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The Geometry of
Our Universe
Castaly Fan
2020/06/15
Local Flatness
Manifold:
Torus
•Topology: Atorus 𝑇!is the product of two circle, that is, 𝑆"× 𝑆".
•Paths on a torus: Looks curved but locally straight.
•Light through a torus: You will see yourself from the rear. Also,
you can see infinitely many copies of yourself in different
directions.
Is our universe one of these flat shapes?
•We haven’t seen infinite copies of ourselves.
•However, due to local flatness, there is no local measurement can distinguish
among these shapes.
•Even if seeing the copies, they are faraway images from the distant past, which
might be hard to be recognized.
•Nevertheless, in 2015, astronomers did not find the evidence of our universe as a
structure such that flat 3-dimensional torus / slab.
(ref: Planck 2015 results. XVIII. Background geometry and topology of the Universe)
Sphere
•Topology: Asphere 𝑆#is ano-boundary, compact topological
manifold.
•The sum of the interior angles of a triangle is larger than 180.
•Parallel transport: A vector’s direction would be changed by
moving it, for example, from equator to pole on a 2-sphere.
•Great circles:The shortest possible paths of traveling light on a
sphere.
•Every points on a 3-dimensional sphere has an opposite point.
Is our universe asphere?
•A spherical universe can be measured locally (e.g. the sum of the angles of a
triangle on asphere).
•From cosmic microwave background (CMB),we can measure the diameter and
the distance between ahot or cold spot with Earth – which forms a triangle by
three sides and look over its curvature.
•Despite several data suggest that our universe is approximately flat, there is a
study in 2018 indicates that the universe is almost spherical.
Hyperbolic structure
•Geometry: Expands outward much more quickly
than flat geometry with negative curvature.
•The sum of the interior angles of a triangle is less
than 180.
•Poincaré disk: A distorted shape of the hyperbolic
plane –the triangles are actually in the same size.
•The boundary circle is infinitely far from any
interior point.
•The visual circle grows exponentially. Therefore, if
aguy walks toward the boundary, he will soon
shrink and disappear in your view.
Is our universe ahyperbolic structure?
•We have not seen the obvious evidence of
hyperbolic geometry of our universe.
•Still, as asphere, we are hard to notice the sum
of the angles of a triangle locally, while our
universe looks almost flat.
•Anti-de Sitter space (AdS): A type of space with
maximal symmetry and constant negative
curvature. One significant duality in physics
called AdS/CFT correspondence suggests that
the quantum gravity theory (i.e. string theory)
inside an AdS can be described by the
conformal field theory (CFT) on its boundary.
This is a crucial tool for understanding some in-
depth theories.
Holographic Principle?
•AdS/CFT correspondence is an
example of the holographic principle.
•The theory implies that the quantum
gravity theory in abulk space can be
encoded in its lower-dimensional
boundary.
•It comes from black hole
thermodynamics, that is, the fact that
entropy is in proportional to ablack
hole’s surface area (𝑆 ∝ 𝐴). It also
provides asolution to the black hole
information paradox.
The shape of the universe - 1
•Friedmann metric:
𝑑𝑠!= 𝑎!𝑡 𝛿$%𝑑𝑥$𝑑𝑥%− 𝑑𝑡!
where 𝑎(𝑡) is the scale factor. Hubble parameter is denoted by 𝐻 ≡ ⁄
3𝑎 𝑎.
•Friedmann equations:
1. From the 00-component of Einstein’s field equation: 𝐻!=&
'
'
!
=()*+,-.!
/−0.!
'!
2. From the trace of Einstein’s field equation: 𝐻!+3
𝐻 = 1
'
'= − 2)*
/𝜌 + /3
.!+-.!
/
The shape of the universe - 2
•The constant 𝑘represents the shape of the universe:
•𝑘 = −1 à3-hyperboloid
•𝑘 = 0 àEuclidean space
•𝑘 = 1 à3-sphere
•The density parameter Ω ≡ ()*+
/4!can also determine
the geometry and even the fate of auniverse:
•Ω < 1 àopen universe
•Ω = 1 àflat universe
•Ω > 1 àclosed universe
•The state of auniverse can also be determined by
the pressure 𝑝and the density 𝜌:
•𝑝 = 0, 𝜌 ∝ 𝑎!" àmatter
•𝑝 = ⁄
𝜌 3 , 𝜌 ∝ 𝑎!# àradiation
•𝑝 = −𝜌, 𝜌 ∝ 𝑐𝑜𝑛𝑠𝑡.àvacuum
Conclusions
•Our universe might be alocally flat geometry, as what we see when
living on Earth.
•However, we still have no idea about what the exact shape of our
universe is.
•It is possibly aflat space, a sphere, a hyperbolic structure, or amore
complicated shape (such as AdS).