A A + B. ra + A + 1. We now want to solve the Einstein equations in the following cases:
|
|
- Alannah Dean
- 5 years ago
- Views:
Transcription
1 Lecture 29: Cosmology Cosmology Reading: Weinberg, Ch A metric tensor appropriate to infalling matter In general (see, eg, Weinberg, Ch ) we may write a spherically symmetric, time-dependent metric in the form (dτ) 2 = B(r, t) (dt) 2 A(r, t) (dr) 2 r 2 (dθ) 2 + r 2 sin 2 θ (dϕ) 2 (29) and from this we deduce R tt = B 2A + B A 4 A A + B B Ȧ r + 2A Ȧ Ȧ 4 A A + B B R rr = B 2B B A 4 B A + B B A ra + A 2B Ȧ Ȧ 4 B A + B B r B R θθ = + 2A B A A + A (292t) (292r) (292θ) R ϕϕ = R θθ sin 2 θ (292ϕ) R tr R rt = Ȧ ra (293) with all other terms = 0 We now want to solve the Einstein equations in the following cases: Empty space Equation 293 implies that Ȧ = Ȧ = 0, hence A is time-independent By the methods used earlier, we find AB =, therefore B is time-independent also Thus we recover the Schwarzschild metric, A(r) = 2MG r (294) This reflects a specific choice of time coordinate, as before 37
2 A metric tensor appropriate to infalling matter This proves Birkhoff s Theorem----a spherically symmetric metric in empty space is time-independent Thus there can be no gravitational radiation from a spherical source 2 Dust to dust Let us redefine r and t to get co-moving coordinates, appropriate to an oberver falling freely with some particular piece of matter: (dτ) 2 = (dt) 2 U(r, t) (dr) 2 (r, t) (dθ) 2 + sin 2 θ (dϕ) 2 (295) then with this metric, U R tt = 2U + U 2 4 U R rr = 2 R θθ = + 2U U U + 2 U + U 2 U 4U 2 2 U 4U 4U U 2 (296t) (296r) (296θ) R ϕϕ = R θθ sin 2 θ R tr = 2 2 U 2U and all other components vanish (296ϕ) (296tr) Dust can be defined as a group of particles with energy density but no pressure An example is a large cluster of well-spaced galaxies The energy-momentum tensor of dust is T µν = ρ U µ U ν (297) The invariant volume element is dt dr dθ dϕ g = dt dr dθ dϕ sinθ U (298) In co-moving coordinates, there is no local motion of a particle, hence U t U µ U r 0 = U θ = 0 U ϕ 0 Of the 4 equations T µν ; ν = 0, (299) the space components are automatically satisfied: T kν ; ν = 0 (290) leaving 38
3 Lecture 29: Cosmology or T 0ν ; ν = 0 (29) g x ν (T tν g) + t σ ν T σν = U t (ρ U) + ρ t t t = 0 (292) and since t t t = 2 gtt t g tt = 0, we have (ρ U) = 0 (293) t The gravitational field equations become U R tt = 2U + U 2 4 U = 4πGρ R rr = 2 R θθ = + 2U (294t) U U + 2 U + U 2 4U U = 4πGρ (294r) 2 U 4U 2 2 U = 4πGρ (294θ) 4U R ϕϕ = R θθ sin 2 θ = 4πGρ sin 2 θ (294ϕ) R tr = 2 2 U 2U = 0 To solve these, multiply Eq 294tr by and divide by, so d dt ln( ) d 2 dt ln(u) = 0 or (294tr) = F(r) U (295) where F(r) is some arbitrary----for the moment----function of r Next add Eq 294r and Eq 294t and subtract twice Eq 294θ, to get 2 2U = 0 (296) Combining Eq 296 with Eq 295 we get 2 F2 (r) = 0 39
4 A metric tensor appropriate to infalling matter We see that 2 2 is independent of t This suggests 2 (r, t) = R 2 (t) Γ(r), and thence 2 Γ(r) 2R 2 + 2ṘṘ R 2 F(r) + 2 = 0 (297) 2 ie, R 2 + 2ṘṘ df = constant = k (298) From Eq 295 we have Γ (r) R(t) = F(r) Γ(r) U (299) which suggests U(r, t) = R(t) f(r) We are at liberty to redefine r so that Γ(r) = r 2 ; thus from Eq 297 F 2 (r) = 4 ( kr 2 ) (2920) and from Eq f(r) = F 2 (r) = ( kr2 ) (292) The metric now has the form (dτ) 2 = (dt) 2 R 2 (t) (dr) 2 This is called the Robertson-Walker metric kr 2 + r2 (dθ) 2 + sin 2 θ (dϕ) 2 (2922) We suppose that the density varies with time only (since the radial and angular velocity components vanish) Then energy conservation becomes t (ρ U) = r2 f(r) t ρ(t) R3 (t) = 0 (2923) from which we deduce ρ(t) = ρ(0) R3 (0) R 3 (t) (2924) 40
5 Lecture 29: Cosmology Time evolution of dust By convention, R(0) = From Eq 294t, U R tt = 2U + U 2 4 U = 4πGρ, we find ṘṘ 4πGρ(0) 3 = 4πGρ(t) = R 3 (t) We also had R 2 + 2ṘṘ = k which when combined with Eq 2925 gives k + R 2 = 8πGρ(0) 3R(t) If Ṙ(0) = 0, then we may re-write Eq 2926 as 2 2 (2925) (2926) Ṙ(t) = 8πGρ(0) 3 R(t) (2927) where we choose the negative root in order to describe collapse That is, the dust, initially in some distribution with ρ(0) 0, falls freely inward To simplify Eq 2927, let R(t) = ( + cosψ) 2 Then 2 ψ sinψ = λ cosψ + cosψ 2 λ sinψ + cosψ (2928) which can be integrated simply to give (note ψ(0) = 0 ) (ψ + sinψ) = λt (2929) 2 Eq 2929 describes a cycloid: The time to collapse is evidently T collapse = π 2λ (2930) Since, if k>0, the Robertson-Walker metric demands r 2 < k = 3 8πGρ(0) we see that the collapse time is 4
