Theoretical Cosmology and Astrophysics Lecture notes - Chapter 7
|
|
- Malcolm Reynolds
- 6 years ago
- Views:
Transcription
1 Theoretical Cosmology and Astrophysics Lecture notes - Chapter 7 A. Refregier April 24, Cosmological Perturbations 1 In this chapter, we will consider perturbations to the FRW smooth model of the universe. For simplicity, we will consider the flat FRW spacetime. 7.1 Metric perturbations Let us consider perturbations h µν to the FRW metric defined through where ḡ µν (x) = g µν = ḡ µν + h µν, (1) a 2 (t) a 2 (t) a 2 (t), (2) denotes the flat, unperturbed FRW metric. We further require that h µν ḡ µν, i.e. that the perturbations are small. In order to decompose the metric perturbation h µν, recall that we can decompose any vector field v( x) in 3-dimensional Euclidean space into a parallel and perpendicular part as v( x) = v ( x) + v ( x), (3) where v = v = 0. We can further write these two parts as v = v and v = A, where v is a scalar and A is a vector. We can check that these indeed satisfy the conditions specified above: v = ( v) = 0 and ( A) = 0. Note that the fields v and A are not uniquely defined by these relations. We can indeed add the quantities v v + const. (4) A A + f, (5) Notes taken by A. Nicola. 1 See chapter 5 in Dodelson, S., Modern Cosmology, 2003, Academic Press and chapter 5 in Weinberg, S., Cosmology, 2008, Oxford University Press. 1
2 where f is a scalar, without affecting either v or v. This property is called gauge freedom. Similarily we can decompose the metric perturbation h µν (x) into h 00 = E, ] F h i0 = a x i + G i, (6) h ij = a 2 Aδ ij + 2 B x i x j + C i x j + C ] j x i + D ij, where A = A(x), B = B(x), etc. and A, B, E, F are scalars C i, G i are divergenceless vectors D ij is a traceless, symmetric and divergenceless tensor, and the perturbations satisfy the conditions C i x i = G i x i = 0, D ij x i = 0, D ii = 0, D ij = D ji. (7) Since h µν is a symmetric 4 4 tensor, it has 10 independent degrees of freedom. It is easy to check that the new perturbation fields amount to the same number of degrees of freedom: no. of components divergenceless A, B, E, F 4 1 = 4 C i, G i 2 3 = 6 2 ( 1) D ij = = 10 The decomposition theorem states that scalar, vector and tensor perturbation modes do not couple to first order, i.e. they evolve independently. This means that Einstein s field equations can be solved for each perturbation mode separately. It can be shown that the amplitude of the vector modes decays as a function of time. The tensor modes correspond to gravitational waves, which are only important for CMB polarisation. Here, we will only consider scalar modes. 7.2 Gauge transformations Let us consider the spacetime coordinate transformation (see Fig.1) x µ x µ = x µ + ɛ µ (x), (8) where ɛ µ (x) is small, just as h µν is small compared to ḡ µν. Under such a coordinate transformation, the metric transforms as g µν(x ) = g λκ (x) xλ x κ x µ x ν. (9) 2
3 Inserting Eq. 8 into Eq. 9 leads to ) ) g µν(x ) = (δ λµ (δ ɛλ κν x µ ɛκ x ν g λκ (x). (10) Writing g λκ (x) = ḡ λκ (x) + h λκ (x) and expanding to first order finally leads to g µν(x ) ḡ µν (x) + h µν (x) ɛλ x µ ḡλν(x) ɛκ x ν ḡµκ(x). (11) M 0 x µ x 0 µ Figure 1: Illustration of coordinate transformations. Such a coordinate transformation affects the coordinates and unperturbed fields as well as the perturbations to the fields. In order to derive the transformations to the perturbed fields which leave the physics invariant we consider gauge transformations. Under a gauge transformation, the metric transforms as g µν (x) g µν(x). (12) From the generic metric transformation law given by Eq. 9 we get to first order in ɛ(x) and h µν (x) g µν(x ) = g µν(x + ɛ) = g µν(x) + g µν(x ) x λ ɛ λ. (13) Solving for g µν(x) and expressing g µν(x ) in terms of g µν (x) = ḡ µν (x) + h µν (x) through Eq. 9, gives us µν(x ) g µν(x) = g µν(x ) g x λ ɛ λ = ḡ µν (x) + h µν (x) ɛλ x µ ḡλν(x) ɛκ x ν ḡµκ(x) ḡ µν(x) x λ ɛ λ, (14) Letting g µν(x) = ḡ µν (x) + h µν(x) and h µν(x) = h µν (x) + h µν (x) finally gives us h µν (x) = h µν(x) h µν (x) = ɛλ x µ ḡλν(x) ɛκ x ν ḡµκ(x) ḡ µν(x) x λ ɛ λ. (15) We thus see that gauge transformations correspond to changes in the metric perturbations. 3
