Third Year: General Relativity and Cosmology. 1 Problem Sheet 1 - Newtonian Gravity and the Equivalence Principle

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1 Third Year: General Relativity and Cosmology 2011/2012 Problem Sheets (Version 2) Prof. Pedro Ferreira: 1 Problem Sheet 1 - Newtonian Gravity and the Equivalence Principle : In Problem Sheet 1, do not worry about covariant and contravariant tensors 1. The Newtonian Potential of spherical masses. The Newtonian Gravitational potential outside an isolated spherically symmetric body of mass M is Φ = GM r. (a) Show that the gravitational field is g = GM r 2 ˆr where ˆr is the unit vector in the direction of r. [you will need to calculate r/ x i. This can be done easily by first calculating (r 2 )/ x i and invoking the product rule] (b) Now assume that the mass has radius R and constant density ρ. The gravitation potential inside the mass is Show that Φ = GMr2 2R 3 +constant 2 Φ = 4πGρ (c) Show that the gravitational field inside the body is g = GM R 3 r 2. The relative acceleration in a spherical well. The gravitational potential outside a single spherically symmetric massive body (such as an idealised version of the Earth) of mass M is Φ = GM r

2 Consider an observer and a set of nearby test-particles that are initially at rest and then are simultaneously released to fall freely in this gravitational field. The separation, i of two nearby particles satisfies i = 2 Φ x i x j j (a) Find the matrix M ij = 2 Φ x i x j. (b) Show that the radius vector x i is an eigenvector of M ij. [hint: calculate the product M ij x j ] (c) Show that any vector orthogonal to x i is also an eigenvector of M ij. [hint: calculate the product M ij a j if a r = 0. (d) Using these results describe the effect that the gravitational field has on the separations of the test particles from the central observer. (e) Use your knowledge of the gravitational field of a spherically symmetric massive body to show how the qualitative result could have been deduced more simply. These are known as gravitational tidal forces 3. Orbits in Newtonian Gravity Consider a massive stationary object of mass M fixed at the origin, and a particle moving in the x y plane subject only to the Newtonian gravitational field of the massive object. (a) Consider the unit vectors, ˆr and ˆθ given in Cartesian coordinates by ˆr = (cosθ,sinθ) and ˆθ = ( sinθ,cosθ). Show that d = θˆθ, dtˆr d dtˆθ = θˆr (b) Writing the vector r = rˆr, show that ṙ = ṙˆr+r θˆθ, r = ( r r θ 2 )ˆr+(2ṙ θ +r θ)ˆθ (c) Newton s equations for the momentum p of the particle of mass m > 0 in this situation are r = (GM/r 2 )ˆr. Show that this reduces, in components, to the two equations: r r θ 2 + GM r 2 = 0, 2ṙ θ+r θ = 0 (d) Deduce that J = mr 2 θ, the angular momentum about the origin is constant. (e) By substitution, r(t) = 1/u(θ(t)), show that ṙ = Ju /m where d/dθ and further that u satisfies u +u = GMm2 J 2 2

3 (f) Show that the general solution for the distance from the origin r, as a function of θ is r = l 1+ecos(θ θ 0 ) What do these orbits look like and what is e? 4. Uniformly Accelerated Reference Frame (a) Transform the line element of special relativity from the usual (t, x, y, z) rectangular coordinates to new coordinates (t,x,y,z ) related by ( ) ( ) c t = g + z gt sinh c c ( ) ( ) c z = c g + z gt cosh c c y = y x = x for a constant g with dimensions of acceleration. (b) For gt /c 1, show that this corresponds to a transformation to a uniformly accelerated frame in Newtonian mechanics. (c) Show that a clock at rest in this frame at z = h runs fast compared to a clock at rest at z = 0 by a factor (1+ gh ). How is this related to the equivalence principle idea? c 2 (d) A laboratory has a bottom at z = 0 and a top at z = h both with extent in the x and y direction. Compute the invariant acceleration a = (a α a α ) 1/2, where a α = d2 x α, and show dτ 2 that it is different for the top and bottom of the laboratory. 5. Gravitational Redshift of the Sun Estimate the gravitational redshift of light from the surface of the Sun. Discuss the possibility of measuring this effect given that the velocities of matter in convection cells at the surface of the Sun is of of order 1 km/s. Is there one part of the surface is better than another for making the observation? 6. Time Dilation The Earth his approximately 5 billion years old. c2 g (a) How much younger are the rocks at the center of the Earth than at the surface? (b) If equal abundances of a radioactive element with a decay time of 4 billion years were present to start, how much more of that element would be present at the center of the surface [Assume the density of the Earth is constant]? 3

