Lots of Calculations in General Relativity Susan Larsen Tuesday, August 20, 2019 1 http://physicssusan.mono.net [email protected]Content 1 Introduction ............................................................................................................................................... 2 1.1 Space-times ....................................................................................................................................... 4 2 The Metric tensor and Vector Transformations. ....................................................................................... 6 3 Four vectors and four velocity ................................................................................................................... 6 4 Christoffel Symbols, Geodesic Equation and Killing Vectors ..................................................................... 6 5 Covariant Derivative, Lie Derivative and Killings Equation ........................................................................ 7 6 The Riemann tensor .................................................................................................................................. 7 7 Cartan’s Structure Equations – a Shortcut to the Einstein equation ........................................................ 7 8 The Einstein Field Equations ...................................................................................................................... 7 8.1 The vacuum Einstein equations......................................................................................................... 7 8.2 The vacuum Einstein equations with a cosmological constant ......................................................... 7 8.3 General remarks on the Einstein equations with a cosmological constant ...................................... 8 8.4 Using the contracted Bianchi identities, prove that: = .................................................... 9 8.5 2+1 dimensions: Gravitational collapse of an inhomogeneous spherically symmetric dust cloud. 10 8.5.1 Find the components of the curvature tensor for the metric in 2+1 dimensions using Cartan’s structure equations ................................................................................................................................. 10 8.5.2 Find the components of the curvature tensor for the metric in 2+1 dimensions using Cartan’s structure equations – alternative solution .............................................................................................. 11 8.5.3 Find the components of the Einstein tensor in the coordinate basis for the metric in 2+1 dimensions............................................................................................................................................... 12 8.5.4 The Einstein equations of the metric in 2+1 dimensions. ....................................................... 15 8.6 Ricci rotation coefficients, Ricci scalar and Einstein equations for a general 4-dimensional metric: = − + , + , + , ........................................................................... 15 8.7 The de Sitter Spacetime................................................................................................................... 20 8.7.1 The Einstein equations ............................................................................................................ 20 8.7.2 Solving the Einstein equations................................................................................................. 21 8.8 The Anti-de Sitter Spacetime ........................................................................................................... 21 9 The Energy-Momentum Tensor .............................................................................................................. 22 9.1 The Einstein equation with source .................................................................................................. 22 9.2 Perfect Fluids ................................................................................................................................... 23 9.3 More examples on stress-energy tensors ....................................................................................... 24 9.3.1 Pure Matter ............................................................................................................................. 24 9.3.2 More complicated fluids .......................................................................................................... 24 9.3.3 The electromagnetic field ........................................................................................................ 24 9.4 The Gödel metric ............................................................................................................................. 27
2 The Metric tensor and Vector Transformations. ....................................................................................... 6
3 Four vectors and four velocity ................................................................................................................... 6
4 Christoffel Symbols, Geodesic Equation and Killing Vectors ..................................................................... 6
5 Covariant Derivative, Lie Derivative and Killings Equation ........................................................................ 7
6 The Riemann tensor .................................................................................................................................. 7
7 Cartan’s Structure Equations – a Shortcut to the Einstein equation ........................................................ 7
8 The Einstein Field Equations ...................................................................................................................... 7
8.1 The vacuum Einstein equations ......................................................................................................... 7
8.2 The vacuum Einstein equations with a cosmological constant ......................................................... 7
8.3 General remarks on the Einstein equations with a cosmological constant ...................................... 8
8.4 Using the contracted Bianchi identities, prove that: 𝛁𝒃𝑮𝒂𝒃 = 𝟎 .................................................... 9
8.5 2+1 dimensions: Gravitational collapse of an inhomogeneous spherically symmetric dust cloud. 10
8.5.1 Find the components of the curvature tensor for the metric in 2+1 dimensions using Cartan’s structure equations ................................................................................................................................. 10
8.5.2 Find the components of the curvature tensor for the metric in 2+1 dimensions using Cartan’s structure equations – alternative solution .............................................................................................. 11
8.5.3 Find the components of the Einstein tensor in the coordinate basis for the metric in 2+1 dimensions............................................................................................................................................... 12
8.5.4 The Einstein equations of the metric in 2+1 dimensions. ....................................................... 15
8.6 Ricci rotation coefficients, Ricci scalar and Einstein equations for a general 4-dimensional metric: 𝒅𝒔𝟐 = −𝒅𝒕𝟐 + 𝑳𝟐𝒕, 𝒓𝒅𝒓𝟐 + 𝑩𝟐𝒕, 𝒓𝒅𝝓𝟐 +𝑴𝟐𝒕, 𝒓𝒅𝒛𝟐 ........................................................................... 15
8.7 The de Sitter Spacetime................................................................................................................... 20
8.7.1 The Einstein equations ............................................................................................................ 20
8.7.2 Solving the Einstein equations................................................................................................. 21
8.8 The Anti-de Sitter Spacetime ........................................................................................................... 21
9 The Energy-Momentum Tensor .............................................................................................................. 22
9.1 The Einstein equation with source .................................................................................................. 22
9.5 The Einstein Cylinder ....................................................................................................................... 30
9.5.1 The line element ...................................................................................................................... 30
9.5.2 The Ricci tensor ....................................................................................................................... 30
9.5.3 The Einstein Equations ............................................................................................................ 32
9.5.4 The Einstein tensor with a cosmological constant .................................................................. 33
9.6 The Newtonian Approximation – The right hand side! ................................................................... 33
10 Null Tetrads and the Petrov Classification........................................................................................... 35
10.1 Weyl scalars and Petrov classification ............................................................................................. 35
10.2 Construct a null tetrad for the flat space-time Minkowski metric .................................................. 36
11 The Schwarzschild Solution ................................................................................................................. 37
12 Black Holes ........................................................................................................................................... 37
General relativity is the description of the smallest changes and the largest entities. The best way to describe this I found in a quote by Roger Penrose: Calculus is built from two basic ingredients: differentiation and integration. Differentiation is concerned with
velocities, accelerations, the slopes and curvature of curves and surfaces… These are the rates at which things change, and they are quantities defined locally, in terms of structure or behavior in the tiniest neigh-borhood of single points. Integration on the other hand, is concerned with areas and volumes, with centers of gravity…These are things which involves measures of totality … and they are not defined merely by what
is going on in the local or infinitesimal neighborhoods of individual points. (Penrose, 2004, s. 103)
1 Introduction We begin with a few important facts:
- The gravitational force is always attractive. - The gravitational force is a long range force without boundaries. - A gravitational field is created from all kinds of masses and (because 𝐸 = 𝑚𝑐2) all kinds of ener-
gies. - A mass/energy creates a curvature of the four-dimensional space-time, where masses (test
Calculus: Working 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 four levels of differential equations are:
1. The metric or line-element: 𝑑𝑠2 = 𝑔𝑎𝑏𝑑𝑥
𝑎𝑑𝑥𝑏 Example: Gravitational red shifta:
𝑑𝜏 = √1 −2𝑚
𝑟𝑑𝑡
Light emitted upward in a gravitational field, from an observer located at some inner radius 𝑟1 to an ob-server positioned at some outer radius 𝑟2
𝛼 =√1 −
2𝑚𝑟2
√1 −2𝑚𝑟1
2. Killing’s equations are conservation equations: ∇𝑏𝑋𝑎 + ∇𝑎𝑋𝑏 = 0 If you move along the direction of a Killing vector, then the metric does not change. This leads to conserved quantities: A free particle moving in a direction where the metric does not change will not fell any forces.
If 𝑋 is a Killing vector,𝑢 = (𝑑𝑡
𝑑𝜏,𝑑𝑟
𝑑𝜏,𝑑𝜃
𝑑𝜏,𝑑𝜙
𝑑𝜏) is the particle four velocity and 𝑝 is the particle four impulse,
then 𝑋 ⋅ 𝑢 = 𝑔𝑎𝑏𝑋𝑎𝑢𝑏 = 𝑐𝑜𝑛𝑠𝑡 and 𝑋 ⋅ 𝑝 = 𝑔𝑎𝑏𝑋
𝑎𝑝𝑏 = 𝑐𝑜𝑛𝑠𝑡 along a geodesicb. Translational symmetry: Whenever 𝜕σ∗𝑔𝜇𝜈 = 0 for some fixed 𝜎∗ (but for all 𝜇 and 𝜈) there will be a sym-
metry under translation along 𝑥𝜎∗c. Example: Killing vectors in the Schwarzschild metricd. The Killing vector that corresponds to the independence of the metric of 𝑡 is 𝜉 = (1,0,0,0) and of 𝜙 is 휂 =
(0,0,0,1). The conserved energy per unit rest mass: 𝑒 = −𝜉 ⋅ 𝑢 = −𝑔𝑎𝑏𝜉𝑎𝑢𝑏 = −𝑔𝑡𝑡 ⋅ 1 ⋅
𝑑𝑡
𝑑𝜏= −(1 −
2𝑚
𝑟)𝑑𝑡
𝑑𝜏. The conserved angular momentum per unit rest mass 𝑙 = 휂 ⋅ 𝑢 = 𝑔𝑎𝑏휂
𝑎𝑢𝑏 = 𝑔𝜙𝜙 ⋅ 1 ⋅𝑑𝜙
𝑑𝜏=
−𝑟2 sin2 휃𝑑𝜙
𝑑𝜏= −𝑟2
𝑑𝜙
𝑑𝜏 for 휃 =
𝜋
2
3. The Geodesic equation leads to equations of motion:
𝐾 =1
2𝑔𝑎𝑏�̇�
𝑎�̇�𝑏
𝜕𝐾
𝜕𝑥𝑎 =
𝑑
𝑑𝑠(𝜕𝐾
𝜕�̇�𝑎)
𝑑2𝑥𝑎
𝑑𝑠2+ Γ 𝑏𝑐
𝑎𝑑𝑥𝑏
𝑑𝑠
𝑑𝑥𝑐
𝑑𝑠 = 0
Example: Planetary orbitse Manipulating the geodesic equations of the Schwarzschild metric leads to the following equation
(𝑑𝑢
𝑑𝜙)2
+ 𝑢2 =𝑘2 − 1
ℎ2+2𝑚
ℎ2𝑢 + 2𝑚𝑢3
Which can be interpreted in terms of elliptic functions, 𝑢 =1
𝑟, and h and k are constants of integration.