4 The FRW metric is given by ḡ µν (x) = Therefore we obtain h 00 = 2 ɛ0 t, 0 a 2 (t) a 2 (t) a 2 (t). (16) h i0 = ɛi t a2 + ɛ0 x i, (17) h ij = ɛi x j a2 ɛj x i a2 2a da dt δ ijɛ 0. We can simplify these equations by noting that ɛ µ = g µν ɛ ν = (ḡ µν + h µν )ɛ ν ḡ µν ɛ ν. Therefore we can use the smooth FRW metric to raise and lower indices on ɛ ν, i.e. ɛ 0 = ɛ 0 and ɛ i = a 2 ɛ i, and we obtain the transformations of the metric perturbations under gauge transformations: h 00 = 2 ɛ 0 t, h i0 = ɛ i t + 2 a h ij = ɛ i x j ɛ j x i + 2ada dt δ ijɛ 0. da dt ɛ i ɛ 0 x i, (18) In order to study how the scalar-vector-tensor components of the metric transform under a gauge transformation, we decompose the spatial part of the 4- vector ɛ µ into the gradient of a scalar plus a divergenceless vector ɛ i = ɛs x i + ɛv i, with ɛv i = 0. (19) xi Inserting this decomposition into Eq. 18, using Eq. 6 and considering only scalar modes gives us A = 2 da a dt ɛ 0, B = 2 a 2 ɛs, E = 2 dɛ 0 dt, (20) F = 1 ( ɛ 0 dɛs a dt + 2 ) da a dt ɛs. Similar identities hold for the vector and tensor perturbations The choice of gauge For the Newtonian gauge we choose ɛ S such that B = 0 and ɛ 0 such that F = 0. We are therefore left with the scalar perturbations A and E, which are relabelled 4
5 as E = 2Ψ, (21) A = 2Φ. (22) The perturbed metric in Newtonian gauge then becomes, keeping only scalar perturbations g 00 = 1 2Ψ, g 0i = 0, g ij = a 2 δ ij 1 + 2Φ]. (23) For the synchronous gauge we choose ɛ 0 such that E = 0 and ɛ S such that F = 0. The perturbed metric in synchronous gauge then becomes, keeping only scalar perturbations ] g 00 = 1, g 0i = 0, g ij = a 2 (1 + A)δ ij + 2 B x i x j. (24) There are other possible gauge choices. An example for a further gauge choice would be the co-moving gauge. It is also possible to perform cosmological perturbation theory solely in terms of gauge-invariant variables; this is the so-called gauge-invariant perturbation theory. In the following, we will choose the Newtonian gauge, which has the advantage that it is the easiest to relate to the Newtonian limit Geometrical interpretation of gauge transformations 2 To define perturbations, we need to compare two manifolds to each other (see Fig. 2): M: perturbed spacetime manifold with metric g µν M: background (unperturbed) spacetime manifold with metric ḡ µν. For this purpose, we need a mapping (diffeomorphism) φ between M and M. Then we can define the metric perturbation h µν as h µν = (φ 1 g) µν ḡ µν, (25) where everything is defined on M. Gauge freedom arises because there are many permissible (i.e. when perturbations are small) mappings φ betwen M and M. M M ḡ µ g µ 1 Figure 2: Illustration of mapping φ between manifolds M and M. For example, consider a mapping Λ ɛ of M onto itself, induced by a vector field ɛ µ on M, i.e. x µ x µ + ɛ µ (x) in a given coordinate system; as illustrated in 2 See chapter 7.1 in Carroll, S. M., Spacetime and Geometry: An Introduction to General Relativity, 2004, Addison Wesley. 5
6 Fig. 3. Then (φ Λ ɛ ) 1 = Λ 1 ɛ φ 1 is a new mapping between M to M, which induces new metric perturbations h (ɛ) µν = (φ Λ ɛ ) 1 g] µν ḡ µν = Λ 1 ɛ (φ 1 g)] µν ḡ µν. (26) All possible infinitesimal Λ ɛ s correspond to the gauge transformations we considered above (see Eq. 12). M M 1 Figure 3: Illustration of gauge freedom Number of degrees of freedom The metric g µν is a symmetric rank 2 tensor, i.e. it has 10 independent components. However, we saw above that there is gauge freedom with gauge transformations generated by vector fields ɛ µ which have 4 components. Therefore the number of physical degrees of freedom in GR is 10 4 = Einstein equations Recall that in Newtonian gauge, the flat perturbed FRW metric, keeping only scalar perturbations, is given by g 00 = 1 2Ψ, g 0i = 0, g ij = a 2 δ ij 1 + 2Φ]. (27) Let us compute the Christoffel symbols for this metric. In general, the Christoffel symbols are defined as Γ µ αβ = gµν gαν 2 x β + g βγ x α g ] αβ x ν. (28) The Christoffel symbols for the perturbed FRW metric in Newtonian gauge are therefore given by Γ 0 00 = Ψ,0, Γ 0 0i = Γ 0 i0 = Ψ,i, Γ 0 ij = δ ij a 2 H + 2H(Φ Ψ) Φ,0 ], Γ i 00 = 1 a 2 Ψ,i, (29) Γ i 0j = Γ i j0 = δ ij (H + Φ,0 ), Γ i jk = δ ij Φ,k + δ ik Φ,j δ jk Φ,i, 6