4 2 Problem Sheet 2 - Geodesics and orbits. 1. Four velocity Show by direct calculation from the geodesic equation that the norm of the four-velocity u α u α is a constant along a geodesic. 2. Time around a Schwarzschild metric A spaceship is moving without power in a circular orbit about an object with mass M. The radius of the orbit is R = 7GM/c 2. (a) Find the relation between the rate of change of the angular position of the spaceship and the the proper time and radius of the orbit. (b) What is the period of the orbit as measured by an observer at infinity. (c) What is the period of the orbit as measured by a clock in the spaceship? 3. The Geometry around the Earth Consider a particle moving on a circular orbit (of radius R) about the Earth. Assume the metric is [ ds 2 = 1+2 Φ(r) ] [ c 2 dt Φ(r) ] (dx 2 +dy 2 +dz 2 ) c 2 c 2 with Φ(r) = GM /r. Let P be the period of the orbit measured in the time t. Consider two space-time events, A and B located at the same spatial position on the orbit but separated in t by the period P. Calculate (to first order in 1/c 2 ) the proper time for an observer that (a) follows the orbit of the particle itself; (b) stays fixed in the same place throughout; (c) the world line of a photon that moves radially away from A and returns to B in a time P. Compare your three results and discuss. 4. Light Deflection Suppose in another theory of gravity (not Einstien s general relativity) the metric outside a spherical star is given by ( ds 2 = 1 2GM ) [ c 2 dt 2 +dr 2 +r 2 (dθ 2 +sin 2 θdφ 2 ) ] c 2 r Calculate the deflection of light by a spherical star in this theory assuming that photons move along null geodesics in this geometry. 4

5 5. Rotating Reference frames Thelineelement of a flatspacetime in aframe(t,x,y,z) thatisrotating with an angular velocity Ω about the z axis of an inertial frame is ds 2 = [c 2 Ω 2 (x 2 +y 2 )]dt 2 +2Ω(ydx xdy)dt+dx 2 +dy 2 +dz 2 (a) Verify this by transforming to polar coordinates and checking that the line element is with the substituion φ φ Ωt. ds 2 = c 2 dt 2 +dr 2 +r 2 (dθ 2 +sin 2 θdφ 2 ) (b) Find the geodesic equation for x, y and z in the rotating frame. (c) Show that, in the non-relativistic limit these reduce to the usual equations of Newtonian mechanics for a free particle in a rotating frame exhibiting the centrifugal force and the Coriolis force. 5

6 3 Problem Sheet 3 -Geometry 1. Twisted Flat Space Consider the following line element ds 2 = c 2 dt 2 +2cdxdt+dy 2 +dz 2 (a) Find the connection coefficients and the Ricci tensor for this metric. (b) Find a coordinate transformation that puts the line element in the usual flat space form. 2. Spherically Symmetric Space-Time In a certain spacetime geometry the metric is ds 2 = c 2 (1 Ar 2 ) 2 dt 2 +(1 Ar 2 ) 2 dr 2 +r 2 (dθ 2 +sin 2 θdφ 2 ) (a) Calculate the proper distance along the radial line from the center r = 0 to a coordinate radius r = R (b) Calculate the area of the sphere of coordinate radius r = R. (c) Calculate the three-volume of a sphere of coordinate radius r = R (d) Calculate the four-volume of a four-dimensional tube bounded by a sphere of coordinate radius R and two t =constant planes separated by time T. 3. The Einstein Tensor of the Universe A spacetime has the metric ds 2 = c 2 dt 2 +a 2 (t)(dx 2 +dy 2 +dz 2 ) (a) Show that the only non-zero connections coefficients are Γ 0 11 = Γ0 22 = Γ0 33 = aȧ/c and Γ1 10 = Γ2 20 = Γ3 30 = ȧ/(ac) (b) Deduce that particles may be at rest in such a spacetime and that for such particles the coordinate t is their proper time. Show further that the non-zero components of the Ricci tensor or R 00 = 3ä/(ac 2 ) and R 11 = R 22 = R 33 = (aä+2ȧ 2 )/c 2 (c) Hence show that the 00-component of the Einstein tensor is G 00 = 3ȧ 2 /(ca) The equations for a relativistic fluid The motion of a relativistic fluid is characterised by its velocity field U µ whixh is time-like everywhere and normalised so that U µ U µ = 1. (a) Show that the acceleration A µ = U ν ν U µ is orthogonal to the velocity U µ. 6

