No TitleAstronomy 640 General Relativity, Spring 2006, Kuhn Notes 1.
1.1 IntroductionQ Why study gravitation? We'll take a non-astronomical and sometimes
empirical approach to this subject. For most of astronomy GR isn't necessary,
so we can afford the luxury of a pedagogical and sometimes philosophical
approach to understanding gravity. We're going to build our theory of
gravity from a vacuum...Q What is the noble goal of doing astrophysics? or, Why do we do what we do?
A Aim should be to find defects in our explanation of physical observations -
process of elimination works but making models of systems to explain observations should be only a secondary goal
C It is not often that we have a chance to confront gravity theories, our best chances are with astrophysical systems, and nowhere else (at least in
this and many more lifetimes).
C The course goal is to gain a fundamental understanding of gravity,
sufficient to begin thinking about where astrophysics can improve
our understanding of gravitation.L I might try to generate summary notes that are accessible from the
web (www.ifa.hawaii.edu /users/kuhn/ast640/ ast640.html) but I haven't
yet found a good way to translate tex to HTML.
You'll see that these notes highlight items with letters Q, A, C,
L, and T. Issues flagged with a Q are questions you should think
about and answer, A notes my answer to a question, C notes a comment, L
notes a logistical course issue (schedule, etc.), and T marks a change
in topic. Questions which are not answered in class will be revived in
the next class, with answers to be provided by students. These notes
will not substitute for lectures but they may provide a useful outline
of what we discuss.
T A qualitative understanding of gravity
C The context is 4 fundamental interactions, gravity is the weakest and wierdest. Just about everything we know about gravity comes from astrophysical measurements. Gravity is weak ® large masses, distance scales.
Q What do we know about gravity that doesn't come from astrophysical systems?
A Perhaps the only useful fact is the value of Newton's constant, big G.
C There are many theories of gravity - neither Newtonian nor General
Relativity (GR) are completely correct - there's almost certainly more
to gravity than will be explained by GR (think quantum mechanics). But
equally important is the observation that GR is more than "just a
theory of gravity." Think about electrodynamics. Special relativity is
the essential framework needed to understand electrodynamics, but
special relativity has much more to tell us than just how charged
particles and fields interact.
C There are two principles that guide progress in understanding gravity
(and hopefully any other fundamental theory):
A proper accounting of (or consistency with) all observations and measurements
A theoretical foundation built on respect for invariance principles and symmetries of nature...
The first point here is fundamental, the second is debatable. Dicke tried
to refine the second point (the "Dicke framework") which we'll get to later.
C Symmetries make theories. Electrostatics and galilean transformations: Fundamental form
x¢a=Gab xb
.
G(v1+v2)=G(v1)+G(v2) characterizes these.
Gab (v)=
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more
explicitly defines a class of transformations between coordinate systems.
Here the primed frame is moving in the +z direction away from the
unprimed and the two coordinate frames are coincident at t=0.
C Electrostatic forces are galilean invariant.
C Note that we use greek letters as indices that run from 0(t) to 3(z)
and we omit an implicit summation sign when indices are repeated up and
down. This notation "Einstein Summation" is standard, some authors
reserve latin indices (i,j,k) for spatial x,y,z markers (others invert
this convention - like Landau and Lifshitz).
C Think about
electrodynamics (i.e. two parallel wires with a current). Galilean
invariance doesn't work.
Lorentz invariance is needed for electrodynamics G® L with
Lab (vz
^
)
=
æ ç ç ç
ç ç è
g
0
0
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0
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g
ö ÷ ÷ ÷
÷ ÷ ø
, x¢a=Lab xb
To simplify equations we'll measure velocities in units where the speed of light (c) is 1. Here we define g = 1/Ö{1-v2}.
Q How is there consistency between the Galilean and Lorentz
invariance physical requirements?
C Physical observations depend on particle trajectories and
"events." Theories which describe particles and their interactions
always include non-observables. Distinguishing what is physically
important from theoretical structure in a theory will engage much of
our attention.
Q What are some of the inconsequential (non-physical) ingredients of
quantum mechanics or electrodynamics?
C The symmetries of particle interactions must be contained in the
theoretical constructions we use. Good constructions make this obvious. Think
about free particle mechanics. Trajectories and interactions are described by
F=ma which is a complete mechanical theory (for many circumstances).
Isotropy (conservation of energy and momentum) isn't obvious. We'll be
using Lagrangian theories, where the theoretical action captures
the fundamental relationship between interactions (the physical aspects of
a theory) and the symmetries (the "fundamental" justification for the theory).
Q What's the action of a free particle of mass m traveling along a
trajectory x(t)? How do we figure out how the particle
gets from x0 to x1? What are the geometric invariants
of this theory?
T What evidence is there that 1/r2-like forces don't work
for gravity?
C Think of two particles of charge and mass e, m separated a
distance r. The electrical force on any one of them is Fe=e2/r2
[repulsive] (in cgs units) and the gravitational force is
Fg=-Gm2/r2 [attractive]. The sign difference has profound
consequences. The pervasive fact that m > 0 also is fundamental.
C It is axiomatic to our physical theories that interactions over
distances imply "fields." In principle these are physical and
observable with "field" meters. We'll generalize Lagrangian particle
theories to include the surrounding field, so that the action
includes contributions beyond the particle trajectories. This field (e.g. due
to charge or gravity( is essential to have particle-particle interactions.
The energy of this field is part of the action.
C Think about our two electrically charged particles. We would compute
the electric field from Ñ·[E\vec] = 4pr. Electric field
meters would work by measuring the interaction force on charged test
particles. The equivalent gravitational "Gausses" law must be
Ñ·[g\vec]=-4pGr where r is now the mass volume
density rather than the electrical charge density in the real Gausses
law. The fields in empty space imply energy density. You know that the
electric field energy density is uE µ E2. Energy
(axiomatically) is a positive quantity. It must be that the
gravitational field energy density is uG µ g2. The field
energy of an isolated particle is Ug µ òe¥ 4pr2 [(m2)/(r4)] dr µ 4pm2/e. This is as it should
be since we expect the energy to diverge as we get closer to the charge
(or mass). Now we have a problem though. Consider two point masses (1
and 2) infinitely separated. Obviously Usep=2U1. Now bring them
together so the system looks like a single point of mass 2m. It is
evident that Ucomb=4U1. But U1 > 0 so the energy of the combined
particles is twice as large as the separate particles. The only way to
increase the energy of this system is to apply an external force
against a repulsive interaction, which would do work on the system,
thus increasing its energy. The problem is that gravitating particles
are attractive so we extract energy from the system by letting
them come together. This simple contradiction implies that gravity is
very different from electrodynamics.
C The gravitational field energy cannot be simply proportional to the
square of the local gravitational field. Our job in this class will be
to construct a theory of gravity from other (non electrodynamic)
principles.
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