HOMEWORK 10. Applications: special relativity, Newtonian limit, gravitational waves, gravitational lensing, cosmology, 1 black holes

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1 General Relativity 8.96 (Petters, spring 003) HOMEWORK 10. Applications: special relativity, Newtonian limit, gravitational waves, gravitational lensing, cosmology, 1 black holes 1. Special Relativity (a) (Illusion of exceeding the speed of light) Suppose that relative to an observer s inertial frame a source S emits material at time t = 0 towards an observer at an angle θ with the observer-source line of sight, where θ << 1. Assume that the speed of the material is v. i. Suppose that the time interval between when the material was ejected from the source and when the material arrived at position P is t P. Show that the arrival time difference (starting at time t = 0) between when light arrives at the observer when from P and from S is given by ( t = 1 v ) t P. c ii. Determine the apparent transverse speed of the material. iii. Suppose that the observer-source distance is 100 times the distance from P to the source, and that the angle θ P which P makes with the observer-source line of sight is 1 arcsecond. Assume that the speed v of the ejected material is slight less than the speed of light. In which interval must the speed v lie in order for the apparent transverse speed of the material to exceed the speed of light. (b) Using the fact that phase space number density is a Lorentz invariant, show that I ν /ν 3, where I ν is the specific intensity (or surface brightness see [PLW, p.83]) of light at frequencey ν, is also invariant under Lorentz transformations. (c) Show that the phase kx ωt of an electromagnetic wave moving in the x-direction in Minkowski spacetime is a Lorentz invariant.. Consider a local region of spacetime covered by Cartesian coordinates x 0 = ct, x 1, x, x 3. The Newtonian limit corresponds to (1) a weak gravitational field, i.e., the spacetime metric g µν is a small perturbation of the Minkowski metric g µν = η µν + h µν, h µν << 1, with h µν considered only up to first order; () a static spacetime metric, which means that g µν / x 0 = 0; and (3) particles move slowly with respect to the speed of light, i.e., dx i /dτ << c = dx 0 /dτ. Show that the geodesic equations yield that the spatial acceleration a of particles under only the influence of gravity is given by the usual Newtonian result: a = Φ, where h 00 Φ/c with Φ the three-dimensional Newtonian potential of the gravitational field. What is the value of the perturbation h 00 from flat spacetime near the surface of the earth? The sun? 1 Unless stated to the contrary, assume that Λ = 0. 1

2 3. For a spacetime with a weak gravitational wave source, the metric has the form g αβ = η αβ + h αβ, where the gravitational wave amplitudes h αβ are small, h αβ << 1. Einstein s equations in the Lorentz gauge transform into the following wave equations: h αβ = 16πG c 4 T αβ, where h αβ = h αβ 1 η αβh with h = h µ µ. The Ricci curvature is given by R αβ = 1 h αβ. (a) Show that h αβ = h αβ 1 η αβ h, where h = h ν ν. It then suffices to solve the Einstein equations for h αβ in order to determine the gravitational wave amplitudes h αβ. (b) For vacuum, show that the Einstein equations for a gravitational wave become h αβ = 0. Note that there is no bar on h αβ. 4. Consider a single-plane gravitational lens system due to g point masses with continuous matter having constant dimensionless density κ c and shear γ from infinity acting along the horizonal axis [PLW, Sec ]. The time delay function is given by T y(x) = x y ψ (x), where ψ (x) = κ c x γ (u v ) + g m i ln x ξ i. Here x = (u, v) and m i = M i /(πd L σ crt, where M i is the mass of the ith point mass at position ξ i and σ crt = c d S /(4πGd L d L,S ) (critical density). The lens equation is y = x α (x), where α = grad ψ. Due to the singularities in T y arising from the positions ξ i of the point masses, the domain of the time delay function is L = R {ξ 1,..., ξ g }. (a) For several lensing situations, the source moves at constant velocity along a straight line. In principle, this means that the individual images of the source will trace out trajectories on the lens plane L. Show that for this lens system, no image can trace out a curve in L that is diffeomorphic to a circle. (b) Suppose that the continuous matter is supercritical, i.e, κ c > 1. Show that no minimum lensed images can occur? Can maximum lensed images occur if κ c is subcritical (i.e., κ c < 1)? 5. Consider a gravitational lens system with time delay function T y. Suppose that i=1 1 κ c + γ > 0, 1 κ c γ > 0. (a) Determine the magnification µ(x) of a minimum lensed image as x. (b) If the number of saddle lensed images increases by one, by how much will the number of minimum lensed images increase? (c) For the case of k-plane lensing by point masses only, suppose that the number of negative parity lensed images increases by one. Determine the increase in the number of positive parity lensed images.