4. The Einstein equations are equations describing the spacetime.
𝐺𝑎𝑏 = 𝑅𝑎𝑏 −1
2𝑔𝑎𝑏𝑅
8𝜋𝐺𝑇𝑎𝑏 = 𝐺𝑎𝑏 ± 𝑔𝑎𝑏Λ If 𝑛 = 4, 𝑅𝑎𝑏𝑐𝑑 has twenty independent component – ten of which are given by 𝑅𝑎𝑏 and the remaining ten by the Weyl tensorf.
Example: The Friedmann equations A homogenous, isotropic and expanding universe described by the Robertson-Walker space-timeg, in this case the Einstein equations becomes the Friedmann equations:
8𝜋𝜌 =3
𝑎2(𝑘 + �̇�2) − Λ
−8𝜋𝑃 = 2
�̈�
𝑎+1
𝑎2(𝑘 + �̇�2) − Λ
1.1 Space-times This document includes many different space-time examples. In order to keep track of them I have made this list- in alphabetical order, so that you can see in which chapter you can find the space-time you are looking for.
5 Covariant Derivative, Lie Derivative and Killings Equation See separate document
6 The Riemann tensor See separate document
7 Cartan’s Structure Equations – a Shortcut to the Einstein equation See separate document
8 The Einstein Field Equations
8.1 hThe vacuum Einstein equations Prove that the Einstein field equations 𝐺𝑎𝑏 = 𝜅𝑇𝑎𝑏 reduces to the vacuum Einstein equations 𝑅𝑎𝑏 = 0 if we set 𝑇𝑎𝑏 = 0. The Einstein tensor
𝐺𝑎𝑏 = 𝑅𝑎𝑏 −1
2𝑔𝑎𝑏𝑅
If 𝐺𝑎𝑏 = 𝜅𝑇𝑎𝑏 = 0:
⇒ 𝐺𝑎𝑏 = 𝑅𝑎𝑏 −1
2𝑔𝑎𝑏𝑅 = 0
⇒ 𝑅𝑎𝑏 =1
2𝑔𝑎𝑏𝑅
Contracting with 𝑔𝑎𝑏
⇒ 𝑔𝑎𝑏𝑅𝑎𝑏 =1
2𝑔𝑎𝑏𝑔𝑎𝑏𝑅
using the definition 𝑅 = 𝑔𝑎𝑏𝑅𝑎𝑏
and that in 4 dimensions 𝑔𝑎𝑏𝑔𝑎𝑏 = 4
⇒ 𝑅 =1
24𝑅 = 2𝑅
Now this can only be true if 𝑅𝑎𝑏 = 0
8.2 The vacuum Einstein equations with a cosmological constant Prove that the Einstein field equations 𝐺𝑎𝑏 = 𝜅𝑇𝑎𝑏 reduces to 𝑅𝑎𝑏 = 𝑔𝑎𝑏Λ and 𝑅 = 4Λ for metrics with positive signature and 𝑅𝑎𝑏 = −𝑔𝑎𝑏Λ and 𝑅 = −4Λ for metrics with negative signature in vacuum with a cosmological constanti. The Einstein equation in vacuum with a cosmological constant and positive signature is
In the non coordinate basis 𝑅�̂��̂� = 휂�̂��̂�Λ In the case of metrics with negative signature the Einstein equation in vacuum with a cosmological con-stant
0 = 𝑅𝑎𝑏 −1
2𝑔𝑎𝑏𝑅 − 𝑔𝑎𝑏Λ
and we can see that 𝑅 = −4Λ Q.E.D. 𝑅𝑎𝑏 = −𝑔𝑎𝑏Λ Q.E.D. In the non coordinate basis 𝑅�̂��̂� = −휂�̂��̂�Λ
8.3 jGeneral remarks on the Einstein equations with a cosmological constant If we demand that the gravitational field equations are (1) generally covariant (2) be of second differential order in 𝑔𝑎𝑏 (3) involve the energy-momentum 𝑇𝑎𝑏 linearly it can be shown that the only equation which meets these requirements is 𝑅𝑎𝑏 + 𝜇𝑅𝑔𝑎𝑏 − Λ𝑔𝑎𝑏 = 𝜅𝑇𝑎𝑏 where 𝜇, Λ, and 𝜅 are constants. The demand that 𝑇𝑎𝑏 satisfies the conservation equation ∇𝑏𝑇
𝑎𝑏 = 0 leads to
𝜇 = −1
2
Proof: ∇𝑏𝑇
𝑎𝑏 = 0
⇒ ∇𝑏(𝑅𝑎𝑏 + 𝜇𝑅𝑔𝑎𝑏 − Λ𝑔𝑎𝑏) = 0
⇒ ∇𝑏𝑅𝑎𝑏 + 𝜇∇𝑏(𝑅𝑔
𝑎𝑏) − Λ∇𝑏𝑔𝑎𝑏 = 0
⇒ 1 ∇𝑏𝑅𝑎𝑏 + 𝜇 ((∇𝑏𝑅)𝑔
𝑎𝑏 + 𝑅(∇𝑏𝑔𝑎𝑏)) = 0
⇒ ∇𝑏𝑅𝑎𝑏 + 𝜇(∇𝑏𝑅)𝑔
𝑎𝑏 = 0 Next we use the Bianchi identity: ∇𝑎𝑅𝑑𝑒𝑏𝑐 + ∇𝑏𝑅𝑑𝑒𝑐𝑎 + ∇𝑐𝑅𝑑𝑒𝑎𝑏 = 0 ⇒ 𝑔𝑑𝑏(∇𝑎𝑅𝑑𝑒𝑏𝑐 + ∇𝑏𝑅𝑑𝑒𝑐𝑎 + ∇𝑐𝑅𝑑𝑒𝑎𝑏) = 0 ⇒ ∇𝑎𝑔
𝑎𝑏 This is a very important result because it leads to the conservation laws of the right hand side of the Ein-stein equation, which we will look into later. ∇𝑎𝑇
𝑎𝑏 = 0
8.5 42+1 dimensions: Gravitational collapse of an inhomogeneous spherically symmetric dust cloud.
