry #Geodesics #RiemannianGeometry #MathematicalPhysics #Curvature #TensorCalculus #AppliedMathematics #Geometry #ScienceAnimation

Ever Heard of a Metric in Differential Geometry? How About a Geodesic? These two ideas are central to how Mathematics and Physics describe Spacetime. A metric is how a surface measures distance and angle. A geodesic is the path that counts as "straight" after that measuring rule is chosen. Once the metric changes from point to point, straightness is no longer something you judge by eye. On a curved surface, the shortest paths are geodesics. They are chosen by the metric. Take a surface written as r(x,y) = (x, y, f(x,y)) Its metric is gᵢⱼ = ∂ᵢr · ∂ⱼr A geodesic q(t) = (x(t), y(t)) obeys q̈ᵏ + Γᵏᵢⱼ q̇ᶦq̇ʲ = 0 where Γᵏᵢⱼ are the Christoffel symbols of the surface metric. Launch many almost-parallel paths. At the start, they look like a clean beam until curvature takes over. Some paths focus, others separate and some slide around ridges. The surface is essentially the rulebook for motion. The shortest path is local. The geometry remembers everything. #DifferentialGeometry #Geodesics #RiemannianGeometry #MathematicalPhysics #Curvature #TensorCalculus #AppliedMathematics #Geometry #ScienceAnimation

Mathelirium

22,170 Aufrufe • vor 1 Monat

White Holes Don’t Push You Way. Spacetime Refuses Your Future. A White Hole appears in the maximally extended Schwarzschild solution of General Relativity, the full mathematical extension of the Spacetime around an eternal, non-rotating Black Hole. The exterior geometry is the same Schwarzschild geometry ds² = −(1 − 2M/r)dt² + (1 − 2M/r)⁻¹dr² + r²dΩ² But the causal direction is reversed. For a Black Hole, future-directed paths can cross the horizon inward. For a White Hole, future-directed paths can emerge from the horizon, but outside particles cannot enter that region. The object has not been observed in nature. But inside the Mathematics of General Relativity, it is a real sector of the extended Schwarzschild Spacetime... the Black Hole’s time-reversed counterpart.

White Holes Don’t Push You Way. Spacetime Refuses Your Future. A White Hole appears in the maximally extended Schwarzschild solution of General Relativity, the full mathematical extension of the Spacetime around an eternal, non-rotating Black Hole. The exterior geometry is the same Schwarzschild geometry ds² = −(1 − 2M/r)dt² + (1 − 2M/r)⁻¹dr² + r²dΩ² But the causal direction is reversed. For a Black Hole, future-directed paths can cross the horizon inward. For a White Hole, future-directed paths can emerge from the horizon, but outside particles cannot enter that region. The object has not been observed in nature. But inside the Mathematics of General Relativity, it is a real sector of the extended Schwarzschild Spacetime... the Black Hole’s time-reversed counterpart.

Mathelirium

30,989 Aufrufe • vor 2 Monaten

String Theory Lecture 2 In Conformal Gauge, the String Becomes a Wave Equation Episode 1 showed the geometric jump point particle -> worldline string -> worldsheet Episode 2 is the dynamical jump. The Nambu-Goto action measures the area of the worldsheet, S = −T ∫ dτ dσ √[−det hₐᵦ] but the square-root determinant is awkward to work with. So we usually rewrite the same classical theory in Polyakov form, S = −(T/2) ∫ dτ dσ √[−γ] γᵃᵇ ∂ₐXᵘ ∂ᵦXᵤ Here Xᵘ(τ,σ) tells us where each point of the string’s worldsheet sits in spacetime, and γₐᵦ is the metric we put on the worldsheet. The power of this form is that we can choose a convenient gauge. In conformal gauge, the equations of motion simplify to (∂²/∂τ² − ∂²/∂σ²) Xᵘ = 0 So the string’s spacetime coordinates behave like waves living on the worldsheet. The τ-derivative measures how the string changes in worldsheet time. The σ-derivative measures how it bends along its own length. For a closed string, σ is periodic Xᵘ(τ, σ + 2π) = Xᵘ(τ,σ) and the wave equation splits into two traveling pieces, Xᵘ(τ,σ) = Fᵘ(τ + σ) + Gᵘ(τ − σ) One family moves one way around the string and the other moves the opposite way. These are the left-moving and right-moving modes. In the render, the bright loop is the string at the present moment. The glowing cylinder behind it is the worldsheet it has swept out. The cyan curves trace one traveling family, and the gold curves trace the other. They are the visual version of τ + σ and τ − σ. Therefore, the theory has an internal wave equation, and its normal modes are the raw material for the string spectrum. #StringTheory #TheoreticalPhysics #ConformalGauge #Worldsheet #WaveEquation #Physics #Mathematics #MathematicalPhysics #QuantumGravity #ScienceVisuals

String Theory Lecture 2 In Conformal Gauge, the String Becomes a Wave Equation Episode 1 showed the geometric jump point particle -> worldline string -> worldsheet Episode 2 is the dynamical jump. The Nambu-Goto action measures the area of the worldsheet, S = −T ∫ dτ dσ √[−det hₐᵦ] but the square-root determinant is awkward to work with. So we usually rewrite the same classical theory in Polyakov form, S = −(T/2) ∫ dτ dσ √[−γ] γᵃᵇ ∂ₐXᵘ ∂ᵦXᵤ Here Xᵘ(τ,σ) tells us where each point of the string’s worldsheet sits in spacetime, and γₐᵦ is the metric we put on the worldsheet. The power of this form is that we can choose a convenient gauge. In conformal gauge, the equations of motion simplify to (∂²/∂τ² − ∂²/∂σ²) Xᵘ = 0 So the string’s spacetime coordinates behave like waves living on the worldsheet. The τ-derivative measures how the string changes in worldsheet time. The σ-derivative measures how it bends along its own length. For a closed string, σ is periodic Xᵘ(τ, σ + 2π) = Xᵘ(τ,σ) and the wave equation splits into two traveling pieces, Xᵘ(τ,σ) = Fᵘ(τ + σ) + Gᵘ(τ − σ) One family moves one way around the string and the other moves the opposite way. These are the left-moving and right-moving modes. In the render, the bright loop is the string at the present moment. The glowing cylinder behind it is the worldsheet it has swept out. The cyan curves trace one traveling family, and the gold curves trace the other. They are the visual version of τ + σ and τ − σ. Therefore, the theory has an internal wave equation, and its normal modes are the raw material for the string spectrum. #StringTheory #TheoreticalPhysics #ConformalGauge #Worldsheet #WaveEquation #Physics #Mathematics #MathematicalPhysics #QuantumGravity #ScienceVisuals

Mathelirium

15,295 Aufrufe • vor 3 Monaten

You're watching spacetime in motion...two spinning black holes orbit, twist, and merge, dragging the very geometry of the universe into a luminous storm. Their union births a single remnant that fires smoke-wreathed jet as gravitational waves ring the cosmic fabric itself. 🤩😍 #Math #Physics #Astrophysics #SciArt #MathArt

You're watching spacetime in motion...two spinning black holes orbit, twist, and merge, dragging the very geometry of the universe into a luminous storm. Their union births a single remnant that fires smoke-wreathed jet as gravitational waves ring the cosmic fabric itself. 🤩😍 #Math #Physics #Astrophysics #SciArt #MathArt

Mathelirium

42,531 Aufrufe • vor 8 Monaten