A A + B. ra + A + 1. We now want to solve the Einstein equations in the following cases:

Size: px
Start display at page:

Download "A A + B. ra + A + 1. We now want to solve the Einstein equations in the following cases:"

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 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 information

General Relativity and Cosmology Mock exam

General 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 information

2.1 Basics of the Relativistic Cosmology: Global Geometry and the Dynamics of the Universe Part I

2.1 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 information

MASSACHUSETTS 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 8.286: 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 information

Introduction to Cosmology

Introduction 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 information

General Birkhoff s Theorem

General 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 information

AST1100 Lecture Notes

AST1100 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 information

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

Third 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 information

Astronomy, Astrophysics, and Cosmology

Astronomy, 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 information

arxiv:gr-qc/ v1 10 Apr 2003

arxiv: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 information

TO 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 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 information

NEWTONIAN COSMOLOGY. Figure 2.1: All observers see galaxies expanding with the same Hubble law. v A = H 0 r A (2.1)

NEWTONIAN 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 information

Uniformity of the Universe

Uniformity 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 information

PHY 475/375. Lecture 5. (April 9, 2012)

PHY 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 information

AST1100 Lecture Notes

AST1100 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 information

Cosmology (Cont.) Lecture 19

Cosmology (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 information

General Relativity Lecture 20

General 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 information

A5682: Introduction to Cosmology Course Notes. 2. General Relativity

A5682: 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 information

Kerr black hole and rotating wormhole

Kerr 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 information

Week 2 Part 2. The Friedmann Models: What are the constituents of the Universe?

Week 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 information

Cosmology ASTR 2120 Sarazin. Hubble Ultra-Deep Field

Cosmology 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 information

FRW 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) 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 information

ASTR 610: Solutions to Problem Set 1

ASTR 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 information

1 Cosmological Principle

1 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 information

New Blackhole Theorem and its Applications to Cosmology and Astrophysics

New 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 information

2.1 The metric and and coordinate transformations

2.1 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 information

Lecture 13 Friedmann Model

Lecture 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 information

D. f(r) gravity. φ = 1 + f R (R). (48)

D. 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 information

Physics 133: Extragalactic Astronomy and Cosmology

Physics 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 information

The Motion of A Test Particle in the Gravitational Field of A Collapsing Shell

The 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 information

THE DARK SIDE OF THE COSMOLOGICAL CONSTANT

THE 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 information

PAPER 73 PHYSICAL COSMOLOGY

PAPER 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 information

Astr 2320 Tues. May 2, 2017 Today s Topics Chapter 23: Cosmology: The Big Bang and Beyond Introduction Newtonian Cosmology Solutions to Einstein s

Astr 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 information

PAPER 71 COSMOLOGY. Attempt THREE questions There are seven questions in total The questions carry equal weight

PAPER 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 information

3.1 Cosmological Parameters

3.1 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 information

Physics 133: Extragalactic Astronomy ad Cosmology

Physics 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 information

Geometry of the Universe: Cosmological Principle

Geometry 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 information

Astronomy 421. Lecture 24: Black Holes

Astronomy 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 information

Cosmology PHYS3170 The angular-diameter-redshift relation

Cosmology 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 information

Chapter 9. Perturbations in the Universe

Chapter 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 information

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

HOMEWORK 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 information

Why is the Universe Expanding?

Why 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 information

PROBLEM SET 6 EXTRA CREDIT PROBLEM SET

PROBLEM 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 information

Gravitational radiation

Gravitational 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 information

Is the Rate of Expansion of Our Universe. Really Accelerating?

Is 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 information

Einstein s Equations. July 1, 2008

Einstein 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 information

You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.

