Courtney James Mewton Page 1 GR, Tensor Analysis & Geometry

Tensor Analysis & Geometry

Spherical Coordinates

φθ cossin1 rx = , φθ sinsin2 rx = , θcos3 rx = , ctx =4

3-dimensional l ine element: ( )222222 sin φθθ ddrdrds ++=

Christoffel Symbols

Christoffel Symbol of the first kind: [ ]

∂∂

−∂∂

+∂∂

==Γr

mnmrn

nrm

rmn x

g

x

g

x

grmn

2

1, .

Christoffel Symbol of the second kind: mnrrzz

mn gmnz Γ=

î=Γ .

Derivation o f the Riemann Curvature TensorIn general, a second order differentiation on a covariant vector is independent of the order in which it is carriedout, i.e.:

jki

kji

xx

V

xx

V

∂∂∂

=∂∂

∂ 22

.

However, the presence of Christoffel symbols can have an effect on this statement. We investigate this by firstfinding the general second derivatives for both permutations of the differentiating parameters:

( ) rirjkjr

rikjkikji VVVV ,,,;; Γ−Γ−= .

But ss

ijjiji VVV Γ−= ,; ,

( ) [ ] [ ]ss

irrirjks

srjjr

rikks

sijsk

sij

jkikji VVVVVVx

VV Γ−Γ−Γ−Γ−Γ−∂Γ∂

−=∴ ,,,,::

( ) ss

irrjkri

rjks

srj

rikjr

rikks

sijsk

sij

jkikji VVVVVVx

VV ΓΓ+Γ−ΓΓ+Γ−Γ−∂Γ∂

−=⇔ ,,,,::

Now we interchange j and k (which is the other possible way of determining this second derivative):

( ) ss

irrkjri

rkjs

srk

rijkr

rijjs

siksj

sik

kjijki VVVVVVx

VV ΓΓ+Γ−ΓΓ+Γ−Γ−∂Γ∂

−= ,,,,:: .

We now find the difference between these two. On the RHS, the first, third, fourth, sixth, and seventh termscancel out, thus giving the result:

ssrk

rij

srj

rikk

sij

j

sik

ssrk

rijsj

sik

ssrj

riksk

sij

kjijki Vxx

VVx

VVx

VV

ΓΓ−ΓΓ+

∂Γ∂

−∂Γ∂=ΓΓ−

∂Γ∂+ΓΓ+

∂Γ∂

−=− :: .

We define the Riemann (or Riemann-Christoffel) Curvature tensor by:

srk

rij

srj

rikk

sij

j

siks

ijk xxR ΓΓ−ΓΓ+

∂Γ∂

−∂Γ∂

= .

The difference between the covariant derivatives can thus be written as ssijkkjijki VRVV =− :: . The Riemann

tensor used in this equation is called the Riemann curvature tensor of the second kind. The curvature tensor ofthe first kind is defined as:

rjklirijkl RgR = .

Symmetry Properties:First skew symmetry jiklijkl RR −=Second skew symmetry ijlkijkl RR −=

GENERAL RELATIVITY, TENSOR ANALYSIS AND GEOMETRY

Courtney James Mewton Page 2 GR, Tensor Analysis & Geometry

Block symmetry klijijkl RR =Bianchi’s identity 0=++ iljkikljijkl RRR

The Ricc i TensorThe Ricci tensor of the first kind is simply a contraction of the Riemann tensor:

kijkij RR = .

The last index can be raised to yield the Ricci tensor of the second kind:

kiikj

i RgR = .

If this tensor is finally contracted by letting I = j, we get the Ricci curvature scalar. If it is zero, the space is flat.From the first of the two equations above the Ricci tensor of the first kind can be calculated directly by:

krk

rij

krj

rikk

kij

j

kikk

ijkij xxRR ΓΓ−ΓΓ+

∂Γ∂

−∂Γ∂

== .

Transformation Of A Geodesic From Parameter u To v, Where v = f(u)Given a particular geodesic in terms of a parameter u, in this section the geodesic equation will be transformedso that it is in terms of a new parameter v.

Start with 02

2

=Γ+=

du

dx

du

dx

du

xd

du

dx

du

D cba

bc

aa

.

