# Gravity Probe B and the Search for the Lense-Thirring Effect

## Keith Bechtol November 30, 2007

### (Submitted as coursework for Physics 210, Stanford University, Fall 2007)

 Fig. 1: The Gravity Probe B experiment will search for two predictions of General Relativity : the geodetic effect and the Lense-Thirring effect. The satellite carries four gyroscopes onboard which are initially set spinning with their axes of rotation pointed towards the guide star of the experiment, IM Pegasi. The geodetetic effect is expected to rotate the axes of gyroscope rotors 6.6 arcseconds per year in the plane of the satellite's polar orbit. The relativistic correction due to the Lense-Thirring effect is predicted to be 0.042 arcseconds per year in the direction of the Earth's rotation.

## Introduction

Gravity Probe B is a satellite mission designed to study two predictions of General Relativity [1]. The first of these predictions, known as the geodetic effect, is a result of the bending of spacetime around a massive but static object. A secondary Lense-Thirring effect is expected around spinning objects. In this case, the rotation of the mass pulls the surrounding spacetime along in the same direction as the rotation. Gravity Probe B will search for both of these uniquely relativistic effects by closely monitoring the angular orientation of four gyroscopes carried onboard the spacecraft. The axes of the gyroscopes are initially aligned with a guide star which serves as a reference point. In flat spacetime, the rotational inertia of the gyroscopes should remain constant so that the axes of the spinning gyroscope rotors are fixed. However, bending and twisting of spacetime around the Earth may deflect the orientation of the gyroscopes relative to the guide star. Fig. 1 shows a schematic of the Gravity Probe B experiment.

Both the geodetic effect and Lense-Thirring effect have long theoretical histories. However, classical tests of General Relativity have focused on the bending of spacetime in non-rotating systems rather than influences specific to spinning masses [2]. Gravity Probe B is the first experiment with sufficient precision to test the Lense-Thirring effect directly. The Gravity Probe B spacecraft was launched in April 2004 and final results of the mission are expected in December 2007.

## Lense-Thirring Effect

General Relativity is a theory of gravity that recognizes a connection between the time and spatial dimensions. According to this view, the acceleration that we experience as gravity is a product of curved spacetime [2, 3]. More precisely, objects travel along paths of shortest distance known as geodesics. But these are not the traditional distances measured in three spatial dimensions. Distances in General Relativity are four-dimensional intervals measured by a metric that describes the geometry of the system. As an example, consider the motion of the Earth about the Sun. In three spatial dimensions, the orbit is an ellipse. However, the path of the Earth is a straight line when including an additional time dimension in the General Relativity equations. The shape of spacetime is governed by the mass and energy content of the system. Massive objects warp surrounding spacetime and even slow the passage of time in their vicinity.

Several predictions of General relativity have already been confirmed experimentally. Gravitational lensing provided the first evidence for the Theory. During the 1919 solar eclipse, Arthur Eddington observed stars located behind the disk of the Sun that should not have been visible without taking relativistic effects into account [2]. Light from the distant stars deflected around the Sun while passing near the solar surface. Gravitational lensing has also been employed to detect the presence of dark matter in galactic clusters. The precession of perihlion of Mercury provided additional support for the theory [2]. Einstein correctly predicted that the orbit of the innermost planet advances by about 43 arcseconds per century. General Relativity has also been used to explain gravitational redshift. In 1959, Pound and Rebka performed an experiment to search for gravitational redshift by sending a beam of gamma rays twenty-two meters up the inside of the Jefferson Physics Laboratory at Harvard University [4]. The two scientists measured a change in frequency that matched the predicted Doppler shift of photons escaping a gravitating body. Each of these classical tests of General Relativity helped establish curved spacetime as an effective model for gravity.

In contrast with the classical tests of General Relativity, the Lense-Thirring effect lingers as an experimental loose end. In 1918, Lens and Thirring theorized that spinning masses can twist spacetime [2, 5, 6]. This effect is sometimes called "frame-dragging" because the the angular momentum of the rotating mass literally pulls the surrounding space into whirlpool pattern. Although the first predictions of the Lense-Thirring effect followed shortly after the introduction of the General Relativity, ideas for how to test the concept proved elusive for decades. At the time, evidence for black holes was non-existent and relativistic corrections near the Earth due to frame-dragging would be imperceptible to any ground-based instruments. Consequently, the Lense-Thirring effect was set aside by both theorists and experimentalists until the 1960s when glimmerings of the Space-Age and a newly predicted class of rotating black holes inspired new hope of a direct observation.

