Elementary Differential Geometry

Question 1

What is spacetime M in GR?

Answer 1

A 4‑dimensional set of events; not necessarily all of ℝ⁴ may be an open subset of ℝ⁴

Question 2

What describes an event in spacetime?

Answer 2

Four coordinates x^μ with μ = 0,1,2,3.

Question 3

What is the key difference between SR and GR regarding frames?

Answer 3

No inertial frames in GR; all choices of coordinate systems are equally valid.

Question 4

What must GR laws be invariant under?

Answer 4

General coordinate transformations.
They have been formulated in such a way that they are.

Question 5

What is a general coordinate transformation?

Answer 5

A smooth, invertible map x^μ → x’^μ = x’^μ (x^μ) = x’^μ (x^0, x^1, x^2, x^3)

Question 6

What is the inverse transformation requirement?

Answer 6

x’^μ → x^μ = x^μ (x’^μ) = x^μ (x^’0, x^’1, x^’2, x^’3)

x’^μ(x(x’)) = x’^μ and x^μ(x’(x)) = x^μ.

Question 7

What is the Jacobian of a coordinate transformation?

Answer 7

J^μ_ν = ∂x’^μ / ∂x^ν.

Question 8

What property must the Jacobian satisfy?

Answer 8

It must be invertible.

Question 9

Proof the Jacobian is invertible

Answer 9

J^-1μ_ν J^v_σ
= ∂x^μ / ∂x’^ν ∂x’^v / ∂x^σ
= ∂x^μ / ∂x^σ
= δ^μ_σ
= identity

Question 10

What is the ‘set up’ for finding tangent and cotangent spaces?

Answer 10

consider parametrised world lines x^μ : ℝ⁴ –> M ,
λ –> x^μ(λ)

fix a point x_0 in M and assume that our world line satisfies x^μ(0) = x^μ_0
ie it passes through our point at parameter λ =0

Question 11

What is the tangent vector to a worldline at λ=0?

Answer 11

t^μ = dx^μ/dλ |_{λ=0}.

Question 12

What is the tangent space TₓM?

Answer 12

The vector space of all tangent vectors at x.

Question 13

What is the abstract definition of tangent vectors?

Answer 13

Equivalence classes of curves with identical derivatives at x.

Question 14

How do tangent vectors transform?

Answer 14

t’^μ = J^μ_ν t^ν

Question 15

What is the cotangent space T*ₓM?

Answer 15

The dual vector space to TₓM.

Question 16

What does a cotangent vector do?

Answer 16

Maps tangent vectors to real numbers.
Each cotangent vector ω_μ in TₓM defines a linear map
ω_μ : R –> TₓM,
t^μ –> ω_μ t^μ

Question 17

How do cotangent vectors transform?

Answer 17

w’_μ = J^{-1^ν}_μ w_ν.

Question 18

Loosely speaking what is a field?

Answer 18

A field A on M is an assignment, x –> A(x) = some physical quantity at x, of a physical quantity A(x) to each point in spacetime x in M

Question 19

What is a scalar field?

Answer 19

x –> φ(x) in R
assignment of a real number φ(x) to each point in spacetime

Question 20

How do scalar fields transform?

Answer 20

φ’(x’) = φ(x).

Question 21

What is a vector field?

Answer 21

x –> A^μ(x) ∈ TₓM
assignment of a tangent vector A^μ(x) at x to each point in spacetime

Question 22

How do vector fields transform?

Answer 22

A^’μ(x’) = J^μ_ν A^v(x)

Question 23

What is a covector field?

Answer 23

x –> B_μ(x) ∈ T*ₓM
assignment of a cotangent vector B_μ(x) at x to each point in spacetime

Question 24

How do covector fields transform?

Answer 24

B_‘μ(x’) = J^-1v_μ B_v(x)

Question 25

What is index contraction?

Answer 25

Summing one upper and one lower index to form a scalar.

Question 26

What is a (p,q)-tensor field?

Answer 26

An object with p upper and q lower indices transforming with p Jacobians and q inverse Jacobians.

Question 27

What are examples of tensor types?

Answer 27

(0,0)=scalar field (1,0)=vector field (0,1)=covector field

Question 28

What is the line element in GR?

Answer 28

ds² = g_{μν}(x) dx^μ dx^ν.

Question 29

What does g_{μν} represent?

Answer 29

A symmetric (0,2) tensor encoding spacetime geometry.

Question 30

What must ds² be under coordinate transformations?

Answer 30

Invariant.

Question 31

What transformation rule must g_{μν} satisfy?

Answer 31

g'_{αβ} = J^{-1μ}_α J^{-1ν}_β g_{μν}.

Question 32

What condition must g_{μν} satisfy for equivalence principle?

Answer 32

At each point, it must be transformable to Minkowski form. This will give us a 'good' g_{μν}

Question 33

What do we assume about g_{μν} at each point in spacetime

Answer 33

the matrix g_{μν} has 1 negative and 3 positive eigen values

Question 34

What is a Lorentzian metric?

Answer 34

A symmetric (0,2) tensor field with 1 negative and 3 positive eigenvalues.

Question 35

A Lorentzian metric g_{μν} allows us to decompose the tangent space TₓM at any point x_0 into...

Answer 35

Time like if g_{μν} t^μ t^ν <0, light like if g_{μν} t^μ t^ν =0, Space like if g_{μν} t^μ t^ν >0

Question 36

What defines timelike/lightlike/spacelike worldlines?