8.5.1 Find the components of the curvature tensor for the metric in 2+1 dimensions using Cartan’s structure equations
The line element: 𝑑𝑠2 = −𝑑𝑡2 + 𝑒2𝑏(𝑡,𝑟)𝑑𝑟2 + 𝑅2(𝑡, 𝑟)𝑑𝜙2 The Basis one forms
𝜔�̂� = 𝑑𝑡
휂𝑖𝑗 = {−1
11
} 𝜔�̂� = 𝑒𝑏(𝑡,𝑟)𝑑𝑟 𝑑𝑟 = 𝑒−𝑏(𝑡,𝑟)𝜔�̂�
𝜔�̂� = 𝑅(𝑡, 𝑟)𝑑𝜙 𝑑𝜙 =1
𝑅(𝑡, 𝑟)𝜔�̂�
Cartan’s First Structure equation and the calculation of the curvature two-forms 𝑑𝜔�̂� = −Γ �̂�
Next we can use the former calculations of the Tolman-Bondi – de Sitter metric to find the Riemann and Einstein tensor for the 2+1 metric. But first we need to find �̇� =
𝑑𝜓(𝑡′, 𝑟′)
𝑑𝑡′= 𝑒−𝜓(𝑡
′,𝑟′)𝑑
𝑑𝑡′(𝑒𝜓(𝑡
′,𝑟′)) = 𝑒𝑏(𝑡,𝑟)𝑑𝑟
𝑑𝑟′𝑑
𝑑𝑡(𝑒−𝑏(𝑡,𝑟)
𝑑𝑟′
𝑑𝑟) = −
𝑑𝑏(𝑡, 𝑟)
𝑑𝑡= −�̇�(𝑡, 𝑟)
�̈� =
𝑑2𝜓(𝑡′, 𝑟′)
𝑑𝑡′2=𝑑
𝑑𝑡(−�̇�) = −�̈�(𝑡, 𝑟)
𝜓′ =
𝑑𝜓(𝑡′, 𝑟′)
𝑑𝑟′= 𝑒−𝜓(𝑡
′,𝑟′)𝑑
𝑑𝑟′(𝑒𝜓(𝑡
′,𝑟′)) = 𝑒−𝜓(𝑡′,𝑟′)
𝑑𝑟
𝑑𝑟′𝑑
𝑑𝑟(𝑒−𝑏(𝑡,𝑟)
𝑑𝑟′
𝑑𝑟) = −𝑒−𝜓(𝑡
′,𝑟′)𝑒−𝑏(𝑡,𝑟)𝑏′(𝑡, 𝑟)
�̇� =
𝑑𝑅(𝑡′, 𝑟′)
𝑑𝑡′=𝑑𝑅(𝑡, 𝑟)
𝑑𝑡= �̇�(𝑡, 𝑟)
�̈� =
𝑑2𝑅(𝑡′, 𝑟′)
𝑑𝑡′2=𝑑2𝑅(𝑡, 𝑟)
𝑑𝑡2= �̈�(𝑡, 𝑟)
𝑅′ =
𝑑𝑅(𝑡′, 𝑟′)
𝑑𝑟′=𝑑𝑟
𝑑𝑟′𝑑𝑅(𝑡, 𝑟)
𝑑𝑟= 𝑒−𝜓(𝑡
′,𝑟′)𝑒−𝑏(𝑡,𝑟)𝑅′(𝑡, 𝑟)
�̇�′ =
𝑑2𝑅(𝑡′, 𝑟′)
𝑑𝑡′𝑑𝑟′=
𝑑
𝑑𝑟′(𝑑𝑅(𝑡′, 𝑟′)
𝑑𝑡′) =
𝑑𝑟
𝑑𝑟′𝑑
𝑑𝑟(�̇�(𝑡, 𝑟)) = 𝑒−𝜓(𝑡
′,𝑟′)𝑒−𝑏(𝑡,𝑟)�̇�′(𝑡, 𝑟)
The Riemann tensor Tolman –Bondi – de Sitter 2+1
R �̂��̂��̂��̂� = [�̈� − (�̇�)
2] ⇒ R �̂��̂��̂�
�̂� (𝐴) = −[�̈� + (�̇�)2]
𝑅 �̂��̂��̂��̂� = −
�̈�
𝑅
𝑅 �̂��̂��̂��̂� = −[(�̇�)
′+ 𝑅′�̇�]
𝑒𝜓(𝑡,𝑟)
𝑅
𝑅 �̂��̂��̂��̂� = −[(𝑅′′ + 𝑅′𝜓′)
𝑒2𝜓(𝑡,𝑟)
𝑅+�̇��̇�
𝑅]
𝑅 �̂��̂��̂�
�̂�= −
�̈�
𝑅 ⇒ 𝑅
�̂��̂��̂�
�̂�(𝐵) = −
�̈�
𝑅
𝑅 �̂��̂��̂�
�̂�= −[(�̇�)
′+ 𝑅′�̇�]
𝑒𝜓(𝑡,𝑟)
𝑅 ⇒ 𝑅
�̂��̂��̂�
�̂�(𝐶) = −[(�̇�)
′− 𝑅′�̇�]
𝑒−𝑏(𝑡,𝑟)
𝑅
𝑅 �̂��̂��̂�
�̂�= −[(𝑅′′ + 𝑅′𝜓′)
𝑒2𝜓(𝑡,𝑟)
𝑅+�̇��̇�
𝑅] ⇒ 𝑅
�̂��̂��̂�
�̂� (𝐷) = −[(𝑅′′ − 𝑅′𝑏′)𝑒−2𝑏(𝑡,𝑟)
𝑅−�̇��̇�
𝑅]
𝑅 �̂��̂��̂�
�̂�= [
1
𝑅2+(�̇�)
2
𝑅2−(𝑅′)2
𝑅2𝑒2𝜓(𝑡,𝑟)]
Where A, B, C and D will be used later to make the calculations easier
8.5.3 Find the components of the Einstein tensor in the coordinate basis for the metric in 2+1 dimensions.
8.5.4 The Einstein equations of the metric in 2+1 dimensions. Given the Einstein equation ( if 𝑐 = 𝐺 = 1): 𝐺�̂��̂� + Λ휂�̂��̂� = 𝜅𝑇�̂��̂� (6.40) with Λ = −𝜆2 you get 𝐺�̂��̂� − 𝜆
2휂�̂��̂� = 𝜅𝑇�̂��̂� and the stress-energy tensor:
𝑇�̂��̂� = 𝜅 {𝜌 0 00 0 00 0 0
}
You can find the Einstein – equations
{
− [(𝑅′′ − 𝑅′𝑏′)
𝑒−2𝑏
𝑅−�̇��̇�
𝑅] − [(�̇�)
′− 𝑅′�̇�]
𝑒−𝑏
𝑅0
𝑆 −�̈�
𝑅0
0 0 − [�̈� + (�̇�)2]}
− 𝜆2 {−1
1
1
} = 𝜅 {𝜌 0 00 0 00 0 0
}
𝐺�̂��̂�: −[(𝑅′′ − 𝑅′𝑏′)𝑒−2𝑏
𝑅−�̇��̇�
𝑅] + 𝜆2 = 𝜅𝜌 p.152
𝐺�̂��̂�: −[(�̇�)′− 𝑅′�̇�]
𝑒−𝑏
𝑅 = 0
⇔ (�̇�)′− 𝑅′�̇� = 0 p.152
𝐺�̂��̂�: −�̈�
𝑅− 𝜆2 = 0
⇔ �̈� + 𝜆2𝑅 = 0 (6.41) 𝐺�̂��̂�: −[�̈� + (�̇�)
2] − 𝜆2 = 0 (6.42)
8.6 5Ricci rotation coefficients, Ricci scalar and Einstein equations for a general 4-dimensional metric: 𝒅𝒔𝟐 = −𝒅𝒕𝟐 + 𝑳𝟐(𝒕, 𝒓)𝒅𝒓𝟐 + 𝑩𝟐(𝒕, 𝒓)𝒅𝝓𝟐 +𝑴𝟐(𝒕, 𝒓)𝒅𝒛𝟐
The line element: 𝑑𝑠2 = −𝑑𝑡2 + 𝐿2(𝑡, 𝑟)𝑑𝑟2 + 𝐵2(𝑡, 𝑟)𝑑𝜙2 +𝑀2(𝑡, 𝑟)𝑑𝑧2 The Basis one forms
5 (McMahon, 2006, pp. 152-53), quiz 6-5, 6-6, 6-7 and 6-8, the answer to quiz 6-5 is (a) and quiz 6-6 is (c), the answer to quiz 6-7 is (a), the answer to quiz 6-8 is (a)
8.7 6The de Sitter Spacetime The line element 𝑑𝑠2 = −𝑑𝑡2 + 𝑎(𝑡)2(𝑑휃2 + sin2 휃 (𝑑𝜙2 + sin2𝜙𝑑𝜓2)) (IV.11.5)
8.7.1 The Einstein equations The de Sitter spacetime is an example of the Robertson Walker metric in vacuum, positive curvature and a cosmological constant. The solution is the Friedman equations7
0 =3
𝑎2(1 + �̇�2) − Λ
6 (Choquet-Bruhat, 2015, s. 96) Problem IV.11.2. 7 See the chapter named: The Einstein tensor and Friedmann-equations for the Robertson Walker metric
8.8 8The Anti-de Sitter Spacetime The line element 𝑑𝑠2 = −𝑑𝑡2 + cos2(𝑡) 𝑑𝑟2 + cos2(𝑡) sinh2(𝑟) 𝑑휃2 + cos2(𝑡) sinh2(𝑟) sin2 휃 𝑑𝜙2 (IV.11.10) To show that this spacetime is a solution to the Einstein vacuum equation with cosmological constant Λ =−3 we can compare to the Tolman-Bondi de Sitter spacetime9 𝑑𝑠2 = 10𝑑𝑡2 − 𝑒−2𝜓(𝑡,𝑟)𝑑𝑟2 − 𝑅2(𝑡, 𝑟)𝑑휃2 − 𝑅2(𝑡, 𝑟) sin2 휃 𝑑𝜙2
8 (Choquet-Bruhat, 2015, s. 97) Problem IV.11.3.1 9 See the chapter named: The Einstein Tensor of the Tolman-Bondi de Sitter Metric 10 Notice the two spacetimes do not have the same signature. Since we are only going to use the Ricci scalar, and the Ricci scalar does not change sign, when the signature changes, this is not a problem.
