2. Differential Geometry Flashcards
What are the two aspects of space?
Geometry and topology
What is a geodesic?
A curve of shortest length between two given points
How is curvature defined?
The difference between a small circle of geodesic radius r with circumference C(r) that differs from 2pi r
What are the 4 ways to represent space?
- Sketch/visualisation
- Parameterise - list all points of space
- Equations - Equation that is satisfied by every point
- Metric - How to measure distances between points in the space
What does the metric represent?
The square of the distance between nearby points
What is the Euclidean metric?
Pythagoras’ theorem
What are the important results of the metric of the 3-sphere?
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
What is the core difference between the metric for Euclidean and Minkowski space?
There is a minus sign infront of the time-direction
What defines Lorentzian space?
There is only one time like direction
What is the signature of Lorentzian space?
n-2
What are Riemann normal coordinates?
Coords in which the metric is Minkowski to second order accuracy at a given point
What is the critical point of the distance function called?
A geodesic
What are the three types of geodesic
Space like, time like and null
What type of geodesic to massless and massive particles travel along?
Massive - time like
Massless- null
How is a vector at point p defined?
As a tangent to a curve passing through p
What is a tangent space at p?
The collection of all vectors at p
What is a tangent bundle?
The collection of all tangent spaces
What is the vector field?
A choice of vector from each tangent space
Define a function f in terms of a manifold
A function f is a rule that assigns a real number to every point p of a manifold
What is a manifold?
A curved space that is locally flat
What is the derivative of the function f?
An object that operates on vectors to obtain directional derivatives
What is a 1 form?
A linear map from the tangent space at point p to the real line
What is the cotangent space and how is it formed?
A linear vector space which is dual to the tangent space formed by 1-forms
What is the cotangent bundle?
The collection of cotangent spaces for every point p of the manifold
What does taking the derivative of the coordinate basis give?
1 forms
Define a tensor
A natural physical quantitiy that generalises the concept of a vector by exhibiting multilinearity
Describe what the (k,l) mean in a (k, l) ranked tensor
k 1 forms
l vectors
What is a great circle?
A geodesic on a sphere where the sphere intersects with planes through the origin
What is the sectional curvature?
The circumference of a circle of geodesic radius
Describe the properties of a circle
- Centre p
- Geodesic radius r
- Choice of 2 plane in tangent space (plane circle lies on)
What is a Gauss map?
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
What is geodesic deviation?
The failure of initially parallel lines to maintain constant separation e.g. two lines on the equator meeting at the north pole
How does geodesic deviation change for spaces of positive and negative curvature?
Positive - Converge
Negative - Diverge
What are tidal forces?
The convergence or divergence of test particles moving along geodesics of space time in GR
What is parallel transport of a vector?
Where the vector changes only in the normal direction, but is unchanged in the tangential direction
What is the Riemann curvature?
The change in a vector under parallel transport around a circle
What is a connection?
A rule for differentiating vectors. (Same as the grad symbol)
What is the covariant derivative?
When you act on a vector Z with the connection (grad)
- The covariant derivative of a function is the ordinary derivative of that function
When is a vector covariantly constant?
When the covariant derivative is equal to 0
- Also the equivalent to parallel transport
