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Here are GIFs on random topics, hosted at Giphy.  Browse and have fun. Copy as is.


In curved space, try to lay out a perfect grid of meter sticks. You can’t. You end up with extra “pixels” that don’t fit. Either shorten some sticks, or let them bulge out in the third dimension.

Einstein Curvature GIF - Find & Share on GIPHY

A 2d closed surface can be viewed at embedded in 3d, or as a distorted disk in 2d (Feynman’s hot plate):

Geodesic: traced by a cart on a 2d spherical surface embedded in 3d:

Geodesic: traced by cart on Feynman’s hot plate, where the wheel sizes (or speeds) vary:

In 3d, a grid of rods make voxels (3d pixels)…:

…and in 4d, a grid of rods (including time rods) make hypervoxels (4d pixels):

Curvature of time. Just like the space dimensions deform or buckle, time stretches in 4d “voxels.” Where the copper “time-stick” is long, clocks run slow.

There’s a conspiracy of nature: When everything runs in slow motion — clocks, your brain, your muscles… — you cannot tell:



Black holes

Schwarzschild spacetime. Increase the mass, spacetime curves:

Static (Schwarzschild) black hole. Particle falls, seems to freeze, then happens to bounce on something (mirror, quantum fluctuation…)

Black holes don’t “suck” in matter any more than a star of the same mass. Orbits of particles are the same. In fact, black holes are less likely to swallow a particle, because it’s smaller, and most particles fly right past the black hole. You need a head-on collision to get destroyed, and for black holes, being very small, that’s rare:

Rotating (Kerr) black hole. Particle falls, is frame-dragged, orbits an infinite number of times before reaching the event horizon:

At the event horizon, light cones tip over, so an outside cannot say: “the falling thing has crossed the horizon”:

At the event horizon, clocks slow to zero. As with the Shapiro delay, you could bounce just outside the event horizon, and reappear after a trillion years:

When a black hole evaporates (via Hawking radiation), you cannot fall across the event horizon (more details here):



Slow light

For the same measured distance, light takes different time to bounce and return. This Shapiro delay can be arbitrarily large when bouncing close to the event horizon of a black hole. Taylor and Wheeler refer to this as “slow light”:

Prism: light bends toward slower light. Half of the light flash (“wave front”) enters the prism first, slowing down, turning the light flash toward the prism:

Light bends toward slower light. Here, we cannot make a square with 90 degree angles:

Measuring the local speed of light always gives c, because your measurement equipment slows along with light.

Einstein Speed Of Light GIF - Find & Share on GIPHY




Travel the quick way, through a wormhole:

Build a worm hole by connecting two “mouths” in the embedding dimension:

Morris-Thorne wormhole. Step into one mouth, step out the other (along yellow rods):

The two wormhole mouths can be attached at different times:

Same skewed attachment with the t coordinate visualized. You can the make a time machine (more here).





The universe looks the same from everywhere, even when it’s closed:

A closed universe is not infinite. Go straight, and you loop back to where you started, just like on planet Earth’s surface.

A closed universe is not infinite. It could be like the surface of the Earth, closing back on itself (in an embedding dimension), or it could be a hot plate (see Feynman’s lecture), where spacetime distorts just so that it behaves like a spherical surface. You get different boundary conditions for a path that hits the edge head on:

In an open (infinite) universe, turn the curvature up and down:

With expansion, space is added between galaxies that sit still. Cosmic time is measured “at rest” and runs the fastest. A moving clock (for example one at fixed distance during expansion) shows less time:

With expansion, matter and radiation is diluted. Dark-energy density stays the same:

With matter (yellow dots) the universe decelerates. With enough dark energy (red dots), the universe accelerates:

Early inflation, before the other eras:

Gravitational wave: A cyclic expansion–contraction of space:

The expansion rate of the universe is measured in [1/s], not [m/s]. It’s not a speed. So please don’t say that “it’s expanding faster than light.” However, the distance [m] between two galaxies changes with time [m/s], so you can call this a recession speed. The further apart, the higher the recession speed. You can always find two points receding faster than light, even when expansion is slow:

During 13 billion years, a galaxy can recede by much more than 13 billion light-years, because of expansion. A photon emitted by the receding galaxy may initially be “expanded away,” then eventually catch up and reach us.




Special Relativity

Video tutorial here.

The three effects at high speed: length contraction, time dilation, desynchronization:

Synchronize clocks while moving: Method 1. Flash light halfway between clocks, reset clocks to zero when they detect the flash. Fails on moving ship). because flash takes longer to catch up to front clock:

Synchronize clocks while moving: Method 2. Synchronize clocks at same location, then move them apart. Fails identically on moving ship, because front clock moves faster, runs slower:

Conspiracy of nature: Moving, contracted ship measures at-rest, long ship to be contracted:

Conspiracy of nature: Moving ship has slow clocks, but measures at-rest fast clocks to be slow! Why? Because moving clocks are desynchronized:

Paradoxes evaporate with you remember that it’s all about measurements, by an observer that may be moving, not about some global, external POV:

Conspiracy of nature:  When you and your meter stick both contract, you cannot tell (locally):

Conspiracy of nature:  When you and your clocks run in slow motion, you cannot tell (locally). Also note tilted rod (clock desynchronization):

For a spinning ring, you get time dilation. For length contraction, it’s the instantaneous velocity that matters. The spokes don’t get shorter, but the yellow rods do, so you get gaps:

For the spinning ring, you cannot synchronize clocks all the way around (blue light flashes propagate):



Breathers (sine-Gordon)

non-linear string can “trap” the energy of a disturbance. Below, the top string oscillates between kinetic and potential energy,  in one location, like a particle at rest. (The boring linear string, bottom: waves rush off at the speed of the medium, c):

Sine-Gordon breathers, like particles, obey Special Relativity: they contract, the oscillations dilate in time, and the front lags behind the rear. Also, the total energy, E = T+U of the non-linear string, increases with a -factor:

Sine-Gordon breathers, like particles, obey General Relativity in one respect: they accelerates toward where c (the speed of the medium) is smaller:





Binding energies:  the more tightly bound, the more energy needed to remove. For quarks, so much energy is input that new quarks are created, bonding with the remnants: