4. Spacetime and General Relativity

Question 1

Minkowski Spacetime Metric

Answer 1

g = -c²dtxdt + a²(t)[dχxdχ + sin²χ(dθxdθ + sin²θ dφxdφ)]

Question 2

Connection One-Forms

Answer 2

ωμν are the connection one forms
-one forms are orthonormal with respect to the metric so, for each component take the square root of the factor in the metric that contains that component:
ωt = cdt
ωχ = a(t)dχ
ωθ = a sinχ dθ
ωφ = a sinχ sinθ dφ

Question 3

Metric in Terms of Connection One-Forms

Answer 3

g = -ωtxωt + ωχxωχ + ωθxωθ + ωφxωφ

Question 4

Applying d to the Connection One-Forms

Answer 4

• recall d²=0

- use product rule on each one form to calculate dωt, dωχ, dωθ and dωφ

Question 5

Cartan’s First Structure Equation

Answer 5

-for each component t, χ, θ and φ,
dωi = -ωij∧wj - ωij∧wj - ωij∧wj
-where j represents each of the other three components
-substitute in the expressions for dωi, then equating each component, calculate, either exactly or as a proportionality relation, all ωij
-there are there for each component, so 12 in total

Question 6

Ricci Curvature

Definition

Answer 6

-Ricci(x1) represents the mean curvature in planes orthogonal to x1

Question 7

Ricci in Time Component

Answer 7

Ricci(et, et) = -3 a’‘/ac²

Question 8

Ricci in Space Components

Answer 8

Ricci(eχ,eχ) = Ricci(eθ,eθ) = Ricci(eφ,eφ)

= a’/ac² + 2/a² (1 + a’²/c²)

Question 9

Scalar Curvature, R

Answer 9

-the average of the Ricci curvature for each component calculated with respect to the metric:
R = -Ricci(et,et) + Ricci(eχ,eχ) + Ricci(eθ,eθ) + Ricci(eφ,eφ)
= 6 a’‘/ac² + 6/a² (1 + a’²/c²)

Question 10

Einstein Tensor

Time

Answer 10

G(et, et) = Ricci(et,et) - 1/2 g(et,et) R

= 3/a² (1 + a’²/c²)

Question 11

Einstein Tensor

Space

Answer 11

G(eχ,eχ) = G(eθ,eθ) = G(eφ,eφ)

= -2 a’‘/ac² - 1/a² (1 + a’²/c²)

Question 12

Energy-Momentum Tensor for Perfect Fluids

Answer 12

-energy-momentum tensor = T
T(et,et) = ρc², where ρ is energy density
T(eχ,eχ) = T(eθ,eθ) = T(eφ,eφ) = P
-where P is the isotropic pressure of the fluid

Question 13

Einstein’s Equation

Answer 13

8πG/c^4 T(ei,ei) = G(ei,ei)

• where G on the LHS is the gravitational constant and G on the RHS is the Einstein tensor
• true for all 4 dimensions and for any spacetime, i=t,χ,θ,φ

Question 14

Friedmann Equation for Positive Curvature

Answer 14

8πGρ/3 = c²/a² 1= a’²/a²

Question 15

Acceleration Equation for Positive Curvature

Answer 15

-4πG/3 (P/3c² + ρ) = a’‘/a

Question 16

Friedmann Equation General Case

Answer 16

8πGρ/3 = ϰc²/a² 1= a’²/a²

Question 17

Acceleration Equation General Case

Answer 17

-4πG/3 (P/3c² + ρ) = a’‘/a

Question 18

Fluid Equation General Case

Answer 18

-a particular linear combination of the general Friedmann and acceleration equations
ρ’ = -3a’/a (ρ + P/c²)
-where a’/a = H(t)