GR NOTES
ALEXANDER J. WEAVER
1. Reading List and Projects
So ive been contacted by around 3 people with reading lists for their projects, I think it
pertinent to give you some good complimentary material, specically the HEP Inspire website.
It has a huge

GR Notes for Dec. 1st
Yaqi Han
Look at the metric g = + h under the condition that the source is a
far away purterbation from the observer(detector), i.e., |h |
1.
To the linear order the variation of the Einstein tensor would be
G =
1
h = 8GT
2
(1)
whe

Notes for Tuesday, November 25th, 2014
C. L. Wang
1
Orbiting Clock
Suppose there is a clock in a circular orbit in the Schwarzschild geometry,
lets gure out how much time elapses while moving along the orbit. First
we know the invariant line element in Sc

GENERAL RELATIVITY NOTES
ALEXANDER J. WEAVER
We start with 2 manifolds M, N and a smooth mapping between them, : M N , which
gives rise to a natural way to write coordinate functions of N in terms of those on M, y (x ) =
1
N (M (x ) where the are the coor

GR Lecture Notes October 27th, 2014
Bobby Bond
October 27, 2014
Schwarschild Solution
G (g) = 8GN T
(1)
Looking for vacuum solutions, set T = 0
1
G = R Rg = 0 R = 0
2
(2)
Looking for Static, Spherically Sym.
ds2 = h(R)dt2 + k(R)dR2 + f (R)d2
(3)
Let r2 =

GR October 13th Lecture
Bobby Bond
October 13, 2014
From the previous lectures we discussed the following equations
a=
(1)
2
= 4G
(2)
a = 4G
(3)
We want to show the connection between these equations to the Einstein equation. In
order to show the conne

Lecture Notes, October 1, 2014
Elisa Todarello
1
Parametrization of geodesics
The equation for a geodesic
dx
d
x
=0
can be derived from an action principle. Two possible actions are:
d g
I1 =
x x
=
g
d
x x
,
1
x x
d g
.
2
I2 is not invariant under a

GENERAL RELATIVITY
ALEXANDER J. WEAVER
So last time we were nishing our analysis of the schwartzchild star, where we had
ds2 = e2 dt2 + e2 dr2 + r2 d2
Where we had a basic uid that was unchanging in time so for pressure we have
p = ( + p)
and with
dm(r)
=

GR
ALEXANDER J. WEAVER
Last time we looked at the minkowski metric perturbed slightly
g = + h
Then we found the zero component was proportional to the eld potential, h00 = 2. What
we went through to deduce this was the minimal requirements necessary such

Lecture note 10/20
Dipsikha Debnath
October 30, 2014
Gravitational Wave
The equivalence of gravitational and inertial mass leads to an understanding of gravity as the physical
manifestation of the curvature of space-time.Lets look at Einstein equation
G =

GR
So last time we talked about the event horizons of the Schwartchild black holes. For a black
hole of mass M we broke up space time into 4 region. We dened the horizons as the boundary
of the set of the time-like curves leading to the scri+ boundary (ca

Lecture note 10/22
Dipsikha Debnath
October 30, 2014
Alternative theories of gravity
U
U
= lR U + l2 U
RU
Now we do dimensional analysis of the above equation U = dimension less
[l] = L
[ ] = L1
[R] = L2
In the second term in right hand side of the abov

General Relativity Notes
October 7th 2014
Recall our discussion from last class, the action we were talking about was:
Where
is the trajectory. We can write this as:
The stress tensor of the particle is:
Where,
Now,
Suppose we have a small volume of parti

Properties of Curvature
The stress tensor is the variation with the matter action with respect to the metric:
Recall our definition of the Riemann Tensor in a set of coordinates:
Another form we use a lot is to lower one index:
For a locally flat coord

GR December 8 Lecture Notes
Bobby Bond
December 8, 2014
We will look at Degrees of Freedom by setting the perturbed metric components to
certain values
g = + h
(1)
We will now set the values of h to the following
h00 = 2
(2)
h0i = +wi
(3)
hij = 2Sij 2ij
(

GR September 29 Lecture
Bobby Bond
September 29, 2014
p-folds
Dierential forms: A dierential p-form is a (0,p) tensor which is completely antisymmetric.
Example:
F = A A
(1)
Where 0 p n. A 0 form is fn and 1 form is a covector.
Wedge Product: p + q n. If

Lecture on Sept 23, 2014
(PHZ 6607 by Prof. B. Whiting)
Sankha Subhra Chakrabarty
Luyi Yan
The Metric
The metric is a (0,2) tensor with basis elements dx dx .
ds2 = g = g dx dx
(1)
Here ds2 is not the square of any dierential, it is the name of the tensor

GR Notes for Wednesday, September 24th, 2014
Peisen Ma
1
Tensor Densities
First lets recall the Levi-Civita symbol which is dened as
1 2 n = +1
(1)
if 1 2 n is an even permutation of 01 (n 1);
1 2 n = 1
(2)
if 1 2 n is an odd permutation of 01 (n 1);
1 2

Nicole Crisosto
PHZ 6607
GR Class Notes
8/29/2014
Last class recap:
A B = Ai B j gij
(A B)k =
Vi =
dxi (t)
dt
Fi =
i j
ijk A A
f (x)
xi
V i Fi =
df (xi (t)
dt
Some class generated denitions,
vector:
-components transform like coordinates (only applies to

Nicole Crisosto
PHZ 6607
GR Class Notes
9/3/2014
Lagrangian density for a scalar eld
1
L = g ij (
2
Scfw_ =
j ) (
i )
gd4 x
we are interested in the variation
S
1
=
2
( ggij
j
i )
+
1
2
i
gg ij
j
cfw_99%of the time g ij = g ji
so for a symmetric matrix,

General Relativity Notes: Sept. 9, 2014
Andy Chilton
1.
LORENTZ TRANSFORMS
Recall that, for a boost in the x-direction, the Lorentz Transform of the coordinates x and t are given by
v
x = (x vt) = x ct
c
v
ct x
t =
c
c
where =
previously,
1
v2
c2
(1)
(2)

Lecture Notes, September 30, 2014
Elisa Todarello
1
Covariant Derivatives
We dene the covariant derivative as
v =
v
+ v .
x
(1)
For this equation to hold, the connection has to transform in a given
way under coordinate transformations. The connection doe

General Relativity Notes
September 15th, 2014
Peisen Ma
1
Covariant Geometric Form of Maxwell Eqs.
We may start by writing the Maxwell Eqs. as the following form:
iB
i
iE
i
ijk
j Ek
ijk
j Bk
=
=
=
=
0
4
B i
E i + 4J i d
(1)
(2)
(3)
(4)
where i is a deriva

General Relativity: Lecture 5
Daniel Brooker and Dustin Tracy
9-08-2014
Last time we discussed a straight forward way for computing the connection
coecients from the lagrangian for a free particle in an arbitrary spacetime
specied by the metric g. Namely

Lecture on Sept 22, 2014
(PHZ 6607 by Prof. B. Whiting)
Sankha Subhra Chakrabarty
Vectors
In a manifold, M , a vector is an object associated with a single point in the
manifold. To be precise, a vector at a point P in the manifold is an element of
the ta

Lecture note 9/16
B. Bond, L. Yan
September 30, 2014
To have a better understanding of the equations in General Realtivity we should
look at similar equations, mainly Maxwells Equations, to get a better idea of
what each term represents and how we can com