In 𝑛 + 1 dimensions with a cosmological constant Λ we have
𝜌𝑎𝑏 = 𝑇𝑎𝑏 +1
𝑛 − 1((𝑛 + 1)Λ − 𝑇𝑐
𝑐)𝑔𝑎𝑏 (IV.3.3)
9.2 12Perfect Fluids The most general form of the stress energy tensor is
𝑇𝑎𝑏 = 𝐴𝑢𝑎𝑢𝑏 + 𝐵𝑔𝑎𝑏 (7.8) In the local frame we know that
𝑇�̂��̂� = {
𝜌 0 0 00 𝑃 0 00 0 𝑃 00 0 0 𝑃
} (7.6)
and
𝑢�̂� = (1,0,0,0)
Then we choose the metric with negative signature
휂�̂��̂� = {
1 0 0 00 −1 0 00 0 −1 00 0 0 −1
}
This we can use to find the constants 𝐴 and 𝐵
𝑇0̂0̂ = 𝐴𝑢0̂𝑢0̂ + 𝐵휂0̂0̂ = 𝐴 + 𝐵 = 𝜌
𝑇 �̂��̂� = 𝐴𝑢�̂�𝑢�̂� + 𝐵휂�̂��̂� = −𝐵 = {𝑃0
𝑖𝑓 𝑖 = 𝑗 𝑖𝑓 𝑖 ≠ 𝑗
⇒ 𝐵 = −𝑃 and 𝐴 = 𝜌 − 𝐵 = 𝜌 + 𝑃 Which leaves us with the most general form of the stress energy tensor for a perfect fluid for a metric with negative signature
𝑇𝑎𝑏 = (𝜌 + 𝑃)𝑢𝑎𝑢𝑏 − 𝑃𝑔𝑎𝑏 (7.11) If we instead choose the metric with positive signature
휂�̂��̂� = {
−1 0 0 00 1 0 00 0 1 00 0 0 1
}
𝑇0̂0̂ = 𝐴𝑢0̂𝑢0̂ + 𝐵휂0̂0̂ = 𝐴 − 𝐵 = 𝜌
𝑇 �̂��̂� = 𝐴𝑢�̂�𝑢�̂� + 𝐵휂�̂��̂� = 𝐵 = {𝑃0
𝑖𝑓 𝑖 = 𝑗
12 (McMahon, 2006, p. 160), (Carroll, 2004, pp. 33-35)
Which leaves us with the most general form of the stress energy tensor for a perfect fluid for a metric with negative signature
𝑇𝑎𝑏 = (𝜌 + 𝑃)𝑢𝑎𝑢𝑏 + 𝑃𝑔𝑎𝑏 (7.12)
9.3 More examples on stress-energy tensors
9.3.1 13Pure Matter In the case of pure matter with no pressure the stress-energy tensor is 𝑇𝑎𝑏 = 𝜌𝑢𝑎𝑢𝑏 (IV.2.7) Where 𝑢 is the unit flow velocity and 𝜌 is the rest-mass density. 14For a co-moving observer the four velocity reduces to 𝑢𝑎 = (1,0,0,0) And the energy-momentum tensor reduces to
𝑇𝑎𝑏 = {
𝜌 0 0 00 0 0 00 0 0 00 0 0 0
} (7.4)
15In the case of a stationary observer the dust particles have a four velocity 𝑢𝑎 = (𝛾, 𝛾𝑢𝑥 , 𝛾𝑢𝑦, 𝛾𝑢𝑧) And the energy-momentum tensor is
𝑇𝑎𝑏 = 𝜌𝛾2
{
1 𝑢𝑥 𝑢𝑦 𝑢𝑧
𝑢𝑥 (𝑢𝑥)2 𝑢𝑥𝑢𝑦 𝑢𝑥𝑢𝑧
𝑢𝑦 𝑢𝑦𝑢𝑥 (𝑢𝑦)2 𝑢𝑦𝑢𝑧
𝑢𝑧 𝑢𝑧𝑢𝑥 𝑢𝑧𝑢𝑦 (𝑢𝑧)2}
(7.5)
9.3.2 16More complicated fluids The most general form of a matter-stress-energy-tensor is the non-perfect fluid with viscosity and shear. 𝑇𝑎𝑏 = 𝜌(1 + 휀)𝑢𝑎𝑢𝑏 + (𝑃 − 휁휃)ℎ𝑎𝑏 − 2휂𝜎𝑎𝑏 + 𝑞𝑎𝑢𝑏 + 𝑞𝑏𝑢𝑎 Where the various quantities are defined as 휀 - Specific energy density of the fluid in its rest frame 𝑃 - Pressure ℎ𝑎𝑏 = 𝑢𝑎𝑢𝑏 + 𝑔𝑎𝑏 – The spatial projection tensor
휂 - shear viscosity 휁 - bulk viscosity 휃 = ∇𝑎𝑢
𝑎 – expansion 𝜎𝑎𝑏 =
1
2(∇𝑐𝑢
𝑎ℎ𝑐𝑏 + ∇𝑐𝑢𝑏ℎ𝑐𝑎) −
1
3휃ℎ𝑎𝑏 – shear tensor
𝑞𝑎 - energy flux tensor
9.3.3 17The electromagnetic field The stress energy tensor of an electromagnetic field is the Maxwell tensor 𝑇𝛼𝛽 = 𝐹𝛼
𝜆𝐹𝛽𝜆 −1
4휂𝛼𝛽𝐹
𝜇𝜈𝐹𝜇𝜈 (IV.2.10)
13 (Choquet-Bruhat, 2015, s. 71) 14 (McMahon, 2006, p. 158) 15 (McMahon, 2006, p. 159) 16 (McMahon, 2006, p. 164) 17 (Choquet-Bruhat, 2015, s. 71)
9.3.3.1 18The Maxwell equations The electromagnetic field tensor 𝐹𝜇𝜈 is defined by
𝐹𝜇𝜈 = {
0 −𝐸1 −𝐸2 −𝐸3
𝐸1 0 −𝜖123𝐵3 −𝜖132𝐵2
𝐸2 −𝜖213𝐵3 0 −𝜖231𝐵1
𝐸3 −𝜖312𝐵2 −𝜖321𝐵1 0
} = {
0 −𝐸1 −𝐸2 −𝐸3
𝐸1 0 −𝐵3 𝐵2
𝐸2 𝐵3 0 −𝐵1
𝐸3 −𝐵2 𝐵1 0
}
i.e. 𝐹0𝑖 = −𝐸𝑖 𝐹𝑖𝑗 = −𝜖𝑖𝑗𝑘𝐵𝑘 Expressed by the vector potential19 𝐴𝜇 𝐹𝜇𝜈 = 𝜕𝜇𝐴𝜈 − 𝜕𝜈𝐴𝜇 Or �̅� = 20 − ∇̅ × �̅�
This we can use to find the four Maxwell equations a)Notice that ∇ ⋅ �̅� = −∇ ⋅ (∇̅ × �̅�)
= −∇ ⋅ {
𝜕2𝐴3 − 𝜕3𝐴
2
𝜕3𝐴1 − 𝜕1𝐴
3
𝜕1𝐴2 − 𝜕2𝐴
1
}
= 𝜕1(𝜕2𝐴3 − 𝜕3𝐴
2) + 𝜕2(𝜕3𝐴1 − 𝜕1𝐴
3) + 𝜕3(𝜕1𝐴2 − 𝜕2𝐴
1) = 0 (II) b)Also notice �̅� = 21 − 𝜕0�̅� − ∇̅𝐴0
⇒ ∇̅ × �̅� = ∇̅ × (−𝜕0�̅� − ∇̅𝐴0)
= −∇̅ × {
𝜕0𝐴1 + 𝜕1𝐴0𝜕0𝐴2 + 𝜕2𝐴0𝜕0𝐴3 + 𝜕3𝐴0
}
= −{
𝜕2(𝜕0𝐴3 + 𝜕3𝐴0) − 𝜕3(𝜕0𝐴2 + 𝜕2𝐴0)
𝜕3(𝜕0𝐴1 + 𝜕1𝐴0) − 𝜕1(𝜕0𝐴3 + 𝜕3𝐴0)
𝜕1(𝜕0𝐴2 + 𝜕2𝐴0) − 𝜕2(𝜕0𝐴1 + 𝜕1𝐴0)}
= −{
𝜕2(𝜕0𝐴3) − 𝜕3(𝜕0𝐴2)
𝜕3(𝜕0𝐴1) − 𝜕1(𝜕0𝐴3)
𝜕1(𝜕0𝐴2) − 𝜕2(𝜕0𝐴1)}
= −𝜕0 {
𝜕2(𝐴3) − 𝜕3(𝐴2)
𝜕3(𝐴1) − 𝜕1(𝐴3)
𝜕1(𝐴2) − 𝜕2(𝐴1)}
= −𝜕0(∇̅ × �̅�)
18 (Michael E. Peskin, 1995, s. 33) 19 Definition: Vector potential: A function �̅� such that �̅� ≡ ∇ × �̅�. The most common use of a vector potential is the representation of a magnetic field. If a vector field has zero divergence, it may be represented by a vector potential http://mathworld.wolfram.com/VectorPotential.html
