gravitational wave physics

Books
1 General Relativity
1.1 Classical
• Gravitation and Cosmology S. Weinberg , Wiley (1972)
• Gravitation Charles W. Misner, Kip S. Thorne and John A. Wheeler, Freeman (1973)
• Problem Book in Relativity and Gravitation A.P. Lightman, W.H. Press, R.H. Price and S.A.Teukolky,
Princeton (1975)
1.2 Textbooks
• A First Course in General Relativity B.F.Schutz, Cambridge (1986)
• Gravitation and Spacetime Hans Ohanian and Remo Ruffini, W.W.Norton (1994)
• GRAVITY : an introduction to Einstein’s General Relativity J.B. Hartle, Addison-Wesley
(2003)
• Relativity: An Introduction to Special and General Relativity Hans Stefani, Cambridge (2004)
(also in German)
• An Introduction to General Relativity: SPACETIME and GEOMETRY S.M. Carroll, Addisson-
Wesley (2004)
• Relativity, Gravitation and Cosmology : A Basic Introduction Ta-Pei Cheng, Oxford (2005)
• Relativity : Special, General and Cosmological W.Rindler, Oxford (2006)
• General Relativity : An Introduction for Physicists M.P. Hobson, G. Efstathiou and A.N.
Lasenby, Cambridge (2006)
2 Neutron Stars, Relativistic Astrophysics
• Compact Stars: Nuclear Physics, Particle Physics and General Relativity Norman K. Glenden-
ning, Springer (2000)
• Black Holes, White Dwarfs and Neutron Stars Stuart L. Shapiro and Saul A. Teukolsky, Willey
(1983)
• NEUTRON STARS I : Equations of State and Structure P. Haensel, A.Y. Potekhin, D.G.
Yakovlev, Springer (2007)
1
Tensor Algebra
A Short Introduction to Tensors
Kostas Kokkotas

May 2, 2007
Kostas Kokkotas A Short Introduction to Tensors
Tensor Algebra
Tensors: Scalars and Vectors
Any physical quantity, e.g. the velocity of a particle, is determined
by a set of numerical values - its components - which depend on
the coordinate system. Studying the way in which these values
change with the coordinate sys tem leads to the concept of tensor.
With the help of this concept we can express the physical laws by
tensor equations, which have the same form in every coordinate
system.

Scalar field : is any physical quantity determined by a single
numerical value i.e. just one component which is independent
of the coordinate system (mass, charge, )

Vector field (contravariant): an example is the infinitesimal
displacement vector, leading from a point A with coordinates
x
µ
to a neighb ouring point A

with coordinates x
µ
+ dx
µ
. The
components of such a vector are the differentials dx
µ
.

Kostas Kokkotas A Short Introduction to Tensors
Tensor Algebra
Tensors: Scalars and Vectors
Any physical quantity, e.g. the velocity of a particle, is determined
by a set of numerical values - its components - which depend on
the coordinate system. Studying the way in which these values
change with the coordinate sys tem leads to the concept of tensor.
With the help of this concept we can express the physical laws by
tensor equations, which have the same form in every coordinate
system.

Scalar field : is any physical quantity determined by a single
numerical value i.e. just one component which is independent
of the coordinate system (mass, charge, )

Vector field (contravariant): an example is the infinitesimal
displacement vector, leading from a point A with coordinates
x
µ
to a neighb ouring point A

with coordinates x
µ
+ dx
µ
. The
components of such a vector are the differentials dx
µ
.
Kostas Kokkotas A Short Introduction to Tensors

Tensor Algebra
Tensors: Vector Transformations
From the infinitesimal vector

AA

with components dx
µ
we can
construct a finite vector v
µ
defined at A. This will be the tangent
vector of the curve x
µ
= f
µ
(λ) where the points A and A

correspond to the values λ and λ + dλ of the parameter. Then
v
µ
=
dx
µ
dλ
(1)
Any transformation from x
µ
to ˜x
µ

(x
µ
→ ˜x
µ
) will be determined
by n equations of the form: ˜x
µ
= f
µ
(x
ν
) where µ , ν = 1, 2, , n.
This means that :
d ˜x
µ
=

ν
∂˜x
µ
∂x
ν
dx
ν
=

ν
∂f
µ
∂x

ν
dx
ν
for ν = 1, , n (2)
˜v
µ
=
d ˜x
µ
dλ
=

ν
∂˜x
µ
∂x
ν
dx
ν
dλ
=

ν
∂˜x
µ
∂x
ν
v
ν
(3)

