Astronomy 640 General Relativity, Spring 2003,

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Astronomy 640 General Relativity, Spring 2003,
Astronomy 640 General Relativity, Spring 2003, Kuhn Notes 3

2.2 Useful Theories of Gravity, Scalar

C    To derive a source equation for scalar gravity we need to find an action that involves the scalar f as a dynamical variable. As before we need to build the action with all of the symmetries we expect the theory to respect.
Q    Describe some simple lagrangian densities that are Lorentz invariant. Find the simplest action that yields a non-trivial field equation for gravity.
A    Taking f as a relativistic scalar field we could write L=f and I=òfdt. This does not yield a useful field equation. Similarly any power of f in the lagrangian density yields an algebraic equation for the field with no dynamical interest. The simplest density with non-trivial dynamics is the quadratic form L=f_,af_,bh^ab. Note that this is a Lorentz scalar invariant because the covariant gradient of the scalar field multiplies the Minkowski tensor. An obvious shorthand notation for this product is f_,af^,a.
C    The full action includes two pieces, one involving masses (the mass-field interaction term) and another involving just the field self-interaction term. We write

I=I₁+I₂= ó
õ ó
õ me^fd⁴(x^a (s)-x^a)dtd⁴x + K ó
õ f_,af^,ad⁴x

This represents the simplest field theory we might construct for gravity based purely on symmetries. Note that we introduce an arbitrary constant K which is needed to relate these equations to our physical system of units. Fields f and units could be chosen to make this arbitrary constant equal unity. We add it explicitly and will compute it by letting the field correspond to ordinary gravitational potentials in the low velocity and low field-strength limit with our units choice.
C    Note that our first equation of motion is unaffected by I₂ since we derived these dynamics by allowing x^a (s) to be our dynamical variables. Since I₂ doesn't depend on particle trajectories the particle equation of motion is unaffected by generalizing the action to include the new term.
C    Each term in the action separates in the Euler equation, that is, I₁[f] and I₂[f_,a ]. Thus we find

¶L
¶f
= ó
õ me^fd⁴(x^a (s)-x^a )dt = ó
õ me^fd⁴(·)dt/g = e^fr₀ (x^a )/g

Here we generalize the definition of mass density to include point masses so that r₀ is the rest frame mass-energy density. For uniform motion characterized by Lorentz factor g we write r = r₀g² (one factor of gamma from the rest mass to moving mass and one factor from the length contraction of the density volume element).
C    The Euler equation for the second term in the action yields

¶L
¶f_,a
=2Kf_,bh^ab=2Kf^,a

so that

d
dx^a
¶L
¶f_,a
=2Kf^,a_,a

The final source equation in terms of the relativistic mass density is then

2Kf^,a_,a=re^f /g³

Q    Is this equation familiar? Reduce f^,a_,a to standard partial differential form. What is the solution of this equation for a point source that pops into existence at the origin at time t=0? See Jackson (Electrodynamics) section 6.5-6 to review solution to the wave equation.

T    Comparing with physical reality: Experiments
Q    Expand these equations in the Newtonian limit. What is the equation of motion for test particles and how do masses generate the scalar field? What is the value of K that yields agreement with our system of units? Does this theory violate any low velocity or low field strength observational constraints?
C    Consider rank 1 tensor (vector) 4 momentum and 4 velocity quantities

u^a = dx^a
dt
,  p^a = mu^a

Notice that we can write the scalar EOM of a "particle" in terms of p^a, so that

d
dt
(p_a e^f )=0

Q    Can you construct an argument showing how p^a remains finite for a photon (mass=0 but finite momentum, and infinite g)? What then does our gravity theory imply for photon deflections by gravitating objects.

N    The first measurements of gravitational deflection of starlight were obtained in 1919 by Dyson and Eddington. Near the edge (limb) of the Sun eclipse experiments yielded 1.98±0.16 arcseconds. The theoretical prediction of General Relativity is 1.75" A recent modern measurement is Jones, A.J., 81, 455 (1976), who obtained 1.66±0.19 arcsec. Compared to many modern measurements this fundamental test is not terribly precise, but good enough to rule out any theory that doesn't predict light deflection.

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On 20 Jan 2003, 17:30.