6 Outside the ball of dust T = π 2 r max c To recapitulate, a ball of dust (ie p=0), initially at rest, will collapse to a point, under its mutual gravitational attraction, in time T Outside the ball of dust It is possible to put the metric outside in Schwarzschild form (dτ) 2 = B(r ) (dt ) 2 A(r ) (dr ) 2 r 2 (dθ) 2 + sin 2 θ (dϕ) 2 (293) To do this, we need to match up at the surface Let θ = θ, ϕ = ϕ and r = r R(t) Then dr = R dr + r Ṙ dt and we define the outside time by t = ka 2 k 2 S(r, t) S(r, t) = kr 2 ka 2 2 dr ka 2 R R R If we fit at r = a (the radius of the dustball) then we have ka 3 B(a, t ) = ar(t) A(a, t ) = ka 3 ar(t) 2 (2932a) ( R(t)) (2932b) (2933) (2934) This matches the outside solution if 2MG = ka 3 But from Eq 2926 we have k = 8πGρ(0) 3 so the condition is M = 4πa3 3 ρ(0), ----not a very surprising result! see Weinberg, p
7 Lecture 29: Cosmology Collapse seen by outside observer We now ask what the collapse looks like to a distant observer We see that if light is emitted radially from the surface of the star at (outside) time t 0, it propagates according to dτ = 0, or dt = A(r ) dr hence r t = t 0 + a R(t) dr 2MG r (2935) We see that as R(t) approaches 2MG a (that is, as the surface approaches the Schwarzschild radius, the time for the light to reach the observer becomes (logarithmically) infinite The gravitational red shift of light reaching the observer becomes df dt z = = dt 0 a Ṙ(t) dt dt 2MG a R(t) (2936) hence as the radius reaches r S, z exp t 2MG (2937) For most of the star s life, r >> r S and t t, ie the redshift is essentially zero But as the end of the collapse approaches, an outside observer sees an exponentially increasing redshift, ie the star disappears into redness, with a time scale of minutes The further collapse to R(T) = 0 is invisible to an outside observer A co-moving observer has no difficulty seeing the collapse to R = 0 His time becomes disconnected from that of the outside world after he passes within the Schwarzschild radius The surface r = r S represents a trapped discontinuity that separates inside from outside Stuff can fall in, but it can never get out again, in classical General Relativity Model universes The most appropriate metric for cosmology is the spatially homogeneous Robertson-Walker metric (dτ) 2 = (dt) 2 R 2 (t) (dr) 2 kr 2 + r2 (dθ) 2 + sin 2 θ (dϕ) 2 (2922) that arises automatically from co-moving coordinates The Robertson-Walker metric embodies the idea that at fixed t (spacelike hypersurface) any point is equivalent to any other point The curvature of the 3-dimensional hypersurfaces t = const is K 3 (t) = k R 2 (t) By rescaling r and R(t) k can be assming tidal forces do not exceed his personal Roché limit 43
8 Positive curvature normalized to ±, if k 0 Thus, a space of positive curvature K 3 kas k = +, and a space of negative curvature has k = Positive curvature When k = +, the proper circumference of the space is L 3 = 2π R(t) and the proper volume is 3 = 2π 2 R 3 (t) (2938a) (2938b) At fixed t the universe is the surface of a 3-sphere of radius R(t) embedded in a Euclidean 4-dimensional manifold, so R(t) is the radius of the universe Space is finite, but unbounded (since (dr) 2 ( kr 2 ) ) Zero curvature When k = 0, we say space is flat (in an average or global sense) Flat space is infinite, since the 3-dimensional hypersurfaces t = const are open Negative curvature When k = space is also infinite because a negatively curved hypersurface K 3 = const < 0 is open Influence of matter In isotropic 3-space T 00 must be scalar with respect to transformations of r, θ, ϕ ; hence T 00 = ρ(t) (2939a) T k0 = 0 T jk = p(t) g jk (2939b) (2939c) We can define a flux of galaxies J G µ : J G 0 = n G (t) (2940a) J G µ = n G U µ (2940b) and then 44
9 Lecture 29: Cosmology T µν = p g µν + (p + ρ) U µ U ν (294) and as before, U 0 = U k = 0 Conservation of galaxies may be written g 2 t g 2 n G = 0 (2942) and with g = R 6 (t) r4 sin 2 θ kr 2 we find, unsurprisingly, n G (t) R 3 (t) = const (2943) Conservation of energy-momentum, T µν ; ν = 0 implies R 3 (t) p t = t R3 (t) (p + ρ) (2944) If pressure is negligible, then as for the dust model of a collapsing star, ρ(t) R 3 (t) = const (2945) Note that 2945 and 2943 are inequivalent unless we neglect pressure Proper distances Imagine observers in galaxies along a line of sight to some distant galaxy at r n, at some cosmic time t Each measures the distance to the next galaxy by----say----the travel time for a light signal Then the sum of the distances along the line of sight would be n ds n = n 2 (dr n ) 2 R 2 (t) kr 2 (2946) or if we assume the observers closely spaced relative to the overall distance, df r D proper (t) = dr g rr r 2 = R(t) 0 dr kr 2 2 (2947) 0 45
