Gravity probe-B

pervect
Staff Emeritus
While I'll certainly agree that the light appears aberrated, I don't see any errors in my argument / calculations that torque-free gyroscopes (modelled as fermi-walker transported tetrads) see no apparent rotation of the fixed stars for the constant acceleration case when the gyroscopes are orthogonal to the direction of motion (ie for the y and z spin axes in the coordinate system I introduced).

The case where the gyroscope points towards the guide star is more difficult mathematically, but it's pretty clear from the above argument that if this gyroscope had no y or z components in its angular momentum initially, it will not have them if it is acclerated. Thus I don't believe that the change in the 4-momentum of a gyroscope with the spin axis in the 'x' direction can't be interepreted as a spatial precession.

So I would say yes to aberration and no to precession for these cases.

I haven't analyzed the case where a gyroscope is at "in-between" angles, I suppose that if the x component of the angular momentum were initially equal to the y component, this would not remain true as the system accelerated - the x component would vary because of the boost, but the y component should remain constant. I suppose this would be regarded as precession.

Garth
Gold Member
Yes, I agree with you that the torque-free Fermi-Walker transported gyroscope does not rotate relative to the Fermi-Walker transported basis vectors of its constantly accelerating frame of reference. What I should have said was that relative to an inertial frame of reference that accelerating frame of basis vectors looks warped, i.e. not orthogonal to each other.

The travelling frame experiences a relativistic aberration of light from the 'fixed stars', which may be a corollary of the relativity of simultaneity.

That aberration would increase as its velocity built up, thus, relative to a fixed star, the gyroscope will appear to suffer a precession.

Let's do some maths:

Take the case of an observer accelerating uniformly at a > 0 but momentarily at zero velocity (to keep the maths simple on these Forums!).

A star at angle $\theta_0$ to the direction of travel is observed at angle $\theta$ due to the relativistic aberration of light where:

$$\cos{\theta} = \frac{\cos{\theta_0} + v/c}{1 + \cos{\theta_0}v/c}$$

After a short time $\delta t$, $\theta$ increases to $\theta + \delta \theta$, and differentiating, i.e. dividing by $\delta t$ and letting $\delta t$ and $\delta \theta \rightarrow 0$, then setting v = 0 and hence $\theta_0 = \theta$ we obtain

$$\frac{d\theta}{dt} = - \frac{a}{c}\sin{\theta}$$

The gyroscope measured relative to this distant star would appear to have a precession with the opposite sense, thus obtaining my formula in post #21.

At Earth g acceleration ~ 103 cm.sec-2 this introduces an precession of a star (where the direction of the star is orthogonal to the acceleration) of 13 milliarcsec/sec, quite significant to GP-B!. Note the acceleration is measured relative to the fixed stars and not GP-B's freely falling frame, but it would be cyclic and therefore easy to detect and eliminate.

Garth

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pervect
Staff Emeritus
Yes, I agree with you that the torque-free Fermi-Walker transported gyroscope does not rotate relative to the Fermi-Walker transported basis vectors of its constantly accelerating frame of reference. What I should have said was that relative to an inertial frame of reference that accelerating frame looks warped.
What I was trying to say was that we adopt a particular orthonormal reference frame for our Minkowski space-time (the t,x,y,z coordinate system I set up) which we regard as the "fixed stars". We can furthermore actually imagine that we have fixed, unmoving stars scattered around at constant unchanging coordinates in this particular coordinate system.

We would measure precession as a change in the spin axis of the gyroscope relative to this particular coordinate system.

We have to go further, actually - we have to separate out the space parts of the precession. I have simply chopped off the time component of the 4-vector to do this, I *think* this is a reasonable approach. (In some of the cases we considered, though, the time component turns out not to vary anyway, making the job easier).

If we imagine a relativistic uniformly accelerating traveller, he would "see" the stars start to cluster ahead of him as he accelerates.

http://www.exo.net/~pauld/stars/PD_images_relativ.html

I would regard this as a visual effect though, a change in the paths light took rather than an actual precession. YMMV.

Garth
Gold Member
I agree, but we were talking about GP-B.

Here the precession of the gyroscope is measured relative to the observed line of direction to IM Pegasi and then IM Pegasi's proper motion relative to a distant quasar. In fact it is the only way of measuring the precession accurately.