3 6. Consider a galaxy lens on macro- and micro-scales. At the macro-scale, the galaxy is treated as smoothed-out with surface mass density κ s and shear Γ s, where Γ s = Γ 1 + Γ with Γ 1 and Γ the components of the shear matrix (see [PLW, p. 96]). Suppose that the smoothed-out galaxy produces multiple images, called macroimages. Let A L be the area of a small region D L in the lens plane about one of the macro-images and let and A S be the area of the small region about the source that D L is mapped onto by the (macro-) lensing map. On the micro-scale, granularity is introduced in the lens by treating the region A L as consisting of stars with a constant shear γ from infinity (due to tidal action of the rest of the galaxy) and continuous matter (interstellar matter, dark matter) of constant density κ c. Note that the areas A L and A S appear large from the micro-scale perspective. Since the positions of the stars about the macro-image are not known, assume that the stars are randomly distributed in D L. The multiple images produced at the micro-scale are called microimages. (a) Show that the magnification of a given macro-image is given approximately by µ macro = A L /A S. (b) Show that the average total magnification of the microimages in D L of the source is related to the magnification of the macroimage at D L as follows: < µ tot >= 1 (1 κ s ) Γ s = µ macro. 7. The stress-energy tensor for a Robertson-Walker universe dominated by electromagnetic radiation can be expressed as T µν = 1 [F µλ F ν λ 14 ] 4π gµν F λσ F λσ, where F µν is the electromagnetic field strength tensor. Show that the equation of state for this model is = ρ 3, where and ρ are the pressure and energy density of the photons, resp. 8. (a) By Homework 7 (3,d), the age of the universe in an Einstein-de Sitter universe, which is a matter-dominated ( = 0, ρ > 0) Roberston-Walker universe with critical cosmic density (Ω 0 = 1) and no cosmological constant (Λ = 0), is given by the formula τ 0 = 3 H 1 0. Hubble s constant is usually expressed in the form H 0 = 100hkms 1 Mpc 1, where the factor h is a dimensionless number. Recent observations indicate H 0 = (67 ± 10)kms 1 Mpc 1. i. Estimate the age of the universe in an Einstein-de Sitter model using h = ii. The age of the oldest known stars, which occur in globular clusters, are roughly 11.5 ± yrs. If one assumes that these globular cluster stars were formed roughly billion years after the Big Bang, then the age of the universe would be between 1 and 15 billion years. How does this compare with the age of the universe estimated via an Einstein-de Sitter model? (b) Recent observations show that the cosmological constant Λ is positive. Roberston-Walker model with = 0, ρ > 0, k = 0, and Λ > 0. 3 For a

4 i. Show that ρ M + ρ Λ = 3H 0 8πG, where ρ M is the present cosmic mass density of matter and ρ Λ = Λc /(8πG). ii. Show that ( ) ( 1/3 ρm a(τ) = a 0 sinh /3 τ ) 3Λc. ρ Λ iii. Show that the age of the universe is given by τ 0 = 3 H 1 0 ( tanh 1 ) Ω Λ, ΩΛ where Ω Λ = c Λ/3H0. iv. Observations of the cosmic microwave background yields that the current cosmic density Ω 0 = Ω M +Ω Λ is near critical Ω 0 = 1. Together with the results of type Ia Supernovae observations, one obtains Ω M 0.3 (contribution from matter, includes dark matter) and Ω Λ 0.7 (contribution from dark energy). Using h = 0.65, estimate the age of the universe in this cosmology. How does it compare with the age of the universe when estimated using the oldest known stars? 9. Let γ(s) = (x 0 (s), x 1 (s), x (s), x 3 (s)) be a freely falling causal particle that is outside the horizon of a Schwarzschild black hole. Express (x 1 (s), x (s), x 3 (s)) in terms of spherical coordinates (r(s), θ(s), φ(s)) and assume that γ is initially equatorial. (a) Show that γ is determined by the following equations: ( 1 r ) sch dt r ds = E = constant, θ(s) = π dφ, r ds = L = constant. (b) For γ a light ray, determine the physical dimensions of c E and L in terms of energy and time. Conclude that the impact parameter b = L /(ce) has physical dimension of length. (c) For γ a light ray that flies by the black hole at perihelion distance r 0, show that the impact parameter b is given by b = r 0. 1 r sch r 0 In other words, the impact parameter and the distance of closest approach are not equal in general: b > r 0 for r 0 > r sch and b r 0 for r 0 >> r sch. 10. Consider a freely falling initially equatorial material particle in a bound nearly-elliptical orbit about the horizon of a Schwarzschild black hole. (a) Show that the radial position r of the particle is governed by the following relativistic orbit equation: d u dφ + u = GM L + 3 GM c u, where u = 1/r and M is the mass of the black hole. How does this compare with the Keplerian orbit equation in the Newtonian limit? 4

5 (b) Assume that L >> (GM/c) and r φ << c. It can be shown that the relativistic orbit equation is solved approximately by r(φ) = (L /GM) 1 + e cos[(1 d 0 )φ], where e is the eccentricity of the orbit and d 0 = 3[GM/(cL)] << 1. i. Show how the radial position r of the orbit compares with the Schwarzschild radius. What is the semi-major axis a of the orbit? ii. To first-order in d 0, determine the values of φ where the perihelions occur. What is the φ-value of the perihelion position after the nth orbit? Comparing this with the φ-value after the (n+1)st orbit, show that the angular perihelion shift δ per orbit is δ = 6πGM c a(1 e ), which is in terms of measurable quantities. iii. The semi-major axis and eccentricity of Mercury s orbit around the sun are a = m and e = Determine the total increase in the angular perihelion shift of Mercury per century. Give your answer in arcseconds per century. 5