Relativity, Black Holes and Cosmology

Question 1

Riemann tensor vanishes to 0 =>

Answer 1

Spacetime is flat

Question 2

How to prove spacetime is flat

Answer 2

Show Riemann tensor vanishes to 0

Question 3

State the properties of 𝐷/𝑑𝜆 for each quantity

Answer 3

• 𝐷/𝑑𝜆 = 𝑑/𝑑𝜆 for scalars
• (𝐷(𝐴^𝛼))/𝑑𝜆 = 0 for parallelly transported vectors and 1-forms

Question 4

Conversion for time/mass

Answer 4

G/c^3, sec

Question 5

Conversion for distance/mass

Answer 5

G/c^2, m

Question 6

Units of c

Answer 6

m/sec

Question 7

Units of G

Answer 7

m^3 / (sec^2 kg)

Question 8

Photon 𝑢^𝛼

Answer 8

𝑢_𝛼 𝑢^𝛼 = 0

Question 9

Equivalent values of 4-velocity

Answer 9

Question 10

Equivalent forms of ∇_𝛼

Answer 10

Question 11

Static observer in Schwarzschild spacetime

Answer 11

Question 12

Proper time

Answer 12

Question 13

Radial =>

Answer 13

Question 14

Answer 14

Question 15

Value of Kronecker delta

Answer 15

Question 16

Covariant derivative

Answer 16

Question 17

∆𝑠^2 for null, timelike, spacelike events

Answer 17

• = 0 for null-seperated
• < 0 for timelike-seperated
• > 0 for spacelike-seperated

Question 18

To show a coordinate is null

Answer 18

For coordinate a,
show g^aa = 0

Question 19

Lorentz factor

Answer 19

Question 20

Spacetime diagram lines for particles

Answer 20

• Particle slower than light: > 45
• Speed of light: 45 degrees
• Particle faster than ligth < 45

Question 21

2D Riemann components

Answer 21

Only one component to consider:

Question 22

How to solve the Euler-Lagrange equations:
* deriving null geodesics
* and finding radial equations for a light ray

Answer 22

Question 23

Estimate values of sinh(x) and cosh(x) for x << 1

Answer 23

sinh(x) ~ x
cosh(x) ~ 1 + x^2