10 Cosmic red shift Cosmic red shift The equation of motion of light is dτ = 0, or dr dt = R(t) kr 2 (2948) hence if the light leaves r at t and arrives at r=0 (Earth) at time t 0 we have t 0 r dt R(t) = 0 t dr kr 2 = f(r ) (2949) The right side of Eq 2949 is independent of time For nearby galaxies, kr 2 << so f(r ) r Assume the next wave crest leaves at t + δt and arrives at t 0 + δt 0 ; then so t 0 + δt 0 t + δt δt 0 R(t 0 ) t 0 dt R(t) = t dt R(t) = f(r ) (2950) δt R(t ) = 0 (295) But since the time between successive wave crests, at a fixed location, is df δt = ν = λ c, we may write λ 0 = R(t 0 ) λ R(t ) and thus the red-shift z defined in 2936 above becomes z = λ 0 λ = R(t 0 ) R(t ) (2952) λ R(t ) In an expanding universe, t < t 0 so that R(t ) < R(t 0 ) and we get a cosmological red-shift, z > 0 For a nearby galaxy, we should say the proper distance is D(t) R(t) r and that its radial velocity is therefore v rad = Ḋ(t) = Ṙ(t) r (2953) But since R(t 0 ) R(t ) Ṙ(t 0 ) t 0 t, and t 0 t R(t 0 ) we find r 46
11 Lecture 29: Cosmology ie z Ṙ(t 0 ) R(t 0 ) t 0 t R (t0 ) R(t 0 ) r R(t 0 ) = v r (2954) v r = Ṙ(t 0 ) R(t 0 ) D(t 0 ) (2955) Thus we may identify Ṙ(t 0 ) R(t 0 ) as the Hubble constant, H 0 Deceleration parameter Assuming the cosmic scale parameter R(t) to be well-enough behaved, we may expand it in Taylor s series, measuring time from the present: R(t) R(t 0 ) Ṙ(t 0 ) + R(t 0 ) t + Ṙ (t 0 ) 2 R(t 0 ) t2 + (2957) Now, the deceleration parameter is defined as df Ṙ (t 0 ) q 0 = H 2 0 R(t 0 ) (note that deceleration corresponds to q 0 > 0 ) Equation 2957 can then be written in standard format R(t) R(t 0 ) + H 0 t 2 2 q 0 H 0 t + (2958) Our isotropic model universe satisfies the equations Ṙ R = 4πG 3 (ρ + 3p) R2 (2959t) Ṙ R + 2R 2 + 2k = 4πG (ρ p) R 2 (2959r) and thence R 2 + k = 8πG 3 ρ R2, (2960) and (from energy-momentum conservation) R 3 (t) p = t t R3 (t) (p + ρ) There are two obvious cases: p << ρ (dust) then R 3 (t) ρ const and from Eq 2959t, Ṙ R 2 (296) 47
12 Deceleration parameter which leads to a power-law behavior for R(t): R t 2 3 (2962) 2 p = ρ (ultrarelativistic gas) 3 Now, from Eq 296, ρ R 4 = const R(t) t 2 (2963) The (dimensionless) deceleration parameter q 0 can be related to the average density of mass-energy in the universe; hence the question of whether the universe is open (k 0) or closed (k > 0) can in principle be answered by measuring q 0 Unfortunately, while the observational evidence that q 0 > 0 is good, we cannot say more than that at present And recently new evidence has been obtained that may indicate q 0 < 0, which would mean the expansion of the universe is accelerating From Eq 2960, we may obtain k R 2 (t 0 ) + H 0 2 = 8πG 3 ρ(t 0 ) (2964) and from Eq 2959t and the definition of q 0 we find q 0 H 0 2 = 4πG 3 ρ 0 + 3p 0 (2965) From Eq 2965 and Eq 2964 we can then derive an expression for the pressure now: k p 0 = 8πG R 2 + H 0 2 ( 2q 0 ) (2966) 0 Moreover, from Eq 2964 we determine that k = R 2 8πG 0 3 ρ 0 ρ crit (2967) so that the criterion for closure of the metric is ρ 0 > ρ crit Moreover, if the present pressure is 0, then setting Eq 2966 to 0 gives ρ 0 2q 0 ρ crit (2968) hence q 0 > 2 ρ 0 > ρ crit and consequently, k > 0 48
3 The Friedmann-Robertson-Walker metric
3 The Friedmann-Robertson-Walker metric 3.1 Three dimensions The most general isotropic and homogeneous metric in three dimensions is similar to the two dimensional result of eq. (43): ( ) dr ds 2 = a
More informationGeneral Relativity and Cosmology Mock exam
Physikalisches Institut Mock Exam Universität Bonn 29. June 2011 Theoretische Physik SS 2011 General Relativity and Cosmology Mock exam Priv. Doz. Dr. S. Förste Exercise 1: Overview Give short answers
More information2.1 Basics of the Relativistic Cosmology: Global Geometry and the Dynamics of the Universe Part I
1 2.1 Basics of the Relativistic Cosmology: Global Geometry and the Dynamics of the Universe Part I 2 Special Relativity (1905) A fundamental change in viewing the physical space and time, now unified
