Alternative theories being tested by Gravity probe B

jgraber

Hi everyone,
Sorry I can only post on coffee breaks when I am near the APS temporary hotspot. As I understand the "glimpses" each ellipse is a one sigma radius, and GR is very near the center of the earliest, crudest glimpse, and about two radii (i.e. approx two sigma) from the smallest, latest "glimpse", which is still based on a very limited amount of not yet fully processed data. As I understand it, the data in that plot are total motions, including nonrelativistic effects as well as the two GR effects. The speakers mentione three or four of order 40-80 mas/y total that add to the Geodetic Effect and at least one of order 40mas/y that adds to the Frame Dragging Effect. Because of these additions, the GR prediction is offset from the values usually quoted. The values on the "glimpse" chart have minus signs and increase in absolute value downward and to the left. They increase (or decrease) by 20 per grid line and the central values are 80 for the vertical (Frame Dragging) axis and 6580 for the horizontal (Geodetic) axis. The latest "Glimpse" is slightly offset in the direction of larger values, so IF you take it at face value, which is wildly over optimistic in my opinion, it would indicate that both effects are slightly larger than the GR prediction by about 10 mas/y compared to 6600 and 40. However, there will be new, much more reliable numbers in December or so, and the only sensible course in my opinion is to wait until then. remember, there are systematic as well as statistical errors, and the experiment is quoting their current overall sigma as 90-100 mas/y. This is small compared to the Geodetic effect, but totally swamps the Frame Dragging effect. They say the Geodetic effect is totally obvious from the rawish data, but the Frame Dragging Effect must be dug out of the noise. Remember, they found two major unexpected noise sources, and for several months were afraid that they would not be able to report any frame dragging result. It is only because of the large amount of redundancy in the data and the fact that the two GR effects and the two unexpected noise sources have four different mathematical characteristics that they expect to be able to recover something close to the originally expected accuracy.
Best, Jim

LeBourdais

Thank you very much Jim !!! It gives me a better picture of what's going on.

Paul

Kris Krogh

Jim,

Thanks very much for your insights! That all seems to add up. Would it be possible to confirm with someone there from GP-B that the best of this data would hint at a slightly larger than expected frame-dragging effect? (If it hints at anything.)

Best wishes,

Kris

Garth

I have just returned from the APS Meeting at Jacksonville and a holiday in Florida.

As has been well discussed the first results have verified the GR geodetic prediction to 1% but there is no handle on the frame-dragging prediction, basically because unexpected signals so far swamp it, except for 'glimpses'.

By the end of the year the correct removal of these effects will give a robust reading to both precessions.

The running now stands:

1. Einstein's General Relativity(GR)
2. Brans-Dicke theory (BD)
3. Barber's Self Creation Cosmology (SCC),
4. Moffat's Nonsymmetric Gravitational Theory (NGT),
5. Hai-Long Zhao's Mass Variance SR Theory (MVSR),
6. Stanley Robertson's Newtonian Gravity Theory (NG),
7. Junhao & Xiang's Flat Space-Time Theory (FST).
8. R. L. Collin's Mass-Metric Relativity (MMR) and
9. F. Henry-Couannier's Dark Gravity Theory (DG).
10. Alexander and Yunes' prediction for the Chern-Simons gravity theory (CS).
11. Kris Krogh's Wave Gravity Theory (WG)
12. Hongya Liu & J. M. Overduin prediction of the Kaluza-Klein gravity theory (KK).
13. Kerr's Planck Scale Gravity: now accepted for publication Predictions of Experimental Results from a Gravity Theory (PSG)

The following are still in the running:

GPB Geodetic precession (North-South)
1. GR = 6.6144 arcsec/yr.
2. BD = [itex](3\omega + 4)/(3\omega + 6)[/itex] 6.6144 arcsec/yr. where now [itex]\omega[/itex] >60.
4. NGT = 6.6144 - a small [itex]\sigma[/itex] correction arcsec/yr.
6. NG = 6.6144 arcsec/yr.
9. DG = 6.6144 arcsec/yr.
10. CS = 6.6144 arcsec/yr.
11. WG = 6.6144 arcsec/yr.
12. KK = (1 + b/6 - 3b² + ...) 6.6144 arcsec/yr. where 0 < b < 0.07.

