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Measurements of Newton's gravitational constant and the length of day

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About a dozen measurements of Newton's gravitational constant, G , since 1962 have yielded values that differ by far more than their reported random plus systematic errors. We find that these values for G are oscillatory in nature, with a period of P = 5.899 +/- 0.062 yr , an amplitude of (1.619 +/- 0.103) x 10^-14 m^3 kg^-1 s^-2, and mean-value crossings in 1994 and 1997. However, we do not suggest that G is actually varying by this much, this quickly, but instead that something in the measurement process varies. Of other recently reported results, to the best of our knowledge, the only measurement with the same period and phase is the Length of Day (LOD - defined as a frequency measurement such that a positive increase in LOD values means slower Earth rotation rates and therefore longer days). The aforementioned period is also about half of a solar activity cycle, but the correlation is far less convincing. The 5.9 year periodic signal in LOD has previously been interpreted as due to fluid core motions and inner-core coupling. We report the G /LOD correlation, whose statistical significance is 0.99764 assuming no difference in phase, without claiming to have any satisfactory explanation for it. Least unlikely, perhaps, are currents in the Earth's fluid core that change both its moment of inertia (affecting LOD) and the circumstances in which the Earth-based experiments measure G . In this case, there might be correlations with terrestrial-magnetic-field measurements.
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Measurements of Newton’s gravitational constant and the length
of day
J. D. Anderson1(a) , G. Schubert2, V. Trimble3and M. R. Feldman4
1Jet Propulsion Laboratory, California Institute of Technology - Pasadena, CA 91109, USA
2Department of Earth, Planetary and Space Sciences, University of California, Los Angeles
Los Angeles, CA 90095, USA
3Department of Physics and Astronomy, University of California Irvine - Irvine CA 92697, USA
4Private researcher - Los Angeles, CA 90046, USA
PACS 04.80.-y – Experimental studies of gravity
PACS 06.30.Gv – Velocity, acceleration, and rotation
PACS 96.60.Q- – Solar activity
Abstract –About a dozen measurements of Newton’s gravitational constant, G, since 1962 have
yielded values that differ by far more than their reported random plus systematic errors. We
find that these values for Gare oscillatory in nature, with a period of P= 5.899 ±0.062 yr, an
amplitude of (1.619 ±0.103) ×1014 m3kg1s2, and mean-value crossings in 1994 and 1997.
However, we do not suggest that Gis actually varying by this much, this quickly, but instead that
something in the measurement process varies. Of other recently reported results, to the best of
our knowledge, the only measurement with the same period and phase is the Length of Day (LOD
- defined as a frequency measurement such that a positive increase in LOD values means slower
Earth rotation rates and therefore longer days). The aforementioned period is also about half of
a solar activity cycle, but the correlation is far less convincing. The 5.9 year periodic signal in
LOD has previously been interpreted as due to fluid core motions and inner-core coupling. We
report the G/LOD correlation, whose statistical significance is 0.99764 assuming no difference in
phase, without claiming to have any satisfactory explanation for it. Least unlikely, perhaps, are
currents in the Earth’s fluid core that change both its moment of inertia (affecting LOD) and
the circumstances in which the Earth-based experiments measure G. In this case, there might be
correlations with terrestrial magnetic field measurements.
Introduction. – Newton’s gravitational constant, G,1
is one of a handful of universal constants that comprise our2
understanding of fundamental physical processes [1] and3
plays an essential role in our understanding of gravitation,4
whether previously in Newton’s attractive gravitational5
force between two massive bodies m1, m2of magnitude [2]6
F=Gm1m2
r2,(1)
where ris their separation distance, or currently as7
the proportionality constant in the interaction between8
energy-momentum content Tab (the stress-energy tensor)9
and space-time curvature Gab (Einstein tensor) in Ein-10
(a)Retired.