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.286: The Early Universe October 27, 2013 Prof. Alan Guth PROBLEM SET 6
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.86: The Early Universe October 7, 013 Prof. Alan Guth PROBLEM SET 6 DUE DATE: Monday, November 4, 013 READING ASSIGNMENT: Steven Weinberg,
More informationIntroduction to Cosmology
Introduction to Cosmology João G. Rosa joao.rosa@ua.pt http://gravitation.web.ua.pt/cosmo LECTURE 2 - Newtonian cosmology I As a first approach to the Hot Big Bang model, in this lecture we will consider
More informationGeneral Birkhoff s Theorem
General Birkhoff s Theorem Amir H. Abbassi Department of Physics, School of Sciences, Tarbiat Modarres University, P.O.Box 14155-4838, Tehran, I.R.Iran E-mail: ahabbasi@net1cs.modares.ac.ir Abstract Space-time
More informationAST1100 Lecture Notes
AST1100 Lecture Notes 23-24: Cosmology: models of the universe 1 The FRW-metric Cosmology is the study of the universe as a whole. In the lectures on cosmology we will look at current theories of how the
More informationThird Year: General Relativity and Cosmology. 1 Problem Sheet 1 - Newtonian Gravity and the Equivalence Principle
Third Year: General Relativity and Cosmology 2011/2012 Problem Sheets (Version 2) Prof. Pedro Ferreira: p.ferreira1@physics.ox.ac.uk 1 Problem Sheet 1 - Newtonian Gravity and the Equivalence Principle
More informationAstronomy, Astrophysics, and Cosmology
Astronomy, Astrophysics, and Cosmology Luis A. Anchordoqui Department of Physics and Astronomy Lehman College, City University of New York Lesson VI March 15, 2016 arxiv:0706.1988 L. A. Anchordoqui (CUNY)
More informationarxiv:gr-qc/ v1 10 Apr 2003
United Approach to the Universe Model and the Local Gravitation Problem Guang-Wen Ma a and Zong-Kuan Guo b a Department of Physics, Zhengzhou University, Zhengzhou, Henan 45005, PR China b Institute of
More informationTO GET SCHWARZSCHILD BLACKHOLE SOLUTION USING MATHEMATICA FOR COMPULSORY COURSE WORK PAPER PHY 601
TO GET SCHWARZSCHILD BLACKHOLE SOLUTION USING MATHEMATICA FOR COMPULSORY COURSE WORK PAPER PHY 601 PRESENTED BY: DEOBRAT SINGH RESEARCH SCHOLAR DEPARTMENT OF PHYSICS AND ASTROPHYSICS UNIVERSITY OF DELHI
More informationNEWTONIAN COSMOLOGY. Figure 2.1: All observers see galaxies expanding with the same Hubble law. v A = H 0 r A (2.1)
M. Pettini: Introduction to Cosmology Lecture 2 NEWTONIAN COSMOLOGY The equations that describe the time evolution of an expanding universe which is homogeneous and isotropic can be deduced from Newtonian
More informationUniformity of the Universe
Outline Universe is homogenous and isotropic Spacetime metrics Friedmann-Walker-Robertson metric Number of numbers needed to specify a physical quantity. Energy-momentum tensor Energy-momentum tensor of
More informationPHY 475/375. Lecture 5. (April 9, 2012)
PHY 475/375 Lecture 5 (April 9, 2012) Describing Curvature (contd.) So far, we have studied homogenous and isotropic surfaces in 2-dimensions. The results can be extended easily to three dimensions. As
More informationAST1100 Lecture Notes
AST1100 Lecture Notes 23-24: Cosmology: models of the universe 1 The FRW-metric Cosmology is the study of the universe as a whole. In the lectures on cosmology we will look at current theories of how the
More informationCosmology (Cont.) Lecture 19
Cosmology (Cont.) Lecture 19 1 General relativity General relativity is the classical theory of gravitation, and as the gravitational interaction is due to the structure of space-time, the mathematical
More informationGeneral Relativity Lecture 20
General Relativity Lecture 20 1 General relativity General relativity is the classical (not quantum mechanical) theory of gravitation. As the gravitational interaction is a result of the structure of space-time,
More informationA5682: Introduction to Cosmology Course Notes. 2. General Relativity
2. General Relativity Reading: Chapter 3 (sections 3.1 and 3.2) Special Relativity Postulates of theory: 1. There is no state of absolute rest. 2. The speed of light in vacuum is constant, independent
More informationKerr black hole and rotating wormhole
Kerr Fest (Christchurch, August 26-28, 2004) Kerr black hole and rotating wormhole Sung-Won Kim(Ewha Womans Univ.) August 27, 2004 INTRODUCTION STATIC WORMHOLE ROTATING WORMHOLE KERR METRIC SUMMARY AND
More informationWeek 2 Part 2. The Friedmann Models: What are the constituents of the Universe?
