Gravitoelectromagnetism : The Lense-Thirring Effect

Vasco Miguel Roldão Manteigas Extended abstract written to Obtain the Master of Science Degree in Physics Engineering Instituto Superior Técnico - Department of Physics

Abstract General Relativity is the major theorical framework that explaind many gravitational phenomena in our Universe, and several experiments was planed and executed to verify the main prediction of this theory. One of lastest tests involve the measure of the Lense-Thirring effect, predicted by GR and explained naturally with a special theorical framework which use a formal analogy of electrodynamics. Using the so called GEM (Gravitoelectromagnetism) framework, it's possible ao ease the derivation and calculation of Lense-Thirring effect. Using the lastest data from Gravity Probe B, so far, the theorical prediction in GR/GEM agrees the experimental data obtained from this probe. Keywords: Gravitoelectromagnetism, Gravity Probe B, Lense-Thirring Effect.

1.Introduction: In this little introductory paper will introduce 4 h00=− the basic mathematical formulation of c2 (1) Gravitoelectromagnetism (GEM) Model, And the new gravitomagnetic potential as: which is a formal analogy of 2 A h =− i (2) Electromagnetism when applied to the 0i c2 Linearized Solution of General Relativity, as described by Mashhoon [1]. It is easy to compute explicitily the GEM potencials in terms of primary mass (M) and GEM framework is usefull to explain the their angular momentum (J). If we admit a geodesic effects recently observed by Gravity round primary (like a star or a planet), then the Probe B [2], using a nice analogy of an analog newtonian potential will value: of magnetic field around a curly source. G M =− (3) The inertial rotating source in primary will R produce a so-called gravitomagnetic force, and And the gravitomagnetic potential is just the any sattelite around the primary should be effect of angular momentum of the primary: G [L×r ] affected by this field, which acordly by RG is A= (4) R3 c no more than the non-diagonal terms of metric.

Applying the equations (3) and (4) on Linear Taking the linear approximation of RG field Einstein Field Equations [3], we can derive a equations [3], we can define a newtonian gravitoelectric field (rougly a small correction potential as: of newtonian force): 1 ∂ A Returning to the RG formalism, this Larmor E =−∇ − (5) 2c ∂t precession is due to the torsion of local metric And the gravitomagnetic field: due to angular momentum of the primary [1]. 1 B = ∇ ×A  (6) Any orbit around the primary will show a tiny 2 variation of the keplerian parameters, and the rate of precession can be calculated using the With this two fields (5,6) we can evaluate RG, and the re-interpreted using the GEM their own version of Maxwell's Equations of terms. GEM model: ∇⋅E=−4 G  (7) Using the correct approuch, the derivation of 1 ∂ B ∇ ×E =− (8) geodesic effects of local metric are based by a c ∂t propriety in RG [5] that it called the Fermi- ∇⋅B =0 (9) Walker paralell transport, which constrains the 1 ∂ E 4 G ∇ ×B= − J (10) conservation laws of physics to the geodesic c ∂t c path of the particles in the local manifold ruled by RG. We can also dedrive the Lorentz Force of Since the gravity filed in RG is equal to a free GEM from the geodesic equation from RG: particle in a curved spacetime, then we can v F =m E 2m ×B  (11) c assume the angular momentum of the satellite around the primary is a constant of motion:

 i 2. Derivation and Explanation of the S S =Si  S =const. (12) Geodesic Effect and Lense-Thirring Effect And the covariant derivative of the angular The Geodesic Effects observed in RG can be momentum will obey the following evaluation: understond in our GEM model as an simple  D S D u =u S (13) concept from the Classic Electrodynamics, the d s   d s   Larmor Frequency derived in many physics Which he angular momentum depends of the fields such the plasma physics. [4] velocity and position of the satellite. From the Lorentz Force (11), we can split the movement of a satellite around the primary, Taking explicity the (13) according to the with a tangential component which is the usual local metric, we obtain an equation of motion keplerian orbit, and a new axial origin vector similar to the gyroscope [5]: derived from the gravitomagnetic force which d S⃗ =Ω⃗˙ ×S⃗ (14) will cause a Larmor-type precession of the d s keplerian orbit. Where the derivative of the Larmor-type frequency of the (14) is equal to: This means that it's possible to derive a ˙ 1 d ⃗v 1 theorical evalution that should be agree if the Ω⃗ =− ⃗v× +⃗v×∇ ϕ− (∇×A⃗ ) (15) 2 d t 2 c RG is correct at the GEM model approximation. By splitting (15) into three terms, we obtain the Thomas Precession (due to kinematics), the For a low orbit satelitte, the average value of De Sitter or geodesic precession and the geodesic precession (De Sitter) would be:

Lense-Thirring precession. [5] 3/ 2 3(G M ) 9 R 2 〈Ω 〉= 1− J (19) G 2c2 r 5/ 2 [ 8 2( r ) ] Since the Thomas Precession is irrelevant in And the Lense-Thirring precession would be: many non-relativistic bound systems (inertial 2π G M R2 219 R 2 〈Ω 〉= 1− J (20) frames, for example), the other two factors of LT 5c2 r 3T [ 392 2( r ) ] (15) can be simplified in our toy model:

Let's assume the source's primary Taking the numerical orbital values of GP-B gravitational filed have a spherical symmetry [2] into (19,20), we get: with some multi-polar expantion corrections. 〈ΩG〉=6,606arc/ yr (21) The multi-polar expantion is needed to some 〈ΩLT 〉=39,2marc / yr (22) precision tests like the Graviy Probe B [2], We talk the experimental results after later in since it's orbit is a low altitude type, and the next setion, to introduce first the GP-B setup. quadrupolar terms of gravitiational field 3. The Gravity Probe B Setup and Brief cannot be ignored. Experimental Results Taking this constraints seriously, the De Sitter Precession rate of a low-orbit satelitte will be: The Gravity Probe B experiment can be ⃗ 3G 3 3 ΩG= M− I (⃗r ×⃗v)+ ( I ⃗r ×⃗v) (16) traced back to early 1960 when two Ph.D. in 2r 3 [( r 2 ) r 2 ] Physics publish a paper, “Possible New And the Lense-Thirring Precession will be: Experimental Test of General Relativity” in ⃗ 2G I 3 ΩLT = (−ω⃗ + (ω⃗⋅⃗r )⃗r ) (17) “Physical Review Letters”, which propose the r 3 r 2 use of a gyroscope in space to measure the geodesic effects predicted by RG. The “I” term in (16) and (17) is the inertial The project gain momentum when it's momentum of the primary. Ignoring oblate sponsered by NASA in the transition form factors of the curvate of the Earth, the inertial 1960's to 1970's, specially in the brand new momentum of a sphere is: space technology and better high precision 2 2 I = M r (18) 5 manufacturing technics. The main prototype of GP-B was designed in 1984, which consists a highlly round sphere Bibliography (to reduce oblatness), a telescope, a helium cooling device and the measurements [1] Bahram Mashhoon : arXiv:gr- hardware support. qc/0011014v1, 3/11/2000, To avoid complications in harmonics analysis, “Gravitoelectromagnetism”. the gyro's sphere should be perfectly smooth (and the final sphere was a oblatness less that [2] “Gravity Probe B in a nutshell”, “The 40 nm in 10 cm of sphere radius!), and this Gravity Probe B Experiment” : Stanford achivement was awarded by the Guiness Book University, Lockhead Martin, NASA official of Records! [2] presentation. The satellite telescope are used to maintain the satellite aligned to a distant star, whose [3] Sean M.Carrol , “Lecture Notes on General relative movement in celestial sphere can be Relativity”. ignored without jeopardize the measurements. (This means the distant star is an “perfect” [4] John David Jackson, “Classical inertial frame in the short period of the GP-B Electrodynamics”, 1962, John Wiley & Sons, mission when collects data.) Inc.

After several years of internal development, [5] Ignazio Ciufolini, David Lucchesi, the GP-B was finally launched in 2004, and Francesco Vespe, Federico Chieppa : arXiv:gr- the main mission occured between 2005 and qc/9704065v1, 23/4/1997, “Detection of 2007, however the data analisys will take Lense-Thirring Effect Due to Earth's Spin”. several years to account. In 2012 [6], the experimental results of [6] C. W. F. Everitt, D. B. DeBra, B. W. geodesic effect and frame-dragging was Parkinson et all: arXiv:1105.3456v1 [gr-qc], publish by NASA, and the values are: 17/5/2011, “Gravity Probe B: Final Results of

〈ΩG〉exp=6,6018±0,0183arc/ yr (23) a Space Experiment to Test General

〈ΩLT 〉exp=37,2±7,2marc/ yr (24) Relativity”

This means the first high precison test of RG show that the gravity is a space-time metric effect, and the results appart of statistical flutuations, are consistent with the prediction of RG.