calculations in general relativity

8/10/2019 Calculations in General Relativity

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Lots of Calculations in General Relativity

Susan Larsen Saturday, August 16, 2014

http://physicssusan.mono.net/9035/General%20Relativity Page 1

Contents 1 Introduktion ............................................................................................................................................... 6

2 Special relativity......................................................................................................................................... 7

2.1 How to calculate : Example ......................................................................................................... 7

2.2 The four velocity and the four force: ............................................................................. 7

3 Metric and Vector Transformations. ......................................................................................................... 7

3.1 Flat Minkowski space: ....................................................................................................................... 7

3.2 Other realizations of the flat space: .................................................................................................. 8

3.2.1 Spherical polar coordinates ....................................................................................................... 8

3.2.2 Flat space with a singularity ...................................................................................................... 9

3.2.3 Coordinate transformations ...................................................................................................... 9

3.2.4 Flat space in two dimensions .................................................................................................. 10

3.2.5 The Penrose Diagram for Flat Space ........................................................................................ 11

3.3 The line-element and metric of an ellipsoid: ................................................................................... 11

3.4 The signature of a metric................................................................................................................. 12

3.5 Three-dimensional flat space in spherical coordinates and vector transformation ....................... 12

3.6 Static Weak Field Metric ................................................................................................................. 13

3.6.1 Rates of Emission and Reception............................................................................................. 13

3.7 NEW - Local inertial frames ............................................................................................................. 14

3.7.1 The metric of a Sphere at the North Pole. ............................................................................... 14

3.8 New - Length, Area, Volume and Four-Volume for Diagonal Metrics ............................................. 16

3.8.1 Area and Volume Elements of a Sphere .................................................................................. 16

3.8.2 Distance, Area and Volume in the Curved Space of a Constant Density Spherical Star or aHomogenous Closed Universe ................................................................................................................. 17

3.8.3 Distance, Area, Volume and four-volume of a metric ............................................................. 18

3.8.4 The dimensions of a peanut .................................................................................................... 19

3.8.5 The dimensions of an egg ........................................................................................................ 20

3.8.6 Length and volume of the Schwarzschild geometry ............................................................... 20

3.8.7 Volume in the Wormhole geometry ........................................................................................ 21

4 Tensor Calculus ........................................................................................................................................ 22

4.1 Christoffel symbols. ......................................................................................................................... 22

4.1.1 and in a diagonal metric ..................................................................................... 22
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4.1.2 Find the Christoffel symbols of the 2-sphere with radius .................................................... 22

4.1.3 Find the Christoffel symbols of the Kahn-Penrose metric (Colliding gravitational waves) ..... 22

4.2 Alternative solution: Show that .................................................................................. 23

4.3 One-forms. ....................................................................................................................................... 23

4.3.1 One-forms: why .......................................................................................................... 23

4.3.2 The exterior derivative of a one-form. .................................................................................... 24

4.4 The geodesic equation. ................................................................................................................... 24

4.4.1 Find the geodesic equations for cylindrical coordinates ........................................................ 24

4.4.2 Use the geodesic equations to find the Christoffel symbols for the Rindler metric. .............. 26

4.4.3 New - Geodesics Equations of the plane in polar coordinates ................................................ 26

4.4.4 New Equations for geodesics in a Wormhole Geometry ..................................................... 274.5 New - Solving the geodesic equation .............................................................................................. 28

4.5.1 New - The travel time through a wormhole ............................................................................ 28

4.5.2 NEW - Geodesics in the Plane Using Polar Coordinates. ......................................................... 29

4.6 Killing Vectors .................................................................................................................................. 30

4.6.1 Show that if the Lie derivative of the metric tensor with respect to vector X vanishes( ), the vector X satisfies the Killing equation. - Alternative version ................................... 30

4.6.2 Prove that ............................................................................................... 31

4.6.3 Constructing a Conserved Current with Killing Vectors Alternative version: ....................... 31

4.6.4 Given a Killing vector the Ricci scalar satisfies : ................................................ 314.7 The Riemann tensor ........................................................................................................................ 31

4.7.1 Independent elements in the Riemann, Ricci and Weyl tensor .............................................. 33

4.7.2 Compute the components of the Riemann tensor for the unit 2-sphere ............................... 34

4.8 Show that the Ricci scalar for the unit 2-sphere .................................................................. 35

4.9 Proof: if a space is conformally flat, i.e. the Weyl tensor vanishes ..................... 35

4.10 The three dimensional flat space in spherical polar coordinates.................................................... 39

4.10.1 Calculate the Christoffel symbols of the three dimensional flat space in spherical polarcoordinates .............................................................................................................................................. 39

4.10.2 The Riemann tensor of the three dimensional flat space in spherical polar coordinates ...... 40

4.10.3 A Lie derivative in the three dimensional flat space in spherical polar coordinates ............... 40

4.11 The Ricci scalar of the Penrose Kahn metric ................................................................................... 41

4.12 A metric example 1: .............................................................. 42

4.12.1 The Christoffel symbols of a metric example .......................................................................... 42


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4.12.2 The Ricci scalar of a metric example ....................................................................................... 42

4.13 Calculate the Christoffel symbols for a metric example 2: ..................................................................................................................................... 43

4.14 A metric example 3: ...................................... 444.14.1 Calculate the Christoffel symbols for a metric example.......................................................... 44

4.14.2 Calculate the Riemann tensor of metric example ................................................................... 45