Week Part The Friedmann Models: What are the constituents of the Universe? We now need to look at the expansion of the Universe described by R(τ) and its derivatives, and their relation to curvature. For
More informationCosmology ASTR 2120 Sarazin. Hubble Ultra-Deep Field
Cosmology ASTR 2120 Sarazin Hubble Ultra-Deep Field Cosmology - Da Facts! 1) Big Universe of Galaxies 2) Sky is Dark at Night 3) Isotropy of Universe Cosmological Principle = Universe Homogeneous 4) Hubble
More informationFRW cosmology: an application of Einstein s equations to universe. 1. The metric of a FRW cosmology is given by (without proof)
FRW cosmology: an application of Einstein s equations to universe 1. The metric of a FRW cosmology is given by (without proof) [ ] dr = d(ct) R(t) 1 kr + r (dθ + sin θdφ ),. For generalized coordinates
More informationASTR 610: Solutions to Problem Set 1
ASTR 610: Solutions to Problem Set 1 Problem 1: The Einstein-de Sitter (EdS) cosmology is defined as a flat, matter dominated cosmology without cosmological constant. In an EdS cosmology the universe is
More information1 Cosmological Principle
Notes on Cosmology April 2014 1 Cosmological Principle Now we leave behind galaxies and beginning cosmology. Cosmology is the study of the Universe as a whole. It concerns topics such as the basic content
More informationNew Blackhole Theorem and its Applications to Cosmology and Astrophysics
New Blackhole Theorem and its Applications to Cosmology and Astrophysics I. New Blackhole Theorem II. Structure of the Universe III. New Law of Gravity IV. PID-Cosmological Model Tian Ma, Shouhong Wang
More information2.1 The metric and and coordinate transformations
2 Cosmology and GR The first step toward a cosmological theory, following what we called the cosmological principle is to implement the assumptions of isotropy and homogeneity withing the context of general
More informationLecture 13 Friedmann Model
Lecture 13 Friedmann Model FRW Model for the Einstein Equations First Solutions Einstein (Static Universe) de Sitter (Empty Universe) and H(t) Steady-State Solution (Continuous Creation of Matter) Friedmann-Lemaître
More informationD. f(r) gravity. φ = 1 + f R (R). (48)
5 D. f(r) gravity f(r) gravity is the first modified gravity model proposed as an alternative explanation for the accelerated expansion of the Universe [9]. We write the gravitational action as S = d 4
More informationPhysics 133: Extragalactic Astronomy and Cosmology
Physics 133: Extragalactic Astronomy and Cosmology Week 2 Spring 2018 Previously: Empirical foundations of the Big Bang theory. II: Hubble s Law ==> Expanding Universe CMB Radiation ==> Universe was hot
More informationThe Motion of A Test Particle in the Gravitational Field of A Collapsing Shell
EJTP 6, No. 21 (2009) 175 186 Electronic Journal of Theoretical Physics The Motion of A Test Particle in the Gravitational Field of A Collapsing Shell A. Eid, and A. M. Hamza Department of Astronomy, Faculty
More informationTHE DARK SIDE OF THE COSMOLOGICAL CONSTANT
THE DARK SIDE OF THE COSMOLOGICAL CONSTANT CAMILO POSADA AGUIRRE University of South Carolina Department of Physics and Astronomy 09/23/11 Outline 1 Einstein s Greatest Blunder 2 The FLRW Universe 3 A
More informationPAPER 73 PHYSICAL COSMOLOGY
MATHEMATICAL TRIPOS Part III Wednesday 4 June 2008 1.30 to 4.30 PAPER 73 PHYSICAL COSMOLOGY Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight. STATIONERY
More informationAstr 2320 Tues. May 2, 2017 Today s Topics Chapter 23: Cosmology: The Big Bang and Beyond Introduction Newtonian Cosmology Solutions to Einstein s
Astr 0 Tues. May, 07 Today s Topics Chapter : Cosmology: The Big Bang and Beyond Introduction Newtonian Cosmology Solutions to Einstein s Field Equations The Primeval Fireball Standard Big Bang Model Chapter
More informationPAPER 71 COSMOLOGY. Attempt THREE questions There are seven questions in total The questions carry equal weight
MATHEMATICAL TRIPOS Part III Friday 31 May 00 9 to 1 PAPER 71 COSMOLOGY Attempt THREE questions There are seven questions in total The questions carry equal weight You may make free use of the information
More information3.1 Cosmological Parameters
3.1 Cosmological Parameters 1 Cosmological Parameters Cosmological models are typically defined through several handy key parameters: Hubble Constant Defines the Scale of the Universe R 0 H 0 = slope at
More informationPhysics 133: Extragalactic Astronomy ad Cosmology
Physics 133: Extragalactic Astronomy ad Cosmology Lecture 4; January 15 2014 Previously The dominant force on the scale of the Universe is gravity Gravity is accurately described by the theory of general
More informationGeometry of the Universe: Cosmological Principle
Geometry of the Universe: Cosmological Principle God is an infinite sphere whose centre is everywhere and its circumference nowhere Empedocles, 5 th cent BC Homogeneous Cosmological Principle: Describes
More informationAstronomy 421. Lecture 24: Black Holes
Astronomy 421 Lecture 24: Black Holes 1 Outline General Relativity Equivalence Principle and its Consequences The Schwarzschild Metric The Kerr Metric for rotating black holes Black holes Black hole candidates
More informationCosmology PHYS3170 The angular-diameter-redshift relation
Cosmology PHYS3170 The angular-diameter-redshift relation John Webb, School of Physics, UNSW Purpose: A fun way of understanding a non-intuitive aspect of curved expanding spaces that the apparent angular-diameter
More informationChapter 9. Perturbations in the Universe
Chapter 9 Perturbations in the Universe In this chapter the theory of linear perturbations in the universe are studied. 9.1 Differential Equations of Linear Perturbation in the Universe A covariant, linear,
More informationHOMEWORK 10. Applications: special relativity, Newtonian limit, gravitational waves, gravitational lensing, cosmology, 1 black holes
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
More informationWhy is the Universe Expanding?