There are several sources of observed precession 'noise' for the GP-B team to eliminate, which is why they are taking so long to publish.

But not long now for the first results!

Garth

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Garth
Gold Member
Continuing this line of thought....

Consider no gravitational field and a laboratory A carried by a rocket with three stars x, y, z conveniently placed so they are each in the direction of A's orthogonal basis vectors x, y, z. The rocket motor now ignites accelerating the lab in the z direction.

The x, and y stars now precess towards z at the rate

$$\frac{d\theta}{dt} = - \frac{a}{c}$$

and relative to those stars x and y a torque-free gyroscope carried in the lab appears to precess.

If its rotational axis is inclined at an angle $\theta$ to z, then it appears to precess at the rate:

$$\frac{d\theta}{dt} = + \frac{a}{c}\sin{\theta}$$

relative to a guide star in the direction of its axis.

Whereas to the accelerating observer in the lab the three basis vectors remain orthogonal and the stars x and y precess, to an inertial observer it is the three stars that remain fixed and it is the lab's orthonormal basis tetrad that appears to warp.

The lab's x-y plane appears in the inertial frame to be warped into a cone of which the sides are inclined to the inertial x-y plane by an angle that increases by the rate a/c.

In this inertial frame it is the gyroscope that appears to precess away from the direction of its acceleration.

This precession is due to the rate of change of the lab's velocity and applies as soon as the acceleration begins even while the velocity is momentarily zero.

IMHO It would not apply to a supported, stationary, lab in a gravitational field of equal acceleration.

It is an interesting reflection on the equivalence principle that by simply observing the fixed stars you can distinguish between 'gravitational' and 'velocity' accelerations, even while the velocity of the latter w.r.t. the fixed stars builds up from zero.

Of course the same could be said of an accelerating observer measuring the rate of change of the frequency of an absorption line (increase of blue shift) emitted by a distant star ahead, in line with her acceleration.

Garth

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pervect
Staff Emeritus
I'm also quite curious as to the GP-B results, though I don't have anything riding on it.

I guess april 15 is the magic date, I imagine some people "in the know" already have an idea what will be presented in the way of the preliminary data analysis at the APS conference but of course they aren't talking publically.

As far as your points on precession goes, I do agree with your recent points about the apparent precession of distant stars from an accelerating observer, and I think any remaining issues are semantic.

Chris Hillman
The GP-B press release in plain English?

The gravity probe-b mission had finished collection date for a long time .Why not announce any result ??

Do you think that the results will consistent with "general relativity"?
I'm also quite curious as to the GP-B results, though I don't have anything riding on it.
After more than forty years (!) of development, GP-B was finally launched in April 2004. It recorded data for more than a year, and the Stanford team has been analyzing this data ever since. They will hold a press conference on the 18th but a press release has been made available today. The total cost of the experiment is said to be some 700 million dollars. For comparison, this is less than the cost of a one mile stretch of a proposed six lane traffic tunnel in an American city.

Recall that geodetic precession or de Sitter precession of the spin axis of a gyroscope in a quasi-Keplerian orbit around a nonrotating massive object (think of a Schwarzschild object, treated in weak-field gtr) has already been confirmed to good accuracy by previous work. See http://science.nasa.gov/headlines/y2000/geodetic.htm [Broken]. The goal of GP-B is to measure a much smaller effect, the frame-dragging precession or gravitomagnetic precession, or Lense-Thirring precession, which takes account of rotation effects when the massive object is spinning about its own axis (think of a Kerr object, treated in weak-field gtr). These effects should not be confused with the precession of the periastria (locus of closest approach) of a small object in quasi-Keplerian orbit around a massive object.

Recall also that there is a large class of metric gravitation theories (certain relativistic classical field theories) which behave much like gtr but are more complicated in various senses. Most of these also predict de Sitter and Lense-Thirring type effects, but with magnitudes differing from the gtr prediction, in any given situation--- but possibly not by very much. So the goal of GP-B and other tests can be understood as asking the question: "Do we need to consider more complicated theories than gtr, or does gtr suffice to account for all the gravitational effects we can directly test?" See http://math.ucr.edu/home/baez/RelWWW/HTML/grad.html#tests [Broken] for background on the subject of testing theories of gravitation.