7 where H = 1 da a dt denotes the Hubble parameter. We can check that we recover the smooth universe results when we only look at the 0 th order terms in the Christoffel symbols: Γ 0 00 = Γ 0 0i = Γ 0 i0 = 0, Γ 0 ij = δ ij a 2 H = δ ij a da dt, Γ i 00 = Γ i jk = 0, (30) Γ i 0j = Γ i 1 da j0 = δ ij a dt. From the Christoffel symbols we can calculate the Ricci tensor for the perturbed FRW metric. The Ricci tensor is given by R µν = Γ α µν,α Γ α µα,ν + Γ α βαγ β µν Γ α βνγ β µα. (31) For the perturbed FRW metric we therefore obtain R 00 = 3 1 d 2 a a dt a 2 Ψ,ii 3Φ,00 + 3H(Ψ,0 2Φ,0 ), R ij = δ ij ( 2a 2 H 2 + a d2 a dt 2 + a 2 Φ,00 Φ,ii ] (Φ,ij + Ψ ij ). We can again check the 0 th order terms R 00 = 3 1 d 2 a a dt 2, ] R ij = δ ij 2a 2 H 2 + a d2 a dt 2 = δ ij 2 ) (1 + 2Φ 2Ψ) + a 2 H(6Φ,0 Ψ,0 ) (32) ( da dt ) ] 2 + a d2 a dt 2. (33) Note that we will not need R 0i, because the perturbed FRW metric is diagonal. The Ricci scalar is given by R = g µν R µν. (34) Inserting the explicit components of both the metric and the Ricci tensor therefore leads to ( R = 6 H d 2 ) a a dt 2 (1 2Ψ) 2 a 2 Ψ,ii +6Φ,00 6H(Ψ,0 4Φ,0 ) 4 a 2 Φ,ii. (35) We can again check that the 0 th order term agrees with that for the FRW metric ( R = 6 H d 2 ) a a dt 2. (36) Using the Ricci tensor and the Ricci scalar we can finally compute the Einstein tensor, which is given by G µν = R µν 1 2 g µνr. (37) 7
8 It is more convenient to work with the Einstein tensor with one index raised, i.e. G µ ν = g µρ G ρν = R µ ν 1 2 gµ νr. (38) For the perturbed FRW metric we therefore obtain the time-time component of the Einstein tensor as G 0 1 d 2 a 0 = 3 a dt 2 1 ( ) ] 2 ( ) 2 da da a ( ) 2 da 1 dt dt a dt Φ da,0 + 6 Ψ + 2 a dt a 2 Φ,ii. (39) We can again check the 0 th order component G 0 0 = ḡ 00 G 00 = 3 1 d 2 a a dt 2 1 a 2 The space-space component of the Einstein tensor is given by ( ) ] 2 da. (40) dt G i j = Aδ ij 1 a 2 (Φ,ij + Ψ,ij ), (41) where A contains almost a dozen terms which are all proportional to δ ij and therefore contribute only to the trace of G i j. Since we will only need to consider the longitudinal and traceless part of the Einstein tensor, we will not need to compute the explicit form of A. Einstein s field equations are then given by G µ ν = 8πGT µ ν. (42) Decomposing the metric into the background and the perturbations, we can write these equations as (Ḡµ ν + G µ ) ( ν = 8πG T µ ν + T µ ) ν. (43) In order to compute the equations governing the evolution of perturbations in the universe, we therefore need to compute the traceless part of T µ ν from the Boltzmann equations. 8
UNIVERSITY OF OSLO Faculty of Mathematics and Natural Sciences
UNIVERSITY OF OSLO Faculty of Mathematics and Natural Sciences Exam for AST5/94 Cosmology II Date: Thursday, June 11th, 15 Time: 9. 13. The exam set consists of 11 pages. Appendix: Equation summary Allowed
More informationUNIVERSITY OF OSLO Faculty of Mathematics and Natural Sciences
UNIVERSITY OF OSLO Faculty of Mathematics and Natural Sciences Exam for AST5220 Cosmology II Date: Tuesday, June 4th, 2013 Time: 09.00 13.00 The exam set consists of 13 pages. Appendix: Equation summary
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 informationmatter The second term vanishes upon using the equations of motion of the matter field, then the remaining term can be rewritten
9.1 The energy momentum tensor It will be useful to follow the analogy with electromagnetism (the same arguments can be repeated, with obvious modifications, also for nonabelian gauge theories). Recall
More informationNotes on General Relativity Linearized Gravity and Gravitational waves
Notes on General Relativity Linearized Gravity and Gravitational waves August Geelmuyden Universitetet i Oslo I. Perturbation theory Solving the Einstein equation for the spacetime metric is tremendously
More informationLecture VIII: Linearized gravity
Lecture VIII: Linearized gravity Christopher M. Hirata Caltech M/C 350-17, Pasadena CA 91125, USA (Dated: November 5, 2012) I. OVERVIEW We are now ready to consider the solutions of GR for the case of
More informationPAPER 52 GENERAL RELATIVITY
MATHEMATICAL TRIPOS Part III Monday, 1 June, 2015 9:00 am to 12:00 pm PAPER 52 GENERAL RELATIVITY Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight.