We await the GPB gravitomagnetic frame dragging precession (East-West) result.

1. GR = 0.0409 arcsec/yr.
2. BD = [itex](2\omega + 3)/(2\omega + 4)[/itex] 0.0409 arcsec/yr.
4. NGT = 0.0409 arcsec/yr.
6. NG = 0.0102 arcsec/yr.
9. DG = 0.0000 arcsec/yr.
10. CS = 0.0409 arcsec/yr. + CS correction
11. WG = 0.0000 arcsec/yr.
12. KK = 0.0409 arcsec/yr.


Those that have fallen by the wayside:

3. SCC = 4.4096 arcsec/yr.
5. MVSR = 0.0 arcsec/yr.
7. FST = 4.4096 arcsec/yr.
8. MMR = -6.56124 arcsec/yr.
13. PSG = 0.0000 arcsec/yr.

Garth

LeBourdais

Hi Garth,

Thank you for this status.

Sorry for your theory.

Best wishes
Paul

Garth

Thank you for your commiserations!

Garth

magnetar

I do not understand why so many "Alternative theories" about gravity???
why so many controversial ??

sylas

Hi Garth,

Can you explain the difference between the 6614.4 and 40.9 being quoted in numerous sources before the APS meeting(including many still available in the GP-B site) and the 6606 and 39 numbers now being used? (numbers being GR expectations for geodetic and framedragging effects in milliarcsec/yr)

I'm guessing it could be a difference in the altitude of the final orbit, but I don't know.

Cheers -- Sylas

Garth

That is a good question that wasn't addressed at the meeting, I have only recently become aware of that anomaly myself.

The posters clearly show the latter (6606 and 39 mas/yr) set of values while all their previous literature showed the former (6614.4 and 40.9 mas/yr) set.

The orbit decreased in SMA by about 350 metres during the lifetime of the experiment, but that should have increased the expected precessions by about one part in 10^-4 in my estimation.

The present measured value of the geodetic precession is 6638 +/- 97 mas/yr. (Francis Everitt APS Plenary Session 14th April 07)

Note however on the Gravity Probe B Science Data Analysis: Filtering Strategy poster (Click on the title), it says of the geodetic measurement:GP-B website:Garth

sylas

OK... I have gone back to first principles, and I think I have sorted this one out.

The information at the GP-B seems pretty sloppy. I have checked out the Fact Sheet, dated February 2005. The information therein is inconsistent.

Here are the orbit characteristics...
The semi-latus rectum (wiki ref) is given as:
[tex]a*(1-e^2) = 7.0274*10^6*(1-0.0014^2) = 7.027386226*10^6[/tex]
which is the same a, up to five figure accuracy.

The formula for geodetic precession is [itex]1.5(GM)^{1.5}c^{-2}R^{-2.5}[/itex], where R is the semi-latus rectum (ref: Gravitation and cosmology, S. Weinberg (1972) [pp237-8]).

Plug in
[tex]
G = 6.6742*10^{-11},
M = 5.976*10^24,
c = 299792458
[/tex]

and we get [itex]1.0155*10^{-12}[/itex] rad/sec, or 6.60559 arcsec/yr.

However, the same press release, with these same orbit parameters, gives 6.6144

I'm guessing they had already calculated 6.6144 from a projected orbit; and then recalculated for the actual orbit, but did not properly update all the recorded predictions.

The value 6.6144 implies an orbit about 3.7 km smaller in radius.

Now… what formulae do I need to use for the Lense-Thirring effect?

