GRAVITATION F10
S. G. RAJEEV
Lecture 6
1. G ENERAL C O - ORDINATES
1.1. The choice of co-ordinate system should be adapted to the system
being studied. For example, curvilinear co-ordinates are useful in solving
the Laplace equation in various geometries.
GRAVITATION F10
S. G. RAJEEV
Lecture 3
1. Maxwells Equations
Read the book by Jackson on Classical Electrodynamics. Or the second volume
of the series by Landau and Lifshitz Classical Theory of Fields.
1.1. All magnetic elds must have zero divergence.
B=0
GRAVITATION F10
S. G. RAJEEV
Lecture 8
1. The Geodesic Equation
1.1. Riemann discovered the essential features of metric geometry in arbitrary dimensions. The key idea is that the distance between nearby points
ds2 = g dx dx
contains all the essential inf
GRAVITATION F10
S. G. RAJEEV
Lecture 13
1. The Vacuum Einsteins Equation
1.1. The metric tensor of space-time satises a nonlinear Partial Dierential Equation which determines it given initial conditions. The metric
tensor describes the gravitational eld.
GRAVITATION F10
S. G. RAJEEV
Lecture 9
1. Parallel Transport
1.1. The partial derivatives of the components of a vector do not transform as a tensor under nonlinear transformations of co-ordinates.
v =
x
v
x
By repeated use of the chain rule,
v
x x v
GRAVITATION F10
S. G. RAJEEV
Lecture 4
1. Variational Principles
1.1. The fundamental laws of classical physics are dierential equations
which arise from variational principles.
1.1.1. Not all dierential equations follow from variational principles. It is
GRAVITATION F10
S. G. RAJEEV
Lecture 11
1. The Wave Equation
1.1. The amplitude of a small wave propagating with speed satises.
1 2
2 = 0
2 2
1.1.1. Plane waves are solutions
() = [kx] ,
2
k2 = 0
2
.
1.2. In Lorentz invariant form the wave equation is.
GRAVITATION F10
S. G. RAJEEV
Lecture 7
1. The Sphere
1.1. The geometry of the sphere was studied by the ancients. There were
two spheres of interest to astronomers: the surface of the Earth and the celestial
sphere, upon which we see the stars. Eratosthen
GRAVITATION F10
S. G. RAJEEV
Lecture 14
1. Gravitational Waves
1.1. There are exact solutions to Einsteins equations that describe gravitational waves. Could it be that gravitational waves are only approximate solutions to Einsteins equations, and not of
Lecture 1
1. R ELATIVITY
1.1. It is an astonishing physical fact that the speed of light is the same
for all observers. This is not true of other waves. For example, the speed
of sound measured by someone on standing on Earth is different from that
measur
GRAVITATION F10
S. G. RAJEEV
Lecture 10
1. Curvature
1.1. If the metric g is a constant, the Christoel symbols vanish and
the geodesics are straightlines. Thus the geometry is locally that of Euclidean
space.
1.1.1. But just because g depends on co-ordina
GRAVITATION F10
S. G. RAJEEV
Lecture 15
1. Centrally Symmetric Metrics
1.1. Schwarschild discovered a centrally symmetric and static solution to
the vacuum Einsteins equations. Originally this was thought of as describing
the exterior of a star: when the
GRAVITATION F10
S. G. RAJEEV
Lecture 5
1. The Principle of Equivalence
1.1. All particles have the same acceleration in a gravitational eld. By
a particle we mean here a body whose mass and size are small. Thus, the Earth
is a particle compared to the Sun
GRAVITATION F10
S. G. RAJEEV
Lecture 2
1. Causality
1.1. We will throughout use units in which the velocity of light is unity.
This means that length is measured in time units (e.g., light seconds).
1.2. There are three kinds of vectors in Minkowski space