## Kerr Black Holes

Black holes are one of the great wonders of General Relativity and proivde an ideal testing ground for numerous physics predictions including the Lens-Thirring effect. A black hole can be completely characterized by its mass, charge, and angular momentum. The simplest case is a black hole with zero charge and zero angular momentum [3]. Karl Schwarzchild discovered this solution to Einstein's field equations while serving on the front lines of World War I in 1915. Schwarzchild's solution is not specific to black holes and can be generalized to describe any gravitating body as long as the gravitational field is not too strong. However, the descripton from General Relativity contains profound new features. If the entire mass of the gravitating body is contained within a small radius of the center of mass known as the Schwarzchild radius, the object becomes a black hole. In this case, the Schwarzchild radius becomes an event horizon meaning that a partition of spacetime has been created. The event horizon can only be crossed in one direction. Gravity is so strong in this region that no particle can escape the volume enclosed by the Schwarzchild radius. The event horizon is the defining characteristic of black holes.

The most extreme case of the Lense-Thirring effect may be found in the spacetime surrounding a rotating black hole. Roy Kerr proposed a theory for spinning black holes in 1963 [2]. Fig. 2 shows the structure of Kerr's rotating black hole. In contrast with the Schwarzchild solution, a rotating black hole has an extended ring-shaped singularity as well as two event horizons [3]. The event horizons mark the boundaries of successive regions where a particle becomes trapped upon entering.

 Fig. 2: The structure of a Kerr black hole consists of an extended sigularity, two event horizons, and a stationary limit surface. Objects in the ergosphere region can escape the gravitational pull of the black holes but are carried along in the direction of rotation.

Kerr also identified an entirely new feature associated with the rotation of a black hole. The stationary limit is the outermost surface where a particle can remain in a fixed position [3]. Outside the event horizon but within the stationary limit lies a region known as the ergosphere. Here, a particle can still escape the gravitational influence of the black hole but is pulled along with the swirling spacetime. Even light cannot advance against the rotation in this region. Powerful frame-dragging effects associated with rotating black holes may explain the formation of accretion disks and also provide a mechanism to produce relativistic jets. Material caught in the whirlpool of a spinning black hole could be boosted along in the direction of rotation to enormous speeds before being ejected in spectacular fashion.

Kerr's black hole model suddenly gave the Lense-Thirring effect a robust theoretical prototype. If frame-dragging could have such remarkable consequences for a rotating black hole, modest but still measurable effects might be oservable closer to home. The simultaneous development of rocket technology made precision experiments in Earth's orbit a promising direction for investigation.

## Gravity Probe B

The Theory of General Relativity thrives across the great expanses of the Universe and in the exotic environments around black holes. Most cosmological studies require a thorough understanding and careful application of the theory. In contrast, relativistic corrections to Newtonian gravity are often more subtle and challenging to observe in the local neighborhood of our own Solar System.

In late 1959 and early 1960, George Pugh and Leonard Schiff independently proposed satellite gyroscope experiments to detect the Lens-Thirring effect produced by the Earth's rotation [5, 6]. According to concept of rotational inertia, the axis of a perfectly constructed gyroscope should always point in the same direction. However, twisting of spacetime around a massive rotating body could effectively rotate the spin axis of a gyroscope. The theorized frame-dragging effect around the Earth would be small - a deflection angle of only 42 milliarcseconds per year for a gyroscope in low obit. At the time, the technology to conduct such an experiment was simply non-existent. Detection of frame-dragging near the Earth requires ultra sensitive equipment capable of measuring changes in spin direction at a rate of 10-11 degrees per hour [7]. However, ground-based gyroscopes are limited to measuring spin direction changes at rates of roughly 10-5 degrees per hour. The gyroscope rotor and housing would have to maintain near torsion-free conditions over the course of a year-long experiment. Pugh and Schiff suggested that gyroscopes free from vibrational noise at Earth's surface could achieve sufficient accuracy. Despite serious technoloigcal challenges, NASA agreed to fund the project starting in 1964.

After decades of technological development and numerous setbacks, the Gravity Probe B mission was finally launched in April 2004. The satellite entered a polar orbit 642 km above Earth's surface and began a four month calibration period prior to collecting a year's worth of data [1, 8].

Many of the experimental components on board Gravity Probe B are great technological feats in their own right. For example, consider the four gyroscope rotors. The surface of each 3.8 centimeter diameter rotor deviates from perfect sphericity by less than a millionth of a centimeter [7]. The gyroscope rotors are composed of pure fused quartz and are coated with a thin layer of niobium. The rotors are held in place by an electrostatic suspension system [9]. Electric field gradients between the rotors and housing electrodes ensures that no contact between components occurs even though they are separated by only 32 microns. The rotors rotate at about 4000 rpm and torsion is minimized so that average measured spin-down time (time for spin rate to reduce to 1/e the initial spin rate) of the rotors ranged from 7000 to 2600 years [10]. The entire gyroscope assembly is housed within a dewar filled with liquid helium. The niobium rotor surfaces become a superconductors at the temperature of liquid helium and the rapid spin of the gyroscope rotors produces a magnetic moment known as the London moment [10]. Fortunately, the London moment is perfectly aligned with the spin axes of the gyroscope rotors. Thus, the rotational orientation of the gyroscopes can be determined by measuring the magnetic field around the rotors. No physical contact with the rotors is required. Superconducting SQUID magnetometers monitor the direction of the rotor spin axes to a precision of 0.1 milliarcseconds.