More informationInequivalence of First and Second Order Formulations in D=2 Gravity Models 1
BRX TH-386 Inequivalence of First and Second Order Formulations in D=2 Gravity Models 1 S. Deser Department of Physics Brandeis University, Waltham, MA 02254, USA The usual equivalence between the Palatini
More informationGRAVITATION F10. Lecture Maxwell s Equations in Curved Space-Time 1.1. Recall that Maxwell equations in Lorentz covariant form are.
GRAVITATION F0 S. G. RAJEEV Lecture. Maxwell s Equations in Curved Space-Time.. Recall that Maxwell equations in Lorentz covariant form are. µ F µν = j ν, F µν = µ A ν ν A µ... They follow from the variational
More informationcarroll/notes/ has a lot of good notes on GR and links to other pages. General Relativity Philosophy of general
http://pancake.uchicago.edu/ carroll/notes/ has a lot of good notes on GR and links to other pages. General Relativity Philosophy of general relativity. As with any major theory in physics, GR has been
More informationProblem 1, Lorentz transformations of electric and magnetic
Problem 1, Lorentz transformations of electric and magnetic fields We have that where, F µν = F µ ν = L µ µ Lν ν F µν, 0 B 3 B 2 ie 1 B 3 0 B 1 ie 2 B 2 B 1 0 ie 3 ie 2 ie 2 ie 3 0. Note that we use the
More informationGENERAL RELATIVITY: THE FIELD THEORY APPROACH
CHAPTER 9 GENERAL RELATIVITY: THE FIELD THEORY APPROACH We move now to the modern approach to General Relativity: field theory. The chief advantage of this formulation is that it is simple and easy; the
More informationGeneral Relativity (225A) Fall 2013 Assignment 8 Solutions
University of California at San Diego Department of Physics Prof. John McGreevy General Relativity (5A) Fall 013 Assignment 8 Solutions Posted November 13, 013 Due Monday, December, 013 In the first two
More information4 Evolution of density perturbations
Spring term 2014: Dark Matter lecture 3/9 Torsten Bringmann (torsten.bringmann@fys.uio.no) reading: Weinberg, chapters 5-8 4 Evolution of density perturbations 4.1 Statistical description The cosmological
More informationhas a lot of good notes on GR and links to other pages. General Relativity Philosophy of general relativity.
http://preposterousuniverse.com/grnotes/ has a lot of good notes on GR and links to other pages. General Relativity Philosophy of general relativity. As with any major theory in physics, GR has been framed
More informationChapter 7 Curved Spacetime and General Covariance
Chapter 7 Curved Spacetime and General Covariance In this chapter we generalize the discussion of preceding chapters to extend covariance to more general curved spacetimes. 145 146 CHAPTER 7. CURVED SPACETIME
More informationGravitational Čerenkov Notes
Gravitational Čerenkov Notes These notes were presented at the IUCSS Summer School on the Lorentz- and CPT-violating Standard-Model Extension. They are based Ref. [1]. There are no new results here, and
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 informationGravitation: Tensor Calculus
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 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 informationGravitation: Gravitation
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 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 informationIs there a magnification paradox in gravitational lensing?
