darikwaA landing page, just for fun
E = mc2
Every landing page promises to save you time. This one explains why time is negotiable.
Push the throttle. The stars ahead bunch up and turn blue, the ones behind thin out and turn red, and your clock runs slow against everyone you left at home. The sky, the clocks and the light paths are not decoration: they all come from Einstein's equations.
- Speed
- 60,647 km/s
- Lorentz factor
- γ = 1.02
- One hour on board
- 1.02 h at home
- A metre on board
- 97.9 cm to home
Special relativity, 1905
Mass is energy, at a terrifying exchange rate.
The most famous equation in physics is a currency conversion. Multiply any mass by the speed of light squared and you get the energy locked inside it. The rate is fixed, enormous, and applies to everything on your desk.
Nothing on Earth converts mass to energy completely. A nuclear reactor manages a fraction of a percent, and only in specific isotopes. The exchange rate is real all the same: the Hiroshima bomb released the energy of about 0.7 g of mass, and the Sun turns 4.3 million tonnes of itself into light every second.
Time dilation
Speed slows your clock. Everyone else's keeps going.
Light always travels at the same speed, whoever measures it. A clock that ticks by bouncing light between two mirrors therefore ticks more slowly when it moves past you, because its light has further to go. The factor is γ, and it turns up in everything that follows.
γ from 1 at rest to 8 as v nears c
General relativity, 1915
Mass bends space, and light follows the bend.
Gravity is not a force pulling things down. Mass tells spacetime how to curve, and curved spacetime tells everything, light included, how to move. Move the mass and watch the rays of light steer around it. Rays that come too close never leave.
Move your pointer over the grid, tap it, or use the arrow keys to place the mass. The gold ring is the event horizon. The dashed ring marks one and a half Schwarzschild radii, the photon sphere of the exact solution; this weak-field sketch swallows rays that stray anywhere near it.
Black holes
Squeeze anything hard enough and it disappears.
Every mass has a radius at which its escape velocity reaches the speed of light. Compress the object inside that radius and nothing, light included, can get out. Karl Schwarzschild worked this out at the end of 1915 while serving on the Russian front. It was published in January 1916 and he was dead by May; the radius still carries his name.
6.0 × 10²⁴ kg
The radius scales with mass and nothing else. Double the mass, double the radius. That is why the black hole at the centre of our galaxy, at 4.3 million solar masses, has a horizon eighteen times wider than the Sun, a fifth of the way out to Mercury, rather than a town.
Gravitational time dilation
Gravity slows time too, and your phone knows it.
Clocks run slower deeper in a gravitational well. A satellite clock therefore runs fast because it is higher up, and slow because it is moving quickly. GPS satellite clocks gain about 38 microseconds a day on the clocks below them, and if nobody corrected for it, satnav would drift by kilometres every day.
Drawn to scale, geostationary orbit at the edge
The two effects cancel at 3,186 km, one and a half Earth radii from the centre. Below that, speed wins and the clock runs slow; above it, altitude wins and the clock runs fast.
The wall
The formulas behind each chapter, in one place.
A baker's dozen of equations. Between them they describe how fast clocks tick, how long rulers are, why starlight bends around the Sun and what happens when a star runs out of room.
Mass and energy are the same thing in different units. The conversion rate is c squared.
How much time stretches and lengths shrink for something moving at speed v.
A moving clock's tick (proper time τ) takes γ times longer as you measure it.
A moving object is shorter along its direction of travel, by the same factor.
How to translate positions and times between two observers in relative motion.
The full version of E = mc². For something at rest, p is zero and it collapses to the famous one.
The one distance every observer agrees on. Space and time trade against each other to keep it fixed.
Why the stars in the hero turn blue ahead and red behind as you accelerate.
Curvature on the left, matter and energy on the right. Ten equations in one line.
Compress a mass inside this radius and it becomes a black hole.
How far a ray of light bends passing a mass at distance b. Confirmed in 1919.
Clocks closer to a mass run slower. Zero at the horizon.
Why the stars in the hero crowd towards the direction you are travelling.
That is the physics
Now, about your business.
The systems I build for clients are considerably less exotic and considerably more useful: a weekly trade report that goes out whatever the numbers say, management accounts that reconcile to the stock file, a seasonal buy planned against actuals. No black holes, unless you count the old spreadsheet.