The orientations of the gyroscopes are calculated relative to the direction of bright reference star, IM Pegasi. Ideally, the satellite would be directed against an extremely distant background object such as a quasar. IM Pegasi was selected as a bright object that would be easy to track. Although this position of this star is known to drift across the sky, the movement can be tracked by other telescopes against background quasars [Limits of Engineering, post-flight analysis]. During the Gravity Probe B mission, IM Pegasi became the most consistently monitored star in history!

In April 2007, NASA and Stanford released a preliminary report on their findings [11]. Although the geodetic effect was clearly observed, the more subtle frame-dragging effect remained unclear. Data analysis had been set back by the discovery a time-dependent wobble obscurring the data. But before going deeper into the unforeseen analysis challenges of the mission, it is helpful to review the podhole motion of rotating solids.

## Podhole Motion

 Fig. 3: Three principle Axes of symmetry are shown for a rectangular block. The moment of inertia is minimized along the long axis of symmetry and maximized along the short axis of symmetry. A spinning object with its rotational axis initially along the long axis of symmetry will tend to shift the axis of rotation towards the short axis of symmetry as kinetic energy thermally dissipates.

Every three-dimensional object has three principle axes of symmetry [12]. One of these axes corresponds to the axis that maximizes the moment of inertia. Another principle axis of symmetry minimizes the moment of inertia. Rigid objects can rotate in a stable motion about either of these two axes. Rotation about the third axis of symmetry has a moment of inertia that lies somewhere in between the two extremes. Fig. 3 shows the three axes of symmetry for a rectangular block of uniform density. Objects spinning about some intermediate axis exhibit a wobbling motion due to the instability of the configuration. The elliptical path traced out by the rotational axis of a spinning opject is called the podhole. If the rotational axis is close to one of the principle axes of symmetry that minimizes or maximizes the moment of inertia, the podhole wobble is small. Conversely, the podhole motion is most exaggerated for axes of rotations between the two extremes. Although the motion may appear highly unstable in these intermediate sitauations, the podhole motion is completely predictable for rigid bodies in which the internal energy of the system remains constant. However, systems with variable internal energy experience a time-dependent podhole motion. Let's examine this arguement more closely.

We can immediately apply two conservation laws to a spinning object. Both the total energy and angular momentum remain fixed throughout the experiment as long as there are no external torques. We can write the total energy of the system as the sum of rotational kinetic energy, potential energy, and internal energy dissipated as heat.

I represents the moment of intertia, &omega is the angular velocity, U is the potential energy, and Q is the energy dissipated as heat. Additionally we may write the angular momentum as the product of the moment of inertia and the angular velocity.

Now rewrite the total energy in terms of the angular momentum.

Observe that only two of the factors in the above equation can vary. The total energy and angular momentum are constant. Also, the potential energy will remain fixed. Therefore, the only the amount of energy dissipated and angular velocity can vary. If the internal energy is converted to heat, then the angular velocity must decrease to balance the equation. However, if the angular velocity decreases, our equation for the conservation of angular momentum stipulates that that the moment of inertia must rise. The axis of rotation will migrate towards the principle axis of symmetry that maximizes the moment of inertia. We conclude that rotating bodies with freedom to dissipate internal energy will experience a time-dependent podhole motion.

## Trouble-shooting

Signals from the Gravity Probe B gyroscopes suffered from two types of distortion. First, each of the four gyroscopes exhibited a unique pattern of time-dependent podhole motion [8]. The observation that each gyroscope behaved differently over the course of the mission pointed investigators towards some individual characteristic of the gyroscopes. In addition to individual podhole motions, the gyroscopes also showed different slow-down rates. The analysis team reasoned that a mechanism for internal energy dissipation must be at work. First, the gyroscopes are not perfectly rigid. The rapid spinning of the rotors causes the surface to bulge about 10nm at around the equator. Therefore, some internal energy is lost as the surface of the rotor expands and contracts during the podhole period. More surprising though was the discovery of large electrostatic patches on the surface of the rotors. Electrostatic patches on the rotor can interact with charges on the gyroscope housing to dissipate energy of the system [9]. During the seventeen month period of data collection, two of the rotors flipped their axes of rotation from near the axis of least moment of inertia to the axis with greatest moment of inertia [7, 8]. The issue became even more complicated when considering the unique regions of trapped magnetic flux on the surface of each rotor. As the niobium coating on the rotors cooled to superconducting temperature, some potions of the magnetic field became fixed on the surface. Irregularities of the magnetic field can normally be accounted for when observing the London moment of the gyroscope rotor as long as the rotation remains steady. However, adding podhole dynamics due to the asymmetric magnetic field confounded measurements of the magnetic moment.