Is there a magnification paradox in gravitational lensing? Olaf Wucknitz wucknitz@astro.uni-bonn.de Astrophysics seminar/colloquium, Potsdam, 26 November 2007 Is there a magnification paradox in gravitational
More informationConnections and geodesics in the space of metrics The exponential parametrization from a geometric perspective
Connections and geodesics in the space of metrics The exponential parametrization from a geometric perspective Andreas Nink Institute of Physics University of Mainz September 21, 2015 Based on: M. Demmel
More informationGeneral Relativity and Differential
Lecture Series on... General Relativity and Differential Geometry CHAD A. MIDDLETON Department of Physics Rhodes College November 1, 2005 OUTLINE Distance in 3D Euclidean Space Distance in 4D Minkowski
More informationLight Propagation in the Averaged Universe. arxiv:
Light Propagation in the Averaged Universe arxiv: 1404.2185 Samae Bagheri Dominik Schwarz Bielefeld University Cosmology Conference, Centre de Ciencias de Benasque Pedro Pascual, 11.Aug, 2014 Outline 1
More informationCurved Spacetime III Einstein's field equations
Curved Spacetime III Einstein's field equations Dr. Naylor Note that in this lecture we will work in SI units: namely c 1 Last Week s class: Curved spacetime II Riemann curvature tensor: This is a tensor
More informationAn introduction to gravitational waves. Enrico Barausse (Institut d'astrophysique de Paris/CNRS, France)
An introduction to gravitational waves Enrico Barausse (Institut d'astrophysique de Paris/CNRS, France) Outline of lectures (1/2) The world's shortest introduction to General Relativity The linearized
More informationVariational Principle and Einstein s equations
Chapter 15 Variational Principle and Einstein s equations 15.1 An useful formula There exists an useful equation relating g µν, g µν and g = det(g µν ) : g x α = ggµν g µν x α. (15.1) The proof is the
More informationCurved Spacetime I. Dr. Naylor
Curved Spacetime I Dr. Naylor Last Week Einstein's principle of equivalence We discussed how in the frame of reference of a freely falling object we can construct a locally inertial frame (LIF) Space tells
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 informationCurved spacetime and general covariance
Chapter 7 Curved spacetime and general covariance In this chapter we generalize the discussion of preceding chapters to extend covariance to more general curved spacetimes. 219 220 CHAPTER 7. CURVED SPACETIME
More informationProblem Sets on Cosmology and Cosmic Microwave Background
Problem Sets on Cosmology and Cosmic Microwave Background Lecturer: Prof. Dr. Eiichiro Komatsu October 16, 2014 1 Expansion of the Universe In this section, we will use Einstein s General Relativity to
More informationPAPER 309 GENERAL RELATIVITY
MATHEMATICAL TRIPOS Part III Monday, 30 May, 2016 9:00 am to 12:00 pm PAPER 309 GENERAL RELATIVITY Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight.
More informationCHAPTER 6 EINSTEIN EQUATIONS. 6.1 The energy-momentum tensor
CHAPTER 6 EINSTEIN EQUATIONS You will be convinced of the general theory of relativity once you have studied it. Therefore I am not going to defend it with a single word. A. Einstein 6.1 The energy-momentum
More informationGeneralized Harmonic Coordinates Using Abigel
Outline Introduction The Abigel Code Jennifer Seiler jennifer.seiler@aei.mpg.de Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut) Oberjoch, Germany 10th June 2006 Outline Introduction
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 informationPAPER 310 COSMOLOGY. Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight.
MATHEMATICAL TRIPOS Part III Wednesday, 1 June, 2016 9:00 am to 12:00 pm PAPER 310 COSMOLOGY Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight. STATIONERY
More informationGravitational Waves. Basic theory and applications for core-collapse supernovae. Moritz Greif. 1. Nov Stockholm University 1 / 21
Gravitational Waves Basic theory and applications for core-collapse supernovae Moritz Greif Stockholm University 1. Nov 2012 1 / 21 General Relativity Outline 1 General Relativity Basic GR Gravitational
More informationTensor Calculus, Part 2
Massachusetts Institute of Technology Department of Physics Physics 8.962 Spring 2002 Tensor Calculus, Part 2 c 2000, 2002 Edmund Bertschinger. 1 Introduction The first set of 8.962 notes, Introduction
More informationEinstein Toolkit Workshop. Joshua Faber Apr
Einstein Toolkit Workshop Joshua Faber Apr 05 2012 Outline Space, time, and special relativity The metric tensor and geometry Curvature Geodesics Einstein s equations The Stress-energy tensor 3+1 formalisms
More informationThe principle of equivalence and its consequences.
