CHAPTER FOUR
where ω
m
and ω
s
are the orbital angular frequencies of the moon and the
sun, respectively, around the earth. Lunar-ranging permits the measure-
ment of the earth-moon distance D with high precision by measuring the
round-trip travel times of laser pulses reflected by the retroreflectors on
the moon. Observations carried out at the McDonald Observatory near
El Paso, Texas (and subsequently at other observatories since 1969) using
3-ns duration ruby-laser pulses, determined D with a precision of a few
centimeters and showed that the Nordtvedt parameter η is equal to zero
with a high degree of accuracy.
59
The lunar laser-ranging experiments result showing that η is zero
with high precision has three theoretical implications: It supports the
thesis that the gravitational self-energy contributes in equal measure
to m
i
and m
p
of massive bodies; it agrees with the general theory of
relativity, according to which η = 0; and it imposes strong constraints
on alternative theories of gravitation. Thus, e.g., the Brans-Dicke scalar-
tensor theory of gravitation, perhaps the best-motivated competitor of
Einstein’s theory, differs from the latter by certain additional terms that
tend to zero as η approaches zero, which implies that the observational
predictions of the two theories become almost identical.
60
We are now in a position to understand the historical issue we dis-
cussed above of crediting Newton with having anticipated, at least in
principle, the problem of the Nordtvedt effect, a claim that has recently
been made primarily by Thibault Damour.
61
Let us first recall the two
corollaries of Newton’s Principia referred to above.
59
J. G. Williams et al., “New Test of the Equivalence Principle from Lunar Laser Rang-
ing,” Physical Review Letters 36, 551–554 (1976) (η = 0± 0.03). I. I. Shapiro, C. C. Counselman
III, and R. W. King, “Verification of the Principle of Equivalence for Massive Bodies,”
Physical Review Letters 36, 555–558 (1976) (η = 0.001 ± 0.015). E. G. Adelberger, B. R. Heckel,
G. Smith, Y. Su, and H. E. Swanson, “E
¨
otv
¨
os Experiments, Lunar Ranging and the Strong
Equivalence Principle, Nature 347, 261–263 (1990). J. M
¨
uller, M. Schneider, M. Soffel, and
H. Ruder, “Testing Einstein’s Theory of Gravity by Analyzing Lunar Laser Ranging Data,”
Astrophysical Journal 382, L101–L103 (1991). J. O. Dickey et al., “Lunar Laser Ranging: A
Continuing Legacy of the Apollo Program,” Science 265, 482–490 (1994) (η =−0.0005 ±
0.0011). J. G. Williams, X. X. Newhall, and J. O. Dickey, “Relativity Parameters Determined
from Lunar Laser Ranging,” Physical Review D 53, 6730–6739 (1996).
60
C. Brans and R. H. Dicke, “Mach’s Principle and a Relativistic Theory,” Physical
Review 124, 925–935 (1961).
61
T. Damour, “The Problem of Motion in Newtonian and Einsteinian Gravity,” in
S. Hawking and W. Israel, eds., Three Hundred Years of Gravity (Cambridge: Cambridge
University Press, 1987, 1990), pp. 128–198. T. Damour and D. Vokrouhlick
´
y, “Equivalence
Principle and the Moon,” Physical Review D 53, 4177–4201 (1996).
118