Astronomy 640 General Relativity, Spring 2003,

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Astronomy 640 General Relativity, Spring 2003,
Astronomy 640 General Relativity, Spring 2003, Kuhn Notes 2

2.1 Useful Theories of Gravity

L    Our approach in this class will be to emphasize the ties between theories of gravity (in particular General Relativity) and observational reality and fundamental concepts. In practice this means learning to compute the physical consequences of GR - I much prefer the style of Weinberg, or Landau and Lifshitz over, for example, Misner, Thorne, and Wheeler for these reasons. Thus we will tend to avoid the machinery of differential geometry, in favor of the algebraic approach. See references below if you want to learn this mathematical machinery.
Q    Starting from a "blank page" how do we construct a useful theory of gravity (that is more powerful than Newton's)?
C    There are lots of theories you've never heard of - between 1905-1960 more than 25 gravity theories were published (cf. Whitrow and Morduch 1965, Vistas in Ast. 6, 1-67). You've heard of GR because it is the most successful - i.e. it most "economically" explains all observations. Nevertheless, reconcile yourself with the likely fact that it is not the last word. The famous problem of how to make a theory like GR work with quantum mechanics is just one issue that we know leaves GR incomplete. There are other nagging problems which many believe leave GR to be fundamentally "problemmatic" (e.g. why no scalar field, local energy concepts no longer make sense in strong gravity...). I (and many others) are convinced that GR and its geometrical foundations are fundamental, but read the preface to your textbook by Weinberg. Perhaps his antiestablishment view is correct, that the geometrical foundation of GR is "accidental" and a consequence of some yet-to-be-found principle of elementary particle physics.
C    Gravity must tell us how masses interact. Because mass enters all other interaction models (strong, weak, and electromagnetic) we'll see that a successful model teaches much more than just how masses behave.
C    The "Dicke Framework" provides a common foundation from which to express and compare (theoretically and experimentally) various models of gravity. Two ingredients to this framework are:
       1) Space-time can be described by a 4-dimensional manifold
      2) Mass interactions can be described by differential equations on this manifold which yield physical solutions which are coordinate independent
C    A manifold is a collection of points (cartesian 4-space R⁴ x,y,z,t is a manifold) with enough continuity to allow derivatives (like defining tangent vectors). Every manifold in a small enough region has a cartesian geometry. Globally the manifold has a more complex structure (like a sphere or a torus...). For more on the mathematics of a manifold see Burke, Applied Differential Geometry (Cambridge U. Press, 1985).
C    Implications of this general mathematics: A proper theory is expressible in terms of tensor fields defined on a manifold. Tensor fields are quantities that can be expressed in terms of equations which are insensitive to any particular choice of coordinate system. Any good theory must follow from an invariant action principle. Example: Geodesic deviation (geometric fact)

D²x^a
Dt²
= R^a_mgbx^g u^m u^b

or physical law

R_ab-1/2 g_ab R=8pG T_ab

Q    How else could we define a fundamental theory?
C    In practical terms what do we need from a useful theory. Think about classical Newtonian gravity. Here f is the field that explains everything about mass interactions. The manifold it is defined over is ordinary cartesian space.
Q    What are the source and dynamical equations for Newtonian theory?
Q    This gravity theory doesn't describe 4-space, and doesn't qualify by Dicke's criteria. What's the simplest generalization to 4-space? Is this a "good" theory (consistent with observations and fundamental principles)?
Q    What is a more realistic and relativistically accurate gravity theory?
A    Electrodynamics as example...retarded potential solutions:

f(r,t) = -Gm/r   [t-r > 0],   r(r,t)=d(r)m   [t > 0]

T    Action Principles
C    Simplest case, no external potential, define theory for path between s₀ and s₁, and now we minimize action, A = ò_s₁^s₂dA to get an equation of motion - but it must be relativistic ® the action must be a Lorentz invariant.
C    Inventing an action is tantamont to devising the theory. There are no rules or prior constraints except for the symmetries we believe the theory must respect and some principle of economy or Occam's razor that guides our choice.

N    Extremum in òL[x^a ,[(x^b)\dot], s] implies

d
ds
¶L
¶
×

x^a

= ¶L
¶x^a

Q    What action principle describes a free relativistic particle?
C    Think about Lorentz invariance, what does it imply for proper intervals.