The principle of equivalence and its consequences. Asaf Pe er 1 January 28, 2014 This part of the course is based on Refs. [1], [2] and [3]. 1. Introduction We now turn our attention to the physics of
More informationarxiv: v2 [astro-ph.co] 5 Feb 2018
Cosmological Backgrounds of Gravitational Waves arxiv:1801.04268v2 [astro-ph.co] 5 Feb 2018 Chiara Caprini Laboratoire Astroparticule et Cosmologie, CNRS UMR 7164, Université Paris-Diderot, 10 rue Alice
More informationLongitudinal Waves in Scalar, Three-Vector Gravity
Longitudinal Waves in Scalar, Three-Vector Gravity Kenneth Dalton email: kxdalton@yahoo.com Abstract The linear field equations are solved for the metrical component g 00. The solution is applied to the
More information2. Lie groups as manifolds. SU(2) and the three-sphere. * version 1.4 *
. Lie groups as manifolds. SU() and the three-sphere. * version 1.4 * Matthew Foster September 1, 017 Contents.1 The Haar measure 1. The group manifold for SU(): S 3 3.3 Left- and right- group translations
More informationColliding scalar pulses in the Einstein-Gauss-Bonnet gravity
Colliding scalar pulses in the Einstein-Gauss-Bonnet gravity Hisaaki Shinkai 1, and Takashi Torii 2, 1 Department of Information Systems, Osaka Institute of Technology, Hirakata City, Osaka 573-0196, Japan
More informationInitial-Value Problems in General Relativity
Initial-Value Problems in General Relativity Michael Horbatsch March 30, 2006 1 Introduction In this paper the initial-value formulation of general relativity is reviewed. In section (2) domains of dependence,
More informationTensor Calculus, Relativity, and Cosmology
Tensor Calculus, Relativity, and Cosmology A First Course by M. Dalarsson Ericsson Research and Development Stockholm, Sweden and N. Dalarsson Royal Institute of Technology Stockholm, Sweden ELSEVIER ACADEMIC
More informationSpecial Relativity - QMII - Mechina
Special Relativity - QMII - Mechina 2016-17 Daniel Aloni Disclaimer This notes should not replace a course in special relativity, but should serve as a reminder. I tried to cover as many important topics
More informationThe Riemann curvature tensor, its invariants, and their use in the classification of spacetimes
DigitalCommons@USU Presentations and Publications 3-20-2015 The Riemann curvature tensor, its invariants, and their use in the classification of spacetimes Follow this and additional works at: http://digitalcommons.usu.edu/dg_pres
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 informationMassive gravitons in arbitrary spacetimes
Massive gravitons in arbitrary spacetimes Mikhail S. Volkov LMPT, University of Tours, FRANCE Kyoto, YITP, Gravity and Cosmology Workshop, 6-th February 2018 C.Mazuet and M.S.V., Phys.Rev. D96, 124023
More informationLecture: General Theory of Relativity
Chapter 8 Lecture: General Theory of Relativity We shall now employ the central ideas introduced in the previous two chapters: The metric and curvature of spacetime The principle of equivalence The principle
More informationEinstein Double Field Equations
Einstein Double Field Equations Stephen Angus Ewha Woman s University based on arxiv:1804.00964 in collaboration with Kyoungho Cho and Jeong-Hyuck Park (Sogang Univ.) KIAS Workshop on Fields, Strings and
More informationContinuity Equations and the Energy-Momentum Tensor
Physics 4 Lecture 8 Continuity Equations and the Energy-Momentum Tensor Lecture 8 Physics 4 Classical Mechanics II October 8th, 007 We have finished the definition of Lagrange density for a generic space-time
More informationChapter 2 General Relativity and Black Holes
Chapter 2 General Relativity and Black Holes In this book, black holes frequently appear, so we will describe the simplest black hole, the Schwarzschild black hole and its physics. Roughly speaking, a
More informationLinear Cosmological Perturbations and Cosmic Microwave Background Anisotropies. Carlo Baccigalupi
Linear Cosmological Perturbations and Cosmic Microwave Background Anisotropies Carlo Baccigalupi January 21, 2015 2 Contents 1 introduction 7 1.1 things to know for attending the course.............. 8
More informationTHE ENERGY-MOMENTUM PSEUDOTENSOR T µν of matter satisfies the (covariant) divergenceless equation
THE ENERGY-MOMENTUM PSEUDOTENSOR T µν of matter satisfies the (covariant) divergenceless equation T µν ;ν = 0 (3) We know it is not a conservation law, because it cannot be written as an ordinary divergence.
More informationMathematical Journal of Okayama University
Mathematical Journal of Okayama University Volume 42 Issue 2000 Article 6 JANUARY 2000 Certain Metrics on R 4 + Tominosuke Otsuki Copyright c 2000 by the authors. Mathematical Journal of Okayama University
More informationVector and Tensor Calculus
Appendices 58 A Vector and Tensor Calculus In relativistic theory one often encounters vector and tensor expressions in both three- and four-dimensional form. The most important of these expressions are
More informationModern Cosmology / Scott Dodelson Contents
Modern Cosmology / Scott Dodelson Contents The Standard Model and Beyond p. 1 The Expanding Universe p. 1 The Hubble Diagram p. 7 Big Bang Nucleosynthesis p. 9 The Cosmic Microwave Background p. 13 Beyond
More informationMulti-disformal invariance of nonlinear primordial perturbations
Multi-disformal invariance of nonlinear primordial perturbations Yuki Watanabe Natl. Inst. Tech., Gunma Coll.) with Atsushi Naruko and Misao Sasaki accepted in EPL [arxiv:1504.00672] 2nd RESCEU-APCosPA
More informationarxiv:physics/ v1 [physics.ed-ph] 21 Aug 1999
arxiv:physics/9984v [physics.ed-ph] 2 Aug 999 Gravitational Waves: An Introduction Indrajit Chakrabarty Abstract In this article, I present an elementary introduction to the theory of gravitational waves.
More informationBRANE COSMOLOGY and Randall-Sundrum model
BRANE COSMOLOGY and Randall-Sundrum model M. J. Guzmán June 16, 2009 Standard Model of Cosmology CMB and large-scale structure observations provide us a high-precision estimation of the cosmological parameters:
More information221A Miscellaneous Notes Continuity Equation
221A Miscellaneous Notes Continuity Equation 1 The Continuity Equation As I received questions about the midterm problems, I realized that some of you have a conceptual gap about the continuity equation.