Substitute du

dv

dv

dx

du

dx aa

= ,

then 02

2

=∂∂+

∂∂Γ+

∂∂∂=

∂∂

du

vd

v

x

du

dx

du

dv

v

x

du

dv

vu

x

du

dv

v

x

du

D acba

bc

aa

2

2

du

vd

v

x

du

dv

du

dx

v

x

du

dv

vu

x acbabc

a

∂∂−=

∂∂Γ+

∂∂∂⇔

∂∂−=

∂∂Γ+

∂∂∂⇔

dv

du

dv

du

du

vd

v

x

dv

du

dv

du

du

dv

du

dx

v

x

dv

du

dv

du

du

dv

vu

x acbabc

a

2

2

2

2

2

2

∂∂−=

∂∂Γ+

∂∂⇔

dv

du

du

vd

v

x

dv

dx

v

x

v

x acbabc

a

.

Since the LHS is now in terms of v, partial differentiation can be replaced by normal differentiation:2

2

2

2

−=Γ+⇔

du

dv

du

vd

dv

dx

dv

dx

dv

dx

dv

dx acbabc

a

.

By letting

2

2

2

−=

du

dv

du

vdλ , we arrive at the final equation:

dv

dx

dv

dx

dv

dx

dv

dx acbabc

a

λ=Γ+2

.

General Relativity

The Metric Tensor for Special Relativity

GENERAL RELATIVITY, TENSOR ANALYSIS AND GEOMETRY

Courtney James Mewton Page 3 GR, Tensor Analysis & Geometry

−−

−=1000

010000100001

µνη .

Einstein's Law of GravitationSimply stated, Einstein's law of gravitation is:

0=µνR .

This condition holds when the local space is completely devoid of all forms of matter and energy.

Derivation o f the Schwarzchild SolutionIn this section the line-element solution to the field equations for a quasi-static gravitational field produced by aspherical body will be derived.We start by setting up a general l ine element employing spherical coordinates:

( )222222222 sin φθθτ ddWrBdrdtAcdc +−−= .

Before we continue, a few assumptions need to be made:

� The space is asymptotically flat. This means that 1→= BA as ∞→r .� The gravitational field only affects time and radial distance, so W = 1.

We can immediately define the metric tensor:2

00 Acg = , Bg −=11 , 222 rg −= and θ22

33 sinrg −= .

Since we are dealing with the empty space surrounding the body, the Ricci tensor needs to equal zero. With thisin mind, the derivation begins. We first calculate the Christoffel symbols. Note that since a Christoffel symbolof the second kind is defined as

mnrrzz

mn gmnz Γ=

î=Γ

we need only calculate them for values when r = z.

[ ] AAgggcAg ′=−+=Γ=Γ −−− 121

0,100,101,0021

21

100000

10

[ ] 2121

1,001

21

1,000,010,101

21

001111

00 cABgBgggBg ′==−+−=Γ=Γ −−−

[ ] BBgBgggBg ′=−=−+−=Γ=Γ −−− 121

1,111

21

1,111,111,111

21

111111

11

[ ] ( ) 1121

1,221

21

1,222,212,121

21

221111

22 2 −−−− −=⋅−=−=−+−=Γ=Γ rBrBgBgggBg

[ ] ( ) ( ) θθ 212121

1,331

21

1,333,313,131

21

331111

33 sinsin2 −−−− −=−=−−=−+−=Γ=Γ rBrBgBgggBg

[ ] ( ) 1221

2,211,222,212

21

212222

21 2 −−− =−⋅−=−+−=Γ=Γ rrrgggrg

[ ] θθθθ sincossincos2 2221

2,333,323,232

21

332222

33 −=⋅−=−+−=Γ=Γ −− rrgggrg

[ ] ( ) 122221

3,133,131,3322

21

133333

13 sin2sinsin −−−−− =−⋅−=−+−=Γ=Γ rrrgggrg θθθ[ ] ( ) θθθθθ cotsincos2sinsin 222

21

3,233,232,3322

21

233333

23 =−⋅−=−+−=Γ=Γ −−−− rrgggrg

We now solve the field equations:

[ ( ) ( ) ( ) ( )][ ( ) ( ) ( ) ( )][ ( ) ( ) ( ) ( )][ ( ) ( ) ( ) ( )]3

33300

330

303

323

200

320

203

313

100

310

103

303

000

300

003

33,00

30,03

232

300

230

302

222

200

220

202

212

100

210

102

202

000

200

002

22,00

20,02

131

300

130

301

121

200

120

201

111

100

110

101

101

000

100

001

11,00

10,01

030

300

030

300

020

200

020

200

010

100

010

100

000

000

000

000

00,00

00,0000

ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+Γ−Γ+

ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+Γ−Γ+

ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+Γ−Γ+

ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+Γ−Γ=R

GENERAL RELATIVITY, TENSOR ANALYSIS AND GEOMETRY

Courtney James Mewton Page 4 GR, Tensor Analysis & Geometry

( ) ( ) ( ) ( )][ ( ) ( ) ( ) ( )]

( ) ( ) ( ) ( )]( ) ( ) ( ) ( )]3

333

11331

313

323

211

321

213

313

111

311

113

303

011

301

013

33,11

31,13

232

311

231

312

222

211

221

212

212

111

211

112

202

011

201

012

22,11

21,12

131

311

131

311

121

211

121

211

111

111

111

111

101

011

101

011

11,11

11,11

030

311

031

310

020

211

021

210

010

111

011

110

000

011

001

010

00,11

01,1011

ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+Γ− Γ+

ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+Γ− Γ+

ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+Γ−Γ+

ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+Γ− Γ=R

[ ( ) ( ) ( ) ( )][ ( ) ( ) ( ) ( )][ ( ) ( ) ( ) ( )][ ( ) ( ) ( ) ( )]3

33322

332

323

323

222

322

223

313

122

312

123

303

022

302

023

33,22

32,23

232

322

232

322

222

222

222

222

212

122

212

122

202

022

202

022

22,22

22,22

131

322

132

321

121

222

122

221

111

122

112

121

101

022

102

021

11,22

12,21

030

322

032

320

020

222

022

220

010

122

012

120

000

022

002

020

00,22

02,2022

ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+Γ−Γ+

ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+Γ−Γ+

ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+Γ−Γ+

ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+Γ−Γ=R

[ ( ) ( ) ( ) ( )][ ( ) ( ) ( ) ( )][ ( ) ( ) ( ) ( )][ ( ) ( ) ( ) ( )]3

33333

333

333

323

233

323

233

313

133

313

133

303

033

303

033

33,33

33,33

232

333

233

332

222

233

223

232

212

133

213

132

202

033

203

032

22,33

23,32

131

333

133

331

121

233

123

231

111

133

113

131

101

033

103

031

11,33

13,31

030

333

033

330

020

233

023

230

010

133

013

130

000

033

003

030

00,33

03,3033

ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+Γ−Γ+

ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+Γ−Γ+

ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+Γ−Γ+

ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+ΓΓ−ΓΓ+Γ−Γ=R

We are left with3

13100

212

100

111

100

100

001

11,0000 ΓΓ−ΓΓ−ΓΓ−ΓΓ+Γ−=R ,

331

313

313

111

31,13

221

212

212

111

21,12

010

111

001

010

01,1011 ΓΓ+ΓΓ−Γ+ΓΓ+ΓΓ−Γ+ΓΓ−ΓΓ+Γ=R ,

332

323

313

122

32,23

122

221

111

122

11,22

010

12222 ΓΓ+ΓΓ−Γ+ΓΓ+ΓΓ−Γ−ΓΓ−=R ,

233

332

212

133

22,33

133

331

111

133

11,33

010

13333 ΓΓ+ΓΓ−Γ−ΓΓ+ΓΓ−Γ−ΓΓ−=R .

These equations must equal zero, thus after substitution, we have:

Br

A

A

AAA

B

ABR

142400

′

−′′

+′′

−′′

= , (1)

Br

AB

B

BAA

A

AR

′−

′′−

′′+

′−=

424

2

11 , (2)

11

22 222 −+′

−′

=BB

rB

AB

rAR , (3)

θθ 222

2233 sinsin1

122

RBB

rB

AB

rAR =

−+

′−

′= . (4)

Equation 2 becomes:

B

BAA

A

A

Br

AB

Br

AB

B

BAA

A

A

4244240

22 ′′−

′′+

′−=

′⇒

′−

′′−

′′+

′−= .

This can be substituted into equation 1 to give:

A

A

B

B

r

A

Br

AB

Br

A

Br

AB ′−=

′⇒

′−=

′⇒

′

−′

−= 10 .

