2. Differential Geometry

Question 1

What are the two aspects of space?

Answer 1

Geometry and topology

Question 2

What is a geodesic?

Answer 2

A curve of shortest length between two given points

Question 3

How is curvature defined?

Answer 3

The difference between a small circle of geodesic radius r with circumference C(r) that differs from 2pi r

Question 4

What are the 4 ways to represent space?

Answer 4

1. Sketch/visualisation
2. Parameterise - list all points of space
3. Equations - Equation that is satisfied by every point
4. Metric - How to measure distances between points in the space

Question 5

What does the metric represent?

Answer 5

The square of the distance between nearby points

Question 6

What is the Euclidean metric?

Answer 6

Pythagoras’ theorem

Question 7

What are the important results of the metric of the 3-sphere?

Answer 7

The metric is singular when r=R which diverges at the equator
- However space itself is not singular at these points
- This is a coordinate singularity

Question 8

What is the core difference between the metric for Euclidean and Minkowski space?

Answer 8

There is a minus sign infront of the time-direction

Question 9

What defines Lorentzian space?

Answer 9

There is only one time like direction

Question 10

What is the signature of Lorentzian space?

Answer 10

n-2

Question 11

What are Riemann normal coordinates?

Answer 11

Coords in which the metric is Minkowski to second order accuracy at a given point

Question 12

What is the critical point of the distance function called?

Answer 12

A geodesic

Question 13

What are the three types of geodesic

Answer 13

Space like, time like and null

Question 14

What type of geodesic to massless and massive particles travel along?

Answer 14

Massive - time like
Massless- null

Question 15

How is a vector at point p defined?

Answer 15

As a tangent to a curve passing through p

Question 16

What is a tangent space at p?

Answer 16

The collection of all vectors at p

Question 17

What is a tangent bundle?

Answer 17

The collection of all tangent spaces

Question 18

What is the vector field?

Answer 18

A choice of vector from each tangent space

Question 19

Define a function f in terms of a manifold

Answer 19

A function f is a rule that assigns a real number to every point p of a manifold

Question 20

What is a manifold?

Answer 20

A curved space that is locally flat

Question 21

What is the derivative of the function f?

Answer 21

An object that operates on vectors to obtain directional derivatives

Question 22

What is a 1 form?

Answer 22

A linear map from the tangent space at point p to the real line

Question 23

What is the cotangent space and how is it formed?

Answer 23

A linear vector space which is dual to the tangent space formed by 1-forms

Question 24

What is the cotangent bundle?

Answer 24

The collection of cotangent spaces for every point p of the manifold

Question 25

What does taking the derivative of the coordinate basis give?

Answer 25

1 forms

Question 26

Define a tensor

Answer 26

A natural physical quantitiy that generalises the concept of a vector by exhibiting multilinearity

Question 27

Describe what the (k,l) mean in a (k, l) ranked tensor

Answer 27

k 1 forms
l vectors

Question 28

What is a great circle?

Answer 28

A geodesic on a sphere where the sphere intersects with planes through the origin

Question 29

What is the sectional curvature?

Answer 29

The circumference of a circle of geodesic radius

Question 30

Describe the properties of a circle

Answer 30

• Centre p
• Geodesic radius r
• Choice of 2 plane in tangent space (plane circle lies on)

Question 31

What is a Gauss map?

Answer 31

A translation between each point on a surface to a unit sphere by using the normal vector of a small area centered at a point p

Question 32

What is geodesic deviation?

Answer 32

The failure of initially parallel lines to maintain constant separation e.g. two lines on the equator meeting at the north pole

Question 33

How does geodesic deviation change for spaces of positive and negative curvature?

Answer 33

Positive - Converge
Negative - Diverge

Question 34

What are tidal forces?

Answer 34

The convergence or divergence of test particles moving along geodesics of space time in GR

Question 35

What is parallel transport of a vector?

Answer 35

Where the vector changes only in the normal direction, but is unchanged in the tangential direction

Question 36

What is the Riemann curvature?

Answer 36

The change in a vector under parallel transport around a circle

Question 37

What is a connection?

Answer 37

A rule for differentiating vectors. (Same as the grad symbol)

Question 38

What is the covariant derivative?

Answer 38

When you act on a vector Z with the connection (grad)
- The covariant derivative of a function is the ordinary derivative of that function

Question 39

When is a vector covariantly constant?

Answer 39

When the covariant derivative is equal to 0
- Also the equivalent to parallel transport