More informationDark Energy and Dark Matter Interaction. f (R) A Worked Example. Wayne Hu Florence, February 2009
Dark Energy and Dark Matter Interaction f (R) A Worked Example Wayne Hu Florence, February 2009 Why Study f(r)? Cosmic acceleration, like the cosmological constant, can either be viewed as arising from
More informationSymmetry Transformations, the Einstein-Hilbert Action, and Gauge Invariance
Massachusetts Institute of Technology Department of Physics Physics 8.962 Spring 2002 Symmetry Transformations, the Einstein-Hilbert Action, and Gauge Invariance c 2000, 2002 Edmund Bertschinger. 1 Introduction
More informationarxiv: v1 [physics.gen-ph] 18 Mar 2010
arxiv:1003.4981v1 [physics.gen-ph] 18 Mar 2010 Riemann-Liouville Fractional Einstein Field Equations Joakim Munkhammar October 22, 2018 Abstract In this paper we establish a fractional generalization of
More informationLinearized Gravity Return to Linearized Field Equations
Physics 411 Lecture 28 Linearized Gravity Lecture 28 Physics 411 Classical Mechanics II November 7th, 2007 We have seen, in disguised form, the equations of linearized gravity. Now we will pick a gauge
More informationImperial College 4th Year Physics UG, General Relativity Revision lecture. Toby Wiseman; Huxley 507,
Imperial College 4th Year Physics UG, 2012-13 General Relativity Revision lecture Toby Wiseman; Huxley 507, email: t.wiseman@imperial.ac.uk 1 1 Exam This is 2 hours. There is one compulsory question (
More informationN-body simulations with massive neutrinos
N-body simulations with massive neutrinos Håkon Opheimsbakken Thesis submitted for the degree of Master of Science in Astronomy Institute of Theoretical Astrophysics University of Oslo June, 2014 Abstract
More informationGeneral Relativity (j µ and T µν for point particles) van Nieuwenhuizen, Spring 2018
Consistency of conservation laws in SR and GR General Relativity j µ and for point particles van Nieuwenhuizen, Spring 2018 1 Introduction The Einstein equations for matter coupled to gravity read Einstein
More informationLOCAL EXISTENCE THEORY OF THE VACUUM EINSTEIN EQUATIONS
LOCAL EXISTENCE THEORY OF THE VACUUM EINSTEIN EQUATIONS TSZ KIU AARON CHOW Abstract. In this paper, we treat the vacuum Einstein equations as a local Cauchy problem. We show that it reduces to a system
More informationarxiv: v5 [gr-qc] 24 Mar 2014
Relativistic Cosmological Perturbation Theory and the Evolution of Small-Scale Inhomogeneities P. G. Miedema Netherlands Defence Academy Hogeschoollaan, NL-4818 CR Breda, The Netherlands (Dated: March
More informationCMB Polarization in Einstein-Aether Theory
CMB Polarization in Einstein-Aether Theory Masahiro Nakashima (The Univ. of Tokyo, RESCEU) With Tsutomu Kobayashi (RESCEU) COSMO/CosPa 2010 Introduction Two Big Mysteries of Cosmology Dark Energy & Dark
More informationIntroduction to General Relativity and Gravitational Waves
Introduction to General Relativity and Gravitational Waves Patrick J. Sutton Cardiff University International School of Physics Enrico Fermi Varenna, 2017/07/03-04 Suggested reading James B. Hartle, Gravity:
More informationOne-loop renormalization in a toy model of Hořava-Lifshitz gravity
1/0 Università di Roma TRE, Max-Planck-Institut für Gravitationsphysik One-loop renormalization in a toy model of Hořava-Lifshitz gravity Based on (hep-th:1311.653) with Dario Benedetti Filippo Guarnieri
More informationQuantum Field Theory Notes. Ryan D. Reece
Quantum Field Theory Notes Ryan D. Reece November 27, 2007 Chapter 1 Preliminaries 1.1 Overview of Special Relativity 1.1.1 Lorentz Boosts Searches in the later part 19th century for the coordinate transformation
More informationDynamics of star clusters containing stellar mass black holes: 1. Introduction to Gravitational Waves
Dynamics of star clusters containing stellar mass black holes: 1. Introduction to Gravitational Waves July 25, 2017 Bonn Seoul National University Outline What are the gravitational waves? Generation of
More informationLecture XXXIV: Hypersurfaces and the 3+1 formulation of geometry
Lecture XXXIV: Hypersurfaces and the 3+1 formulation of geometry Christopher M. Hirata Caltech M/C 350-17, Pasadena CA 91125, USA (Dated: May 4, 2012) I. OVERVIEW Having covered the Lagrangian formulation
More informationKonstantin E. Osetrin. Tomsk State Pedagogical University
Space-time models with dust and cosmological constant, that allow integrating the Hamilton-Jacobi test particle equation by separation of variables method. Konstantin E. Osetrin Tomsk State Pedagogical
More informationLecture X: External fields and generation of gravitational waves