Answer 36

Sign of g_{μν} dx^μ/dλ dx^ν/dλ Time like if <0, light like if =0, Space like if >0

Question 37

What is the matrix for Friedmann-Robertson-Walker spacetime which has the line element ds^2 = -c^2 dt^2 + a^2 (t) [(dx^1)^2 (dx^2)^2 (dx^3)^2]

Answer 37

g_{μν} = [(-1,0,0,0) (0,a^2(t),0,0) (0,0,a^2(t),0) (0,0,0,a^2(t)]

Question 38

What is the inverse metric g^{μν}?

Answer 38

The (2,0) tensor satisfying g^{μν} g_{νρ} = δ^μ_ρ.

Question 39

What does the inverse metric allow?

Answer 39

Raising indices B^μ = g^{μν} B_v

Question 40

What does the metric allow?

Answer 40

Lowering indices A_μ = g_{μν} A^v

Question 41

What is the inner product of two vectors A and A~?

Answer 41

=g_{μν} A^μ A~^ν = A_v A~^v or = A^μ A~_μ

Question 42

What is the inner product of two covectors B and B~?

Answer 42

=g^{μν} B_μ B~_ν = B^v B~_v or = B_μ B~^μ

Question 43

What is the partial derivative of a scalar field?

Answer 43

∂_μ Φ(x) = ∂/∂x^μ Φ defines a (0,1) tensor field

Question 44

What is the problem with partial derivatives of vector fields?

Answer 44

∂_μ A^ν does not transform as a tensor.

Question 45

What fixes this?

Answer 45

The covariant derivative.

Question 46

What is the covariant derivative of a vector field?

Answer 46

∇_v A^μ = ∂_v A^μ + Γ^μ_{vρ} A^ρ.

Question 47

What are Γ^μ_{vρ}?

Answer 47

Christoffel symbols.

Question 48

How do covariant derivatives of covectors differ?

Answer 48

∇_μ B_ν = ∂_μ B_ν − Γ^ρ_{μν} B_ρ.

Question 49

∂_μ under g c t is

Answer 49

∂'_μ = J^{-1v}_μ ∂_v

Question 50

Γ^μ_{vρ} under g c t is

Answer 50

Γ'^μ_{vρ} = J^μ_σ J^{-1α}_v J^{-1β}_ρ Γ^σ_{αβ} - J^{-1α}_v J^{-1β}_ρ (∂_α J^μ_β)

Question 51

By construction the covarient derivative is a ... tensor field?

Answer 51

(1,1)

Question 52

How does the covarient derivative transform under g c t?

Answer 52

∇'_v A'^μ = J^{-1ρ}_v J^μ_σ ∇_ρ A^σ

Question 53

For larger tensor fields ie (p,q) tensor fields what is the covarient derivative

Answer 53

the partial derivative of the field minus a Christoffel contraction for each lower index plus a Christoffel contraction for each upper index eg ∇_μ C^{αβ}_σ = ∂_μ C^{αβ}_σ + Γ^α_{μρ} C^{ρβ}_σ + Γ^β_{μγ} C^{γα}_σ - Γ^δ_{μσ} C^{αβ}_δ

Question 54

What two conditions define the Levi‑Civita connection?

Answer 54

Unique covariant derivative that satisfies Metric compatibility ∇_μ g_{νρ}=0 and symmetry Γ^μ_{ρν}=Γ^μ_{νρ}.

Question 55

What are the Christoffel symbols using the Levi‑Civita connection?

Answer 55

Γ^ρ_{μν} = ½ g^{ρσ}[∂_μ g_{νσ} + ∂_ν g_{μσ} − ∂_σ g_{μν}]

Question 56

What is Tor ^ρ _{μv}

Answer 56

Torsion tensor Tor ^ρ _{μv} = Γ^ρ_{μν} - Γ^ρ_{vμ} 0 if Levi Civita

Question 57

In Minkowski spacetime in any inertial frame g_{μv} = η_{μv} so what are the Christoffel symbols?

Answer 57

Γ^ρ_{μν} = 0

Question 58

In every inertial frame the covariant derivative of Minkowski spacetime is? In the case that Γ^ρ_{μν} = 0 (Levi-Civita)

Answer 58

∇_μ = ∂_μ

Question 59

In Levi-Civita what is ∇_μ g^{ρν} ?

Answer 59

∇_μ g^{ρν} = 0

Question 60

What is the Riemann curvature tensor?

Answer 60

R^β_{ρνμ} = ∂_v Γ^β_{μρ} − ∂_μ Γ^β_{vρ} + Γ^σ_{μρ} Γ^β_{vσ} − Γ^σ_{vρ} Γ^β_{μσ}

Question 61

What type of tensor is Riemann?

Answer 61

A (1,3) tensor.

Question 62

What is the Riemann tensor dependent on?

Answer 62

∂Γ , Γ ∂^2 g, ∂g, g

Question 63

What are key symmetries of Riemann?

Answer 63

R_{ρσμν} = −R_{σρμν}, and R_{ρσμν} = R_{μνρσ}.

Question 64

What are the Bianchi identities?

Answer 64

R^β_{ρνμ} + R^β_{vμρ}+ R^β_{μρv} = 0 and ∇_σ R^β_{ρνμ} + ∇_v R^β_{ρμσ} + ∇_μ R^β_{ρσν} = 0

Question 65

What is curvature in flat spacetime?

Answer 65

Riemann tensor vanishes.