4.14.3 Calculate the Riemann tensor of metric example Alternative version ................................ 46

5 Cartans Structure Equations ................................................................................................................... 46

5.1 Ricci rotation coefficients for the Tolman-Bondi- de Sitter metric (Spherical dust with acosmological constant) ................................................................................................................................ 46

5.2 The curvature two forms and the Riemann tensor ......................................................................... 49

5.3 Find the Ricci scalar using Cartans str ucture equations of the 2-sphere ....................................... 49

5.4 The three dimensional flat space in spherical polar coordinates.................................................... 50

5.4.1 Ricci rotation coefficients of the three dimensional flat space in spherical polar coordinates 50

5.4.2 Transformation of the Ricci rotation coefficients into the Christoffel symbols of the three dimensional flat space in spherical polar coordinates .......................................... 515.5 Ricci rotation coefficients of the Rindler metric .............................................................................. 52

5.6 The Einstein tensor for the Tolman-Bondi- de Sitter metric ........................................................... 52

5.7 Calculate the Ricci rotation coefficients for a metric example 3: ..................................................................................................................................... 56

6 The Einstein Field Equations .................................................................................................................... 58

6.1 The vacuum Einstein equations ....................................................................................................... 58

6.2 The vacuum Einstein equations with a cosmological constant ....................................................... 58

6.3 General remarks on the Einstein equations with a cosmological constant .................................... 59

6.4 2+1 dimensions: Gravitational collapse of an inhomogeneous spherically symmetric dust cloud. 60

6.4.1 Find the components of the curvature tensor for the metric in 2+1 dimensions using Cartansstructure equations ................................................................................................................................. 60

6.4.2 Find the components of the curvature tensor fo r the metric in 2+1 dimensions using Cartansstructure equations alternative solution .............................................................................................. 62

6.4.3 Find the components of the Einstein tensor in the coordinate basis for the metric in 2+1dimensions............................................................................................................................................... 63

6.4.4 The Einstein equations of the metric in 2+1 dimensions. ....................................................... 65

6.5 Using the contracted Bianchi identities, prove that: .................................................. 65


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6.6 Ricci rotation coefficients, Ricci scalar and Einstein equations for a general 4-dimensional metric:, ........................................................................... 667 The Energy-Momentum Tensor .............................................................................................................. 71

7.1 Perfect Fluids Alternative derivation ............................................................................................ 717.2 The Gdel metric ............................................................................................................................. 72

8 Null Tetrads and the Petrov Classification............................................................................................... 75

8.1 Construct a null tetrad for the flat space Minkowski metric........................................................... 75

8.2 The Brinkmann metric (Plane gravitational waves) ........................................................................ 77

9 The Schwarzschild Solution ..................................................................................................................... 87

9.1 The Riemann and Ricci tensor of the general Schwarzschild metric ............................................... 87

9.2 The Riemann tensor of the Schwarzschild metric ........................................................................... 90

9.3 Calculation of the scalar in the Schwarzschild metric ............................................ 91

9.4 Geodesics in the Schwarzschild Spacetime ..................................................................................... 91

9.5 The meaning of the integration constant: The choice of ......................................................... 92

9.6 Time Delay ....................................................................................................................................... 93

9.7 Use the geodesic equations to find the Christoffel symbols for the general Schwarzschild metric. 95

9.8 The Ricci tensor for the general time dependent Schwarzschild metric. ........................................ 97

9.9 The Schwarzschild metric with nonzero cosmological constant. .................................................. 101

9.9.1 The Ricci rotation coefficients and Ricci tensor for the Schwarzschild metric with nonzerocosmological constant. .......................................................................................................................... 101

9.9.2 The general Schwarzschild metric in vacuum with a cosmological constant: The Ricci scalar 102

9.9.3 The general Schwarzschild metric in vacuum with a cosmological constant: Integrationconstants 103

9.9.4 The general Schwarzschild metric in vacuum with a cosmological constant: The spatial part of

the line element. ................................................................................................................................... 1049.9.5 The effect of the cosmological constant over the scale of the solar system ........................ 105

9.10 The Petrov type of the Schwarzschild spacetime .......................................................................... 106

9.11 The deflection of a light ray in a Schwarzschild metric with two different masses ...................... 113

9.12 The non-zero Weyl scalars of the Reissner-Nordstrm spacetime ............................................... 113

10 Black Holes ......................................................................................................................................... 123

10.1 The Path of a Radially Infalling Particle ......................................................................................... 123

10.2 The Schwarzschild metric in Kruskal Coordinates. ........................................................................ 126


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10.3 The Kerr metric .............................................................................................................................. 129

10.3.1 The Kerr-Newman geometry ................................................................................................. 129

10.3.2 The inverse metric of the Kerr Spinning Black Hole .............................................................. 131

11 Cosmology ......................................................................................................................................... 133

11.1 Light travelling in the Universe ...................................................................................................... 133

11.2 Spaces of Positive, Negative, and Zero Curvature ......................................................................... 133

11.3 The Robertson-Walker metric ....................................................................................................... 135

11.3.1 Find the components of the Riemann tensor of the Robertson-Walker metric (Homogenous,isotropic and expanding universe) using Cartans structure equations ................................................ 135

11.3.2 The Einstein tensor and Friedmann-equations for the Robertson Walker metric ................ 137

11.3.3 The Einstein tensor for the Robertson Walker metric Alternative version. ....................... 139

11.4 Manipulating the Friedmann equations. ....................................................................................... 140