stein’s general relativity [3,4] 11
Gab =Rab 1
2gabR= 8πGTab,(2)
in units where the local speed of light in vacuum c= 1. 12
Yet, experimental determination of Newton’s gravitational 13
constant remains a challenging endeavor. As reviewed in 14
[5], several measurements over the last thirty years appear 15
to give inconsistent values for G, of course an issue for our 16
understanding of this universal constant. Our purpose 17
with this letter is to inform the reader of a one-to-one 18
correlation between an apparent temporal periodicity in 19
measurements of G, generally thought to result from in- 20
consistency in measurements, with recently reported oscil- 21
latory variations in measurements of LOD [6]. LOD refers 22
to the excess of the duration of the day (observed period 23
p-1
J. D. Anderson et al.
1985
1990
1995
2000
2005
2010
2015
6.671
6.672
6.673
6.674
6.675
6.676
Date HyrL
GH10-11 m3s-2kg-1L
NIST-82
TR&D-96
LANL-97
HUST-05
UWash-00
BIPM-01
UWup-02
JILA-10
UZur-06
HUST-09
BIPM-13
LENS-14
Fig. 1: Result of the comparison of the CODATA set of G
measurements with a fitted sine wave (solid curve) and the 5.9
year oscillation in LOD daily measurements (dashed curve),
scaled in amplitude to match the fitted Gsine wave. The
acronyms for the measurements follow the convention used by
CODATA, with the inclusion of a relatively new BIPM re-
sult from Quinn et al. [11] and another measurement LENS-14
from the MAGIA collaboration [18] that uses a new technique
of laser-cooled atoms and quantum interferometry, rather than
the macroscopic masses of all the other experiments. The green
filled circle represents the weighted mean of the included mea-
surements, along with its one-sigma error bar, determined by
minimizing the L1 norm for all 13 points and taking into ac-
count the periodic variation.
of rotation of the Earth) relative to a standard unit and24
is calculated by taking the difference between atomic time25
(TAI) and universal time (UT1) divided by the aforemen-26
tioned standard unit of 86400 SI s [7]. Variations in LOD27
can be used to determine changes in the Earth’s rotation28
rate effectively providing a means to examine geophysical29
and atmospheric processes [8].30
For the following discussion, we emphasize that our G31
analysis and LOD analysis (a verification of the proce-32
dures employed in [6]) are very much independent of one33
another with the determined fitting parameters for both34
the period and phase of the periodicities in these measure-35
ments coinciding in near perfect agreement. Although we36
recognize that the one-to-one correlation between the fit37
to the Gmeasurements and the LOD periodicity of 5.938
years could be fortuitous, we think this is unlikely, given39
the striking agreement shown in Fig. 1. Furthermore, after40
taking into account this fitted oscillatory trend in the G41
measurements, we obtain agreement amongst the different42
experiments mentioned in [5] with a weighted mean value43
for Gof (6.673899 ±0.000069) ×1011 m3kg1s2.44
Methods. – In the July 2014 issue of Physics To-45
day, Speake and Quinn [5] lay out the problem and review46
the history of seemingly inconsistent measurements of the47
gravitational constant G. They plot twelve Gdetermina-48
tions, along with one-sigma error bars, extending from an49
experiment by Luther and Towler at the National Bureau50
of Standards (NBS) in 1982 [9] to their own at BIPM in51
2001 and 2007 (the latter of which was published in 2013) 52
[10, 11], two measurements in good agreement with each 53
other, but not with the other 10 measurements. Though 54
the vertical scale of years when the measurements were 55
made is not linear, there is a striking appearance of a pe- 56
riodicity running through these values, characterized by 57
a linear drift which suddenly reverses direction and then 58
repeats more than once. 59
With this pattern in mind, we compute a periodogram 60
for the measured Gvalues versus estimated dates of when 61
the experiments were run. A single clear period of 5.9 62
years emerges. The data for our Ganalysis were ob- 63
tained directly from Table XVII in the 2010 CODATA 64
report published in 2012 [1]. There are 11 classical mea- 65
surements made at the macroscopic level. To those we 66
added two more recent data points, another macroscopic 67
measurement, which we label BIPM-13, and the first ever 68