9.4 25The Gödel metric The Gödel metric is an exact solution of the Einstein field equations in which the stress-energy tensor contains two terms, the first representing the matter density of a homogeneous distribution of swirling dust particles, and the second associated with a nonzero cosmological constant.26 The line element:
𝑑𝑠2 =1
2𝜔2((𝑑𝑡 + 𝑒𝑥𝑑𝑧)2 − 𝑑𝑥2 − 𝑑𝑦2 −
1
2𝑒2𝑥𝑑𝑧2)
=
1
2𝜔2(𝑑𝑡2 + 2𝑒𝑥𝑑𝑡𝑑𝑧 − 𝑑𝑥2 − 𝑑𝑦2 +
1
2𝑒2𝑥𝑑𝑧2)
The metric tensor
𝑔𝑎𝑏 =1
2𝜔2
{
1 0 0 𝑒𝑥
0 −1 0 00 0 −1 0
𝑒𝑥 0 01
2𝑒2𝑥
}
and its inverse
𝑔𝑎𝑏 = 2𝜔2 {
−1 0 0 2𝑒−𝑥
0 −1 0 00 0 −1 0
2𝑒−𝑥 0 0 −2𝑒−2𝑥
}
The stress energy tensor
𝑇𝑎𝑏 =𝜌
2𝜔2{
1 0 0 𝑒𝑥
0 0 0 00 0 0 0𝑒𝑥 0 0 𝑒2𝑥
}
The Einstein equation for a metric with a negative signature 8𝜋𝐺𝑇𝑎𝑏 = 𝐺𝑎𝑏 − 𝑔𝑎𝑏Λ
⇒ 27 8𝜋𝐺𝑔𝑎𝑏𝑇𝑎𝑏 = 𝑔𝑎𝑏𝐺𝑎𝑏 − 𝑔𝑎𝑏𝑔𝑎𝑏Λ
8𝜋𝐺𝜌 = 𝑔𝑎𝑏 (𝑅𝑎𝑏 −1
2𝑔𝑎𝑏𝑅) − 4Λ
= 𝑔𝑎𝑏𝑅𝑎𝑏 −1
2𝑔𝑎𝑏𝑔𝑎𝑏𝑅 − 4Λ
= 𝑅 −1
24𝑅 − 4Λ
= −𝑅 − 4Λ
⇒ Λ = −1
4(8𝜋𝐺𝜌 + 𝑅)
To find 𝑅 we work in the non-coordinate basis
23 Recall, if a charge is present the equation is Gauss law: ∇ ⋅ �̅� =
𝜌
0
24 Recall, if a charge is present the equation is Amperes law: ∇ × �̅� = 𝜇0𝑗̅ + 𝜇0휀0𝜕�̅�
𝜕𝑡
25 (McMahon, 2006, p. 326), final exam 14, the answer to Final Exam quiz 14 is (a). 26 http://en.wikipedia.org/wiki/G%C3%B6del_metric 27 𝑔𝑎𝑏𝑇𝑎𝑏 =
4 represents the matter density of a homogeneous distribution of swirling dust particles,
and the second term Λ = −𝜔2
2 is associated with a nonzero cosmological constant.
9.5 29The Einstein Cylinder
9.5.1 The line element The Einstein cylinder has the line element 𝑑𝑠2 = −𝑑𝑡2 + (𝑎0)
2(𝑑휃2 + sin2 휃 (𝑑𝜙2 + sin2𝜙𝑑𝜓2)) (IV.11.1)
Make the coordinate transformation 𝑟 = sin휃 ⇒ 𝑑𝑟 = 𝑑(sin휃) = cos 휃 𝑑휃
⇒ 𝑑𝑠2 = −𝑑𝑡2 + (𝑎0)2 (
𝑑𝑟2
cos2 휃+ 𝑟2(𝑑𝜙2 + sin2𝜙𝑑𝜓2))
= −𝑑𝑡2 + (𝑎0)
2 (𝑑𝑟2
1 − sin2 휃+ 𝑟2(𝑑𝜙2 + sin2𝜙𝑑𝜓2))
= −𝑑𝑡2 + (𝑎0)
2 (𝑑𝑟2
1 − 𝑟2+ 𝑟2(𝑑𝜙2 + sin2𝜙𝑑𝜓2)) (IV.11.2)
9.5.2 The Ricci tensor To find the Ricci tensor we can compare with calculations made for the Robertson Walker and find a proper transformation of the coordinates. The Einstein cylinder
𝑑𝑠2 = −𝑑𝑡2 +(𝑎0)
2
1 − 𝑟2𝑑𝑟2 + (𝑎0𝑟)
2𝑑𝜙2 + (𝑎0𝑟)2 sin2𝜙𝑑𝜓2
The Robertson Walker line element
𝑑𝑠2 = −𝑑𝑡′2+
𝑎2(𝑡′)
1 − 𝑘𝑟′2𝑑𝑟′
2+ 𝑎2(𝑡′)𝑟′
2𝑑휃′
2+ 𝑎2(𝑡′)𝑟′
2sin2 휃′ 𝑑𝜙′
2
Comparing the line elements 𝑑𝑡 = 𝑑𝑡′
28 http://en.wikipedia.org/wiki/Geometrized_unit_system 29 (Choquet-Bruhat, 2015, s. 95) Problem IV.11.1.
9.5.3 31The Einstein Equations To find the Einstein tensor we once more copy the result from the Robertson Walker metric
𝐺�̂��̂� = 3�̇�2 + 𝑘
𝑎2=3
𝑎02
𝐺�̂��̂� = −(2
�̈�
𝑎+𝑎2̇ + 𝑘
𝑎2) = −
1
𝑎02
𝐺�̂��̂� = −(2
�̈�
𝑎+�̇�2 + 𝑘
𝑎2) = −
1
𝑎02
𝐺�̂��̂� = −(2
�̈�
𝑎+�̇�2 + 𝑘
𝑎2) = −
1
𝑎02
Summarized in a matrix
𝐺�̂��̂� =
{
3
𝑎02 0 0 0
0 −1
𝑎02 0 0
0 0 −1
𝑎02 0
0 0 0 −1
𝑎02}
Where 𝑎 refers to column and 𝑏 to row In case of a perfect fluid, the stress energy tensor is 𝑇�̂��̂� = 𝜇𝑢�̂�𝑢�̂� + 𝑝(휂�̂��̂� + 𝑢�̂�𝑢�̂�) (IV.2.8) = 𝜇휂�̂��̂�𝑢
9.5.4 The Einstein tensor with a cosmological constant A negative pressure is problematic in classical physics and we include a cosmological constant. The Einstein equation with a cosmological constant 𝐺�̂��̂� = 𝑇�̂��̂� − 휂�̂��̂�Λ (IV.3.2)
⇒
{
3
𝑎02 0 0 0
0 −1
𝑎02 0 0
0 0 −1
𝑎02 0
0 0 0 −1
𝑎02}
= {
𝜇 0 0 00 𝑝 0 00 0 𝑝 00 0 0 𝑝
} − Λ{
−11
11
}
⇒ 𝜇 =3
𝑎02 − Λ
𝑝 = −
1
𝑎02 + Λ
9.6 33The Newtonian Approximation – The right hand side! The newtionian approximation is characterized by a weak gravitational field34 and bodies of low masses and velocities.
The Einstein space-time in a weak gravitational field can be described by the linearized metric 𝑑𝑠2 = 𝑔𝑎𝑏𝑑𝑥
𝑎𝑑𝑥𝑏 𝑔𝑎𝑏 = 휂𝑎𝑏 + 𝜖ℎ𝑎𝑏 𝜖 ≪ 1 (13.1)35
We look at a particle moving along 𝑥1 with velocity 𝑣 =𝑑𝑥1
𝑑𝑥0≪ 𝑐
⇒ 𝑑𝑠2 = 𝑔𝑜𝑜𝑑𝑥0 + (𝑔01 + 𝑔10)𝑑𝑥1𝑑𝑡 + 𝑔11𝑑𝑥1
2
⇒ (𝑑𝑠
𝑑𝑥0)2
= 𝑔𝑜𝑜 + (𝑔01 + 𝑔10) (𝑑𝑥1𝑑𝑥0
) + 𝑔11 (𝑑𝑥1𝑑𝑥0
)2
= 𝑔𝑜𝑜 + (𝑔01 + 𝑔10)𝑣 + 𝑔11𝑣2
= (−1 + 𝜖ℎ00) + (𝜖ℎ01 + 𝜖ℎ10)𝑣 + (1 + 𝜖ℎ11)𝑣2
→ {−1 + 𝜖ℎ00 𝑖𝑓 𝑣 → 0
𝜖(ℎ00 + ℎ01 + ℎ10 + ℎ11) 𝑖𝑓 𝑣 → 1
(I)
i.e. in the approximation where 𝑣 ≪ 𝑐, the time component ℎ00 is dominant with respect to the space-components (ℎ01, ℎ10, ℎ11).