7 (b) Show that the tensor h µ ν = δ µ ν +U µ U ν satisfies (c) Assume that space-time has a tensor h µ νu ν = 0 h µ νh ν ρ = h µ ρ h µ µ = 3 T µν = (ρ+p)u µ U ν +g µν P where ρ and P are scalar functions. Show that the conservation of energy-momentum equation µ T µν = 0 leads to the relativistic continuity equation U µ µ ρ+(p +ρ) µ U µ = 0 (d) Show that if P = 0, i.e. the fluid is a collection of dust particles following trajectories x µ (τ) with U µ = Dx µ /Dτ, then the particles follow geodesics, that is that 5. The curvature of the Schwarzchild metric D dx µ Dτ dτ Uν ν U µ = 0 This is quite a lengthy calculation, good practice for heavy lifting (a) Evaluate the connection coefficients of the Schwarzschild metric using the geodesic method (b) Calculate all the elements of the Ricci tensor for this metric and show that R αβ = 0. This means the Schwarzschild metric satisfied the vacuum Einstein equations. 6. Exact gravitational plane wave (Optional) Consider the metric ds 2 = (dudv+dvdu)+a 2 (u)dx 2 +b 2 (u)dy 2 where a and b are unspecified function of u. For particular choices of these functions, this metric represents an exact gravitational plane wave. (a) Calculate the connection coefficients and the Ricci tensor for this metric. (b) Use Einstein s equation in vacuum to derive the equations obeyed by a(u) and b(u). (c) Show that an exact solution can be found, in which both a and b are determined in terms of an arbitrary function, f(u). 7

8 4 Problem Sheet 4 - The FRW equation 1. Closed Universe (a) Show that a(η) = C(1 cos η η ) t(η) = C(η η sin η η ) satisfies the closed, matter dominated FRW equation and find an expression for C in terms of H 0, Ω the curvature parameter, k and the scale factor today, a 0. (b) If the parameter η that occurs there is used as a time coordinate, show that the metric takes the form 2. Linearly Expanding Universe ds 2 = a 2 (η)[ c 2 dη 2 +dχ 2 +sin 2 χ(dθ 2 +sin 2 θdφ 2 ] Consider a homogeneous, isotropic, cosmological model described by the line element where t is constant. (a) Is this model open, closed or flat? ( ) t [dx ds 2 = c 2 dt dy 2 +dz 2] (b) Is this a matter dominate Universe? Explain. (c) Assuming the Friedmann equation holds for this Universe, find ρ(t). 3. Hubble parameter Assume that the Universe is dust-dominated. Take H 0 = 100kms 1 Mpc 1. (a) Give a rough estimate of the age of the Universe. (b) How far can light have travelled in this time? t (c) The microwave background has been travelling towards us uninterrupted since decoupling, when the Universe was 1/1000 of its current size. Compute the value of the Hubble parameter H at the time of decoupling. (d) How far could light have travelled in the time up to decoupling (assume that the Universe was dominated by radiation until then)? (e) Between decoupling and the present, the distance that light travelled up to the time of decoupling has been stretched by the subsequent expansion. What would be its physical size today? 8

9 (f) Assuming that the distance to the last-scattering surface is given by part b of this question, what angle is subtended by the distance light could have travelled before decoupling? (g) What is the physical significance of this value? 4. Conformal time We can define conformal time, η, in terms of physical time, t, through dt = adη where a is the scale factor, which is a function of t or η. (a) Show that η a 1 2 in a matter dominated universe and a in one dominated by radiation (b) Consider a universe with only matter and radiation, with equality at a eq. Show that η = 2 ( a+a eq a eq ) Ω M H0 2 (c) What is the conformal time today? And at recombination? 5. Contributions to the dynamics of the Universe (a) Suppose the Universe contains four different contributions to the Friedmann equation namely dust, radiation, a cosmological constant and negative curvature. What is the behaviour of each as a function of the scale-factor a(t)? (b) Which will dominate at early times and which will dominate at late times? 9