quantum measurement with cold atoms, labeled LENS-14. 69
Next we used our best estimates of when the experiments 70
were run, not the publication dates, for purposes of gen- 71
erating a measured Gvalue versus date data file, with 72
one-sigma errors included too. These dates were obtained 73
from the respective articles. This gives us the best data 74
set possible, defined by the measured Gvalues used for 75
the CODATA recommendation plus two more published 76
after 2012. 77
We fit with the raw standard errors, σi, provided with 78
each of the Gmeasurements and used a numerical mini- 79
mization of the L1 and L2 norms of the weighted residu- 80
als, rii, where the residuals are about a fitting model 81
of a single sine wave, a0+a1cos ωt +b1sin ωt, four pa- 82
rameters in all with 13 measurements. Results for the fit 83
to the 13 measured Gvalues are summarized in Fig. 1. 84
The L2 minimization is equivalent to a weighted least 85
squares fit, yet the L1 minimization (solid line in Fig. 1) 86
is a more robust estimator that discriminates against out- 87
liers. Both yield excellent fits with a suggestion that two 88
measurements at Moscow [12] and from the MAGIA col- 89
laboration [18] are outliers. However, the Moscow value is 90
known to suffer from an unexplained temporal drift [12] 91
and the cold-atom value could be fundamentally different 92
(Gat the quantum level). Still, we refrain from specu- 93
lating further on the cold-atom outlier until more micro- 94
scopic measurements of Gare obtained by different exper- 95
imental groups. The other 11 measurements are consis- 96
tent with the L1 fitting curve at the one-sigma level or 97
better. Figure 1 appears to provide convincing evidence 98
that there exists a 5.9 year periodicity in the macroscopic 99
determinations of Gin the laboratory with variations at 100
the level of ∆G/G 2.4×104about a mean value of 101
6.673899 ×1011 m3kg1s2, close to the value recom- 102
mended by CODATA in 2010 [1] but with a much smaller 103
standard error of 10.3 ppm instead of the CODATA rec- 104
ommended error of 120 ppm. 105
The most accurate determination by the Washington 106
group [13] with a standard error of 14 ppm now falls 107
squarely on the fitting curve. Because the two BIPM mea- 108
p-2
Measurements of Newton’s gravitational constant and the length of day
surements were made at the peak of the fitting curve, they109
now not only agree, but they are consistent with all other110
measurements. Notably, the measurement with a simple111
pendulum gravity gradiometer at JILA is no longer bi-112
ased to an unacceptably small value, but like the BIPM113
measurements it falls right on the fitting curve, but at the114
minimum of the sine wave. The Huazhong measurement115
is also at the minimum of the curve.116
Results. – With the 5.9 year periodicity in the G117
measurements accepted, the question arises as to what118
could be the cause and what does it mean. The only thing119
we can think of is a correlation with a 5.9 year periodicity120
in the Earth’s LOD, published by Holme and de Viron last121
year [6]. The International Earth Rotation and Reference122
Systems Service (IERS), established in 1987, maintains123
downloadable data files containing daily values of sev-124
eral parameters related to Earth orientation and rotation.125
The files extend from 1962 January 01, when the Consul-126
tative Committee on International Radio (CCIR) estab-127
lished Universal Time Coordinated (UTC) as the standard128
for time keeping, to the most current date available. We129
extract two rotation files, the first is the difference UT1-130
UTC in seconds and the second the LOD, also expressed131
in seconds, along with daily estimates of standard errors132
for both. There is also a piecewise constant file in integer133
seconds for the standard of atomic time TAI minus UTC.134
By differencing these two files the phase of the Earth ro-135
tation is obtained as measured against a uniform atomic136
time. This difference can be thought of as a continuous137
phase function φ(t) in radians sampled once per day at the138
beginning of the day. It can be expressed in SI seconds, the139
units on the IERS files, by multiplying by the conversion140
factor 86400/2π. It essentially provides the time gained or141
lost over the years by a poor mechanical clock, the Earth,142