Cheers -- Sylas

Garth

Concur; does that mean their first set of values was simply a mistake?Here it is:

[tex]\Omega_{f-d} = \frac{GI}{c^2R^3}[\frac{3\underline{R}}{R^2}(\omega.\underline{R}) - \omega][/tex]

Garth

sylas

Thanks Garth... yes, I know that formula. It is a vector equation, and it varies over the whole orbit. So we have some work to try and get a magnitude from it. I was hoping for a straight formula for the magnitude.

I also need a value for I, which is the moment of inertia for the Earth. I can calculate assuming a solid sphere; but distributions of mass are not uniform, so this is only an approximation.

The vectors are [itex]\omega[/itex], which is in a fixed direction along the Earth's axis, and R, which is the location of the probe. This part has a maximum value over the poles, the dot product is just a product of magnitudes. The direction is the opposite of [itex]\omega[/itex] by the sign differences, so at the poles but in brackets has magnitude [itex]2\omega[/itex]. But over the equatot, the dot product drops to zero and the magnitude is [itex]\omega[/itex] in the opposite direction.

The suggests a mean magnitude of [itex]\omega/2[/itex]. I think.

Using [itex]I = 0.4MR_e^2[/itex] as a solid sphere, I get

[tex] 0.2*GMR_e^2R^{-3}c^{-2}\omega [/tex]

as a magnitude. My spreadsheet gets very roughly in the ball park with this, at 0.049

But I'm still out by much too much.

Any GP-B experts can help me out?

Thanks -- Sylas

sylas

Followup to my previous post! I have now found a figure for the moment of inertia of the Earth, and am able to get close to the OLD value for the framedragging effect. But I'll go through it all from the beginning with new numbers.

To get accurate values, I have tried to use data to five or six figures of accuracy, or whatever is available. I use SI units, unless explicitly given otherwise.

I've checked out the following sources.

• A NASA Planetary fact sheet for the Earth.
• The paper A General Treatment of Orbiting Gyroscope Precession, by Ronald J.Adler and Alexander S. Silbergleit (arXiv:gr-qc/9909054)
• The International Earth Rotation and Reference Systems Service: Useful constants.

Geodetic effect

The geodetic effect is 1.5*(GM)^1.5c^-2r^-2.5, where r is the semilatus-rectum of the probe orbit. This is the semimajor axis times (1-e²), as indicated in a previous post.

The radius of the orbit of GP-B is measured as 7027.4 km (semi-major axis). The eccentricity is e = 0.0014, so the semilatus rectum is 7027.386 km. Adler and Silbergeit use 7028 km with a circular orbit; this was calculated before launch.

The value of GM (gravitational constant times earth mass) is known with great precision; much more than either G or M individually. The value of G*M is 3.986004418*10¹⁴ m³s^-2 in IERS.

NIST currently gives G as 6.67428*10^-11, which would give M as 5.9721864*10²⁴. The NASA page gives Earth mass as 5.9736*10²⁴; this corresponds to G = 6.6727*10^-11.

To convert from radians per second to milliarcseconds per year, the factor is 6.50908*10¹⁵. That uses a tropical year of 365.24219 days (IERS). The speed of light is 299792458

The only meaningful source of error here is in r, the radius of the orbit. The largest value for the geodetic precession requires r to be small, and the conversion factor to be large. The conversion factor for milliarcseconds/year uses the length of a tropical year, being 365.24219 days; there's no basis for using anything greater.

The calculation is

[tex] 1.5 * (3.986004418 * 10^{14})^{1.5} * 299792458^{-2} * (7.027386 * 10^6)^{-2.5} * 365.24219 * 86400 * 360 * 3600 * 1000 / 2 / \pi [/tex]

This gives the geodetic precession as 6603.77 milliarcseconds/year. I can't see any possible way to make this any bigger.

To make matters worse, Adler and Silbergeit also give a correction factor to take account of the Earth's oblate shape. This factor is given as:
[tex]1-\frac{9}{8}*J_2*(R/r)^2)[/tex]

Here R is the radius of the earth and J₂ is the quadrupole moment.