Why is the Universe Expanding? In general relativity, mass warps space. Warped space makes matter move, which changes the structure of space. Thus the universe should be dynamic! Gravity tries to collapse
More informationPROBLEM SET 6 EXTRA CREDIT PROBLEM SET
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.286: The Early Universe May 3, 2004 Prof. Alan Guth PROBLEM SET 6 EXTRA CREDIT PROBLEM SET CAN BE HANDED IN THROUGH: Thursday, May 13,
More informationGravitational radiation
Lecture 28: Gravitational radiation Gravitational radiation Reading: Ohanian and Ruffini, Gravitation and Spacetime, 2nd ed., Ch. 5. Gravitational equations in empty space The linearized field equations
More informationIs the Rate of Expansion of Our Universe. Really Accelerating?
Adv. Studies Theor. Phys., Vol. 5, 2011, no. 13, 633-638 Is the Rate of Expansion of Our Universe Really Accelerating? Nathalie Olivi-Tran 1,2 1 Laboratoire Charles Coulomb CNRS, UMR 5221, place Eugene
More informationEinstein s Equations. July 1, 2008
July 1, 2008 Newtonian Gravity I Poisson equation 2 U( x) = 4πGρ( x) U( x) = G d 3 x ρ( x) x x For a spherically symmetric mass distribution of radius R U(r) = 1 r U(r) = 1 r R 0 r 0 r 2 ρ(r )dr for r
More informationYou may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
MATHEMATICAL TRIPOS Part III Friday 8 June 2001 1.30 to 4.30 PAPER 41 PHYSICAL COSMOLOGY Answer any THREE questions. The questions carry equal weight. You may not start to read the questions printed on
More informationKinetic Theory of Dark Energy within General Relativity
Kinetic Theory of Dark Energy within General Relativity Author: Nikola Perkovic* percestyler@gmail.com University of Novi Sad, Faculty of Sciences, Institute of Physics and Mathematics Abstract: This paper
More informationLecture 2: Cosmological Background
Lecture 2: Cosmological Background Houjun Mo January 27, 2004 Goal: To establish the space-time frame within which cosmic events are to be described. The development of spacetime concept Absolute flat
More informationSchwarzschild s Metrical Model of a Liquid Sphere
Schwarzschild s Metrical Model of a Liquid Sphere N.S. Baaklini nsbqft@aol.com Abstract We study Schwarzschild s metrical model of an incompressible (liquid) sphere of constant density and note the tremendous
More informationLecture 1 General relativity and cosmology. Kerson Huang MIT & IAS, NTU
A Superfluid Universe Lecture 1 General relativity and cosmology Kerson Huang MIT & IAS, NTU Lecture 1. General relativity and cosmology Mathematics and physics Big bang Dark energy Dark matter Robertson-Walker
More informationThe homogeneous and isotropic universe
1 The homogeneous and isotropic universe Notation In this book we denote the derivative with respect to physical time by a prime, and the derivative with respect to conformal time by a dot, dx τ = physical
More information3 Friedmann Robertson Walker Universe
28 3 Friedmann Robertson Walker Universe 3. Kinematics 3.. Robertson Walker metric We adopt now the cosmological principle, and discuss the homogeneous and isotropic model for the universe. This is called
More informationEscape velocity and Schwarzschild s Solution for Black Holes
Escape velocity and Schwarzschild s Solution for Black Holes Amir Ali Tavajoh 1 1 Amir_ali3640@yahoo.com Introduction Scape velocity for every star or planet is different. As a star starts to collapse
More informationGravitation: Cosmology
An Introduction to General Relativity Center for Relativistic Astrophysics School of Physics Georgia Institute of Technology Notes based on textbook: Spacetime and Geometry by S.M. Carroll Spring 2013
More informationExact Solutions of the Einstein Equations
Notes from phz 6607, Special and General Relativity University of Florida, Fall 2004, Detweiler Exact Solutions of the Einstein Equations These notes are not a substitute in any manner for class lectures.
More information3 Friedmann Robertson Walker Universe
3 Friedmann Robertson Walker Universe 3. Kinematics 3.. Robertson Walker metric We adopt now the cosmological principle, and discuss the homogeneous and isotropic model for the universe. This is called
More informationÜbungen zu RT2 SS (4) Show that (any) contraction of a (p, q) - tensor results in a (p 1, q 1) - tensor.