Kostas Kokkotas A Short Introduction to Tensors
Tensor Algebra
Tensors: Contravariant and Covariant Vectors
Contravariant Vector: is a quantity with n components
depending on the coordinate system in such a way that the
components a
µ
in the coordinate system x
µ
are related to the
components ˜a
µ
in ˜x
µ
by a relation of the form
˜a
µ
=

ν
∂˜x
µ
∂x
ν
a
ν
(4)
Covariant Vector: eg. b
µ
, is an object with n components which

depend on the coordinate system on such a way that if a
µ
is any
contravariant vector, the following sums are scalars

µ
b
µ
a
µ
=

µ
˜
b
µ
˜a
µ
for any x
µ
→ ˜x
µ
[Scalar Product] (5)
The covariant vector will transform as:
˜
b
µ
=

ν

∂x
ν
∂˜x
µ
b
ν
(6)
Kostas Kokkotas A Short Introduction to Tensors
Tensor Algebra
Tensors: at last
A conravariant tensor of order 2 is a quantity having n
2
components T
µν
which transforms (x
µ
→ ˜x
µ
) in such a way that,
if a
µ
and b
µ
are arbitrary covariant vectors the following sums are
scalars:
T
λµ
a
µ
b

λ
=
˜
T
λµ
˜a
λ
˜
b
µ
≡ φ (7)
Then the transformation formulae for the components of the
tensors of order 2 are (why?):
˜
T
αβ
=
∂˜x
α
∂x
µ
∂˜x
β
∂x
ν
T
µν
,
˜
T

α
β
=
∂˜x
α
∂x
µ
∂x
ν
∂˜x
β
T
µ
ν
&
˜
T
αβ
=
∂x
µ
∂˜x
α
∂x
ν
∂˜x
β
T
µν
The Kronecker symbol δ

λ
µ
is a mixed tensor having fram e
independent values for its components.
Kostas Kokkotas A Short Introduction to Tensors
Tensor Algebra
Tensors: tensor algebra
Tensor multiplication : The product of two vectors is a tensor of
order 2, because
˜a
α
˜
b
β
=
∂˜x
α
∂x
µ
∂˜x
β
∂x
ν
a
µ
b
ν
(8)
in general:
T

µν
= A
µ
B
ν
or T
µ
ν
= A
µ
B
ν
or T
µν
= A
µ
B
ν
(9)
Contraction: for any mixed tensor of order (p, q) leads to a tensor
of order (p − 1, q − 1)
T
λµν
λα
= T
µν
α
(10)
Symmetric Tensor : T
λµ

= T
µλ
orT
(λµ)
, T
νλµ
= T
νµλ
or T
ν(λµ)
Antisymmetric : T
λµ
= −T
µλ
or T
[λµ]
, T
νλµ
= −T
νµλ
or T
ν[λµ]
No of independent components :
Symmetric : n(n + 1)/2, Antisymmetric : n(n − 1)/2
Kostas Kokkotas A Short Introduction to Tensors
Tensor Algebra
Tensors: Differentiation
The simplest tensor field is a scalar field φ = φ(x
α
) and its

derivatives are the c omponents of a covariant tensor!
∂φ
∂˜x
λ
=
∂x
α
∂˜x
λ
∂φ
∂x
α
we will use:
∂φ
∂x
α
= φ
,α
(11)
i.e. φ
,α
is the gradient of the scalar field φ.
The derivative of a contravariant vector field A
µ
is :
A
µ
,α
≡
∂A

µ
∂x
α
=
∂
∂x
α

∂x
µ
∂˜x
ν
˜
A
ν

=
∂˜x
ρ
∂x
α
∂
∂˜x
ρ

∂x
µ
∂˜x
ν
˜

A
ν

=
∂
2
x
µ
∂˜x
ν
∂˜x
ρ
∂˜x
ρ
∂x
α
˜
A
ν
+
∂x
µ
∂˜x
ν
∂˜x
ρ
∂x
α
∂
˜

A
ν
∂˜x
ρ
(12)
Without thefirst term in the right hand side this equation would be
the transformation formula for a covariant tensor of order 2.
Kostas Kokkotas A Short Introduction to Tensors
Tensor Algebra
Tensors: Connections
The transformation (x
µ
→ ˜x
µ
) of the derivative of a vector is:
A
µ
,α
=
∂x
µ
∂˜x
ν
∂˜x
ρ
∂x
α
(
˜
A