11.5 Parameters in an flat universe with positive cosmological constant: Starting with use a change of variables ........................................................................................ 141

12 Gravitational Waves .......................................................................................................................... 142

12.1 Gauge transformation - The Einstein Gauge ................................................................................. 142

12.2 Plane waves ................................................................................................................................... 144

12.2.1 The Riemann tensor of a plane wave .................................................................................... 144

12.2.2 The line element of a plane wave in the Einstein gauge ....................................................... 148

12.2.3 The line element of a plane wave.......................................................................................... 149

12.2.4 The Rosen line element ......................................................................................................... 150

12.3 Colliding gravity waves - coordinate transformation .................................................................... 153

12.4 The delta and heavy-side functions: prove that .................................... 15512.5 Impulsive gravitational wave Region III ......................................................................................... 156

12.6 Two interacting waves ................................................................................................................... 161

12.7 The Nariai spacetime ..................................................................................................................... 16612.8 Collision of a gravitational wave with an electromagnetic wave The non-zero spin coefficients 180

12.9 The Aichelburg-Sexl Solution The passing of a black hole .......................................................... 184

12.10 Observations: The Future Gravitational Wave detectors. ......................................................... 185

Bibliografi ....................................................................................................................................................... 186


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1 IntroduktionWorking with GR means working with differential equations at four different levels. It can be very useful -whenever one comes across a GR calculation - to keep in mind, on which level you are working. The fourlevels of differential equations are:

1. The metric or line-element:

Example: Gravitational red shift 1:

12 Light emitted upward in a gravitational field, from an observer located at some inner radius to an ob-server positioned at some outer radius

12 12

2. Killings equations are conservation equations:

0 If you move along the direction of a Killing vector, then the metric does not change. This leads to conservedquantities: A free particle moving in a direction where the metric does not change will not fell any forces.

If is a Killing vector, , , , is the particle four velocity and is the particle four impulse,then and along a geodesic 2.Translational symmetry: Whenever 0 for some fixed (but for all and ) there will be a sym-metry under translation along 3.Example: Killing vectors in the Schwarzschild metric 4.

The Killing vector that corresponds to the independence of the metric of is

1,0,0,0 and of is

0,0,0,1. The conserved energy per unit rest mass: 1 1. The conserved angular momentum per unit rest mass 1sin for 3. The Geodesic equation leads to equations of motion: 12

0

Example: Planetary orbits 5 Manipulating the geodesic equations of the Schwarzschild metric leads to the following equation

12 2 Which can be interpreted in terms of elliptic functions, , and h and k are constants of integration.1 (McMahon, p. 234)2 (McMahon, p. 168)3

Carroll,s.134

4

(McMahon, p. 220)5 (A.S.Eddington, pp. 85-86)


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4. The Einstein equations are equations describing the spacetime.

12 8

In case of a cosmological constant 6: If 4, has twenty independent component ten of which are given by and the remainingten by the Weyl tensor 7.

Example: The Friedmann equationsA homogenous, isotropic and expanding universe described by the Robertson-Walker metric 8, in this casethe Einstein equations becomes the Friedmann equations:

8 3 8

21

2 Special relativity

2.1 9How to calculate : ExampleIn flat space calculate for the following pair of events: 1,3,2,4 and 4,0, 1, (1.11)1 43 02 14 1

5333

2

2.2 10 The four velocity and the four force: The four velocity , , , , the four impulse , the four force

. Because we can calculate

12 12 12 is an invariant and 0 3 Metric and Vector Transformations.

3.1 11 Flat Minkowski space:Flat Minkowski spacetime is the mathematical setting in which Einsteins special theory of relativity is mostconveniently formulated. In Cartesian coordinates with 1 the line element is

6 (McMahon, p. 138)7 (d'Inverno, p. 87)8 (McMahon, p. 161)9 (McMahon, 2006, p. 323), final exam 1. The answer to FE-1 is (c)10

(McMahon, 2006, p. 324), final exam 4, and the answer to FE-4 is (a)11 (McMahon, p. 186)


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and the metric

1 0 0 00 1 0 00 0 1 00 0 0 1

0, space-like, outside the light cone.3.2 Other realizations of the flat space:

3.2.1 12 Spherical polar coordinatesThe spherical part of the metric can be transformed into spherical polar coordinates by

sin cos (7.2) sin sin

cos

sin cos cos co sin sin cos sin cos sin (7.3) sin cos cos cos sincos2 sin cos cos2 sinsin coscoscos2sin cos sinsinsin sin sin cos sin sinsin2 sin cos sin2 sinsin cos

cossin2cos sin cossincos cos sin cos2 cos sinsin sincos2 sin cos cos2 sinsin coscoscos2sin cos sinsinsinsinsin2 sin co2 sinsin coscossin2cos sin cos sinsincoscos2 cos sinsin

sincos2 sin cos coscoscossinsinsinsin2 sin cocossinsincoscos2 cos sinsin sincossinsincos2 sin cos cos2 cos sincoscoscossinsinsinsinsincos sincos sincos2 sin coscos sin2 cos sincoscos sinsinsinsin cos 12 (Hartle, 2003, p. 135)


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sin cos2 sin cos 2 ccos sinsin sin The transformed line element is

sin (7.4)

3.2.2 13 Flat space with a singularityLook at the line element of the two-dimensional plane in polar coordinates 0 (7.6)and make the transformation, for some constant

(7.7)

