Can you build a wormhole?
You start knowing nothing. You finish having derived, by hand, the equation that tells physicists why a traversable wormhole needs matter that nobody has ever bottled.
One honest thing before we start: physicists have not proved that wormholes are impossible. Einstein's equations allow certain wormhole shapes on paper. What the math shows is that keeping one open, so a person could actually walk through, would demand a kind of energy that ordinary matter doesn't have. Whether nature has any of it is still an open question. By the end, you'll see exactly where that demand comes from.
Before you read anything, guess.
Your guesses are locked in and we'll come back to them at the end. Researchers who study learning keep finding the same thing: predicting first, even wrongly, makes the real answer stick.
Throw a ball hard enough and it never comes back.
Throw a ball up and gravity slows it, stops it, and pulls it back down. Throw it faster and it goes higher. On Earth, at about 11.2 km/s, it never returns. That speed is called escape velocity.
Escape velocity depends on how much mass there is and how close you are to its center. Keep the mass the same, squeeze it smaller, and the escape speed climbs. Squeeze far enough and escape needs more than the speed of light. Since nothing goes faster than light, nothing gets out. That's a black hole.
The radius where escape speed hits light speed is the Schwarzschild radius. For a non-rotating, uncharged mass:
is Newton's gravitational constant, is the mass, is the speed of light. Double the mass and the horizon doubles. Pick something and find out how small you'd have to crush it:
About the size of a marble. That's how small all of Earth would have to be.
The horizon isn't a surface you'd bump into. It's a boundary of no return: signals from inside can't reach anyone outside. Keep that word horizon in mind. A wormhole you can travel through must not have one.
Gravity isn't a pull. It's a shape.
Newton described gravity as a force between objects. Einstein replaced that with something stranger: space and time form a single four-dimensional fabric, spacetime, and mass and energy bend it. Objects then follow the straightest possible paths through that bent fabric, and those paths look like orbits and falls.
The physicist John Wheeler compressed it into two halves: matter tells spacetime how to curve, and spacetime tells matter how to move.
At zero mass the sheet is flat and the marble rolls straight. Add mass and the marble curves into an orbit. Nothing is pulling it; it's following the shape.
Einstein's field equation is the precise version of Wheeler's sentence. Tap each piece:
Left side: the shape of spacetime. Right side: the stuff in it. Tap a term.
Here's the move that powers the rest of this issue. Normally you'd put stuff in and ask what shape comes out. But you can run the equation backwards: draw the shape you want, then calculate what stuff it requires. That's exactly how physicists test wormholes.
Mission: connect Earth to Proxima b.
Proxima b is 4.24 light-years away. You're the chief engineer, and the board wants a shortcut. Every choice you make is one a physicist has actually considered. Collect lessons by failing in interesting ways; three lessons clears the level.
Sculpt a throat and see what it costs.
In 1988, Michael Morris and Kip Thorne asked a sharp question: if a person really could pass through a wormhole, what would its geometry have to look like? Their answer uses a metric, a formula that says how to measure distances and times:
Two dials control everything. The redshift function sets how time flows; it must stay finite so there's no horizon. The shape function sets how space bends. The narrowest point, the throat, sits at where .
For the tunnel to open outward on both sides, it must satisfy the flare-out condition, . Try breaking it.
Did you notice? As long as the throat flares, the bottom bar never goes positive. And shrinking the throat makes it worse: the cost grows like . Small wormholes are the most expensive. Level 5 proves why.
The proof, one step at a time.
Physicists check whether matter is "reasonable" with tests called energy conditions. The one that matters here is the null energy condition (NEC). In plain terms: measured along the path of a light ray, energy density plus pressure can't be negative. Every ordinary material we know obeys it. For our matter, it reads:
Now we feed the wormhole metric into Einstein's equation and see what it demands.
One component of Einstein's equation gives the energy density in terms of the shape function. At the throat:
Another component gives the pressure along the tunnel:
At the throat , so the bracket is zero and the second term disappears:
Negative pressure means tension: the throat is pulled like a stretched rope. Strange, but not yet fatal.
Add the two together. Given flare-out, , what sign will have?
The bottom is always positive. The top, , is negative whenever the throat flares. So:
The NEC says . The wormhole demands . Both can't be true.
Read that carefully. We didn't assume exotic matter; the geometry forced it on us. And we proved a conditional result. Quantum fields can produce small, brief pockets of negative energy (the Casimir effect is a measured example), and modified theories of gravity change the rules. None of that has yet produced a usable, human-sized wormhole, but it keeps the question open.
Your turn: test a real wormhole.
This one has a name in the literature, the Ellis wormhole. Its shape function and redshift function are:
Time flows normally everywhere and the space goes flat far from the throat. It looks perfect. Use the equations from Level 5 to find out what it costs.
How did your gut do?
Even with exotic matter, four walls remain
That's the whole method of theoretical physics in miniature: propose a shape, run Einstein's equation backwards, then check whether the required stuff could exist.