Download PHYM432 Relativity and Cosmology 6. Differential Geometry

PHYM432 Relativity and Cosmology
6. Differential Geometry
The central idea of general relativity is that gravity arrises from the
curvature of spacetime.
Gravity is Geometry
matter tells space how to curve
curved space tells matter how to move
Geometry: in mathematics can be described several ways
a small number of postulates (or axioms) can be given, which the
other results of geometry can be derived.
Euclid gave 5 axioms which fully describe Euclidean geometry
1) 2 points determine a unique line
2) parallel lines never intersect
3) all right angles are congruent...
Another way to specify Geometry is Differential Geometry where
distances between nearby points are specified, and integral calculus is
used, which fully describes the most general geometry.
The results of Euclidean geometry include:
Triangle: sum of angles = 180 degrees
Circle: C = 2πR
Sphere: A = 4πR2
Euclidean geometry (with flat space) was long thought to be the only
one possible up until the early 1800ʼs, when people started to realise
other geometries were possible.
For instance, like the curved 2D surface on a sphere (surface of Earth)
When people realised more than flat Euclidean geometry was possible,
the question of what geometry the Universe has became an Empirical
question, subject to hypothesis and tests.
In other geometries space can be curved, meaning (for instance) the
sum of the angles of a triangle can be different than 180 degrees.
A curved
2D space
triangle=270 degrees.
Diagrams like this one are 2D geometries projected in 3D.
Any N-dimensional geometry is a surface in a higher dimension,
thought that extra dimension is completely superfluous. The full
mathematics to describe the 2D surface only requires 2 dimensions.
In other geometries space can be curved, meaning (for instance) the
sum of the angles of a triangle can be different than 180 degrees.
A curved
2D space
triangle=270 degrees.
When trying to visualise curved space, itʼs probably best to limit
yourself to 2D surfaces embedded in 3D diagrams. Imagining higher
dimensions is extremely difficult (if not impossible) for people other
than S. Hawking.
Differential geometry
Begins with a systematic way of labeling points (cartesian coordinates,
polar coordinates, ect). Nearby points then have nearby coordinates.
dS = distance between nearby points (line element)
dS
(x, y)
dS =
dS
(r, θ)
dS =
dS is only valid if the increments (dx,dy) are very small, but large
differences can be built up with integration
Example: Circle
x +y =R
2
2
Calculate circumference C
C=
C=
�
�
dS
[dx2 + dy 2 ]1/2
C = 2πR
2
All geometry can be reduced to distances between two points
All distances can be reduced to integrals of
The exact form of
used
dS
dS
dS will vary depending on the coordinate system
As Gravity is geometry, we will be able to fully describe this
fundamental force with dS as well.
Non-Euclidean geometry of a 2D sphere
use angles (θ, φ) of 3D polar coordinates
r=a
dS 2 = a2 (dθ2 + sin2 θdφ2 )
Calculate ratio of circumference to radius
Can orient the polar axis at the centre of the sphere for simplicity
a
θ=0
constant theta
θ = Θ Defines the circle
dS 2 = a2 (dθ2 + sin2 θdφ2 )
dS 2 = a2 sin2 Θdφ2
C=
�
dS
C = 2πa sin Θ
R=
�
Θ
adθ
0
R = aΘ
C = 2πa sin(R/a) Relationship between circumference and radius
C = 2πa sin(R/a)
a is a fixed number characterising the geometry, measuring the scale
at which the geometry is curved.
when a is very large, the 2D sphere looks flat locally
R << a
C=
C ∼ 2πR
sin(x) =
Map making: trying to show the curved 2D surface of the earth on a
flat 2D map
Mercator projection
Equirectangular
dS 2 =
� πa �2
L
[cos2 (πy/L)dx2 + dy 2 ]
dS 2 =
�
π cos(λ(y))a
L
�2
[dx2 + dy 2 ]
Manifold: a mathematical smooth space of any number of dimensions
that on small enough scales resembles flat Euclidean space. In
special relativity, the three spacial dimensions are combined with time
to form a 4-D manifold representing spacetime (Lorentzian manifold).
curved space
flat space
The Essence of GR is transforming our frame of reference from local
inertial reference frames (where space on that small scale is
approximately flat) to accelerated frames, where matter is seen to
curve spacetime.
Back to the 2D sphere
dS 2 = a2 (dθ2 + sin2 θdφ2 )
It is conventional to call this the metric for a sphere.
A metric exists for any manifold which has a rule for computing
distances. So our 2D sphere is our smooth surface of 2-dimensions (a
manifold) and as we can compute an incremental distance for any two
close by points on the sphere, a metric exists.
Once you know the metric, the geometry of the space is entirely
defined. However, there are different ways to write the metric for a
given geometry, corresponding to the different choices of coordinate
systems.
For an n-dimensional Riemann space, the line element has the general
n �
n
form
�
(dS)2 =
gαβ dX α dX β
α=0 β=0
Parallel Transport, Curvature, and the Affine Connection
Curved space can change the direction of a vector
flat space
curved space
Which gives a further way to characterise the curvature of spacetime.
Parallel Transport, Curvature, and the Affine Connection
Curvature is also intimately related to parallel transport of a vector.
Comparing vectors at different points in curves space is not so
straightforward, as the coordinate basis vectors themselves change
direction
For a vector field �
v in a space with coordinates
The position along the curve specified by parameter u
Curve is described by coordinate functions xα = xα (u)
�v (u) = v α (u)êα (u)
In order to parallel transport a vector and preserve itʼs direction, we need
to know precisely how the coordinate basis vectors change along the
curve
�v (u) = v α (u)êα (u)
d�v
=
du
d�v
=
du
∂êα represents the rate of change of the coordinate basis vectors êα
β
β
to
the
coordinate
functions
and is a vector itself
x
x
∂êα
λ
We can rewrite this as
=
Γ
αβ êλ
β
x
λ
Γ αβ are the connection coefficients
α=1 β=2
∂ê1
λ
=
Γ
12 êλ =
x2
e.g.
So the x̂ coordinate basis vector is changing by Γ 12 in the êλ direction
as you move along the y-coordinate direction on the curve
λ
λ
Γ
As the Affine Connection αβ is determined only by unit
vectors and the coordinates, it is uniquely determined by the
metric gαβ which fully describes our curved space
Γλαβ
1 λσ
= g
2
�
∂gσβ
∂gασ
∂gαβ
+
−
α
β
∂X
∂X
∂X σ
�
d�v
=0
we can ensure parallel transporting by specifying
du
which required for each component
dv α
=
du
k
dx
Thus a parallel transport of a vector: v i (u + du) = ui − Γi jk v j
du
du
λ
The definition of Γ αβ introduces the dual metric g αβ which is the inverse
of the metric gαβ
g αβ gβγ = δ αγ
When space is curved
can change.
Γi jk = 0 the initial and final direction of a vector
When space is flat Γi jk = 0 and a parallel transported vector retains
its direction
flat space
curved space
The affine connection is used to calculate the curvature of a manifold
via the Riemann curvature tensor, which we will see later
It is also used to calculate ʻstraight linesʼ in curved space via the
geodesic equation, and thus is extensively used in GR
Lambourne
begin reading ch 4
2.4, 2.6, 2.8, 2.9, 2.14
3.4, 3.5