GENERAL RELATIVITY. MATH
b b
3443
KOMISSAROVbS.S
2009
,2
,Contents
Contents 2
1 Introduction 7
2 Fromb Euclideanb spaceb tob surfacesb andb metricb manifolds 13
2.1 Metricb form ................................................................................................................................ 13
2.1.1 Theb notionb ofb metricb form ............................................................................................. 13
2.1.2 Metricb formsb ofb surfaces: ................................................................................................. 15
2.1.3 Lengthsb ofb curves............................................................................................................. 16
2.1.4 Coordinateb transformations: ........................................................................................... 16
2.2 Vectors,b bases,b andb componentsb ofb vectors ............................................................................... 17
2.2.1 Coordinateb bases .............................................................................................................. 17
2.2.2 Coordinateb transformations ............................................................................................ 18
2.3 Metricb formb andb theb scalarb product .......................................................................................... 18
2.4 Geodesicsb andb theb variationalb principle ..................................................................................... 19
2.4.1 Euler-Lagrangeb Theorem ................................................................................................ 19
2.4.2 Geodesics ........................................................................................................................... 19
2.4.3 Examplesb ofb geodesics: .................................................................................................... 20
2.5 Non-Euclideanb geometryb ofb ab Euclideanb sphere ...................................................................... 22
2.6 Manifolds ........................................................................................................................................ 23
2.7 Vectorsb asb operators ..................................................................................................................... 24
2.7.1 Basicb idea.......................................................................................................................... 24
2.7.2 Coordinateb transformations ............................................................................................ 25
2.7.3 Magnitudesb ofb vectorsb andb theb scalarb product ............................................................ 25
3 Tensors 27
3.1 Tensorsb asb operators ..................................................................................................................... 27
3.1.1 1-formsb asb operatorsb actingb onb vectors ......................................................................... 27
3.1.2 Vectorsb asb operatorsb actingb onb 1-forms ........................................................................ 28
3.1.3 Tensorsb asb operatorsb actingb onb vectorsb andb 1-forms ................................................... 29
3.1.4 Metricb tensor .................................................................................................................... 29
3.1.5 Constructingb higherb rankb tensorsb viab outerb multiplicationb ofb vectorsb andb 1-formsb 30
3.2 Basesb andb componentsb ofb tensors ............................................................................................... 30
3.2.1 Inducedb basisb ofb 1-forms ................................................................................................. 30
3.2.2 Inducedb basesb ofb tensors ................................................................................................. 31
3.2.3 Indexb notationb ofb tensors ................................................................................................ 31
3.2.4 Coordinateb bases .............................................................................................................. 32
3.2.5 Coordinatebcomponentsbofb d˜f ....................................................................................... 32
3.2.6 Metricb formb andb metricb tensor ...................................................................................... 32
3.3 Basicb tensorb operationsb andb tensorb equations .......................................................................... 32
3.4 Basisb transformation .................................................................................................................... 34
3.4.1 Transformationb ofb inducedb bases .................................................................................. 34
3.4.2 Transformationb ofb components....................................................................................... 35
3
, 3.5 Theb operationsb ofb raisingb andb loweringb indexesb ofb tensors ..................................................... 35
3.6 Symmetricb andb antisymmetricb tensors ...................................................................................... 36
3.6.1 Symmetryb withb respectb tob ab pairb ofb indexes ................................................................ 36
3.6.2 Antisymmetryb withb respectb tob ab pairb ofb indexes ......................................................... 36
4 Geometryb ofb Riemannianb manifolds 37
4.1 Parallelb transportb andb Connectionb onb metricb manifolds ......................................................... 37
4.1.1 Parallelb transportb ofb vectors.b Connection ..................................................................... 37
4.1.2 Connectionb ofb Euclideanb space ...................................................................................... 38
4.1.3 Riemannianb Connection ................................................................................................... 39
4.2 Parallelb transportb ofb tensors ....................................................................................................... 39
4.2.1 Scalars ................................................................................................................................ 39
4.2.2 1-forms............................................................................................................................... 39
4.2.3 Generalb tensors ................................................................................................................ 39
4.2.4 Metricb tensor .................................................................................................................... 40
4.3 Absoluteb andb covariantb derivatives ............................................................................................. 40
4.3.1 Absoluteb andb covariantb derivativesb ofb vectorb fields ..................................................... 41
4.3.2 Absoluteb andb covariantb derivativesb ofb 1-formb fields .................................................... 42
4.3.3 Absoluteb andb covariantb derivativesb ofb generalb tensorb fields ....................................... 42
4.3.4 Absoluteb andb covariantb derivativesb ofb scalarb fields ..................................................... 43
4.3.5 Generalb propertiesb ofb covariantb differentiation ............................................................. 43
4.3.6 Theb fieldb ofb metricb tensor ............................................................................................... 43
4.4 Geodesicsb andb parallelb transport ................................................................................................. 43