ν
,ρ
+
∂
2
x
κ
∂˜x
σ
∂˜x
ρ
∂x
ν
∂˜x
κ
  
˜
Γ
ν
σρ
˜
A
σ
) (13)
in another coordinate (x
µ
→ x
µ
) we get again
A

µ
,α
=
∂x
µ
∂x
ν
∂x
ρ
∂x
α

A
ν
,ρ
+ Γ
ν
σρ
A
σ

. (14)
Suggesting that the transformation (˜x
µ
→ x
µ
) will be:
˜
A
µ

,α
+
˜
Γ
µ
αλ
˜
A
λ
=
∂˜x
µ
∂x
ν
∂x
ρ
∂˜x
α

A
ν
,ρ
+ Γ
ν
σρ
A
σ

(15)
The necessary and sufficie nt condition for A

µ
,α
to be a tensor is:
Γ
λ
ρν
=
∂
2
x
µ
∂x
ν
∂x
ρ
∂x
λ
∂x
µ
+
∂x
κ
∂x
ρ
∂x
σ
∂x
ν
∂x
λ

∂x
µ
Γ
µ
κσ
. (16)
Γ
λ
ρν
is the called the connection of the space.
Kostas Kokkotas A Short Introduction to Tensors
Tensor Algebra
Tensors: Covariant Derivative
According to the previous assumptions, the following quantity
transforms as a tensor of order 2
A
µ
;α
= A
µ
,α
+ Γ
µ
αλ
A
λ
(17)
and is called covariant derivative of the contravariant vector A
µ
.

In similar way we get (how?) :
φ
;λ
= φ
,λ
(18)
A
λ;µ
= A
λ,µ
− Γ
ρ
µλ
a
ρ
(19)
T
λµ
;ν
= T
λµ
,ν
+ Γ
λ
αν
T
αµ
+ Γ
µ
αν

T
λα
(20)
T
λ
µ;ν
= T
λ
µ,ν
+ Γ
λ
αν
T
α
µ
− Γ
α
µν
T
λ
α
(21)
T
λµ;ν
= T
λµ,ν
− Γ
α
λν
T

µα
− Γ
α
µν
T
λα
(22)
T
λµ···
νρ··· ;σ
= T
λµ···
νρ··· ,σ
+ Γ
λ
ασ
T
αµ···
νρ···
+ Γ
µ
ασ
T
λα···
νρ···
+ · · ·
− Γ
α
νσ
T

λµ···
αρ···
− Γ
α
ρσ
T
λµ···
να···
− · · · (23)
Kostas Kokkotas A Short Introduction to Tensors
Tensor Algebra
Tensors: Parallel Transport
Γ
λ
µν
helps to determine a vector A
λ
, at a point P

, which has to be
considered as equivalent to the vector a
λ
given at P.
δ
(1)
a
µ
= a
µ
(P


) − a
µ
(P) = a
µ
(P) + a
µ,ν
dx
ν
− a
µ
(P) = a
µ,ν
dx
ν
a
µ
(P

) − A
µ
(P

)
  
vector
= a
µ
+ δ
(1)

a
µ
  
at point P
− (a
µ
+ δa
µ
)
  
at point P
= δ
(1)
a
µ
− δa
µ
  
vector
= a
µ,ν
dx
ν
− δa
µ
  
vector
=

a

µ,ν
− C
λ
µν
a
λ

dx
ν
i.e. δa
µ
= C
λ
µν
a
λ
dx
ν
δa
µ
= Γ
λ
µν
a
λ
dx
ν
for covariant vectors (24)
δa
µ

= −Γ
µ
λν
a
λ
dx
ν
for contravariant vectors (25)
The connection Γ
λ
µν
allows to define the transport of a vector a
λ
from a point P to a neighbouring point P

(Parallel Transport).
The parallel transport of a scalar field is zero! δφ = 0 (why?)
Kostas Kokkotas A Short Introduction to Tensors
Tensor Algebra
Tensors: Curvature Tensor
The ”trip” of a parallel transported vector along a closed path
leads to the following total change:
δa
λ
= −
1
2
a
β
R

λ
βνσ
(dx
σ
δx
ν
− dx
ν
δx
σ
) (26)
where
R
λ
βνσ
= −Γ
λ
βν,σ
+ Γ
λ
βσ,ν
− Γ
µ
βν
Γ
λ
µσ
+ Γ
µ
βσ

Γ
λ
µν
(27)
is the curvature tensor.
Kostas Kokkotas A Short Introduction to Tensors
Tensor Algebra
Tensors: Geodesics

For a vector u
λ
at point P we apply the parallel transport
along a curve on an n-dimensional space which will be given
by n equations of the form: x
µ
= f
µ
(λ); µ = 1, 2, , n

If u
µ
=
dx
µ
d λ
is the tangent vector at P the parallel transport of
this vector will determine at another point of the curve a
vector which will not be in general tangent to the curve.