10 5 Problem Sheet 5 -Cosmology 1. Redshift Consider the Friedmann Robertson Walker metric for a homogeneous and isotropic Universe is given by [ ] dr ds 2 = c 2 dt 2 +a(t) kr 2 +r2 (dθ 2 +sin 2 θdφ 2 ), where ds is the proper time interval between two events, t is the cosmic time, k measures the spatial curvature, r, θ and φ are radial, polar and azimuthal co-ordinates respectively. (a) Explain what is meant by a(t) and discuss its physical significance. (b) Describe what is meant by redshift and how spectroscopic observations of extragalactic objects may be used to deduce their redshifts. (c) What does the above expression become in the case of a light-ray? Hence derive an integral expression for a light-ray which leaves the origin at time t em and reaches a comoving distance r 0 at time t obs. A second ray is emitted a time dt after the first. By considering the two intervals as corresponding to successive wave crests, derive the relation λ obs λ em = a(t obs) a(t em ) 1+z, where z is the redshift and λ em and λ obs are the emitted and observed wavelengths respectively. (d) How does the separation of galaxies today compare with the separation of galaxies when light left the galaxies we observe at redshift 1? 2. Horizons (a) Describe the concept of our past and future light-cone. Explain the meaning of the terms particle horizon distance, event horizon distance and world-line and discuss the difference between time-like and space-like locations. (b) Show that in an Einstein-de Sitter Universe in which the scale-factor a(t) at time t follows a(t) t 2/3, the particle horizon is at 3ct and the event horizon is at infinity. (c) Suppose that the scale-factor were given by a(t) exp(mt) where m is a positive constant. Show that the event horizon is finite and that the particle horizon grows exponentially when t 1/m. (d) Explain how such behaviour of the particle horizon might be useful in explaining observations of the cosmic microwave background. 3. The size of the Universe Assume the universe today is flat with both matter and a cosmological constant but no radiation. 10

11 (a) Compute the horizon of the Universe as a function of Ω M and sketch it. (You will need a computer or calculator to do this.) (b) What is the current horizon size for a universe with Ω M = 1/3 and h = 1/ 2? (c) What is the mass contained within the current horizon in solar masses? If all objects were h 1 M in mass, how many are in the observable universe? 4. The Big Bang and the acceleration of the Universe (a) Give an account of the observational evidence for the hot Big Bang model of the Universe. (b) The Friedmann and fluid equations respectively are given by (ȧ a ) 2 8πG kc2 = ρ 3 a 2 and ρ+3ȧ (ρ+ p ) = 0, a c 2 whereaisthescalefactor, ρisthedensity andpisthepressure. (ȧand ρarethederivatives of these quantities with respect to time.) Use these equations to derive the acceleration equation for the Universe. (c) Hence demonstrate that if the Universe is homogeneous and the strong energy condition ρc 2 +3p > 0 holds, then the Universe must have undergone a Big Bang. 5. Recombination and the Surface of Last Scattering (a) What is the surface of last scattering? Would the same surface be seen by any other observer on a different galaxy? (b) Estimate the radius of the last scattering surface, using the age of the Universe. Why might this underestimate the true value? (c) The present number density of electrons in the Universe is the same as that of protons, namely about 0.2m 3. Consider a time long before decoupling when the Universe was a year old and the scale factor was one millionth of its present value. Estimate the number density of electrons at that time and comment on whether the electrons would be relativistic or non-relativistic then. (d) Given that the mean free path of photons through an electron gas of number density n e is d 1/[n e σ e ], where the Thompson scattering cross-section σ e = m 2, determine the mean free path for photons when the scale factor was one millionth its present value. (e) From the mean free path, calculate the typical time between interactions between the photons and electrons. (f) Compare the interaction time with the age of the Universe at that time. What is the significance of this comparison? 11