This can also be written as:

GENERAL RELATIVITY, TENSOR ANALYSIS AND GEOMETRY

Courtney James Mewton Page 5 GR, Tensor Analysis & Geometry

11 =⇒=⇒−=⇔−= ABA

BA

dA

B

dB

Adr

dA

Bdr

dB,

upon solving the simple differential equation. We use this solution to simplify equation 3:

( )′=′+′=+′=⇔−+′

+′

= ArrArAArAAA

rArA11

220

We now integrate:

( ) ArkrdrArdr

ddr =+⇒= ∫∫ ,

where k is an integration constant. The equation can be rearranged to find A:

r

kA +=1 .

Since 1−= AB ,1

1−

+=

r

kB .

The value of k is the last thing to obtain. In the next section on the approximation of Newtonian gravitation,

0000 1 hg += . It can immediately be seen that the 00h is equivalent to k. In the Newtonian approximation, it

is required that the Newtonian gravitational potential 002

21 hcV = . Using the Newtonian potential

rGMV −= , this gives a value of 22 rcGMk −= . By substituting A and B back into the original li ne-

element equation at the start of this section, we have the Schwarzchild solution:

( )222221

222

222 sin

21

21 φθθτ ddrdr

rc

GMdtc

rc

GMdc +−

−−

−=

−

.

Utili sing Geodesic Equation to Find GR Approximation o f Newtonian GravityA particle travels through spacetime along a geodesic, given by the equation:

02

=Γ+τττ

σνµ

νσ

µ

d

dx

d

dx

d

dx.

τ is the time experienced relative to the particle. The equation simply states that relative to a free particle, itexperiences no net acceleration (though other objects appear to accelerate if a gravitational field is present).

We wish to determine the motion of the particle relative to coordinate time, denoted by t. The equation wouldgive the path of the particle in accordance to what other observers would see if they thought they were in agravitational field.

With the above said, the following equation can be immediately written, transforming from proper time tocoordinate time:

dt

dx

d

dt

d

td

dt

dx

dt

dx

dt

xd µσνµ

νσ

µ

ττ

=Γ+

2

22

2

2

2

.

First, we expand and consider its spatial components:

dt

dx

d

dt

d

td

dt

dx

dt

dx

dt

dx

dt

dx

dt

dx

dt

dx

dt

xd ii

kik

kjijk

i

=Γ+Γ+Γ+

2

22

200

00

0

02

2

2ττ

.

One of the Christoffel symbols has a coefficient of two since dt

dx

dt

dx

dt

dx

dt

dx kik

jij

0

0

0

0 Γ=Γ . We simpli fy the

equation:

dt

dx

d

dt

d

tdc

dt

dxc

dt

dx

dt

dx

dt

xd ii

kik

kjijk

i

=Γ+Γ+Γ+

2

22

2

002

02

2

2ττ

.

GENERAL RELATIVITY, TENSOR ANALYSIS AND GEOMETRY

Courtney James Mewton Page 6 GR, Tensor Analysis & Geometry

We assume that the gravitational field is quasi-static, i.e. that it doesn't change with respect to time. Therefore,any derivatives of the metric tensor with respect to time can be left out. Now we evaluate the connectioncoeff icients:

∂∂

−∂∂

+∂∂

=Γ=Γ ρρρ

ρ

ρρ

x

g

x

g

x

ggg jk

j

k

k

ji

jkii

jk 22

1. These derivatives are quite small , and can be neglected.1

( )

∂∂

−∂∂

+∂∂

+≈

∂∂

−∂∂

+∂∂

=Γ ρρρρρ

ρρρ

ρ

ηx

h

x

h

x

hh

x

g

x

g

x

gg kk

k

iikk

k

iik

00

000

00 2

22

( )

∂∂

−∂∂

−≈

∂∂

−∂∂

+∂∂

+≈sk

ksiskk

k

ii

x

h

x

h

x

h

x

h

x

hh 000

0

0 δη ρρρρρ on neglecting terms involving

0x

h

∂∂ µν

.

z

izii

x

h

x

g

x

g

x

gg

∂∂

≈

∂∂

−∂∂

+∂∂

=Γ 00000

000

00 22

δκ

κκκ

.