Unexpectedly large classical misalignment torques created a second major concern for data analysis. The misalignment torques are particularly alarming because they can easily fake relativistic effects by steadily shifting the rotational axes of the gyroscopes [10]. At the start of the mission, each gyroscope rotor was set spinning with its axis pointing towards IM Pegasai. Over the course of the mission, the angle between the axis of gyroscope rotation and the direction towards the guide star grew. Now recall that the Gravity Probe B spacecraft always remains pointed towards the guide star and the spacecraft is constantly rotating about this axis. The direction towards the guide star is thermefore called the roll axis. When the axis of gyroscope rotation coincides with the roll axis, any interactions between the rotor and gyroscope housing can increase or decrease the angular velocity but won't change the orientation of the rotation axis [9]. Once a misalignment occurs, classical torques can alter the spin direction of the rotor. Once again, large electrostatic patches on the gyroscope rotor appeared to be the root cause of the problem. This time, interactions between charges on the superconducting surfaces of the rotor and housing created the troublesome torque.

Once large electrostatic patches were identified as the culprit, Gravity Probe B scientists puzzled over how the surface potential differences could have been created. A thin coating of niobium had been applied to each of the rotors using a sputtering technique [13]. Certain surface irregularities are known to exist. For example, niobium crystals may be oriented in different directions, the concentration of elements may not be perfectly uniform in a thin coating, and some contaminant particles could be stuck to the surface. Each of these defects can create slight charge irregularities. However, engineers thoroughly tested the niobium sputtering method prior to launch and found that the electrostatic patches were small (on the order of microns in size) and randomly distributed over the surface of samples so as to cancel any net effect [9]. Scientists proceeded to apply sixty-four layers of niobium to each of the four gyroscope rotors, turning the rotor with each layer to achieve an even coating. Unfortunately, each layer only covers the half of the rotor surface facing the sputtering device. Therefore, the final niobium layer essentially divided the sphere of the rotor into two halves [13]. Large electrostatic patch formation on three-dimensional objects had not been previously considered because all the test samples had been flat.

Gravity Probe B scientists and engineers remain hopeful despite of the the added challenges created by time-dependent podhole motion and misalignment torques. The analysis of podhole dynamics traced back the motion of each gyroscope rotation by rotation to piece together the history of podhole motion [7]. By September 2007, maps of trapped magnetic flux on each of the four gyroscope rotors had been completed. Also, the analysis team successfully identified the podhole phase at each stage of the mission and determined the anglar orientation of the podhole to within six degrees of accuracy. The gyroscope rotors exhibit such a high degree of spherical symmetry that the principle axes of rotation could only be reconstructed by closely analyzing the resultant podhole dynamics as oppose to direct measurements of the rotors themselves [7]. Misalignment torques had to be confronted by other methods. Fortunately, the onboard liquid helium supply lasted longer than expected and allowed for time after the completion of data collection for additional calibration techniques. During that time, the spacescraft was deliberately pointed towards other stars to measure misalignment torques as a function of the angle between spacecraft roll axis and gyroscope spin axis [11]. The final step of data anaylsis will be to extract the relativistic signals from the background of unrelated interactions.

Gravity Probe B may soon resolve nearly a century of questions about the Lens-Thirring effect. A direct measurement of the phenomena would provide entirely new evidence for General Relativity. Even a null result would mark a tremendous leap in technological capability and would motivate serious theoretical investigation to understand the disagreement between prediction and observation.

For more background information and news about the progress of data analysis, check out the experiment website "Gravity Probe B: Testing Einstein's Universe" (avaialable at http://einstein.stanford.edu/index.html).

© 2007 Keith Bechtol. The author grants permission to copy, distribute and display this work in unaltered form, with attributation to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

## References

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[6] G. E. Pugh, "Proposal for a Satellite test of the Coriolis Prediction of General Relativity," Pentagon Weapons System Evaluation Group (WSEG) memo #11 (1959).

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[9] G. M. Kaiser et al., "The Expected Performance of Gravity Probe B Electrically Suspended Gyroscopes as Differential Accelerometers," excerpt from CP456 Laser Interferometer Space Antenna ed. by William Folkner, (American Institute of Physics, 1998).

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[11] B. Kahn, Stanford University Press Release, April 14, 2007.

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