N    define spacetime intervals, causal separations, Minkowski nomenclature,

h^ab = æ
ç
ç
ç
ç
ç
ç
ç
è

-1

0

0

0
0

1

0

0
0

0

1

0
0

0

0

1
ö
÷
÷
÷
÷
÷
÷
÷
ø

derive

ó
õ s2

s₁
dt = ó
õ s₂

s₁
(-h_ab ¶x^a
¶s
¶x^b
¶s
)^1/2 ds = I

Derive equation of motion (a differential equation) from

d ó
õ (·)ds = 0

N    Free particle EOM:

d
ds

-h_ab
×

x^b

(·)^1/2
  ®   d
ds
(h_ab dx^b
dt
)=0

.
Q    How is free particle EOM related to physical (lab frame) velocity (dx^a /ds)?
A

dt
ds
=(-h_ab ¶x^a
¶s
¶x^b
¶s
)^1/2

C    To generate a relativistic theory of gravity we now simply generalize the action to include interactions with a "field".
Q    If gravity is described by a Lorentz scalar potential f, how would you choose the action?
A    Think about mf or me^f
C    Consider

I= ó
õ e^fm dt

N    Derive

e^-f d
dt
æ
è e^f dx_a
dt
ö
ø = - ¶f
¶x^a

This is dynamical equation (only half of our formal problem).
Q    Does this theory reduce to correct weak gravity, non-relativistic limit?

N    dt = dt /g, potential f = [(-GM)/r] in rationalized units c=1.

T    Lorentz Transformation "tensors."
C    Notice that we've subtly introduced a new quantity x_a with indices down. The class of transformations defined by arbitrary Lorentz transforms x^a® x¢^a = L^a _b x^b has many important symmetry properties intimately related to index up and down quantities and to h. We take as definition that x_a = x^bh_ab. We call an index quantity with indices up contravariant, and with indices down, covariant. The terms contra- and covariant refer to different symmetry properties under the class of coordinate transformations (in this case Lorentz). The quantity which raises indices is the inverse of h_ab^-1=h^ab Note that numerically each term of the inverse is equal to the h matrix. Summing over repeated indices is assumed (as always).
C    One-index quantities we call rank 1 tensors (or vectors). They transform as x¢^a = x^bL^a _b. Two index tensor quantities transform as T¢^ab=T^mnL^a _mL^b _n Thus h¢^ab=h^mnL^a _mL^b _n. Because columns and rows of L are Lorentz orthogonal (the dot product of a column is ±1 or 0) you can show by demonstration that h¢^mn=h^mn, i.e. that h is a Lorentz invariant rank 2 tensor. Note that we write the inverse of L^a _b as L_a ^b .
C    Important other tensors are formed from derivatives. If f is a Lorentz scalar then [(df)/(dx^a )] is a covariant vector.
Q    Prove that f_,a=[(df)/(dx^a )] is covariant. Show that a·b=a^a b_a is indeed a Lorentz scalar.

T    Dynamical Variables and Lagrangian Densities
C    Our EOM for a free particle assumed we knew f and our effort went to finding the trajectory x^a (s). Thus we allowed x^a (s) to vary in the action, or, x^a was a dynamical variable. This action isn't going to give us a means to compute the gravitational potential f which must result in a source equation for the field. We must find an action which allows f to be a dynamical variable.
C    We define a Lagrangian Density, L similar to the Lagrangian but as a quantity defined everywhere in space such that the action is a 4-space integral over this density. In this case

I= ó
õ L d⁴x

where we use d⁴x=dtdxdydz.
C    We can express our free particle Lagrangian in terms of a Lagrangian density by using a 4-space Dirac delta density, d⁽⁴⁾(x^a -x^a (s))=d(t-t(s))d(x-x(s))d(y-y(s))d(z-z(s)) being careful to note that x^a are coordinate variables and x^a(s) are trajectory functions (the particle world line).
Q    Find the Lagrangian density for a free particle.
Q    For example, if f is a scalar dynamical variable and L[f, f_,a] (the density depends on f and all of its spatial derivatives). Show that the extremum property of the action

d ó
õ L d⁴x = 0

leads to the equation

¶L
¶f
= d
dx^a
¶L
¶f_,a

Note that summation over a is implicit in this equation so that the RHS consists of 4 terms.

T    Lagrangian Density for a Lorentz scalar field, Source Equation
Q    Can you guess a form for the action that will lead to a simple scalar gravity theory that generalizes Newton's gravity?

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On 20 Jan 2003, 17:32.