PHYM432 Relativity and Cosmology6. 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 curvecurved 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 geometry1) 2 points determine a unique line2) parallel lines never intersect3) all right angles are congruent...

The results of Euclidean geometry include:Triangle: sum of angles = 180 degreesCircle: Sphere:

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)

C = 2πRA = 4πR2

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.

In other geometries space can be curved, meaning (for instance) the sum of the angles of a triangle can be different than 180 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.

A curved2D spacetriangle=270 degrees.

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.

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.

A curved2D spacetriangle=270 degrees.

Differential geometryBegins 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)

(r, θ)(x, y)

dS

dS = dS =

dS is only valid if the increments (dx,dy) are very small, but large differences can be built up with integration

dS

Example: Circlex2 + y2 = R

Calculate circumference C

C =�

dS

C =�

[dx2 + dy2]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 will vary depending on the coordinate system used

dS

dS

dS

As Gravity is geometry, we will be able to fully describe this fundamental force with as well. dS

Non-Euclidean geometry of a 2D sphere

use angles of 3D polar coordinates (θ,φ)

r = a

dS2 = a2(dθ2 + sin2 θdφ2)

Calculate ratio of circumference to radiusCan orient the polar axis at the centre of the sphere for simplicity θ = 0

θ = Θ constant thetaDefines the circle

C =�

dS R =� Θ

0adθ

dS2 = a2 sin2 Θdφ2

C = 2πa sinΘ R = aΘ

C = 2πa sin(R/a)

a

Relationship between circumference and radius

dS2 = a2(dθ2 + sin2 θdφ2)

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 sin(x) =

C =

C ∼ 2πR

Equirectangular

Mercator projection

dS2 =�

π cos(λ(y))aL

�2

[dx2 + dy2]dS2 =�πa

L

�2[cos2(πy/L)dx2 + dy2]

Map making: trying to show the curved 2D surface of the earth on a flat 2D map

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).

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.

curved space

flat space

Back to the 2D sphere

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 form

(dS)2 =n�

α=0

n�

β=0

gαβdXαdXβ

dS2 = a2(dθ2 + sin2 θdφ2)

Parallel Transport, Curvature, and the Affine Connection

Curved space can change the direction of a vector

flat spacecurved space

Which gives a further way to characterise the curvature of spacetime.

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

The position along the curve specified by parameter

�v(u) = vα(u)eα(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

Parallel Transport, Curvature, and the Affine Connection

�vu

Curve is described by coordinate functions

For a vector field in a space with coordinates

xα = xα(u)

�v(u) = vα(u)eα(u)d�v

du=

d�v

du=

∂eα

xβ

represents the rate of change of the coordinate basis vectors to the coordinate functions and is a vector itself

eαxβ

We can rewrite this as ∂eα

xβ= Γλ

αβ eλ

Γλαβ are the connection coefficients

So the coordinate basis vector is changing by in the direction as you move along the y-coordinate direction on the curve

e.g. ∂e1

x2= Γλ

12eλ =

α = 1 β = 2

x Γλ12 eλ

vi(u + du) = ui − Γijkvj dxk

duduThus a parallel transport of a vector:

As the Affine Connection is determined only by unit vectors and the coordinates, it is uniquely determined by the metric which fully describes our curved space

Γλαβ

gαβ

Γλαβ =

12gλσ

�∂gσβ

∂Xα+

∂gασ

∂Xβ− ∂gαβ

∂Xσ

�

we can ensure parallel transporting by specifyingwhich required for each component

d�v

du= 0

dvα

du=

The definition of introduces the dual metric which is the inverse of the metric

Γλαβ

gαβ

gαβ

gαβgβγ = δαγ

flat space curved space

Γijk = 0When space is flat and a parallel transported vector retains

its direction

When space is curved Γijk = 0 the initial and final direction of a vector

can change.

The affine connection is used to calculate the curvature of a manifold via the Riemann curvature tensor, which we will see laterIt is also used to calculate ʻstraight linesʼ in curved space via the geodesic equation, and thus is extensively used in GR

Lambournebegin reading ch 4

2.4, 2.6, 2.8, 2.9, 2.143.4, 3.5