This line element blows up at 0. Not because something physically interesting happens here, butsimply because the coordinate transformation has mapped all the points at into 0.14We can show that that the distance between 0 and a point with any finite value of is infinite,which corresponds to the distance between some finite value of and :

11 1 0

3.2.3 15 Coordinate transformations

The following line element corresponds to flat spacetime 2 with the metric 1 1 0 01 0 0 00 0 1 00 0 0 1

Find a coordinate transformation that puts the line element in the usual flat space form

13 (Hartle, 2003, p. 136)14

(Hartle, 2003, p. 163), problem 7.115 (Hartle, 2003, p. 164), problem 7.2


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We want to find the matrix, that transforms into we have

1 1 0 01 0 0 00 0 1 00 0 0 11 0 0 00 1 0 00 0 1 00 0 0 1

1 1 0 01 0 0 00 0 1 00 0 0 1

2 2 2 2 2 We check

1 1 0 01 0 0 00 0 1 00 0 0 11 0 0 00 1 0 00 0 1 00 0 0 1

1 1 0 01 0 0 00 0 1 00 0 0 1

3.2.4 16 Flat space in two dimensionsYet another realization of flat space in two dimensions is the line element

(7.20)This can be found from the coordinate transformation

sinh cosh sinh cosh cosh sinh sinh coshcosh sinh sinhcosh 2 sinhcosh cosinh 2 coshsinh 16 (Hartle, 2003, p. 143), example 7.3


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cosh sinhcoshsinh 3.2.5 17 The Penrose Diagram for Flat SpaceA Penrose diagram is a method to map the infinite coordinates such as

, with the range

2

2 1cosh4 (11.5) 2 1sinh4 where

21 (11.8)We calculate

14 2 1sinh44 14 2 1cosh44 14 2 1cosh4 12 12 2 1cosh4 4 112

14

2

1sinh4

12

12 2 1

sinh44

112

Now we can use the chain rule

Written as a matrix

147 (McMahon, 2006, p. 242)


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With the inverse 148

1 1

4 112 4 112{4 112 4 1124 4 }

4 12 12

4

16 2

4 12 16 12 2 Next we find

12 12 16 2

12

12 16 2

Inserting into the Schwarzschild metric

12 12 sin 12 16 sin 12 16 sin 1612 2 1

sin

32 sin (11.7)The Kruskal coordinates


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12cosh4 where

2 1 (11.8)

We calculate

14 12cosh44 14 12sinh44 14 12sinh4 12 12 12sinh44 12 1

14 12cosh4 12 12 12cosh4 4 12 1

As before we find

1

14 12 14 12 1{

4 12 14 12 14 4 }

4 2 12 1 4 16 2 4 2 1

162 1 2

Next we find

12 12 16 2 12 12 16 2 Inserting into the Schwarzschild metric we find as before

12

12

sin


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32 sin (11.7)10.3 The Kerr metric

10.3.1 149 The Kerr-Newman geometryA more general metric is the Kerr-Newman geometry, corresponding to a simultaneously rotating andelectrically charged black hole of mass , charge and angular momentum .

sinsin (33.2) sin 2 sinsin 2 1sin1 2 sin2sin 1 sinsin 1sin2 sin sinsin 1 2sin sin2 sin 2sinsin

sin 2sinsinsin

12 4 sin sinsin 2sinsin sin 2sin 12 4 sin sin sin 2sin 12 4 sin sinsin2 sin 12 4 sin 2 sin sin 2 cos

149 (C.W.Misner, 1973) chapter 33


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sin 2sin We look at three special cases:10.3.1.1 In the case of 0 we see immediately that the Kerr-Newman geometry reduces to the Kerr geometrydescribing a non-charged rotating black hole.

12 4 sin 2 sin sin 2

cos

sin

2sin 10.3.1.2 In the case of 0 we see immediately that the Kerr-Newman geometry reduces to the Reissner-Nordstrm geometry describing a charged non-rotating black hole.

12 4 sin 2 sin sin

12 sin 12 112 2 10.3.1.3 and In the case of 0 and 0 we see immediately that the Kerr-Newman geometry reduces to theSchwarzschild geometry describing a non-charged non-rotating black hole.

12 4 sin 2 sin sin 12 112 2


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10.3.2 150 The inverse metric of the Kerr Spinning Black HoleThe Kerr metric of a spinning black hole with mass and angular momentum .

12

4 sin

2 sin

sin (11.9)where

2 cos sin 2sin

the metric tensor

12 2 sin 2 sin 2 sin sin

with the inverse

{1 }

{1 sin 212 sinsin}

where we can calculate , , from the inverse 151 1

First we calculate the common factor

1

12 2 sin sin2 sin

12 2 sin 2sin1sin 2 sin 2 2 sin 2sin1sin 150 (McMahon, 2006, p. 246)

151


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1sin 2sin 2 sinsin

(11.13)

11 Cosmology

11.1 152 Light travelling in the UniverseLight travelling in the universe can be described by the line element , where and the are commoving coordinates. Light travel along null geodesic i.e. 0.We can now write for the total commoving distance light emitted at time can travel by time . Ifwe multiply this by the value of the scale factor at time , then we will have calculated the physicaldistance that the light has traveled in this time interval. This algorithm can be widely used to calculate howfar light can travel in any given time interval, revealing whether to points in space , for example are in causalcontact. As you can see, for accelerated expansion, even for arbitrarily large the integral is bounded, show-ing that the light will never reach arbitrarily distant commoving locations. Thus, in a universe with acceleratedexpansion, there are locations with which we can never communicate.