Übungen zu RT2 SS 2010 (1) Show that the tensor field g µν (x) = η µν is invariant under Poincaré transformations, i.e. x µ x µ = L µ νx ν + c µ, where L µ ν is a constant matrix subject to L µ ρl ν ση
More informationGeneral Relativity ASTR 2110 Sarazin. Einstein s Equation
General Relativity ASTR 2110 Sarazin Einstein s Equation Curvature of Spacetime 1. Principle of Equvalence: gravity acceleration locally 2. Acceleration curved path in spacetime In gravitational field,
More informationYou may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
MATHEMATICAL TRIPOS Part III Thursday 3 June, 2004 9 to 12 PAPER 67 PHYSICAL COSMOLOGY Attempt THREE questions. There are four questions in total. The questions carry equal weight. You may not start to
More informationAstrophysics Lecture Notes
Astrophysics Lecture Notes by: Eric Deir April 11, 2017 Pop Quiz!! Is the universe homogeneous? Roughly! If averaged over scales of 10 9 pc. Is the universe isotropic? Roughly! If you average over large
More informationTHE UNIVERSITY OF SYDNEY FACULTY OF SCIENCE INTERMEDIATE PHYSICS PHYS 2913 ASTROPHYSICS AND RELATIVITY (ADVANCED) ALL QUESTIONS HAVE THE VALUE SHOWN
CC0937 THE UNIVERSITY OF SYDNEY FACULTY OF SCIENCE INTERMEDIATE PHYSICS PHYS 2913 ASTROPHYSICS AND RELATIVITY (ADVANCED) SEMESTER 2, 2014 TIME ALLOWED: 2 HOURS ALL QUESTIONS HAVE THE VALUE SHOWN INSTRUCTIONS:
More informationNewtonian Gravity and Cosmology
Chapter 30 Newtonian Gravity and Cosmology The Universe is mostly empty space, which might suggest that a Newtonian description of gravity (which is valid in the weak gravity limit) is adequate for describing
More informationTuesday: Special epochs of the universe (recombination, nucleosynthesis, inflation) Wednesday: Structure formation
Introduction to Cosmology Professor Barbara Ryden Department of Astronomy The Ohio State University ICTP Summer School on Cosmology 2016 June 6 Today: Observational evidence for the standard model of cosmology
More informationGeneral Relativity and Cosmology. The End of Absolute Space Cosmological Principle Black Holes CBMR and Big Bang
General Relativity and Cosmology The End of Absolute Space Cosmological Principle Black Holes CBMR and Big Bang The End of Absolute Space (AS) Special Relativity (SR) abolished AS only for the special
More informationMATHEMATICAL TRIPOS Part III PAPER 53 COSMOLOGY
MATHEMATICAL TRIPOS Part III Wednesday, 8 June, 2011 9:00 am to 12:00 pm PAPER 53 COSMOLOGY Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight. STATIONERY
More informationA Bit of History. Hubble s original redshiftdistance
XKCD: April 7, 2014 Cosmology Galaxies are lighthouses that trace the evolution of the universe with time We will concentrate primarily on observational cosmology (how do we measure important cosmological
More informationModeling the Universe Chapter 11 Hawley/Holcomb. Adapted from Dr. Dennis Papadopoulos UMCP
Modeling the Universe Chapter 11 Hawley/Holcomb Adapted from Dr. Dennis Papadopoulos UMCP Spectral Lines - Doppler λ λ em 1+ z = obs z = λ obs λ λ em em Doppler Examples Doppler Examples Expansion Redshifts
More informationAy1 Lecture 17. The Expanding Universe Introduction to Cosmology
Ay1 Lecture 17 The Expanding Universe Introduction to Cosmology 17.1 The Expanding Universe General Relativity (1915) A fundamental change in viewing the physical space and time, and matter/energy Postulates
More informationPHYM432 Relativity and Cosmology 17. Cosmology Robertson Walker Metric
PHYM432 Relativity and Cosmology 17. Cosmology Robertson Walker Metric Cosmology applies physics to the universe as a whole, describing it s origin, nature evolution and ultimate fate. While these questions
More informationBlack holes, Holography and Thermodynamics of Gauge Theories
Black holes, Holography and Thermodynamics of Gauge Theories N. Tetradis University of Athens Duality between a five-dimensional AdS-Schwarzschild geometry and a four-dimensional thermalized, strongly
More informationSet 3: Cosmic Dynamics
Set 3: Cosmic Dynamics FRW Dynamics This is as far as we can go on FRW geometry alone - we still need to know how the scale factor a(t) evolves given matter-energy content General relativity: matter tells
More informationEnergy and Matter in the Universe
Chapter 17 Energy and Matter in the Universe The history and fate of the Universe ultimately turn on how much matter, energy, and pressure it contains: 1. These components of the stress energy tensor all
More informationUn-Darkening the Cosmos: New laws of physics for an expanding universe
Un-Darkening the Cosmos: New laws of physics for an expanding universe William K George 1 Visiting Professor Imperial College of London London, UK georgewilliamk@gmail.com www.turbulence-online.com 1 Professor
More informationArvind Borde / MTH 675, Unit 20: Cosmology
Arvind Borde / MTH 675, Unit 20: Cosmology 1. Review (1) What do we do when we do GR? We try to solve Einstein s equation. (2) What is Einstein s equation? and R ab = e[ 1 2 ged ( a g bd + b g ad d g ab
More informationIntroduction to Cosmology
1 Introduction to Cosmology Mast Maula Centre for Theoretical Physics Jamia Millia Islamia New Delhi - 110025. Collaborators: Nutty Professor, Free Ride Mast Maula (CTP, JMI) Introduction to Cosmology
More information3 Spacetime metrics. 3.1 Introduction. 3.2 Flat spacetime
3 Spacetime metrics 3.1 Introduction The efforts to generalize physical laws under different coordinate transformations would probably not have been very successful without differential calculus. Riemann
More informationκ = f (r 0 ) k µ µ k ν = κk ν (5)
1. Horizon regularity and surface gravity Consider a static, spherically symmetric metric of the form where f(r) vanishes at r = r 0 linearly, and g(r 0 ) 0. Show that near r = r 0 the metric is approximately
More information8 Cosmology. December 1997 Lecture Notes on General Relativity Sean M. Carroll
December 1997 Lecture Notes on General Relativity Sean M. Carroll 8 Cosmology Contemporary cosmological models are based on the idea that the universe is pretty much the same everywhere a stance sometimes
More informationOlbers Paradox. Lecture 14: Cosmology. Resolutions of Olbers paradox. Cosmic redshift
Lecture 14: Cosmology Olbers paradox Redshift and the expansion of the Universe The Cosmological Principle Ω and the curvature of space The Big Bang model Primordial nucleosynthesis The Cosmic Microwave
More informationOn the occasion of the first author s seventieth birthday
METHODS AND APPLICATIONS OF ANALYSIS. c 2005 International Press Vol. 12, No. 4, pp. 451 464, December 2005 006 HOW INFLATIONARY SPACETIMES MIGHT EVOLVE INTO SPACETIMES OF FINITE TOTAL MASS JOEL SMOLLER
More informationScott A. Hughes, MIT SSI, 28 July The basic concepts and properties of black holes in general relativity
The basic concepts and properties of black holes in general relativity For the duration of this talk ħ=0 Heuristic idea: object with gravity so strong that light cannot escape Key concepts from general
More informationReconstructed standard model of cosmology in the Earth-related coordinate system
Reconstructed standard model of cosmology in the Earth-related coordinate system Jian-Miin Liu Department of Physics, Nanjing University Nanjing, The People s Republic of China On leave. E-mail: liu@phys.uri.edu
More informationChapter 4. COSMOLOGICAL PERTURBATION THEORY
Chapter 4. COSMOLOGICAL PERTURBATION THEORY 4.1. NEWTONIAN PERTURBATION THEORY Newtonian gravity is an adequate description on small scales (< H 1 ) and for non-relativistic matter (CDM + baryons after
More informationPROBLEM SET 10 (The Last!)