which runs slow with a loss of about 33 s over the 52 years143
of the downloaded file. Because of its name and units of144
seconds only, the second file LOD is more difficult to in-145
terpret. It is also the gain or loss of time by the Earth,146
but only over the current day, and because of definitions147
there is a reversal in sign. When expressed as a contin-148
uous function of the Earth’s rotational frequency ν(t), it149
is simply ν0˙
φ/2π, where ν0is an adopted frequency of150
rotation with sidereal period of 86164.098903697 s. The151
quantity ˙
φ/(2πν0) is small and can be taken to the first152
order in all calculations.153
Formally, the spectral density of frequency is re-154
lated to the spectral density of phase by SLOD(f) =155
(2πf )2SUT1 (f), where fis the Fourier frequency. How-156
ever, a separate computation of the spectrum for each file157
shows that before 1994 either file can be used for analy-158
sis, but after the introduction of Global Positioning (GPS)159
data in 1993, the LOD data become more accurate by a160
factor of seven or more. This conclusion is consistent with161
the standard errors included with the data files of LOD162
and UT1-UTC. We show our estimate of the spectral den-163
sity for the LOD data in Fig. 2, obtained by weighted least164
-2.0 -1.5 -1.0 -0.5 0.0
-4
-3
-2
-1
0
1
Log f (yr -1 )
Log LOD2(ms2d-2 )
UT1 -TAI
Fig. 2: One-sided power spectral density per unit frequency for
LOD data over the years 1962 to 2014. The white-noise floor
is indicated by the horizontal solid line and corresponds to a
standard deviation of 0.54 ms d1, achieved by introduction
of GPS data in 1993 and consistent with the daily estimates
of standard error archived with the LOD data. The upper
dashed curve corresponds to mean spectral density for the nu-
merical time derivative of the UT1 data, dependent on VLBI
data from radio sources on the sky. For the low end of the
spectrum the LOD and UT1 data both indicate a f2random
walk, which with only 52 years of data can be confused with a
drift in the Earth’s rotation. At the high end, the underlying
spectrum indicates white LOD noise, but with a rich spectrum
from tidal torques and atmospheric loading at higher frequen-
cies not plotted. Although there is power in the region, there
is no suggestion of a single spectral line from the 5.9 year oscil-
lation, a term which must be extracted by analysis in the time
domain [6].
squares and SVD, but this time with 850 Fourier coeffi- 165
cients, 430 degrees of freedom, and 19169 observations. 166
The spectral resolution is 0.019 yr1, which we oversam- 167
ple by a factor of four, and the frequency cut off is 2 yr1,168
far short of the Nyquist frequency of 0.5 d1. A window 169
function is not applied to the data. It introduces unde- 170
sirable artifacts into the low-frequency noise spectrum of 171
interest and does little to isolate spectral lines. The Gaus- 172
sian window produces a hint of a line at 5.9 yr, but only 173
a hint. We proceed to an analysis of the data in the time 174
domain. 175
The 5.9 year periodicity in the LOD data is plotted by 176
Holme and de Viron in Figure 2 of their paper [6]. Their 177
plot looks in phase with the fit to the 13 Gvalues, but 178
in order to obtain an independent check on the reality 179
of the signal and for purposes of having a numerical sine 180
wave extending into 2014, we first smooth the LOD data 181
with a Gaussian filter with a radius of 600 days and a stan- 182
dard deviation of 200 days. As a result, the high-frequency 183
noise at a period of one year and shorter is practically elim- 184
inated, and with little effect on the low-frequency noise 185
spectrum. Next we fit a cubic spline to the smoothed 186
data with a selection of knots or segments for the cubic 187
polynomials done by eye, such that the fitting curve is suf- 188
ficiently smooth but with a negligible effect on the 5.9 year 189
p-3
J. D. Anderson et al.
1985
1990
1995
2000
2005
2010
2015
6.671
6.672
6.673
6.674
6.675
6.676
Date HyrL
GH10-11 m3s-2kg-1L
NIST-82
TR&D-96
LANL-97
HUST-05
UWash-00
BIPM-01
UWup-02
JILA-10
UZur-06
HUST-09
BIPM-13
LENS-14
Fig. 3: Result of the comparison of our Gdata set with
the monthly mean of the total sunspot number, appropriately
scaled. The black curves represent solar activity as reflected in
the international sunspot number.
