Relativity and Cosmology

As promised, here are some calculations of frame dragging effects. Since strong frame dragging is pretty complex, I will start with the contemporary case of the frame dragging measurements of Gravity Probe B (GPB) in the weak gravitational field around Earth.

Look at this beautiful artwork of GPB, released by news-service.stanford.edu. What you see is one gyro (in this case a perfectly spherical mass) in polar orbit with its spin axis perpendicular to Earth's spin axis. The star IM Pegasi serves as a fixed reference frame to measure against.

Figure 1: GPB

(GRAPHIC COURTESY OF GRAVITY PROBE B IMAGE ARCHIVES)

There are two effects of interest: (i) the frame dragging effect, oriented with Earth's spin and (ii) the geodetic precession effect, oriented with the direction of orbital rotation. Note the extremely tiny values of the effects, especially the frame dragging, which will be discussed first.

In the weak gravity field of Earth, the rate of inertial frame dragging, as measured by the perpendicular "drift" of a near-perfect gyro on a polar orbit, is approximated to good accuracy by:

Here J is Earth's average angular (spin) momentum, R is the average radius of the satellite's orbit from Earth's center, and G and c have their standard meanings. The angular momentum of Earth is J ~ 6.87x10³³ kg m² s^-1, pretty large, as might be expected!

However, the factor G/c² ≈ 7.41x10^-28 m kg^-1 (pretty small by any standard), negates most of the angular momentum. Then throw in the inverse cube of orbital radius R ≈ 7.3x10⁶ meters for an average orbital altitude of ≈650 km and we have a very, very small frame dragging effect on an orbiting gyro.

It works out to: Ω ≈ 6.45x10^-15 radians/second, which, converted to the generally used units, approximates to 0.04 arc seconds per year. GPB has apparently successfully verified this, although the first official results are only expected in April this year.

An interesting side effect is that although the net frame dragging effect on the GBP gyro is in the direction of Earth's spin, its direction changes smoothly every 90° of latitude in the polar orbit. When around the equator, the precession effect is opposite to Earth's spin, because the one end of the gyro axis is farther from Earth than the other end and the frame dragging rate drops with the inverse cube of the distance. The factor 0.44 in the above equation is the result of the averaging effect. If static above the the poles, the factor would have been roughly 2.

The second effect tested by GPB was geodetic precession, which is a somewhat larger effect. It has nothing to do with frame dragging and simply measures the effect of movement through the curved space-time created by Earth's gravity field. The approximate equation for the low field limit is:

Here M is the mass of Earth, R_e is Earth's radius and R is the orbital radius. Plugging in the values, it gives: Ω ≈ 1.02x10^-12 radians/second, or 6.6 arc seconds per year. Here M is the mass of Earth, Re is Earth's radius and R is the orbital radius. This is essentially the same effect that makes the inside angles of a triangle, when drawn on a curved surface, to not add up to 180°.

In a future post I will give some gory formulae and interesting results for the strong field regime. But before that, I am preparing a bit of fun with a "free energy generator" using a spinning black hole and the garbage of a futuristic "space city" - watch this space...

The above equations were adapted from Gravitation (Misner, Thorne, Wheeler), equations 40.34 to 40.37'.

Regards, Jorrie