Lecture X: External fields and generation of gravitational waves Christopher M. Hirata Caltech M/C 350-17, Pasadena CA 91125, USA (Dated: November 12, 2012) I. OVEVIEW Having examined weak field gravity
More information2 Post-Keplerian Timing Parameters for General Relativity
1 Introduction General Relativity has been one of the pilars of modern physics for over 100 years now. Testing the theory and its consequences is therefore very important to solidifying our understand
More informationand Zoran Rakić Nonlocal modified gravity Ivan Dimitrijević, Branko Dragovich, Jelena Grujić
Motivation Large cosmological observational findings: High orbital speeds of galaxies in clusters.( F.Zwicky, 1933) High orbital speeds of stars in spiral galaxies. ( Vera Rubin, at the end of 1960es )
More information1 Introduction. 1.1 Notations and conventions
The derivation of the coupling constant in the new Self Creation Cosmology Garth A Barber The Vicarage, Woodland Way, Tadworth, Surrey, England KT206NW Tel: +44 01737 832164 e-mail: garth.barber@virgin.net
More informationNewton s Second Law is Valid in Relativity for Proper Time
Newton s Second Law is Valid in Relativity for Proper Time Steven Kenneth Kauffmann Abstract In Newtonian particle dynamics, time is invariant under inertial transformations, and speed has no upper bound.
More informationStability Results in the Theory of Relativistic Stars
Stability Results in the Theory of Relativistic Stars Asad Lodhia September 5, 2011 Abstract In this article, we discuss, at an accessible level, the relativistic theory of stars. We overview the history
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 informationCosmology. April 13, 2015
Cosmology April 3, 205 The cosmological principle Cosmology is based on the principle that on large scales, space (not spacetime) is homogeneous and isotropic that there is no preferred location or direction
More information3 Parallel transport and geodesics
3 Parallel transport and geodesics 3.1 Differentiation along a curve As a prelude to parallel transport we consider another form of differentiation: differentiation along a curve. A curve is a parametrized
More informationVectors. Three dimensions. (a) Cartesian coordinates ds is the distance from x to x + dx. ds 2 = dx 2 + dy 2 + dz 2 = g ij dx i dx j (1)
Vectors (Dated: September017 I. TENSORS Three dimensions (a Cartesian coordinates ds is the distance from x to x + dx ds dx + dy + dz g ij dx i dx j (1 Here dx 1 dx, dx dy, dx 3 dz, and tensor g ij is
More informationScalar perturbations of Galileon cosmologies in the mechanical approach in the late Universe
Scalar perturbations of Galileon cosmologies in the mechanical approach in the late Universe Perturbation theory as a probe of viable cosmological models Jan Novák Department of physics Technical University
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 informationRank Three Tensors in Unified Gravitation and Electrodynamics
5 Rank Three Tensors in Unified Gravitation and Electrodynamics by Myron W. Evans, Alpha Institute for Advanced Study, Civil List Scientist. (emyrone@aol.com and www.aias.us) Abstract The role of base
More informationA GENERALLY COVARIANT FIELD EQUATION FOR GRAVITATION AND ELECTROMAGNETISM. Institute for Advanced Study Alpha Foundation
A GENERALLY COVARIANT FIELD EQUATION FOR GRAVITATION AND ELECTROMAGNETISM Myron W. Evans Institute for Advanced Study Alpha Foundation E-mail: emyrone@aol.com Received 17 April 2003; revised 1 May 2003
More informationWeek 9: Einstein s field equations
Week 9: Einstein s field equations Riemann tensor and curvature We are looking for an invariant characterisation of an manifold curved by gravity. As the discussion of normal coordinates showed, the first
More informationAstro 321 Set 3: Relativistic Perturbation Theory. Wayne Hu
Astro 321 Set 3: Relativistic Perturbation Theory Wayne Hu Covariant Perturbation Theory Covariant = takes same form in all coordinate systems Invariant = takes the same value in all coordinate systems
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 informationSolving Einstein s Equation Numerically III
Solving Einstein s Equation Numerically III Lee Lindblom Center for Astrophysics and Space Sciences University of California at San Diego Mathematical Sciences Center Lecture Series Tsinghua University
More informationProblem Set #4: 4.1, 4.3, 4.5 (Due Monday Nov. 18th) f = m i a (4.1) f = m g Φ (4.2) a = Φ. (4.4)
Chapter 4 Gravitation Problem Set #4: 4.1, 4.3, 4.5 (Due Monday Nov. 18th) 4.1 Equivalence Principle The Newton s second law states that f = m i a (4.1) where m i is the inertial mass. The Newton s law
More information