Does the Universe have Higher Dimensions? Part 2

Sabine Hossenfelder

Background and Previous Work

Continues previous week's video on history of extra dimensions
Historical context: Adding 7 dimensions of space to normal 3 dimensions allows describing all fundamental forces geometrically
Promising idea for unified theory of physics

Edward Witten (early 1980s) found intriguing that 7 additional dimensions = maximum for supergravity
Numerical coincidence led nowhere
Kaluza-Klein theory (geometric construction of fundamental forces) has unsolved problems:
- Radii of extra dimensions unstable (could grow/shrink)
- Incompatible with observation
- Chirality problem: particles come in left-handed/right-handed versions behaving differently
- Observational fact doesn't fit Kaluza-Klein idea
Witten worried about chirality in 1981 paper

Same issue as Kaluza-Klein theory century ago: If extra dimensions exist, where are they?
String theorists' answer: Can't see them because curled up to small radii
Extra dimensions curled into complicated geometrical shapes called Calabi-Yau manifolds

Curling up creates higher harmonics in strings (same as Kaluza-Klein theory)
With enough energy, strings oscillate at frequencies matching extra dimension radii
String theory does make predictions: higher harmonics should exist
Problem: Need extremely high energies to create them
Small radii = high frequencies = high energies
String theory doesn't specify exact energy requirements
Extra dimensions must be small (haven't seen them yet)
Could be just out of reach - next bigger particle collider might create higher harmonics

Newton's 1/r² law is geometrical
Gravitational force of point mass falls with 1/r² because sphere surface grows with r²
With additional dimensions (3 + n where n = positive integer):
- Sphere surface increases with r^(2+n)
- Gravitational force drops with 1/r^(2+n)
More dimensions = force drops much faster with distance
Geometric consideration remains valid in Einstein's general relativity

Extra dimensions are curled up (otherwise we'd have noticed them)
Force lines can only spread out to distance comparable to extra dimension radius
After this, force lines continue in three large directions only
On distances much larger than extra dimension radius: returns to usual 1/r² law (which we observe)

Normal 3D space: Cannot create black holes (gravity too weak)
With extra dimensions: Gravitational force falls faster going away from mass
Getting closer to mass = force gets much stronger than in 3D only
Makes black hole creation much easier
If extra dimensions large enough: could create black holes at Large Hadron Collider
In practice: LHC did not produce black holes
Means extra dimensions (if they exist) are really small
Size depends on number of extra dimensions: roughly below 1 micrometer

People think extra dimensions provide shortcuts due to illustrations
Idea: Universe like bent sheet, go perpendicular to arrive at distant point faster
Reality: Don't need extra dimensions for that
These are called wormholes - can exist in ordinary general relativity with 3D space
Embedding space in illustrations doesn't actually exist in general relativity
Explains confusion about universe expansion ("expands into what?" - doesn't expand into anything, just expands)

3rd place: Hyperdrives
- Nice idea but makes no sense scientifically
2nd place: Wormholes
- Exist mathematically
- No one knows how to create them
1st place: Warp drives
- Mathematics works out
- In principle possible to create
- Must stay below speed of light limit
Faster-than-light travel: Still don't know how

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