11.2 153 Spaces of Positive, Negative, and Zero CurvatureAccording to (12.5) the spatial part of a homogenous, isotropic metric is

1 sin (12.5)rewriting it in a more general form

sin (12.6)we see that

1 1 and in order to identify the metric for the different -values we solve the latter differential equation.0:

and the metric sin >0: 1

1 1 1154 1 sin 152 (Greene, s. 516) note 10153 (McMahon, 2006, p. 262)154

sin (14.237) (Spiegel, 1990)


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1 sin

1 sin

1 sin

1sin

if 1 we get sin and the metric sin sin sin


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11.3 157 The Robertson-Walker metric

11.3.1 158 Find the components of the Riemann tensor of the Robertson-Walker metric (Homog-enous, isotropic and expanding universe) using Cartans structure equations

The metric: 1 sin The Basis one forms

1 1

1 1111

sin 1sin Cartans First Structure equation and the calculation of the curvature one forms:

0

1 1 1 sin sin s

sin1 sin sin1 1 sin cos11

sin

1 cot

The curvature one-forms summerized in a matrix

157

(McMahon, 2006, p. 161), example 7-2158 (McMahon, 2006, p. 116), example 5-3


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0

01 1

1 0 cot 1 cot 0

Where refers to the column and the row

Curvature two forms:

12 (5.27), (5.28) : 1 1 1 11 0

: 1 1 1 1 1

:

(

sin sin

sin sin 1 cot 1 cot

:

1 1

1


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1

: 1 1 sin 1sin 1 cos 1cot

1cot : cot cot sincos sin 1 1

Summarized in a matrix

0 0

0 0

Now we can read off the elements in the Riemann tensor in the non-coordinate basis

11.3.2 The Einstein tensor and Friedmann-equations for the Robertson Walker metric The Ricci scalar:


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3 8 (7.17)2 1 8 (8.18)can be manipulated into: 0

Rewriting (7.17):

8 3 8 3 a 8 3 a3 6 3 Rewriting (7.18):

8 21 8 21 21 3 6 3 3 8 6 3 3 Now adding8 8 3 6 36 3 3

0 Q.E.D.

11.5 160 Parameters in an flat universe with positive cosmological constant: Start-ing with use a change of variables We have 3

3 and 23

2 3 3 Rearranging we get13 233

160 (McMahon, 2006, p. 278), quiz 12-6. The answer to quiz 12-6 is (a)


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3 23 3

3 2

3 2

161 1 32ln 2 1 3ln 2 1 3ln2 2 2 2 1 3ln 2 2 2 2 1 3ln 2 2 2 11

1 3ln2 1 111 3ln2 ln 1 11 162 1 3ln2 cosh 1 cosh 3 ln21 3C

2[cosh 3 ln21]

Leaving out the constants of integration

3ln2 we get

3C2[cosh 31] 12 Gravitational Waves

12.1 163 Gauge transformation - The Einstein GaugeRequiring that , and are unchanged under a gauge-transformation of first order in , showthat this is fulfilled by the coordinate transformations

(13.11)

, ,

, , ,

where is a function of position and ,1. We have 12 (13.4) 12 (13.5)

161 + + ln (14.280) (Spiegel, 1990)162

cosh ln 1 (8.56) (Spiegel, 1990)163 (McMahon, 2006, p. 286)


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(13.6)

12

(13.8)

The Einstein gauge transformation is a coordinate transformation that leaves , and unchanged.

The coordinate transformation that will do this is (13.11)In order to show this you only have to convince yourself that the line element is unchanged. Checking

, ,

, , , , , , , , , , , , Renaming the dummy variables , ,

, , if , , Q.E.DNext we are going to investigate the transformation of the derivative of the trace reverse , , , , ,

,12

,

,12 ,

,12 , , , , 12 , , , ,12 , , ,12 , , ,12 , , ,12 , ,


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,12, ,12

, , , ,12 , , , , 12 , , , , 12 , , Renaming the dummy variables

, , 12 , , ,

, Q.E.D.

P.288: The choice of , 0 leads to

164

, , ,12, ,12, ,12, 0

12.2 Plane waves

12.2.1 165 The Riemann tensor of a plane waveHere we want to show that the Riemann tensor only depends on

,, and. For symmetryreasons it is only necessary to show that the Riemann tensor does not depend on and.The Riemanntensor 12 (13.4)For plane waves we have

0 We also need

and

164

However I dont know how t o show that the Riemann-tensor keeps the same form if we make this choice165 (McMahon, 2006, pp. 288,13)


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12 12 : 12 12

: 12 12 0 : 12 12

:

12 0

: 12 0 : 12 0 The dependence on , :

12 12

: 0 : 0 : 12 , :

12 12

: 12 0 : 12 12 : 12 1

2

:


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12 12 0 , : 12 12

: 12 0 : 12 12 : 12 1

2

:

12 12 0 , : 12 12 The nonzero calculated elements of the Riemann tensor, from which we can conclude that the Rie-

mann tensor only depends on

,, and

.