If the transported vector is tangent to any point of the curve

then this curve is a geodesic curve of this space and is given
by the equation :
du
ρ
dλ
+ Γ
ρ
µν
u
µ
u
ν
= 0 . (28)

Geodesic curves are the shortest curves connecting two points
of a curved space.
Kostas Kokkotas A Short Introduction to Tensors
Tensor Algebra
Tensors: Metric Tensor
The distance ds of two points P(x
µ
) and P

(x
µ
+ dx
µ
) is given by
ds
2

=

dx
1

2
+

dx
2

2
+

dx
3

2
(29)
In another coordinate system, ˜x
µ
, we will get
dx
ν
=

α
∂x
ν
∂˜x

α
d ˜x
α
(30)
which leads to: ds
2
= ˜g
µν
d ˜x
µ
d ˜x
ν
= g
αβ
dx
α
dx
β
.
which gives the following transformation relation:
˜g
µν
=
∂˜x
µ
∂x
α
∂˜x
ν
∂x

β
g
αβ
(31)
suggesting that the quantity g
µν
is a symmetric tensor, the so
called metric tensor.
Properties:
g
µν
A
µ
= A
ν
, g
µν
T
µα
= T
α
ν
, g
µν
T
µ
α
= T
αν
, g

µν
g
ασ
T
µα
= T
νσ
Kostas Kokkotas A Short Introduction to Tensors
Tensor Algebra
Tensors: Metric Tensor - Christoffel Symbols

Metric element for Minkowski spacetime
ds
2
= −dt
2
+ dx
2
+ dy
2
+ dz
2
(32)
ds
2
= −dt
2
+ dr
2
+ r

2
dθ
2
+ r
2
sin
2
θdφ
2
(33)

For a sphere with radius R : ds
2
= R
2

dθ
2
+ sin
2
θdφ
2


The metric element of a torus with radii a and b
ds
2
= a
2
dφ

2
+ (b + a sin φ)
2
dθ
2
(34)
The contravariant form of the metric tensor:
g
µα
g
αβ
= δ
β
µ
where g
αβ
=
1
det |g
µν
|
G
αβ
(35)
The angle between two infinitesimal vectors d
(1)
x
α
and d
(1)

x
α
is:
cos(ψ) =
g
αβ
d
(1)
x
α
d
(2)
x
β

g
ρσ
d
(1)
x
ρ
d
(2)
x
σ

g
µν
d
(1)

x
µ
d
(2)
x
ν
. (36)
Kostas Kokkotas A Short Introduction to Tensors
Tensor Algebra
Tensors: Christoffel Symbols
The covariant derivative of the metric tensor is
g
µν;ρ
= g
µν,ρ
− g
µσ
Γ
σ
νρ
− g
σν
Γ
σ
µρ
= 0 . (37)
i.e. g
µν
is covariantly constant.
This leads to a unique determination of the connections of the

space (Riemannian space)
Γ
α
µρ
=
1
2
g
αν
(g
µν,ρ
+ g
νρ,µ
− g
ρµ,ν
) . (38)
which are called Christoffel Symbols .
It is obvious that Γ
α
µρ
= Γ
α
ρµ
.
Kostas Kokkotas A Short Introduction to Tensors
Tensor Algebra
Tensors: Riemann Tensor
When in a space we define a metric then is called metric space or
Riemann space. For such a space the curvature tensor
R

λ
βνµ
= −Γ
λ
βν,µ
+ Γ
λ
βµ,ν
− Γ
σ
βν
Γ
λ
σµ
+ Γ
σ
βµ
Γ
λ
σν
(39)
is called Riemann Tensor.
R
κβνµ
= g
κλ
R
λ
βνµ
=

1
2
(g
κµ,βν
+ g
βν,κµ
− g
κν,βµ
− g
βµ,κν
)
+ g
αρ

Γ
α
κµ
Γ
ρ
βν
− Γ
α
κν
Γ
ρ
βµ

Properties of the Riemann Tensor:
R
κβνµ

= −R
κβµν
, R
κβνµ
= −R
βκνµ
, R
κβνµ
= R
νµκβ
, R
κ[βµν]
= 0
Thus in an n-dimensional space the number of independent
components is:
n
2
(n
2
− 1)/12 (40)
For a 4-dimensional space only 20 independent components
Kostas Kokkotas A Short Introduction to Tensors
Tensor Algebra
Tensors : Ricci and Einstein
The contraction of the Riemann tensor leads to Ricci Tensor
R
αβ
= R
λ
αλβ