Now we need to evaluate the RHS of the equation. We start by first looking at the following line element:

( ) ( )dt

dx

dt

dxh

cdt

dx

dt

dxh

cdt

ddxdxgdc

νµ

µν

νµ

µνµννµ

µν ηττ +=+=

⇔= 1

1122

222 . Neglecting

terms involving the spatial components, which are small in comparison to the temporal components, we get:

( ) ( ) ( )002121

0000

2

111 hhdt

dh

dt

d +≈+=⇒+=

ττ

.

Now 00000

2

2

dx

dhc

dt

dh

dt

d ==τ, so the RHS of our major equation becomes:

( )dt

dxh

dx

dhc

dt

dx

hdx

dhc

d

dt

d

td

dt

dx iii

0021

000

0021

000

2

22

2

11

−≈+

≈

ττ.

This is negligible since it involves temporal derivatives of the gravitational field.

By plugging everything into our equation, we get:

02

002002

2

=∂∂

+

∂∂

−∂∂

−z

izk

sk

ksis

i

x

hc

dt

dx

x

h

x

h

dt

xd δδ .

Through multiplying by mass m and rearranging the equation, we get:

dt

dx

x

h

x

hm

x

hmc

dt

xdm

k

sk

ksis

z

izi

∂∂

−∂∂

+∂∂

−= 000022

2

2δδ

.

The term on the left is the force that the particle appears to experience. The first term on the right is some kindof potential of the field, since its temporal component is involved. The last term, which involves perpendicularvelocities, is indicative of some sort of Coriolis force. We are not interested in the Coriolis effects, so we shallassume that we are in a non-rotating frame, so we get:

z

izi

x

hmc

dt

xdm

∂∂

−= 0022

2

2

δ.

If we denote a potential by 002

21 hcV = , then the equation simply becomes VmF ∇−= , or

z

izi

x

V

dt

xd

∂∂−= δ

2

2

. We want the metric tensor to be flat when there is no gravitational field present. The

1 The metric tensor µνµνµν η hg +≈ , where µνη is the famili ar metric tensor of Special Relativity, and

µνh are small terms which include the action of any gravitational fields which may be present, and are small in

comparison to the µνη in weak gravitational fields, as opposed to the awesome sucking power of a black hole!

GENERAL RELATIVITY, TENSOR ANALYSIS AND GEOMETRY

Courtney James Mewton Page 7 GR, Tensor Analysis & Geometry

equation for the potential leads us to the expression 2000000 21 cVhg +=+=η . To finally get the actual

expression for the potential in terms of mass, we need to use the line element from the Schwarzchild solution.

The temporal component gives rGMV −= . With this expression, we can easily obtain an approximation of

the gravitational force:2rGMmVmF −=∇−= .

Field Equations in the Presence of Matter: The Poisson ApproximationLet us write the equation:

µνµνµν κTgR =− 21 ,

or, more compactly as:µνµν κTG = ,

where G is the Einstein tensor.

As a test for General Relativity, at velocities which are small in comparison to the speed of light, there must be

an approximation to Poisson’s equation: ρπGV 42 =∇ . To achieve this requires the weak field

approximation by leaving out negligible terms in the Ricci tensor.2

µνµν κTG =µνµνµν κTRgR =− 2

1

µνµνµν

µνµν

µν κ gTRggRg =− 21

µνµνκ gTRR =− 2

µνµνκ gTR −=∴

We substitute this back into our original equation:

µνµνµνµνµνµνµν κκκ gTgTRgTR 2

121 −=+=

We are approximating that the material energy tensor has a negligible value in for all µ, ν except when µ = ν =0, so we get:

0021

000000

210000 TgTgTR κκκ =−= .

We now make the Ricci tensor covariant:

0021

00000000

2100

0000 TRggTRgg κκ =⇔=Weak field approximation.

( ) ααµνααµνναµαµαναµνααµν ,21

,,,,21 gggggR ≈+−−≈

Vc

ggR 2200

221

,0021

00

1 ∇=∇≈≈∴ αα

0021

00212

2

1vvTV

cκρκ ==∇∴

The particles are traveling at the travelling at the speed of light through time, so we get:4

212 cV κρ=∇

This must equal Poisson’s equation, stated earlier

This gives a value 4

8

c

Gπκ = .

2 Note that this particular calculation would be shorter if we took the Einstein tensor and the material energytensor to be covariant as opposed to contravariant, but due to the actual form of the material energy tensor, Iprefer it to be contravariant.