12 12 12 12 12 12 12 12 12 12 12 12 12

12

12.2.2 166 The line element of a plane wave in the Einstein gaugeThe perturbation

( 12 12 )

(13.16)

the perturbation in the Einstein gauge

166 (McMahon, 2006, pp. 290,12)


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1 1 1

Now we are ready to calculate the curvature two-forms

0 0 1 1

0

1 1 0 1 1 1 1 0

0

Summarized in a matrix:

0 01 1 0 01 1 0 00 0

Where refers to column and to row

Now we can write down the independent elements of the Riemann tensor in the non-coordinate basis:

R 1 R 1 R 1 R 1 R 1 R 1 The Ricci tensor:


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0

21

2 21 21 4 1 21 0 1 21

0 21 Collecting the results0

0

1 1

0 21 0 21 We can now find the Christoffel symbols:

1 1 1 1 The Petrov type

The line element 21 1 The metric tensor:

11 1 1

and its inverse:

11 11 11

The basis one forms

Finding the basis one forms is not so obvious, we write:

21 1


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0 cos sin cosh sinh :

0

2

2

0 : 0

2cos

4 cos sin 2c

0

2 cos sin co

0 2 tan :

0

2cosh

4 cosh sinh 0 2 cosh sinh

0

2 tanh

Collecting the results0 0 cos sin cosh sinh 0 2 tan 0 2 tanh We can now find the Christoffel symbols:

cos sin

cosh sinh

tan

tan

tanh tanh The Petrov type

The line element 2 coscosh The metric tensor: 11

coscosh


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0 1cos1 2cos1 21cos

1cosh1 2 cosh1 21cosh

Collecting the results 1, 0 , 0 , 0 0 ,1 ,0 ,0 0 ,1 ,0 ,0 1, 0 , 0 , 0 1 20, 0, cos , cosh 1 20 ,0 ,1cos, 1cosh 1 20, 0, cos , cosh 1 20 ,0 ,1cos, 1cosh

The spin coefficients calculated from the orthonormal tetrad

12 (9.15)

12 12

12

Calculating the spin-coefficients 0 0 0 0 0

0

cos sin1 2cos

cosh sinh1 2cosh

1 2cosh

2tan tanh


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2 2 2

2tan tanh22tan tanh2tan tanh

21 tan 1 tanh2tan tanh 0 0 0 0 0

: This is a Petrov type N, which means there is a single principal null direction ( of multiplicity 4.

12.7 183 The Nariai spacetime The line element: 21 12 The metric tensor

and its inverse:

11 11

11

The Christoffel symbols

12 (4.15) (4.16) 12

1

2

12 12 12 12 12

183 (McMahon, 2006, p. 318), example 13-3


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12 Collecting the results we find the non-zero Christoffel symbols

The basis one forms

Finding the basis one forms is not so obvious, we write:

21

21 1 2 1 1

12 1

12 1 121 121 1 1 1 11

1

The orthonormal null tetrad

Now we can use the basis one-forms to construct a orthonormal null tetrad

1 21 1 0 01 1 0 00 0 10 0 1 1 2 1 2(

2111

1)

(9.10)

Written in terms of the coordinate basis


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0 0

0 0 0 0

0 0 12 12 12 0 12

12 12 12 12 121 22 2

1 2

12 12 12 12 12 [ ] [ ][ ]


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12 1 2 1 2 2

2

12 21 21 2 21 21 2 21 21 2 21 21 2 12 1 2 1 2 1 21 2 12 2 2 2 2 2 2 2 2 2 2

12 2 2 2 2 2 2 2 2

2 2 12 12 2 2 1 2 1 2 12 2 2 2 2 2 1 2 2 21 2 2 2

12 2 22 2 2 22 2 2 2 Collecting the results

0

0

0

0

0

1 2


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12 NP NP 0

2NP

12 (13.65)

NP 14 (13.64) 14 (13.65) Checking and

14 (9.22)

14 14 14 14 14 1

141 22 211 2 1 21 2 141 1 0 And we can conclude that Instead we will look at a generalized Nariai spacetime

The line element: Finding the basis one forms is not so obvious, we write:


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0 2 0 02 0 0 00 0 0 2 0 0 2 0

Where refers to column and to row

The curvature two forms:

12 (5.27), (5.28)

First we see that 0 for all combinations 2 2 2

2 11 11

4 4 2 2

24 24 2 4 2 4 4 4 4 4


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4 4 Summarized in a matrix:

0 4 0 04 0 0 00 0 040 0 4 0

Now we can write down the independent elements of the Riemann tensor in the non-coordinate basis:

R 4 R 4 R 4 R 4 The Ricci tensor:

(4.46) 4 0 0 0 4 0

0

4

0 4

Summarized in a matrix:


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0 0 00 0 00 0 00 0 0

0 0 00 0 00 0 00 0 0

Where refers to column and to rowCompared with 184 we can see the that we can choose the coefficients are 1;2; 1, 1 and , which corresponds to a Nariai line element consistent with

21 14 So let s copy the Christoffel, spin coefficient and Newman-Penrose identity calculations with this new 1

. The null tetrad is unchanged.