You 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 information

Kinetic Theory of Dark Energy within General Relativity

Kinetic 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 information

Lecture 2: Cosmological Background

Lecture 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 information

Schwarzschild s Metrical Model of a Liquid Sphere

Schwarzschild 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 information

Lecture 1 General relativity and cosmology. Kerson Huang MIT & IAS, NTU

Lecture 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 information

The homogeneous and isotropic universe

The 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 information

3 Friedmann Robertson Walker Universe

3 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 information

Escape velocity and Schwarzschild s Solution for Black Holes

Escape 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 information

Gravitation: Cosmology

Gravitation: 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 information

Exact Solutions of the Einstein Equations

Exact 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 information

3 Friedmann Robertson Walker Universe

3 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 information

Übungen 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 (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 information

General Relativity ASTR 2110 Sarazin. Einstein s Equation

General 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 information

You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.

You 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 information

Astrophysics Lecture Notes

Astrophysics 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 information

THE UNIVERSITY OF SYDNEY FACULTY OF SCIENCE INTERMEDIATE PHYSICS PHYS 2913 ASTROPHYSICS AND RELATIVITY (ADVANCED) ALL QUESTIONS HAVE THE VALUE SHOWN

THE 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 information

Newtonian Gravity and Cosmology

Newtonian 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 information

Tuesday: Special epochs of the universe (recombination, nucleosynthesis, inflation) Wednesday: Structure formation

Tuesday: 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 information

General 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 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 information

MATHEMATICAL TRIPOS Part III PAPER 53 COSMOLOGY

MATHEMATICAL 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 information

A Bit of History. Hubble s original redshiftdistance

A 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 information

Modeling 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 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 information

Ay1 Lecture 17. The Expanding Universe Introduction to Cosmology

Ay1 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 information

PHYM432 Relativity and Cosmology 17. Cosmology Robertson Walker Metric

PHYM432 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 information

Black holes, Holography and Thermodynamics of Gauge Theories

Black 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 information

Set 3: Cosmic Dynamics

Set 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 information

Energy and Matter in the Universe

Energy 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 information

Un-Darkening the Cosmos: New laws of physics for an expanding universe

Un-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 information

Arvind Borde / MTH 675, Unit 20: Cosmology

Arvind 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 information

Introduction to Cosmology

Introduction 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 information

3 Spacetime metrics. 3.1 Introduction. 3.2 Flat spacetime

3 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 information

κ = f (r 0 ) k µ µ k ν = κk ν (5)

κ = 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 information

8 Cosmology. December 1997 Lecture Notes on General Relativity Sean M. Carroll

8 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 information

Olbers Paradox. Lecture 14: Cosmology. Resolutions of Olbers paradox. Cosmic redshift

Olbers 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 information

On the occasion of the first author s seventieth birthday

On 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 information

Scott A. Hughes, MIT SSI, 28 July The basic concepts and properties of black holes in general relativity

Scott 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 information

Reconstructed standard model of cosmology in the Earth-related coordinate system

Reconstructed 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 information

Chapter 4. COSMOLOGICAL PERTURBATION THEORY

Chapter 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 information

PROBLEM SET 10 (The Last!)

PROBLEM 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 information

Theory. V H Satheeshkumar. XXVII Texas Symposium, Dallas, TX December 8 13, 2013

Theory. 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 information

Lecture IX: Field equations, cosmological constant, and tides

Lecture 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 information

The Metric and The Dynamics

The 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 information

Einstein s Theory of Gravity. December 13, 2017

Einstein 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 information

The cosmic horizon. Fulvio Melia Department of Physics and Steward Observatory, The University of Arizona, Tucson, Arizona 85721, USA

The 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 information

Introduction to Inflation

Introduction 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 information

Backreaction as an explanation for Dark Energy?

Backreaction 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 information

Deflection. Hai Huang Min

Deflection. 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 information

The early and late time acceleration of the Universe

The 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 information

Conserved Quantities in Lemaître-Tolman-Bondi Cosmology

Conserved 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 information

General relativity, 3

General 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 information

Lecture XIX: Particle motion exterior to a spherical star

Lecture 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 information

Schwarschild Metric From Kepler s Law

Schwarschild 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 information

General Relativity. on the frame of reference!

General 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 information

VU lecture Introduction to Particle Physics. Thomas Gajdosik, FI & VU. Big Bang (model)

VU 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