36The Ricci tensor: In this linearized theory we have the Christoffel symbols
33 (Choquet-Bruhat, 2015, s. 75) Exercise IV.5.1. 34 For a detailed calculation of the Christoffel symbols, the Riemann and Ricci tensors, the Ricci scalar and the Einstein equation (The left hand side!) see a later chapter named “Linearized metric” 35 (McMahon, 2006, p. 280) 36 (Choquet-Bruhat, 2015, s. 75) Chapter IV.5.1.
Assuming39 the time derivatives 𝜕0ℎ𝑎𝑏 are small compared to the space derivatives 𝜕𝑖ℎ𝑎𝑏 ⇒ 𝑅00 = 𝜕𝑐Γ 00
𝑐
= 𝜕0Γ 000 + ∑ 𝜕𝑖Γ 00
𝑖
1,2,3
𝑖
= ∑ 𝜕𝑖 (1
2𝜖휂𝑖𝑑 (
𝜕ℎ0𝑑𝜕𝑥0
+𝜕ℎ0𝑑𝜕𝑥0
−𝜕ℎ00𝜕𝑥𝑑
))
1,2,3
𝑖
=1
2𝜖 ∑ −
𝜕2ℎ00𝜕𝑥𝑖𝜕𝑥𝑖
1,2,3
𝑖
= 40 −
1
2𝜖∇2ℎ00
Notice
Γ 00𝑖 = −
1
2𝜖𝜕ℎ00𝜕𝑥𝑖
In another chapter41 we found out that
𝑅𝑎𝑏 = 𝜅 (𝑇𝑎𝑏 −1
2𝑔𝑎𝑏𝑇𝑎𝑏𝑔𝑎𝑏) = 𝜅𝜌𝑎𝑏 (IV.2.5)
The stress-energy tensor is 𝑇𝑎𝑏 = 𝜇𝑢𝑎𝑢𝑏
In the simplest case, pure matter no pressure42, 𝑢𝑎 = (𝑑𝑥0
𝑑𝑠,𝑑𝑥𝑖
𝑑𝑠) = (−1,0,0,0) and the only non-zero ele-
ment in the stress energy tensor is 𝑇00 = 𝜇(𝑢0)
2 = 𝜇 And
𝑇 = 𝑔𝑎𝑏𝑇𝑎𝑏 = 𝑔00𝜇
⇒ 𝜌00 = 𝑇00 −1
2𝑇𝑔00 = 43𝜇 −
1
2𝜇𝑔00𝑔00 =
1
2𝜇
(IV.5.2)
Now, because
𝑅00 = −1
2𝜖∇2ℎ00 = 𝜅𝜌00 =
1
2𝜅𝜇
⇒ 𝜖∇2ℎ00 = 44 −1
2𝐺𝐸𝜇
45Which is the equivalent of Poisson’s equation for gravity
∇2𝜙(𝑟) = 46 − 4𝜋𝐺𝑁𝜌
37 (McMahon, 2006, p. 282) 38 (McMahon, 2006) 39 I don’t know how to prove this except to state, that we in the Newtonian approximation work in a regime where frequencies are low and wavelengths long.
40 ∇2= ∑𝜕2
𝜕𝑥𝑖𝜕𝑥𝑖1,2,3𝑖
41 ”The Einstein equation with source.” 42 Chapter “Pure matter” 43 𝑔00𝑔00 ≈ 1 because the off-diagonal elements are ≪ 1. 44 Renaming 𝜅 = 𝐺𝐸 45 http://mathworld.wolfram.com/PoissonsEquation.html 46 If 𝜖ℎ00 = 𝜙 and 𝐺𝐸 = 8𝜋𝐺𝑁
The equation of motion: We can also find the equation of motion in this approximation. In GR a test particle follows geodesics described by
𝑑2𝑥𝑎
𝑑𝑠2+ Γ 𝑏𝑐
𝑎𝑑𝑥𝑏
𝑑𝑠
𝑑𝑥𝑐
𝑑𝑠 = 0
(4.34)47
⇒ 𝑑2𝑥𝑎
𝑑𝑠2 = −Γ 𝑏𝑐
𝑎𝑑𝑥𝑏
𝑑𝑠
𝑑𝑥𝑐
𝑑𝑠
⇒ 48 𝑑2𝑥𝑖
(𝑑𝑥0)2 = −Γ 00
𝑖 (𝑑𝑥0
𝑑𝑠)
2
≅ 49 − Γ 00𝑖 (−1)2 =
1
2𝜖𝜕ℎ00𝜕𝑥𝑖
(IV.5.4)
Again we see the equivalence to Newton’s laws, where the left side represent the force and the right side the gradient of the potential. Recall
�̅� = ∇𝑈 with
𝑈 = −𝐺𝑀𝑚
𝑟
leads to Newton’s famous equation
𝐹 =𝐺𝑀𝑚
𝑟2
10 Null Tetrads and the Petrov Classification
10.1 Weyl scalars and Petrov classification Petrov Classifica-tion50
Weyl scalars51 PND52
Type I: Alg. gen-eral
[1,1,1,1] Ψ0, Ψ1, Ψ2, Ψ3, Ψ4≠ 0
Type II [2,1,1] Ψ0, Ψ1 = 0 Ψ2, Ψ3, Ψ4 ≠ 0
𝑙𝑎 , 𝑙𝑎
Type D or de-gener-ate
[2,2] Ψ0, Ψ1,Ψ3, Ψ4 = 0
Ψ2 ≠ 0 𝑙𝑎 , 𝑙𝑎 , 𝑛𝑎, 𝑛𝑎 Gravitational field of a star or black hole
(Schwarzschild or Kerr vacuum). The two princi-pal null directions correspond to ingoing and outgoing congruences of light rays. 53The fact that the spacetime contains Ψ2 and not Ψ4 or Ψ0 indicates that this spacetime de-scribes electromagnetic fields and not gravita-tional radiation. This spacetime represents a vacuum universe that contains electromagnetic fields with no matter.
47 (McMahon, 2006, p. 82)
48 𝑑2𝑥𝑖
𝑑𝑠2=
𝑑
𝑑𝑠(𝑑𝑥𝑖
𝑑𝑠) =
𝑑
𝑑𝑠(𝑑𝑥𝑖
𝑑𝑥0
𝑑𝑥0
𝑑𝑠) = −
𝑑
𝑑𝑠(𝑑𝑥𝑖
𝑑𝑥0) = −
𝑑
𝑑𝑥0(𝑑𝑥𝑖
𝑑𝑠) = −
𝑑
𝑑𝑥0(𝑑𝑥𝑖
𝑑𝑥0
𝑑𝑥0
𝑑𝑠) =
𝑑
𝑑𝑥0(𝑑𝑥𝑖
𝑑𝑥0)
49 Eq. (I) 50 (Scolarpedia, n.d.), (McMahon, 2006) 51 If 𝑛 = 4, 𝑅𝑎𝑏𝑐𝑑 has twenty independent component – ten of which are given by 𝑅𝑎𝑏 and the remaining ten by the Weyl tensor (d'Inverno, 1992, p. 87) 52 Principal Null Directions 53 (McMahon, p. 321)
𝑙𝑎 , 𝑙𝑎 , 𝑙𝑎 Longitudinal gravity waves with shear due to tidal effects.
Type IV or N
[4] Ψ0, Ψ1, Ψ2, Ψ3= 0 Ψ4 ≠ 0
𝑙𝑎,𝑙𝑎 , 𝑙𝑎 , 𝑙𝑎 Transverse gravity waves, single principal null di-rection of multiplicity 4
54Ψ1, Ψ2, Ψ3, Ψ4= 0 Ψ0 ≠ 0
𝑛𝑎,𝑛𝑎, 𝑛𝑎 , 𝑛𝑎 Gravitational wave region III
Type O Ψ0, Ψ1, Ψ2, Ψ3, Ψ4= 0
In the context of gravitational radiation, we have the following interpretations: Ψ0 transverse wave component in the 𝑛𝑎 direction Ψ1 longitudinal wave component in the 𝑛𝑎 direc-
tion Ψ2 denotes a coulomb component Ψ3 transverse wave component in the 𝑙𝑎 direction Ψ4 longitudinal wave component in the 𝑙𝑎 direc-
tion
10.2 55Construct a null tetrad for the flat space-time Minkowski metric The line element:
a) The line element: 𝑑𝑠2 = 𝑎2[(cos2 휃 + 4 sin2 휃)𝑑휃2 + sin2 휃 𝑑𝜙2] 𝑖𝑓 𝑎 = 𝑏 =1
2𝑐
6) 59Cylindindrical Coordinates.