periodicity. The resulting LOD residuals are fit with a sine190
wave of fixed 5.9 year period which is then subtracted from191
the smoothed data. The same procedure is applied to the192
new smoothed data and the procedure repeated four times193
with the knots for the spline at closer spacing with each194
iteration. The final result is the pure sine wave plotted195
as a dashed curve in Fig. 1. It agrees with the periodic196
signal found by Holme and de Viron. A removal of the fit-197
ted spline representation of the random walk, and also the198
sine wave, from the smoothed data is all that is needed in199
order to reduce the LOD residuals about the fit to a one-200
sigma noise level of 4.8µs d1. The amplitude of the fitted201
periodic signal is 92.64 ±0.18 µs d1, reduced from the202
amplitude of 150 µs d1[6] by the Gaussian smoothing,203
but with a well-determined period of 5.90076 ±0.00074 yr.204
With 99% confidence the period lies between 5.898 and205
5.903 yr. The phasing of the sine wave is as shown in206
Fig. 1 with a standard error of 0.25 yr.207
The correlation between LOD and Gmeasurements in208
Fig. 1 is most likely of terrestrial origin, but the period of209
5.9 years is also close to one-half the principal period of210
solar activity. References [14] and [15] discuss in greater211
detail that a possible correlation between solar activity212
and LOD measurements is not unexpected. Solar activity213
has an effect on mass distribution in the atmosphere which214
ultimately affects the Earth’s axial moment of inertia. It215
is feasible that this effect occurs at longer periods in the216
5.9-year range, as well as at much shorter periods, on the217
order of days, for which models exist [6].218
Consequently, we plot in Fig. 3 the monthly mean of219
the total sunspot number and also a 13-month smoothing220
curve, both shown in black. The two curves, again scaled221
to the magnitude of the Gdata, are taken directly from222
freely available downloads of data archived at www.sidc.be223
by WDC-SILSO, Royal Observatory of Belgium, Brussels.224
The smoothing is done by a standard tapered-boxcar ap-225
proach and is generally regarded as a good measure of226
solar activity. Although the Gmeasurements show a gen-227
eral agreement with solar cycle 23, which peaked around 228
2002, the long and unexpected minimum that followed, 229
and lasted until about 2010, is at odds with the rise in 230
Gvalues during that minimum. There is also a negative 231
correlation between the measurement from 1982 at the 232
National Bureau of Standards, labeled NIST-82, and the 233
sunspot number. It seems that solar activity can be dis- 234
regarded as a cause of the variations in Gmeasurements. 235
Conclusions. – Over the relatively short time span 236
of 34 years considered here, variations in the rotation of 237
the Earth can be considered either a random walk or pos- 238
sibly a drift. Over much longer time scales the rotation 239
must be slowing because of the transfer of spin angular 240
momentum to orbital angular momentum caused by tidal 241
friction of the Moon. Similarly, a real increase in Gshould 242
pull the Earth into a tighter ball with an increase in an- 243
gular velocity and a shorter day due to conservation of 244
angular momentum, contrary to the correlation shown in 245
Fig. 1. Thus, we do not expect that this behavior neces- 246
sarily points to a real variation in Gbut instead to some 247
yet-to-be determined mechanism affecting both measure- 248
ments in a similar manner. 249
Importantly, if the observed effect is connected with a 250
centrifugal force acting on the experimental apparatus, 251
changes in LOD are too small by a factor of about 105252
to explain the changes in Gfor the following reason. The 253
Earth’s angular velocity ωEis by definition 254
ωE=ω0(1 LOD),(3)
where ω0is an adopted sidereal frequency equal to 255
72921151.467064 picoradians per second and the LOD is in 256
ms d1(www.iers.org). The total centrifugal acceleration 257
is given by 258
ac=rsω2
012Asin 2π
P(tt0),(4)
where Ais the amplitude 0.000150/86400 of the 5.9 year 259
sinusoidal LOD variation and rsis the distance of the ap- 260
paratus from the Earth’s spin axis. The maximum per- 261
centage variation of the LOD term is 3.47 ×109of the 262
steady-state acceleration, while ∆G/G is 2.4×104, hence 263
even the full effect of the acceleration with no experimen- 264
tal compensation changes Gby only 105of the amplitude 265
in Fig. 1. Perhaps instead, the effect is connected with 266
changing torques on the Earth’s mantle due to changing 267
motions in the core. Changes of circulation in the core 268
must be accompanied by changes in density variations in 269
the core causing variations in the gravitational accelera- 270