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.286: The Early Universe December 8, 2016 Prof. Alan Guth PROBLEM SET 10 (The Last!) DUE DATE: Wednesday, December 14, 2016, at 4:00 pm.
More informationTheory. V H Satheeshkumar. XXVII Texas Symposium, Dallas, TX December 8 13, 2013
Department of Physics Baylor University Waco, TX 76798-7316, based on my paper with J Greenwald, J Lenells and A Wang Phys. Rev. D 88 (2013) 024044 with XXVII Texas Symposium, Dallas, TX December 8 13,
More informationLecture IX: Field equations, cosmological constant, and tides
Lecture IX: Field equations, cosmological constant, and tides Christopher M. Hirata Caltech M/C 350-17, Pasadena CA 91125, USA (Dated: October 28, 2011) I. OVERVIEW We are now ready to construct Einstein
More informationThe Metric and The Dynamics
The Metric and The Dynamics r τ c t a () t + r + Sin φ ( kr ) The RW metric tell us where in a 3 dimension space is an event and at which proper time. The coordinates of the event do not change as a function
More informationEinstein s Theory of Gravity. December 13, 2017
December 13, 2017 Newtonian Gravity Poisson equation 2 U( x) = 4πGρ( x) U( x) = G ρ( x) x x d 3 x For a spherically symmetric mass distribution of radius R U(r) = 1 r U(r) = 1 r R 0 r 0 r 2 ρ(r )dr for
More informationThe cosmic horizon. Fulvio Melia Department of Physics and Steward Observatory, The University of Arizona, Tucson, Arizona 85721, USA
Mon. Not. R. Astron. Soc. 382, 1917 1921 (2007) doi:10.1111/j.1365-2966.2007.12499.x The cosmic horizon Fulvio Melia Department of Physics and Steward Observatory, The University of Arizona, Tucson, Arizona
More informationIntroduction to Inflation
Introduction to Inflation Miguel Campos MPI für Kernphysik & Heidelberg Universität September 23, 2014 Index (Brief) historic background The Cosmological Principle Big-bang puzzles Flatness Horizons Monopoles
More informationBackreaction as an explanation for Dark Energy?
Backreaction as an explanation for Dark Energy? with some remarks on cosmological perturbation theory James M. Bardeen University of Washington The Very Early Universe 5 Years On Cambridge, December 17,
More informationDeflection. Hai Huang Min
The Gravitational Deflection of Light in F(R)-gravity Long Huang Feng He Hai Hai Huang Min Yao Abstract The fact that the gravitation could deflect the light trajectory has been confirmed by a large number
More informationThe early and late time acceleration of the Universe
The early and late time acceleration of the Universe Tomo Takahashi (Saga University) March 7, 2016 New Generation Quantum Theory -Particle Physics, Cosmology, and Chemistry- @Kyoto University The early
More informationConserved Quantities in Lemaître-Tolman-Bondi Cosmology
1/15 Section 1 Section 2 Section 3 Conserved Quantities in Lemaître-Tolman-Bondi Cosmology Alex Leithes - Blackboard Talk Outline ζ SMTP Evolution Equation: ζ SMTP = H X + 2H Y 3 ρ Valid on all scales.
More informationGeneral relativity, 3
General relativity, 3 Gravity as geometry: part II Even in a region of space-time that is so small that tidal effects cannot be detected, gravity still seems to produce curvature. he argument for this
More informationLecture XIX: Particle motion exterior to a spherical star
Lecture XIX: Particle motion exterior to a spherical star Christopher M. Hirata Caltech M/C 350-7, Pasadena CA 95, USA Dated: January 8, 0 I. OVERVIEW Our next objective is to consider the motion of test
More informationSchwarschild Metric From Kepler s Law
Schwarschild Metric From Kepler s Law Amit kumar Jha Department of Physics, Jamia Millia Islamia Abstract The simplest non-trivial configuration of spacetime in which gravity plays a role is for the region
More informationGeneral Relativity. on the frame of reference!
General Relativity Problems with special relativity What makes inertial frames special? How do you determine whether a frame is inertial? Inertial to what? Problems with gravity: In equation F = GM 1M
More informationVU lecture Introduction to Particle Physics. Thomas Gajdosik, FI & VU. Big Bang (model)
Big Bang (model) What can be seen / measured? basically only light _ (and a few particles: e ±, p, p, ν x ) in different wave lengths: microwave to γ-rays in different intensities (measured in magnitudes)
More information