57 (Hartle, Gravity - An introduction to Einstein's General Relativity, 2003, p. 26) 58 (Hartle, Gravity An Introduction to Einstein's General Relativity, 2003) problem 2-8 59 (McMahon, p. 83)
a) Line element: 𝑑𝑠2 = 𝑑𝜓2 + sinh2𝜓 𝑑휃2 + sinh2𝜓 sin2 휃 𝑑𝜙2
b) The basis one-forms: 𝜔�̂� = 𝑑𝜓, 𝜔�̂� = sinh𝜓𝑑휃, 𝜔�̂� = sinh𝜓 sin휃 𝑑𝜙
c) Ricci rotation coefficients: Γ �̂��̂�
�̂�= −coth𝜓, Γ �̂��̂�
�̂� = coth𝜓, Γ �̂��̂�
�̂�=
coth𝜓
sin𝜃, Γ �̂��̂�
�̂�= −
coth𝜓
sin𝜃,
Γ �̂��̂��̂� = −
cot𝜃
sinh2𝜓, Γ �̂��̂�
�̂�=
cot𝜃
sinh2𝜓
10) 62Metric Example 3.
a) The line element: 𝑑𝑠2 = (𝑢2 + 𝜈2)𝑑𝑢2 + (𝑢2 + 𝜈2)𝑑𝜈2 + 𝑢2𝜈2𝑑휃2
b) The Riemann tensor: 𝑅𝑎𝑏𝑐𝑑 = 0
11) 63General 4-dimensional space-time.
a) The line element: 𝑑𝑠2 = −𝑑𝑡2 + 𝐿2(𝑡, 𝑟)𝑑𝑟2 + 𝐵2(𝑡, 𝑟)𝑑𝜙2 +𝑀2(𝑡, 𝑟)𝑑𝑧2
b) The basis one-forms: 𝜔�̂� = 𝑑𝑡, 𝜔�̂� = 𝐿(𝑡, 𝑟)𝑑𝑟, 𝜔�̂� = 𝐵(𝑡, 𝑟)𝑑𝜙, 𝜔�̂� = 𝑀(𝑡, 𝑟)𝑑𝑧
c) The Einstein tensor: 𝐺�̂��̂� =𝐵′𝐿′−𝐵′′𝐿
𝐵𝐿3+�̇��̇�
𝐵𝐿+𝑀´𝐿´−𝑀´´𝐿
𝑀𝐿3+�̇��̇�
𝑀𝐿+�̇��̇�
𝐵𝑀−
𝐵′𝑀′
𝐿2𝐵𝑀, 𝐺�̂��̂� =
𝐵´�̇�
𝐵𝐿2−
𝐵′̇
𝐵𝐿+
𝑀´�̇�
𝑀𝐿2−
�̇�′
𝑀𝐿,
𝐺�̂��̂� = −�̈�
𝐵−�̈�
𝑀+�̇��̇�
𝐵𝑀−
𝐵′𝑀′
𝐿2𝐵𝑀, 𝐺�̂��̂� = −
�̈�
𝐿−�̈�
𝑀
𝑀´𝐿´−𝑀´´𝐿
𝑀𝐿3+�̇��̇�
𝑀𝐿, 𝐺�̂��̂� = −
�̈�
𝐿−�̈�
𝐵+𝐵′𝐿′−𝐵′′𝐿
𝐵𝐿3+�̇��̇�
𝐵𝐿
12) 64The Plane in polar coordinates.
a) The line element: 𝑑𝑆2 = 𝑑𝑟2 + 𝑟2𝑑𝜙2
b) The geodesic equations:
i) �̈� = 𝑟�̇�2
ii) �̈� = −2
𝑟�̇��̇�
13) 65Mathematical singularity: The two-dimensional plane in polar coordinates 𝑑𝑆2 = 𝑑𝑟2 + 𝑟2𝑑𝜙2 can
blow up in a singularity by making the transformation 𝑟 = 𝑎2/𝑟′ for some constant 𝑎 𝑑𝑆2 =
60 (McMahon, p. 92) 61 (McMahon, 2006, p. 325) 62 (McMahon, 2006, p. 324) 63 (McMahon, p. 153) 64 (Hartle, Gravity An Introduction to Einstein's General Relativity, 2003, p. 171) Example 8-1 65 (Hartle, Gravity An Introduction to Einstein's General Relativity, 2003, p. 136)
66 (Hartle, Gravity An Introduction to Einstein's General Relativity, 2003, p. 184) Problem 8-12 67 (McMahon, p. 170) 68 (McMahon, 2006, p. 91) 69 (McMahon, p. 186) 70 (McMahon, p. 91)
21) 75Static Weak Field Metric: In this model the flat spacetime geometry of special relativity is modified to
introduce a slight curvature that will explain geometrically the behavior of clocks. Further, the world
lines of extremal proper time in this modified geometry will reproduce the predictions of Newtonian
71 (Hartle, Gravity An Introduction to Einstein's General Relativity, 2003, p. 137) 72 (Hartle, Gravity - An introduction to Einstein's General Relativity, 2003) Problem 8-11 73 (Hartle, Gravity - An introduction to Einstein's General Relativity, 2003, p. 164), problem 2 74 (Hartle, Gravity An Introduction to Einstein's General Relativity, 2003, p. 143) 75 (Hartle, Gravity An Introduction to Einstein's General Relativity, 2003, p. 126)
mechanics for motion in a gravitational potential for nonrelativistic velocities. Φ(𝑥𝑖) is a function of
position satisfying the Newtonian field equation76 ∇2Φ(�⃗�) = 4𝜋𝐺𝜇(�⃗�)and assumed to vanish at infin-
ity. For example outside Earth Φ(𝑟) = −𝐺𝑀⊕
𝑟. This line element is predicted by general relativity for
small curvatures produced by time-independent weak sources, and it is a good approximation to the
curved spacetime geometry produced by the Sun.
a) Line element: 𝑑𝑠2 = −(1 +2Φ(𝑥𝑖)
𝑐2) (𝑐𝑑𝑡)2 + (1 −
2Φ(𝑥𝑖)
𝑐2) (𝑑𝑥2 + 𝑑𝑦2 + 𝑑𝑧2)
77Δ𝜏𝐵~(1 +Φ𝐵−Φ𝐴
𝑐2)Δ𝜏𝐴 tells us the observed fact, that when the receiver 𝐵 is at a higher gravitational
potential that the emitter 𝐴, the signals will be received more slowly than they were emitted and vice versa. 22) 78Rindler metric: The Rindler coordinate system or frame describes a uniformly accelerating frame of
reference in Minkowski space.
a) Line element: 𝑑𝑠2 = 𝜉2𝑑𝜏2 − 𝑑𝜉2
b) Basis one-forms: 𝜔�̂� = 𝑑𝜉, 𝜔�̂� = 𝜉𝑑𝜏
c) Geodesic equations:
i) �̈� + 𝜉�̇�2 = 0
ii) �̈� +2
𝜉�̇��̇� = 0
23) 79The Einstein Cylinder:
a) Line element: 𝑑𝑠2 = −𝑑𝑡2 + (𝑎0)2(𝑑휃2 + sin2 휃 (𝑑𝜙2 + sin2𝜙𝑑𝜓2))
b) Coordinate transformation: 𝑟 = sin 휃
i) Line element: 𝑑𝑠2 = −𝑑𝑡2 + (𝑎0)2 (
𝑑𝑟2
1−𝑟2+ 𝑟2(𝑑𝜙2 + sin2𝜙𝑑𝜓2))
c) Ricci scalar: 𝑅 = 3 ⋅2
𝑎02
d) Einstein equations:
i) 𝜇 =3
𝑎02 − Λ
ii) 𝑝 = −1
𝑎02 + Λ
24) 80Classical Anti-de Sitter Space-time
a) Line element: 𝑑𝑠2 = −cosh2(𝑟) 𝑑𝑡2 + 𝑑𝑟2 + sinh2(𝑟) 𝑑휃2 + sinh2(𝑟) sin2 휃 𝑑𝜙2
b) Coordinate transformation: cosh(𝑟) =1
cos𝜓
i) Line element: 𝑑𝑠2 = −1
cos2𝜓(𝑑𝑡2 + 𝑑𝜓2 + sin2𝜓𝑑휃2 + sin2𝜓 sin2 휃 𝑑𝜙2) - which is confor-
mally related to the Einstein cylinder.
c) Killing vectors and conservation equations
i) 𝝃 = (𝜉𝑡 , 𝜉𝑟, 𝜉𝜃, 𝜉𝜙) = (1,0,0,0) ⇒ �̇� =𝐾1
cosh2(𝑟)
76 (Hartle, Gravity An Introduction to Einstein's General Relativity, 2003, p. 40) eq. (3.18) 77 (Hartle, Gravity - An introduction to Einstein's General Relativity, 2003, p. 127) 78 http://en.wikipedia.org/wiki/Rindler_coordinates, (McMahon, p. 84) 79 (Choquet-Bruhat, 2015, s. 95) 80 (Choquet-Bruhat, 2015, s. 97)
iv) 89No matter, radiation, early universe: 𝑎(𝑡) ∝ √𝑡
v) 90No matter, no radiation, no curvature 𝑘 = 0, cosmological constant: 𝑎(𝑡) = 𝐶𝑒√Λ
3𝑡
vi) 91 A particle of mass 𝑚, sitting on a surface of a ball of radius 𝑅 and mass density 𝜌, experiences
an acceleration, 𝑑2𝑅
𝑑𝑡2 given by
4𝜋
3𝑅3𝐺𝜌
𝑅2, and so
1
𝑅
𝑑2𝑅
𝑑𝑡2=
4𝜋
3𝐺𝜌. If we formally identify 𝑅 with the
radius of the Universe, and 𝜌 with the mass density of the Universe, this is Einstein’s equation
for how the size of the Universe evolves, assuming the absence of pressure.