= g
λµ
R
λαµβ
= Γ
µ
αβ,µ
− Γ
µ
αµ,β
+ Γ
µ
αβ
Γ
ν
νµ
− Γ
µ
αν
Γ
ν
βµ
(41)
which is symmetric R
αβ
= R
βα
. Further contraction leads to the
Ricci Scalar
R = R

α
α
= g
αβ
R
αβ
= g
αβ
g
µν
R
µανβ
. (42)
The following combination of Riemann and Ricci tensors is called
Einstein Tensor
G
µν
= R
µν
−
1
2
g
µν
R (43)
with the very important property:
G
µ
ν;µ
=


R
µ
ν
−
1
2
δ
µ
ν
R

;µ
= 0 . (44)
Kostas Kokkotas A Short Introduction to Tensors
Tensor Algebra
Tensors : Flat and Empty Spacetimes
R
αβµν
= 0 flat spacetime
R
µν
= 0 empty spacetime
Prove that :
a
λ
;µ;ν
− a
λ
;ν;µ

= −R
λ
κµν
a
κ
Kostas Kokkotas A Short Introduction to Tensors
Tensor Algebra
Tensors : Euler-Lagrange Eqns vs Geodesic Eqns
The Lagrangian for a freely moving particle is: L = g
µν
u
µ
u
ν
and
the Euler-Lagrange equations:
d
ds

∂L
∂u
µ

−
∂L
∂x
µ
= 0
are equivalent to the geodesic equation
du

ρ
ds
+ Γ
ρ
µν
u
µ
u
ν
= 0 or
d
2
x
ρ
ds
2
+ Γ
ρ
µν
dx
µ
ds
dx
ν
ds
= 0
If we know the tangent vector u
ρ
at a given point of a known
space we can determine the geodesic curve. Which will be called:


|

u|
2
< 0 timelike

|

u|
2
= 0 null where |

u|
2
= g
µν
u
µ
u
ν

|

u|
2
> 0 spacelike
Kostas Kokkotas A Short Introduction to Tensors
Tensor Algebra
Tensors : An example for parallel transport I

A vector

A = A
1

e
θ
+ A
2

e
φ
be parallel transported along a closed
line on the surface of a sphere with metric ds
2
= dθ
2
+ sin
2
dφ
2
and Christoffel symbols Γ
1
22
= − sin θ cos θ and Γ
2
12
= cot θ . The
eqns δA
α

= −Γ
α
µν
A
µ
dx
ν
for parallel transport will be written as:
∂A
1
∂x
2
= −Γ
1
22
A
2
⇒
∂A
1
∂φ
= sin θ cos θA
2
∂A
2
∂x
2
= −Γ
2
12

A
1
⇒
∂A
2
∂φ
= − cot θA
1
Kostas Kokkotas A Short Introduction to Tensors
Tensor Algebra
Tensors : An example for parallel transport II
The solutions will be:
∂
2
A
1
∂φ
2
= − cos
2
θA
1
⇒ A
1
= α cos(φ cos θ) + β sin(φ cos θ)
⇒ A
2
= − [α sin(φ cos θ) − β cos(φ cos θ)] sin
−1
θ

and for a unit vector (A
1
, A
2
) = (1, 0) at (θ, φ) = (θ
0
, 0) the
intebration constants will be α = 1 and β = 0. The solution is:

A = A
1

e
θ
+ A
2

e
φ
= cos(2π cos θ)

e
θ
−
sin(2π cos θ)
sin θ

e
φ
i.e. different components but the measure is still the same

|

A|
2
= g
µν
A
µ
A
ν
=

A
1

2
+ sin
2
θ

A
2

2
= cos
2
(2π cos θ) + sin
2
θ
sin

2
(2π cos θ)
sin
2
θ
= 1
Kostas Kokkotas A Short Introduction to Tensors
Tensor Algebra
A Short Introduction to Tensors
Kostas Kokkotas
May 2, 2007
Kostas Kokkotas A Short Introduction to Tensors
Tensor Algebra
Tensors: Scalars and Vectors
Any physical quantity, e.g. the velocity of a particle, is determined
by a set of numerical values - its components - which depend on
the coordinate system. Studying the way in which these values
change with the coordinate sys tem leads to the concept of tensor.
With the help of this concept we can express the physical laws by
tensor equations, which have the same form in every coordinate
system.

Scalar field : is any physical quantity determined by a single
numerical value i.e. just one component which is independent
of the coordinate system (mass, charge, )

Vector field (contravariant): an example is the infinitesimal
displacement vector, leading from a point A with coordinates
x
µ

to a neighb ouring point A

with coordinates x
µ
+ dx
µ
. The
components of such a vector are the differentials dx
µ
.
Kostas Kokkotas A Short Introduction to Tensors