The Christoffel symbols

12 (4.15) (4.16) 12

1

2

12 2 2 12 2 2 12 2 2 12 2 2 12 2 2

12 2

2

Collecting the results we find the non-zero Christoffel symbols 2 2 The spin coefficients calculated from the orthonormal tetrad

184 Page 138


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121 22 2 1 2

12 12 12 12 12 12

1 2

1 2 2 2 12 2 221 221 22 221 221 2

2 22

1 2

2

1 22 221 221 2 12 21 2 21 2 21 221 2 12 2 2 2 2 2 2 2 2 2 2 2

2 2 2

12 4 2 4 2 4 2 4 2 4 2 12 12 21 2 21 2 21 221 2


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12 2 2 2 2 2 2 2 21 2 2 221 2 2 2

12 4 24 2 4 24 2 4 2 Collecting the results

0 0 0

0

0

1 2

0 0 4 2 0 0 4 2 Newman-Penrose identities

NP (13.58)

2 NP (13.59)

2NP (13.60)Where (9.13)Reduces to NP 2 NP 0 2NP These we can solve

NP 1 2

1 2

1 2

1 2

12

NP 2

4 2 4 2 4 2 4 2 24 2 4 2 4 2 4 2 32 32 232 4 2 4 2

4 2

4 2


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2

2

0

2

: 2 cos sin

2

2 0 cos sin :

0

2cos

4 cos sin

0 2 cos sin 0 2 tan : 0

2cos

4 cos sin

0 2 cos sin 0 2 tan Collecting the results0 0 cos sin 0 2 tan 0

2 tan

We can now find the Christoffel symbols: cos sin cos sin tan tan tan tan The basis one forms:

2 cos


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2 2 cos cos

1 2 1 2 1 111

1 2 1 2 cos 1cos cos 1cos

The orthonormal null tetrad:Now we can use the basis one-forms to construct a orthonormal null tetrad

1 21 1 0 01 1 0 00 0 10 0 1 1 2 1 2( 2 2coscos )

(9.10)

Written in terms of the coordinate basis 1, 0 , 0 , 0 0 ,1 ,0 ,0 1 20, 0, cos , cos 1

20, 0, cos ,

Next we use the metric to rise the indices 10 0 11 1 0 11 1 10 0 0 0 1cos1 2cos1 21cos

1cos1 2 cos1 21cos

Collecting the results: 1, 0 , 0 , 0 0 ,1 ,0 ,0 0 ,1 ,0 ,0 1, 0 , 0 , 0 1 20, 0, cos , cos 1 20 ,0 ,1cos, 1cos 1 20, 0, cos , cos 1 20 ,0 ,1cos, 1cos

The spin coefficients calculated from the orthonormal tetrad:


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12 (9.15)

12

12

12 Calculating the spin-coefficients

0 0

0

0 0 0

cos sin1 2cos cos sin1 2cos1 2cos tan cos sin1 2cos cos sin1 2cos

0

12

12 12 12 12 1 2cos 1 2cosh

12 1 2sin tan1 2cos 1 2sin tan1 2cos


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0 12 12

12 0 12 12 12 0 12 0

Collecting the results 0 0 0 0 0 0 0 tan 0 0 0 0 This means that 0 and there is expansion (or pure focusing=divergence).

12.9 186 The Aichelburg-Sexl Solution The passing of a black holeThe line element

4 log2

Comparing with the Brinkmann metric

, ,2 We see that we can copy the results from the Brinkmann calculations p.195 if , ,4 log

The only non-zero spin-coefficient is:

12 2 (9.30)12 24 log4 log2 2 42 22 2

2 2

186 (McMahon, 2006, p. 322), quiz 13-2. The answer to quiz 13-2 is (b)


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12.10 Observations: The Future Gravitational Wave detectors 187 . GWs are faint deformations of the space-time geometry,propagating at the speed of light and generated by cata-strophic events in the Universe, in which strong gravita-

tional fields and sudden acceleration of asymmetric distri-bution of large masses are involved. GWs have a quadrupo-lar nature and have two polarizations, + and, where is the so called space-time strain | | , the rela-tive dimensional 8distortion of an extended mass distribu-tion. The effect of these two polarizations on a circular massdistribution is shown in the figure. GWs are created by ac-celerating masses, but because gravity is the weakest of thefour fundamental forces, GWs are extremely small. For this reason, only extremely massive and compactobjects having intense and asymmetric gravitational fields, like neutron star and black hole binary systems,are expected to be able to generate detectable GW emission. The direct detection of GWs is still missingand it is quite easy to understand why. For example, the expected amplitude on Earth of the GW emittedby a coalescing binary system of neutron star located in the Virgo cluster is of the order of ~10. Thismeans that a detector having a dimension of a meter experiences an oscillating deformation of 10 ,an astonishingly small quantity. In the 1960s, the first GW detectors were based on a (multi)-ton resonantbar, that should resonate when excited by the passage of a GW. These detectors evolved, operating atcryogenic temperature to minimize the disturbance of the thermal Brownian vibration and being read byvery low noise transducers. These detectors reached a sensitivity of the order of a few 10 aroundthe main resonant mode frequency, which is of the order of one kHz. Although two of these detectors arestill operating, it is worth stating that their era has ended due to the realization of a new kind of GW de-tector: giant interferometers, operating since the first years of the 2000 decade. These instruments profitfrom two key elements of the GW; (i) the tidal nature of a GW: the expected metric deformation of abody traversed by a GW is proportional to its size ~. Hence, if the expected space-time defor-mation is of the order of 10, the effect on a multi-km detector will be a deformation ~1010 . (ii) the quadrupolar nature of the GW. A Michelson interferometer is sensitive to the differencein optical path length between its two arms, and it can match the metric deformation imposed by the GW.The first operative GW interferometric detector has been the Japanese TAMA, a 300m Michelson interfer-ometer that opened the path to this new family of instruments, but had a sensitivity limited by its reducedlength and by its location, in the center of Tokyo, affected by too high environmental disturbance. In Eu-rope two interferometric GW detectors have been realized; GEO600, a 600 m Michelson interferometer,built close to Hannover and Virgo, a Michelson interferometer having Fabry-Perot resonating cavities in-serted in the 3 km long arms, built close to Pisa. The longest interferometric GW detectors in the Worldare the two Laser Interferometer Gravitational wave Observatory (LIGO) detectors, having 4 km longarms, realized in Louisiana and Washington State (USA) with a topology similar to Virgo. Thanks to the longFabry-Perot cavities in the arms of the Virgo and LIGO detectors, the photons are forced to bounce back-and forth between the suspended mirrors, thus squeezing a hundreds km long optical path in the few kmlong detector infrastructure, increasing the sensitivity to the space-time deformation. The length limita-tion, dictated by technical constrains and affecting the terrestrial GW detector infrastructures, will obvi-ously disappear in the space-based observatories, like the eLISA/NGO detector. This project of a multi-million km GW interferometer is to be launched at the end of the 2020 decade, in a heliocentric orbit, andis devoted to the observation of ultra-low-frequency sources 10 10 , like hypermassive black