4.5 Geodesicb coordinatesb andb Fermib coordinates ............................................................................ 44
4.5.1 Geodesicb coordinates ........................................................................................................ 44
4.5.2 Fermib coordinates ............................................................................................................. 46
4.6 Riemannb curvatureb tensor ........................................................................................................... 47
4.7 Propertiesb ofb theb Riemannb curvatureb tensor ............................................................................ 49
4.8 Riccib tensor,b curvatureb scalarb andb theb Einsteinb tensor ........................................................... 50
5 Spaceb andb timeb inb theb theoryb ofb relativity 51
5.1 Physicalb Spaceb andb Timeb inb Newtonianb Physics ...................................................................... 51
5.2 Physicalb Spaceb andb Timeb inb Specialb Relativity ......................................................................... 52
5.3 Relativisticb equationsb ofb motionb ofb particleb dynamics ............................................................. 54
5.4 Conservationb laws ......................................................................................................................... 55
5.5 Relativisticb continuityb equation.................................................................................................57
5.6 Stress-energy-momentumb tensor ................................................................................................. 57
5.6.1 Energy-momentumb vector............................................................................................... 57
5.6.2 Stress-energy-momentumb tensorb ofb dust ....................................................................... 58
5.6.3 Energy-momentumb conservation .................................................................................... 59
5.6.4 Stress-energy-momentumb tensorb ofb perfectb fluid ........................................................ 59
5.7 Spaceb andb Timeb inb Generalb Relativity ........................................................................................ 61
5.8 Einstein’sb equationsb ofb gravitationalb field ................................................................................. 62
5.9 Newtonianb limit ............................................................................................................................. 65
6 SchwarzschildbSolution 69
6.1 Schwarzschildb Solution.................................................................................................................. 69
6.1.1 Schwarzschildb Solutionb inb Schwarzschildb coordinates .................................................. 69
6.1.2 Schwarzschildb Solutionb inb Kerrb coordinates.................................................................. 71
6.1.3 Eventb horizon .................................................................................................................... 72
6.2 Gravitationalb b redshift ................................................................................................................... 73
6.3 Integralsb ofb motionb ofb freeb testb particlesb inb Schwarzschildb spacetime ................................... 74
6.4 Orbitsb ofb testb particlesb inb theb Schwarzschildb geometry ........................................................... 77
6.5 Perihelionb shiftb ofb planets ............................................................................................................ 80
CONTENTS 5
6.6 Bendingb ofb light............................................................................................................................. 82
b b
3443
KOMISSAROVbS.S
2009
,2
,Contents
Contents 2
1 Introduction 7
2 Fromb Euclideanb spaceb tob surfacesb andb metricb manifolds 13
2.1 Metricb form ................................................................................................................................ 13
2.1.1 Theb notionb ofb metricb form ............................................................................................. 13
2.1.2 Metricb formsb ofb surfaces: ................................................................................................. 15
2.1.3 Lengthsb ofb curves............................................................................................................. 16
2.1.4 Coordinateb transformations: ........................................................................................... 16
2.2 Vectors,b bases,b andb componentsb ofb vectors ............................................................................... 17
2.2.1 Coordinateb bases .............................................................................................................. 17
2.2.2 Coordinateb transformations ............................................................................................ 18
2.3 Metricb formb andb theb scalarb product .......................................................................................... 18
2.4 Geodesicsb andb theb variationalb principle ..................................................................................... 19
2.4.1 Euler-Lagrangeb Theorem ................................................................................................ 19
2.4.2 Geodesics ........................................................................................................................... 19
2.4.3 Examplesb ofb geodesics: .................................................................................................... 20
2.5 Non-Euclideanb geometryb ofb ab Euclideanb sphere ...................................................................... 22
2.6 Manifolds ........................................................................................................................................ 23
2.7 Vectorsb asb operators ..................................................................................................................... 24
2.7.1 Basicb idea.......................................................................................................................... 24
2.7.2 Coordinateb transformations ............................................................................................ 25
2.7.3 Magnitudesb ofb vectorsb andb theb scalarb product ............................................................ 25
3 Tensors 27
3.1 Tensorsb asb operators ..................................................................................................................... 27
3.1.1 1-formsb asb operatorsb actingb onb vectors ......................................................................... 27
3.1.2 Vectorsb asb operatorsb actingb onb 1-forms ........................................................................ 28
3.1.3 Tensorsb asb operatorsb actingb onb vectorsb andb 1-forms ................................................... 29
3.1.4 Metricb tensor .................................................................................................................... 29
3.1.5 Constructingb higherb rankb tensorsb viab outerb multiplicationb ofb vectorsb andb 1-formsb 30
3.2 Basesb andb componentsb ofb tensors ............................................................................................... 30
3.2.1 Inducedb basisb ofb 1-forms ................................................................................................. 30
3.2.2 Inducedb basesb ofb tensors ................................................................................................. 31