85 (McMahon, p. 161) 86 (McMahon, p. 270) 87 (McMahon, p. 274) 88 (McMahon, p. 271) 89 (McMahon, p. 272) 90 (McMahon, p. 273) 91 (Greene, 2004, s. 515) note 6
light between Earth and a planet in the solar system, we integrate between 𝑟0 (distance of closets
approach) to 𝑟𝑣 (planet orbit radius) and 𝑟0 to 𝑟𝑒 (Earth orbit radius): 𝑡𝑑𝑒𝑙𝑎𝑦 =
𝑚𝐺
𝑐3[2 ln (
(𝑟𝑣+√𝑟𝑣2−𝑟0
2)(𝑟𝑒+√𝑟𝑒2−𝑟0
2)
𝑟02 ) −
√𝑟𝑣2−𝑟0
2
𝑟𝑣−√𝑟𝑒
2−𝑟02
𝑟𝑒]. The ordinary flat space term is given by
√𝑟𝑣2 − 𝑟0
2 +√𝑟𝑒2 − 𝑟0
2.
n) 103Gravitational Red Shift: 𝑑𝜏 = √1 −2𝑚
𝑟𝑑𝑡. Light emitted upward in a gravitational field, from an
observer located at some inner radius 𝑟1 to an observer positioned at some outer radius 𝑟2. 𝛼 =
√1−2𝑚
𝑟2
√1−2𝑚
𝑟1
o) 104The Path of a Radially Infalling Particle.
i) Line element: From infinity with vanishing initial velocity: 𝑑휃 = 𝑑𝜙 = 0 ⇒ 1 −2𝑚
𝑟=
(1 −2𝑚
𝑟)2(𝑑𝑡
𝑑𝜏)2− (
𝑑𝑟
𝑑𝜏)2
ii) Killing equation: From Killings equation we know that (1 −2𝑚
𝑟)𝑑𝑡
𝑑𝜏 is a constant ⇒
(1 −2𝑚
𝑟)𝑑𝑡
𝑑𝜏= 1 and (
𝑑𝑟
𝑑𝜏)2=
2𝑚
𝑟 ⇒ 𝑡 − 𝑡0 =
2
3√2𝑚(𝑟0
3
2 − 𝑟3
2 + 6𝑚√𝑟0 − 6𝑚√𝑟) +
2𝑚 ln√𝑟0−√2𝑚
√𝑟0+√2𝑚 √𝑟+√2𝑚
√𝑟−√2𝑚
32) 105Schwarzschild Space-time with 휃 =𝜋
2.
a) Line element: 𝑑𝑠2 = −(1 −2𝑚
𝑟) 𝑑𝑡2 + (1 −
2𝑚
𝑟)−1𝑑𝑟2 + 𝑟2𝑑𝜙2
b) Geodesic equations:
i) �̈� +2𝑚
𝑟(𝑟−2𝑚)�̇��̇� = 0
ii) �̈� +𝑚
𝑟3(𝑟 − 2𝑚)�̇�2 −
𝑚
𝑟(𝑟−2𝑚)�̇�2 − (𝑟 − 2𝑚)�̇�2 = 0
iii) �̈� +2
𝑟�̇��̇� = 0
33) 106The Schwarzschild Metric in Kruskal Coordinates.
a) Line element: 𝑑𝑠2 =32𝑚3
𝑟𝑒−
𝑟
2𝑚(𝑑𝑣2 − 𝑑𝑢2) − 𝑟2(𝑑휃 + sin2 휃 𝑑𝜙)
i) 𝑟 > 2𝑚:
(1) 𝑢 = 𝑒𝑟
4𝑚 √𝑟
2𝑚− 1cosh
𝑡
4𝑚
(2) 𝑣 = 𝑒𝑟
4𝑚 √𝑟
2𝑚− 1 sinh
𝑡
4𝑚
ii) 𝑟 < 2𝑚:
102 (McMahon, p. 229) 103 (McMahon, p. 234) 104 (McMahon, p. 238) 105 (Hartle, Gravity - An introduction to Einstein's General Relativity, 2003, p. 183) problem 8-3 106 (McMahon, 2006, p. 242)
k) 124Constant Linear Polarization in Vacuum: 𝐻(𝑢, 𝑥, 𝑦) = ℎ(𝑢)[cos 𝛼 (𝑥2 − 𝑦2) + 2 sin𝛼 𝑥𝑦, where
𝛼 is the angle between the polarization vector and the 𝑥-axis.
i) Einstein tensor: 𝐺𝑢𝑢 = 0
46) 125The Aichelburg-Sexl Solution: A black hole passing near by.
119 (McMahon, 2006, p. 92) 120 (McMahon, p. 195) 121 According to the Weyl scalar calculation the sign is wrong 122 (McMahon, p. 301) 123 (McMahon, p. 301) 124 (McMahon, p. 303) 125 (McMahon, p. 303)
52) 134The Gödel Metric: The Gödel metric is an exact solution of the Einstein field equations in which the
stress-energy tensor contains two terms, the first representing the matter density of a homogeneous
distribution of swirling dust particles, and the second associated with a nonzero cosmological con-
stant.135
a) Line element: 𝑑𝑠2 =1
2𝜔2((𝑑𝑡 + 𝑒𝑥𝑑𝑧)2 − 𝑑𝑥2 − 𝑑𝑦2 −
1
2𝑒2𝑥𝑑𝑧2)
b) Ricci scalar: 𝑅 = 2𝜔2
53) 136Warp-Drive Space-time.
a) Line element: 𝑑𝑠2 = −𝑑𝑡2 + [𝑑𝑥 − 𝑉𝑠(𝑡)𝑓(𝑟𝑠)𝑑𝑡2]2 + 𝑑𝑦2 + 𝑑𝑧2
54) 137Worm Hole Geometry.
a) Line element: 𝑑𝑠2 = −𝑑𝑡2 + 𝑑𝑟2 + (𝑏2 + 𝑟2)(𝑑휃2 + sin2 휃 𝑑𝜙2)
b) Conservation equation: The travel time through a worm-hole: Δ𝜏 =2𝑅
𝑈
c) Geodesic equations.
i) �̈� = 0
ii) �̈� = 𝑟휃̇2 + 𝑟 sin2 휃 �̇�2
iii) 휃̈ = sin휃 cos 휃 �̇�2 −2𝑟
(𝑏2+𝑟2)�̇�휃̇
iv) �̈� = −2𝑟
(𝑏2+𝑟2)�̇��̇� − 2 cot 휃 휃̇�̇�
Bibliografi A.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. Carroll, S. M. (2004). An Introduction to General Relativity, Spacetime and Geometry. San Fransisco, CA:
Addison Wesley. Choquet-Bruhat, Y. (2015). Introduction to General Relativity, Black Holes and Cosmology. Oxford: Oxford
University Press. 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. Hartle, J. B. (2003). Gravity An Introduction to Einstein's General Relativity. San Fransisco, CA: Addison
Wesley. Kay, D. C. (1988). Tensor Calculus. McGraw-Hill. McMahon, D. (2006). Relativity Demystified. McGraw-Hill Companies, Inc. Penrose, R. (2004). The Road to Reality. New York: Vintage Books. Scolarpedia. (n.d.). Retrieved from http://www.scholarpedia.org/article/Spin-coefficient_formalism. Spiegel, M. R. (1990). SCHAUM'S OUTLINE SERIES: Mathematical Handbook of FORMULAS and TABLES.
McGraw-Hill Publishing Company. Weinberg, S. (1979). De første tre minutter. Gyldendal.
a (McMahon, p. 234)
134 (McMahon, 2006, p. 326) Final exam 14 135 http://en.wikipedia.org/wiki/G%C3%B6del_metric 136 (Hartle, Gravity An Introduction to Einstein's General Relativity, 2003, p. 144) Example 7.4 137 (Hartle, Gravity An Introduction to Einstein's General Relativity, 2003) eq (7.39)
b (McMahon, p. 168) c (Carroll, 2004) d (McMahon, p. 220) e (A.S.Eddington, pp. 85-86) f (d'Inverno, p. 87) g (McMahon, p. 161) h (McMahon, 2006, p. 138) i An excellent qualitative explanation of the cosmological constant, you can find in (Greene, 2004, s. 273-279) j (d'Inverno, 1992, p. 172)