187 This is an extract of the article: Opening a New Window on the Universe The Future Gravitational Wave detectors

http://www.europhysicsnews.org/articles/epn/abs/2013/02/epn2013442p16/epn2013442p16.html (Michele Punturo- 2013).
http://physicssusan.mono.net/9035/General%20Relativityhttp://physicssusan.mono.net/9035/General%20Relativityhttp://www.europhysicsnews.org/articles/epn/abs/2013/02/epn2013442p16/epn2013442p16.htmlhttp://www.europhysicsnews.org/articles/epn/abs/2013/02/epn2013442p16/epn2013442p16.htmlhttp://www.europhysicsnews.org/articles/epn/abs/2013/02/epn2013442p16/epn2013442p16.htmlhttp://physicssusan.mono.net/9035/General%20Relativity


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holes. Virgo, GEO600 and LIGO detectors operated in a network during the second half of the 2000 decade,listeningto the sky in the 10 -10000 Hz frequency range. Even though, no detection of GW signal has beenobtained so far but, relevant scientific targets have been reached, putting constraints on potential GWemission by some astrophysical source. For example, thanks to the joint LIGO and Virgo data, an upper

limit has been set to the possible GW emission of the Vela and Crab pulsars. These pulsars, remnants ofsupernovae explosions, are compact neutron stars rotating about 11 and 30 times per second, respec-tively. The pulsars are expected to emit GWs at a frequency double their rotation rate, and at an amplitudedepending on many (unknown) parameters characterizing these stars. Through the radio signal, it is wellknown that these pulsars are slowing down because of emission of energy, due to several possible mech-anisms. LIGO and Virgo have been able to set an upper limit to the fraction of that energy emission due toGWs, stating that no more than few per cent of the energy loss can be due to GW radiation. The Virgo andLIGO detectors are currently offline, being upgraded toward the 2nd generation. In the period 2011-2015several parts of the detectors will be replaced to improve the sensitivity by a factor of ten. An improvementby a factor of ten in sensitivity corresponds to an increase by a factor of a thousand in detection rate: inone year of operation of the advanced detectors at the nominal sensitivity, about 40 coalescences of neu-

tron star binary systems are expected to be detected. The advanced detectors capability to detect a coa-lescence of a binary neutron star system at a distance of about 140 Mpc, and a coalescence of a binarysystems of black holes at a distance of about 1 Gpc, will open up a gravitational-wave astrophysics era. Itwill be possible, for example, to compare the signal detected from the coalescence of a binary system ofneutron stars with the general relativity prediction. Or it will be possible to investigate the nature of anisolated neutron star by looking at its GW emission. Few years later the completion of the Advanced Virgodetector in Europe and of the Advanced LIGO detectors in USA, new nodes will enter the network of GWobservatories: a very innovative 3km interferometer (KAGRA), underground and cryogenic, is under con-struction in Japan. Furthermore, a 3rd Advanced LIGO site is under evaluation in India. European scientistsare attempting to drive the evolution of this research field and the conceptual design of a 3rd generationGW observatory has been realized, able to compete and collaborate with the most sensitive optical tele-

scopes: the Einstein GW Telescope (ET). This new infrastructure, aimed to be operative in the 2020 decade,will test the cosmological model of the universe using GW signals, thanks to its capability to see manysources at large red-shift; ET will be a wonderful proofing tool of the general relativity predictions in allradiative processes involving intense gravitational fields, like in the presence of intermediate-mass blackholes ~10 1000. It will allow detailed investigations of the nature of isolated neutron starslooking both to the continuous emission of the pulsars and to the explosion of supernovae.

BibliografiA.S.Eddington. (1924). The Mathematical Theory of Relativity. Cambridge: At the University Press.

C.W.Misner, K. a. (1973). Gravitation. New York: W.H.Freeman and Company.

d'Inverno, R. (1992). Introducing Einstein's Relativity. Oxford: Clarendon Press.

Greene, B. (2004). The Fabric of the Cosmos. Penguin books.

Hartle, J. B. (2003). Gravity - An introduction to Einstein's General Relativity. Addison Wesley.

McMahon, D. (2006). Relativity Demystified. McGraw-Hill Companies, Inc.

Spiegel, M. R. (1990). SCHAUM'S OUTLINE SERIES: Mathematical Handbook of FORMULAS and TABLES. McGraw-Hill Publishing Company.


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calculations in general relativity

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