Dr. Gyula Bene Department for Theoretical Physics, Loránd Eötvös University Pázmány Péter sétány 1/A, 1117 Budapest 12. week
Experimental evidence of general relativity II.
Precession of a free symmetric top: the Gravity Probe B experiment
As we have seen earlier, the axis of a free top (i.e., in the absence of any
torque) is slowly rotating along the superface of a cone (precession) in stacionary but
non-static gravitational fields. It was also shown in the section on weak
gravitational fields that due to its rotation the Earth's gravity has nonzero
\(g_{0\alpha}\) components. In a weak gravitation field we have in leading order
\[\begin{align}
\frac{dn^\alpha}{dx^0}=\frac{1}{2}\left(g_{0\alpha,\beta}-g_{0\beta,\alpha}\right)n^\beta\;.\phantom{probe1}
\end{align}\]
Hence, if a free top is placed in the outer space near the Earth, its axis is
precessing (frame dragging or Lense-Thirring effect). Indeed, inserting the
expression of the metric around a rotating sphere we have
\[\begin{eqnarray}
\dot{\bf n}&=&\frac{k}{c^2r^5}\left\{r^2\left({\bf n}\times{\bf
J}\right)-3{\bf n}\times\left[\left({\bf r}\cdot {\bf J}\right){\bf r}\right]
\right\}
\;.\phantom{eqnarray}
\end{eqnarray}\]
The same effect is also present if the top
is on the Earth, but then the top also rotates together with the Earth, and
that rotation in itself leads to a precession (geodetic effect or Thomas precession) as we have
seen before. Quantitatively, it is
\[\begin{align}
\dot{\bf n}=-\frac{r^2\omega^3}{2c^2}\left({\bf s}\times{\bf n}\right)
\end{align}\]
where \(\omega\) stands for the angular velocity of revolution, and \({\bf
s}\) for the axis of revolution. This latter effect is much larger, and could
not be distinguished
from frame dragging precession experimentally because both precessions would share the
same axis. In order to observe frame dragging the top should be
placed on a satellite whose plane of orbit does not coincide with the
equatorial plane. In the actual experiment, called Gravity Probe B (2004-2005)
the plane of orbit was perpendicular to the equatorial plane (polar orbit), hence the two
kinds of precession led to a superposition of two pendicular displacements
of the top's axis.
There were four identical gyroscopes used, made of fused quartz with niobium
coating. They were roughly 4 cm big, and perfectly spherical within 10 nm.
In operation, the balls were spinning at 4000 revolution/minute in ultrahigh
vacuum and at temperature 1.8 K. The low temperature is necessary for the
niobium coating to become superconducting. A telescope is rigidly connected to
the gyroscope housings.
In flight, the
telescope always points to the same guide
star. Initially, the gyroscopes’ spin axes are
aligned through the bore sight of the
telescope to this guide star. A set of
superconducting readout systems, namely
SQUIDs
(Superconducting
QUantum
Interference
Devices)
detect
minute
changes in each gyroscope’s spin axis
orientation. Note that a spinning superconductor creates an axially symmetric
magnetic field (London effect) and this magnetic field is measured.
The evaluation of gathered data lasted until 2011. The theoretical value for
the geodetic effect is \(6.6\) angular second per year, which was verified
experimentally with 0.2 % accuracy. Precession due to frame dragging amounted
to \(0.037\)
angular second per year, this was verified with 19 % accuracy.
Radiation of gravitational waves: the Hulse-Taylor pulsar
In 1974 J.H.Taylor and R.A.Hulse discovered a new pulsar by the 300 m radio
telecope in Arecibo, Puerto Rico. Pulsars are neutron stars, their masses are
slightly larger than the Sun, but their radii are around only 10 km. They emit
strong microwave radiation in a narrow beam along their magnetic
axes. The axis of revolution usually does not coincide with the magnetic axis,
therefore the radio beam sweeps a conical surface. If the Earth is by chance
on this surface, then a periodic radio pulse can be observed, the time period
being that of the revolution. The Hulse-Taylor pulsar was observed at the
frequency 430 Mhz, and the time period of of the radio pulses were 59 ms,
which corresponrd to 17 revolutions per second. This time period was, however,
periodically fluctuating. The revolution period of pulsars were stable, so this fluctuation
was attributed to an invisible satellite. Hence, the pulsar was a member of a
binary system, and the time period was changing due to the Doppler
effect. Analyzing the pulses, it was possible to determine the orbital
parameters of the binary. It turned out that both stars were neutron stars with
almost equal mass (around 1.4 solar mass each). They were moving on elongated
elliptic orbits so that their smallest distance was 746 600 km while the
largest 3 153 600 km. The plane of orbit inclined to 45 degree to the direction
of observation. The orbital period was only 7.75 hours. Under such
circumstances the periastron wandering amounted to 4 degree per year
(cf. Mercur's perihelion wandering of 43 angular seconds per hundred years).
Even more importantly, the orbital period was slightly decreasing, meaning
that the binary was loosing energy. Indeed, according to general relativity,
binary systems emit gravitational radiation, and the amount of the radiated
energy can be predicted. The predicted rate of decrease of the orbital period
was 76.5 microseconds per year, which, according to the latest data analysis
(Taylor and Weisberg, 2004), agreed with the observation to 0.2 %.
Hulse and Taylor were awarded for their results the Nobel Prize in Physics in 1993.
Terrestrial observation of gravitational waves
As of December 2019, LIGO has made 50 detections of gravitational waves.
In 2017, the Nobel Prize in Physics was awarded to Rainer Weiss, Kip Thorne
and Barry C. Barish "for decisive contributions to the LIGO detector and the
observation of gravitational waves."
bene@arpad.elte.hu