For the radius of the Earth, Adler and Silbergeit use 6378 km, NASA gives 6378.1 km, and the IERS gives 6378.1366 km.

For J₂, Adler and Silbergeit use 1.083*10^-3, and the IERS gives 1.0826359*10^-3. The accuracy here will not matter much.

This factor reduces the geodetic prediction, by (1-1.00*10^-3), to give a final prediction of 6597.14 milliarcsec/year

How anybody ever got 6614.4 I don't know.

There is an additional precession due to the Sun; but this is in a different plane, and is going to have more effect on framedragging. It is discussed in Adler and Silberguit as well, with a magnitude of 19 milliarcsec/yec, but mostly perpendicular to geodetic precession.

Frame dragging

As derived above (and Adler and Silbergeit confirm) the magnitude of the effect works out to be [itex]GJr^{-3}c^{-2}\omega / 2[/itex]

The moment of inertia for a solid body is [itex]J = kMR^2[/itex], where k is a "moment of inertia ratio". This ratio is 0.4 for a uniform sphere, but it will be more if the mass density is greater near the surface, and less if the mass density is greater near the center.

Adler and Silbergeit use k = 1/3.024 = 0.3307. The NASA fact sheet gives k = 0.3308.

Using the value of GM from IERS, GJ would be kGMR², which is about 5.3640*10²⁷, using the NASA value for the radius R of the Earth.

The IERS gives a value for J directly, which is 8.0365*10³⁷; and with their value of G as 6.6742*10^-11, this gives GJ = 5.3638*10²⁷.

The length of a sidereal day 23.9345 hours (NASA sheet) so the rotation velocity ω is 7.2921 * 10^-5 rad/sec. In IERS it is 7.292115*10^-5.

The framedragging effect is
[tex] GJr^{-3}c^{-2}\omega/2 [/tex]

This works out to
[tex] 5.3638*10^{27} * (7.0274 * 10^6)^{-3} * 299792458^{-2} * 7.292115*10^{-5} * 365.24219 * 86400 * 360 * 3600 * 1000 / 2 / \pi / 2[/tex]

which gives 40.81 milliarcsec/year,

As before, there is a correction factor; this time equal to
[tex]1+\frac{9}{8}*J_2*(R/r)^2(1-\frac{3}{7}*MR^2/J))[/tex]

This works out to [itex]1-2.97*10^{-4}[/itex], which brings the prediction back down to 40.80 milliarcseconds/year.

This close to the 40.9; but now I don't know how they are obtaining 39.

Cheers -- Sylas

Garth

That's a very impressive piece of work, thank you Sylas.

Why don't you e-mail Alex Silbergleit at: gleit@relgyro.stanford.edu with these questions? And then let us know the answer of course.

Garth

henryco

Hi everybody,

Thank you for your efforts to help clarify several points. I still cant decode the axis informations from the plot named Torque modeling example: motion of gyro 3 in the L10028 poster. Can someone who attended the APS conf or GP-B expert help us ?

regards,

F H-C

Garth

The unexpected torques on the rotors were from:
1) A time dependent polhode precession due to the gyros not being exactly spherical. The time dependency is modelled by a dissipation of kinetic energy over time.
2) A misalignment torque due to a variation of electric potential over the surface, which can arise due to the polycrystalline structure.
It can be affected by presence of contaminants and is modelled as dipole layer. The patch fields are present on rotor and housing walls and cause forces and torques between these surfaces.

On the Torque modeling example: motion of gyro 3 in the L10028 poster the legend is very unclear, but the x-axis is E-W orientation milliarcsec/yr and the y-axis I believe is Polhode phase angle error.

Garth

henryco

Thank you garth. I have asked other questions to the GP-B website curator
and were told that an upgrade will soon be available on their site with clearer
plots... For the time being i dont see how i could read the amount of relativistic motion from this plot where the dashed line represents the estimated relativistic motion.
I also asked how the error plot was obtained on the error poster ...
They also told me that the audio file of their presentations will soon be available online

Best regards,

F H-C


Regards,

F H-C