3.2.3 Indexb notationb ofb tensors ................................................................................................ 31
3.2.4 Coordinateb bases .............................................................................................................. 32
3.2.5 Coordinatebcomponentsbofb d˜f ....................................................................................... 32
3.2.6 Metricb formb andb metricb tensor ...................................................................................... 32
3.3 Basicb tensorb operationsb andb tensorb equations .......................................................................... 32
3.4 Basisb transformation .................................................................................................................... 34
3.4.1 Transformationb ofb inducedb bases .................................................................................. 34
3.4.2 Transformationb ofb components....................................................................................... 35
3
, 3.5 Theb operationsb ofb raisingb andb loweringb indexesb ofb tensors ..................................................... 35
3.6 Symmetricb andb antisymmetricb tensors ...................................................................................... 36
3.6.1 Symmetryb withb respectb tob ab pairb ofb indexes ................................................................ 36
3.6.2 Antisymmetryb withb respectb tob ab pairb ofb indexes ......................................................... 36
4 Geometryb ofb Riemannianb manifolds 37
4.1 Parallelb transportb andb Connectionb onb metricb manifolds ......................................................... 37
4.1.1 Parallelb transportb ofb vectors.b Connection ..................................................................... 37
4.1.2 Connectionb ofb Euclideanb space ...................................................................................... 38
4.1.3 Riemannianb Connection ................................................................................................... 39
4.2 Parallelb transportb ofb tensors ....................................................................................................... 39
4.2.1 Scalars ................................................................................................................................ 39
4.2.2 1-forms............................................................................................................................... 39
4.2.3 Generalb tensors ................................................................................................................ 39
4.2.4 Metricb tensor .................................................................................................................... 40
4.3 Absoluteb andb covariantb derivatives ............................................................................................. 40
4.3.1 Absoluteb andb covariantb derivativesb ofb vectorb fields ..................................................... 41
4.3.2 Absoluteb andb covariantb derivativesb ofb 1-formb fields .................................................... 42
4.3.3 Absoluteb andb covariantb derivativesb ofb generalb tensorb fields ....................................... 42
4.3.4 Absoluteb andb covariantb derivativesb ofb scalarb fields ..................................................... 43
4.3.5 Generalb propertiesb ofb covariantb differentiation ............................................................. 43
4.3.6 Theb fieldb ofb metricb tensor ............................................................................................... 43
4.4 Geodesicsb andb parallelb transport ................................................................................................. 43
4.5 Geodesicb coordinatesb andb Fermib coordinates ............................................................................ 44
4.5.1 Geodesicb coordinates ........................................................................................................ 44
4.5.2 Fermib coordinates ............................................................................................................. 46
4.6 Riemannb curvatureb tensor ........................................................................................................... 47
4.7 Propertiesb ofb theb Riemannb curvatureb tensor ............................................................................ 49
4.8 Riccib tensor,b curvatureb scalarb andb theb Einsteinb tensor ........................................................... 50
5 Spaceb andb timeb inb theb theoryb ofb relativity 51
5.1 Physicalb Spaceb andb Timeb inb Newtonianb Physics ...................................................................... 51
5.2 Physicalb Spaceb andb Timeb inb Specialb Relativity ......................................................................... 52
5.3 Relativisticb equationsb ofb motionb ofb particleb dynamics ............................................................. 54
5.4 Conservationb laws ......................................................................................................................... 55
5.5 Relativisticb continuityb equation.................................................................................................57
5.6 Stress-energy-momentumb tensor ................................................................................................. 57
5.6.1 Energy-momentumb vector............................................................................................... 57
5.6.2 Stress-energy-momentumb tensorb ofb dust ....................................................................... 58
5.6.3 Energy-momentumb conservation .................................................................................... 59
5.6.4 Stress-energy-momentumb tensorb ofb perfectb fluid ........................................................ 59
5.7 Spaceb andb Timeb inb Generalb Relativity ........................................................................................ 61
5.8 Einstein’sb equationsb ofb gravitationalb field ................................................................................. 62
5.9 Newtonianb limit ............................................................................................................................. 65
6 SchwarzschildbSolution 69
6.1 Schwarzschildb Solution.................................................................................................................. 69
6.1.1 Schwarzschildb Solutionb inb Schwarzschildb coordinates .................................................. 69
6.1.2 Schwarzschildb Solutionb inb Kerrb coordinates.................................................................. 71
6.1.3 Eventb horizon .................................................................................................................... 72
6.2 Gravitationalb b redshift ................................................................................................................... 73
6.3 Integralsb ofb motionb ofb freeb testb particlesb inb Schwarzschildb spacetime ................................... 74
6.4 Orbitsb ofb testb particlesb inb theb Schwarzschildb geometry ........................................................... 77
6.5 Perihelionb shiftb ofb planets ............................................................................................................ 80
CONTENTS 5
6.6 Bendingb ofb light............................................................................................................................. 82
