Any two-dimensional metric satisfiesR_λρµν = R/2 (g_λµ g_ρν − g_λν g_ρµ).Show that the vacuum Einstein equations (with zero cosmological constant) are satisfied for any2D metric.
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Any two-dimensional metric satisfies
R_λρµν = R/2 (g_λµ g_ρν − g_λν g_ρµ).
Show that the vacuum Einstein equations (with zero cosmological constant) are satisfied for any
2D metric.
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- a)Define the term “standard candle” as used in cosmology. b)The flux is defined asf(Dlum) = L/4πD^2lumwhere L is the absolute luminosity and Dlum is the distance to the radiation source (youmay assume z ≪ 1).Assume that we have measured the flux to be f = 7.234 10^−23 Wm^−2 and the absoluteluminosity is given by L = 3.828 x10^26W. Calculate the luminosity distance D lum to the objectin Mpc.Consider a cosmological spacetime in which the line element is given by ds? = a²(t)(-dt + dr² + dy² + dz²), where a(t) > 0 is the scale factor. Two light rays tangent to l = (1,1,0, 0) and l = (1,0, 1,0) are received at time t = u" = (a-'(to), 0, 0, 0). Compute the observed angle between the correspond- ing images. to by someone with 4-velocityAny two-dimensional metric satisfies R Κιλρμν (9xu9pv - 9xv9pμ). 2 Show that the vacuum Einstein equations (with zero cosmological constant) are satisfied for any 2D metric.
- The Friedmann equation in a matter-dominated universe with curvature is given by 87G -pR² – k , 3 where R is the scale factor, p is the matter densi, and k is a positive constant. Demonstrate that the parametric solution 4G po 4тG Po R(0) (1 – cos 0) 3 k and t( (e – sin 0) 3 k3/2 solves this equation, where 0 is a variable that runs from 0 to 27 and the present-day scale factor is set to Ro = 1. %3DThe geometry of spacetime in the Universe on large scales is determined by the mean energy density of the matter in the Universe, ρ. The critical density of the Universe is denoted by ρ0 and can be used to define the parameter Ω0 = ρ/ρ0. Describe the geometry of space when: (i) Ω0 < 1; (ii) Ω0 = 1; (iii) Ω0 > 1. Explain how measurements of the angular sizes of the hot- and cold-spots in the CMB projected on the sky can inform us about the geometry of spacetime in our Universe. What do measurements of these angular sizes by the WMAP and PLANCK satellites tell us about the value of Ω0?Consider a cosmological spacetime in which the line element is given by ds? = a°(t)(-dt + da² + dy° + dz?), where a(t) > 0 is the scale factor. Two light rays tangent to l4 = (1, 1, 0,0) and = (1,0, 1,0) are received at time t = to by someone with 4-velocity u" = (a-'(to), 0, 0, 0). Compute the observed angle between the correspond- ing images.
- (a) Let L be the diameter of our galaxy.Suppose that a person in a spaceship of massm wants to travel across the galaxy at constantspeed, taking proper time τ. Find the kineticenergy of the spaceship. (b) Your friend is impa-tient, and wants to make the voyage in an hour.For L = 105 light years, estimate the energy inunits of megatons of TNT (1 megaton=4×109 J).Evaluate the Larmor frequency for orbital motion in a field of B=1T.When two spiral galaxies collide, the stars generally do not run into each other, but the gas clouds do collide, triggering a burst of new star formation. (a) Estimate the probability that our Sun would collide with another star in the Andromeda Galaxy, if a collision between the Milky Way and Andromeda were happening at the present time. To simplify the problem, assume that each galaxy has 100 billion stars exactly like the Sun spread evenly over a circular disk with a radius of 100,000 light-years. (Hint: First calculate the total area of 100 billion circles with the radius of the Sun and then compare that total area to the area of the galactic disk.) (b) Estimate the probability that a gas cloud in our galaxy could collide with another gas cloud in the Andromeda Galaxy. To simplify the problem, assume that each galaxy contains 100,000 gas clouds of warm hydrogen gas, that each cloud has a radius of 300 light-years, and that these clouds are spread evenly over a circular disk with a…
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