tion gin the laboratory. At least this mechanism links 271
both LOD and gravitational changes to changes in the 272
core although we do not immediately see how either of 273
these mechanisms could affect measurements of Gin the 274
laboratory given the torsion balance schemes employed. 275
The least likely explanation is a new-physics effect that 276
could make a difference in the macroscopic and micro- 277
scopic determinations of G. Perhaps a repetition of the 278
p-4
Measurements of Newton’s gravitational constant and the length of day
single 2014 quantum measurement over the next decade279
or so can show consistency with a constant value, al-280
though if the variations in Gmeasurements are caused281
by an unknown inertial or frame effect, not by systematic282
experimental error, it likely applies at both the macro-283
scopic and the microscopic levels. The gravitational pa-284
rameter for the Sun, GM, is known to ten signifi-285
cant figures from orbital motions in the Solar System286
(ssd.jpl.nasa.gov/?constants). The universal constant G287
does not vary at that scale, although Krasinsky and Brum-288
berg [16, 17] report a detection of an unexplained secular289
increase in the astronomical unit (AU) over the years 1976290
to 2008, which can be interpreted as an increase in GM
291
proportional to the cube of the AU. However the effect on292
G, if real, is at the level of an increase of 3 parts in 1012
293
per year and undetectable with laboratory measurements294
of G. Nevertheless, the increase in GMis not explainable295
as an increase of the solar mass by accretion as opposed to296
the mass radiated away by solar luminosity [17]. Appar-297
ently, there does seem to be a secular or very long period298
(greater than 20000 years) Gvariation in the Solar Sys-299
tem, but of order 106smaller than the variation shown300
in Fig. 1.301
∗∗∗
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p-5
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... Their absence in space probes allows, for example, the mission eLISA to be sensitive to low frequency gravitational waves, which are impossible to detect on Earth because seismic noise cannot be completely removed at frequencies under 10 Hz. [3] A further essential advantage of using deep space probes is that terrestrial laboratories are subject to the complex dynamics of our planet, including numerous geological and astronomical phenomena, from the movement of continental plates to body tides, to the periodic variation of the length of the day. This latter phenomenon, as signaled by a recent article, could possibly be related to the mystery of the variability of the values of G [4]. According to the article, the variation of the locally measured value of G is fit by a 5.9 years sinusoid. ...
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The study of the Earth's rotation occupies an especially important position in the geophysical literature, a consequence of the fact that it is influenced by such a broad range of processes, involving not only the solid Earth but also its overlying atmosphere and oceans as well as the Earth's core, both the liquid outer core and the solid inner core. Not only are processes within each of these fundamental divisions of the planet involved in determining its rotational state, but the timescales on which measurable interactions occur extend from milliseconds to hundreds of millions and even billions of years. Through careful analysis of the mechanisms involved in such interactions, it has proven possible to learn a great deal concerning the planetary interior that might otherwise have remained hidden from view. The purpose of this chapter is to review a substantial fraction of the progress in understanding that has been documented in the recent literature of this subject, and to highlight those areas that seem ripe for further advance. Although considerable effort has been expended in attempting to make the coverage of this subject as complete as possible, certain specific topics that one could well expect to find covered in such a review have been purposefully omitted from the discussion or treated in a cursory fashion, the most important of which omissions concerns the issue of the evolution of the rate of planetary rotation over the timescale of a billion years on which, as a consequence of the long timescale exchange of spin angular momentum of the Earth with the orbital angular momentum of the Moon, the length of day has changed significantly. The discussion to follow has been designed on the basis of increasing timescale of the observed variations in rotation and has been organized in such a way that theoretical ideas and observational evidence are introduced only as they are required to extend understanding.