To forestall possible misunderstanding:

In the (embargoed) press release at http://einstein.stanford.edu/, the GP-B team says:

1. The preliminary data analysis did confirm the de Sitter precession formula to higher accuracy than ever before.

2. It uncovered two unforeseen sources of error involving classical physics of the experimental equipment--- basically, a telescope in an Earth-orbiting satellite. As I understand it, both new error sources arise from a kind of electrostatic friction in the telescope mounting. Figuring this out took a lot of work. The effect is well known in physics, but when the experiment was originally designed, it was thought that the resulting errors would be self-canceling, but this turns out not to be true.

3. These errors need to be carefully modeled before they can be removed in order to test the Lense-Thirring precession formula.

4. The Stanford team hopes to complete this analysis by Dec 2007.

For a more detailed discussion, try http://twistedphysics.typepad.com/cocktail_party_physics/2007/04/the_not_so_frie.html

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marcus
Gold Member
Dearly Missed
Thanks for the plain English summary! It saves an interested audience member like myself much possible confusion and bother.

rbj
Recall that geodetic precession or de Sitter precession of the spin axis of a gyroscope in a quasi-Keplerian orbit around a nonrotating massive object (think of a Schwarzschild object, treated in weak-field gtr) has already been confirmed to good accuracy by previous work. ... The goal of GP-B is to measure a much smaller effect, the frame-dragging precession or gravitomagnetic precession, or Lense-Thirring precession, which takes account of rotation effects when the massive object is spinning about its own axis (think of a Kerr object, treated in weak-field gtr). ...
Recall also that there is a large class of metric gravitation theories (certain relativistic classical field theories) which behave much like gtr but are more complicated in various senses. Most of these also predict de Sitter and Lense-Thirring type effects, but with magnitudes differing from the gtr prediction, in any given situation--- but possibly not by very much. So the goal of GP-B and other tests can be understood as asking the question: "Do we need to consider more complicated theories than gtr, or does gtr suffice to account for all the gravitational effects we can directly test?"
...
1. The preliminary data analysis did confirm the de Sitter precession formula to higher accuracy than ever before.
so, just to be clear, Chris: does the GP-B results, at least so far, confirm the GTR quantitative prediction of the amount of de Sitter precession over the quantitation preditions of the other theories in this "large class of metric gravitation theories"? did they say as much or are they still hedging this?

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Chris Hillman
The "Acceptable Sliver" in PPN Space

Hi, rbj,

does the GP-B results, at least so far, confirm the GTR quantitative prediction of the amount of de Sitter precession over the quantitation preditions of the other theories in this "large class of metric gravitation theories"? did they say as much or are they still hedging this?
Regarding "confirm the GTR quantitative prediction of the amount of de Sitter precession over the quantitative preditions of the other theories", this phraseology seems to miss one of the points I was trying to emphasize. See the review paper by Will on the webpage I cited and look for the diagrams showing regions of "allowed theories" parameterized by the PPN formalism. Observe that in the PPN picture, gtr sits smack dab in the center of a very thin "acceptable sliver" sitting inside a large (but finite dimensional) space of alternative gravitation theories. This sliver represents the theories which are in agreement with current observation/experiment. (Modulo inevitable quibbles over what observations to accept at face value.) However, there are still points in this sliver which are arbitrarily close to the point representing gtr.

The "acceptable sliver" has just been reduced in one dimension, if you like, because GP-B has measured the de Sitter precession to greater accuracy than ever before. Now, gtr still sits in the center of the newly reduced "sliver", but given the fact that limits of accuracy will always be with us, you can never hope to reduce this sliver to a single point! By Occam's razor, gtr is preferred theoretically because it happens to be the simplest of all these theories, which is one reason for paying so much more attention to it than to the other theories in the "acceptable sliver".

As for "hedging", the whole point of my post was to explain why there is no "hedging" from the Stanford team! Rather, they said that they are not yet ready to say anything about the more interesting Lense-Thirring precession. While this is certainly disappointing, they went on to carefully explain why that is the case. They carefully explained why they think they know what they have to do in order to "safely remove" the unanticipated errors in order to finally obtain the desired clean test, and they said they hope to complete this work by December. Science is hard, and sometimes you just have to be patient while people work out some wrinkles.