1
Exercise 3
S = 2 = :=0
In these problems, assume a holonomic frame system and zero torsion so that
[ ]
1.1
Problem 3.1
The initial value structure of Maxwell Field Equations of Electromagnetism s may be understood from the identity [g gF
;
]; = 0
for an
1
1.1
Initial Value Structure of Einstein Equations s
Standard System
The traditional form of Einstein eld equations consists of the structure equas tions =0 (1) = 0; T Q that are usually solved for the connection coe cients in terms of the metric tensor
1
1.1
Equivalence Principals: The Weak, The Strong, and The Grand
Inertial Reference Frames
Einstein' theory of gravitation, also called general relativity, is more than just a s theory of gravity. It is a complete theory of physical interactions in the p
1
1.1
1.1.1
Basic Dierential Geometry
Tensorial Derivatives
Starting from Derivatives of Basis Vectors Dv e = e Dv w = v (w ) + w (v ) = Dv ' = v Dv Q = v eQ Q e ' +Q e
;
(v ) (v ) e v ' e e +Q ! +Q ! ! Q
Dv Q = v Q Q 1.1.2
;
e
=e
Q
+Q
Dierentiating Argum
1
1.1
The Classical Embedding Problem
Surfaces vs Manifolds
The currently accepted way to represent a non-Euclidean geometry is to start with a manifold, dened only in terms of overlapping coordinate charts. That method was not how non-Euclidean geometry
1
1.1
Tangent Space Vectors and Tensors
Representations
At each point P of a manifold M , there is a tangent space TP of vectors. Choosing a set of basis vectors e 2 TP provides a representation of each vector u 2 TP in terms of components u . u=u e = u0
1
1.1
The View from 1915 (or thereabouts)
Recovering the Old Formulas
Einstein Theory of Gravitation came into being before many of the tools of s modern dierential geometry. One way to appreciate what these tools have done for us is to work our way back
1
1.1
Examples of Projection Tensor Fields
Describing Hypersurfaces
We often .nd ourselves describing spacetime in terms of submanifolds such as the constant-t surfaces that make up "space at a particular time." Usually we know something about the geometr
1
1.1
1.1.1
The Linearized Einstein Equations
The Assumption
Simplest Version
The simplest version of the linearized theory begins with Minkowski spaceat time with basis vectors @ @= @x and metric tensor components 8 = =0 < 1 for 0 for 6= = : 1 for = = 1;
1
Exercise 4
For both of these problems you can either use the connection coe cients for 3-space in spherical coordinates or you can rewrite the metric in Cartesian at coordinates x; y; z: Either approach has its di culties.
1.1
Problem 4.1
ds2 = dt2 + dr
1
This Course
The basic ideas of dierential geometry with applications to both special and general relativity were covered in an earlier course. Here, I will review the essential tools of dierential geometry for completeness but the main focus will be on
1
1.1
Exercise 5
Problem 5.1
Show that, if u is the tangent vector to a geodesic, and k is a Killing vector eld (actually it is a form-eld), then the scalar u k is conserved along the geodesic.
Answer 5.1
Let be an a ne parameter along the geodesic so tha
1
Exercise 3
S = 2 = :=0
In these problems, assume a holonomic frame system and zero torsion so that
[ ]
1.1
Problem 3.1
The initial value structure of Maxwell Field Equations of Electromagnetism s may be understood from the identity [g gF
;
]; = 0
for an
1
Exercise 4
For both of these problems you can either use the connection coe cients for 3-space in spherical coordinates or you can rewrite the metric in Cartesian at coordinates x; y; z: Either approach has its di culties.
1.1
Problem 4.1
ds2 = dt2 + dr
1
Problems with the Schwarzschild Metric
The spacetime outside of a non-rotating star (or planet or whatever) of total mass m is described by the metric tensor ds2 = 1 2m r dt2 + 1 1
2m r
dr2 + r2 d
2
This metric solves the vacuum Einstein equations and,