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This Letter describes new work on the determination of the Newtonian constant of gravitation, G, carried out at the BIPM since publication of the first results in 2001. The apparatus has been completely rebuilt and extensive tests carried out on the key parameters needed to produce a new value for G. The basic principles of the experiment remain the same, namely a torsion balance suspended from a wide, thin Cu-Be strip with two modes of operation, free deflection (Cavendish) and electrostatic servo control. The result from the new work is: G=6.67545(18)×10-11 m3 kg-1 s-2 with a standard uncertainty of 27 ppm. This is 21 ppm below our 2001 result but 241 ppm above The CODATA 2010 value, which has an assigned uncertainty of 120 ppm. This confirms the discrepancy of our results with the CODATA value and highlights the wide divergence that now exists in recent values of G. The many changes made to the apparatus lead to the formal correlation between our two results being close to zero. Being statistically independent and statistically consistent, the two results taken together provide a unique contribution to determinations of G.
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Variations in Earth's rotation (defined in terms of length of day) arise from external tidal torques, or from an exchange of angular momentum between the solid Earth and its fluid components. On short timescales (annual or shorter) the non-tidal component is dominated by the atmosphere, with small contributions from the ocean and hydrological system. On decadal timescales, the dominant contribution is from angular momentum exchange between the solid mantle and fluid outer core. Intradecadal periods have been less clear and have been characterized by signals with a wide range of periods and varying amplitudes, including a peak at about 6 years (refs 2, 3, 4). Here, by working in the time domain rather than the frequency domain, we show a clear partition of the non-atmospheric component into only three components: a decadally varying trend, a 5.9-year period oscillation, and jumps at times contemporaneous with geomagnetic jerks. The nature of the jumps in length of day leads to a fundamental change in what class of phenomena may give rise to the jerks, and provides a strong constraint on electrical conductivity of the lower mantle, which can in turn constrain its structure and composition.
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The universal Newtonian gravitational constant is being redetermined at the National Bureau of Standards with use of the method of Boyes in which the period of a torsion pendulum is altered by the presence of two 10.5-kg tungsten balls. The difference in the squares of the frequencies with and without the balls is proportional to $G$. The resulting value of $G$ is (6.6726\ifmmode\pm\else\textpm\fi{}0.0005)\ifmmode\times\else\texttimes\fi{}${10}^{$-${}11}$ ${\mathrm{m}}^{3}$\ifmmode\cdot\else\textperiodcentered\fi{} ${\mathrm{sec}}^{$-${}2}$\ifmmode\cdot\else\textperiodcentered\fi{} ${\mathrm{kg}}^{$-${}1}$\ifmmode\cdot\else\textperiodcentered\fi{}
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Length-of-day (LOD) estimates from the seven Global Positioning System (GPS) analysis centers of the International GPS Service for Geodynamics have been compared to values derived from very long baseline interferometry (VLBI) for a recent 16-month period. All GPS time series show significant LOD biases which vary widely among the centers. Within individual series the LOD errors show time-dependent correlations which are sometimes large and periodic. Clear correlations between ostensibly independent analyses are also evident. In the best case the GPS LOD errors, after bias removal, approach Gaussian with an intrinsic scatter estimated to be as small as -21 gs/d and a correlation time constant of perhaps 0.75 day. Integration of such data to determine variations in UT1 will have approximately random walk errors which grow as the square root of the integration time. For the current best GPS performance, UT1 errors exceed those of daily 1-hour VLBI observations after integration for -3 days. Assuming the stability of LOD biases can be reliably controlled, GPS-derived UT1 can be useful for near real time applications where otherwise extrapolations for several days from the most current VLBI data can be inaccurate by up to -1 ms.
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This book is intended to provide a thorough introduction to the theory of general relativity. It is intended to serve as both a text for graduate students and a reference book for researchers. According to general relativity, as formulated by Einstein in 1915, the intrinsic, observer-independent, properties of spacetime are described by a spacetime metric, as in special relativity. The structure of spacetime is related to the matter content of spacetime. Manifolds and tensor fields are considered along with curvature, Einstein's equation, isotropic cosmology, the Schwarzschild solution, methods for solving Einstein's equation, causal structure, singularities, the initial value formulation, asymptotic flatness, black holes, spinors, and quantum effects in strong gravitational fields. Attention is also given to topological spaces, maps of manifolds, Lie derivatives, Killing fields, conformal transformations, and Lagrangian and Hamiltonian formulations of Einstein's equation.