1. Normal gravity field in relativistic geodesy
Sergei Kopeikin
University of Missouri-Columbia, USA

2. ISSI Workshop on Clocks and Spacetime Metrology (Bern, Switzerland)
Published: Phys. Rev. D 97, , 39 pp. (2018)
March 19-23, 2018
ISSI Workshop on Clocks and Spacetime Metrology (Bern, Switzerland)

3. ISSI Workshop on Clocks and Spacetime Metrology (Bern, Switzerland)
Outline:
Normal gravity in the Newtonian theory
Post-Newtonian metric
Post-Newtonian spheroid and coordinate freedom
Solving Einstein’s equations
Scalar potential and mass multipole moments
Vector potential and spin multipole moments
Normal gravity field potential in relativistic geodesy
The post-Newtonian Somigliana formula
Kerr metric for relativistic geodesy?
Conclusions
March 19-23, 2018
ISSI Workshop on Clocks and Spacetime Metrology (Bern, Switzerland)

4. Geoid, global and local ellipsoids
March 19-23, 2018
ISSI Workshop on Clocks and Spacetime Metrology (Bern, Switzerland)

5. Normal Gravity in Newtonian Theory
Earth’s body is modeled as a uniformly rotating, homogeneous ellipsoid made of an ideal fluid with kinematic equation of state 𝑝=𝑝(𝑥,𝑦,𝑧) Z 𝜔 𝒃
𝜌 0 Θ Y 𝒂 Φ X
March 19-23, 2018
ISSI Workshop on Clocks and Spacetime Metrology (Bern, Switzerland)

6. ISSI Workshop on Clocks and Spacetime Metrology (Bern, Switzerland)
Coordinate Charts:
Spherical coordinates: 𝑋=𝑅 sinΘ cosΦ 𝑌=𝑅 sinΘ sinΦ 𝑍=𝑅 cosΘ Ellipsoidal Coordinates: 𝛼= 𝑎 2 − 𝑏 2 𝑋=𝛼 1+ 𝜎 2 sin𝜃 cos𝜙 𝑌=𝛼 1+ 𝜎 2 sin𝜃 sin𝜙 𝑍=𝛼𝜎 cos𝜃
March 19-23, 2018
ISSI Workshop on Clocks and Spacetime Metrology (Bern, Switzerland)

7. Newtonian potential of normal gravity field
𝑊 𝑁 =𝑉+𝑍
𝑍= 1 2 𝜔 2 𝛼 𝜎 2 sin 2 𝜃 a centrifugal potential
𝑉= 𝐺𝑀 𝛼 𝑞 0 𝜎 + 𝑞 2 𝜎 𝑃 2 ( cos 𝜃) - gravitational potential
𝑞 0 𝜎 = arccot 𝜎 ; 𝑞 2 𝜎 = 𝑝 2 𝜎 arccot 𝜎 − 3 2 𝜎; 𝑝 2 𝜎 = 3 2 𝜎 ;
are the Legendre functions/polynomials of the imaginary argument,
𝑃 2 ( cos 𝜃)= cos 2 𝜃 − 1 2
Distrurbing potential: 𝑇=𝑈− 𝑊 𝑁
March 19-23, 2018
ISSI Workshop on Clocks and Spacetime Metrology (Bern, Switzerland)

8. The Somigliana formula
𝛾 𝑎 - normal gravity at the equator
𝛾 𝑏 - normal gravity at the pole
March 19-23, 2018
ISSI Workshop on Clocks and Spacetime Metrology (Bern, Switzerland)

9. Normal gravity in relativistic geodesy
Newton → Einstein’s general relativity
Which solution to use to model the normal gravity – an exact one (Kerr, Thorne- Hartle,…) or PN approximations?
What reference figure to use – an ellipsoid or spheroid?
Coordinate charts and coordinate freedom
Level and equipotential surfaces
March 19-23, 2018
ISSI Workshop on Clocks and Spacetime Metrology (Bern, Switzerland)

10. Post-Newtonian Metric
March 19-23, 2018
ISSI Workshop on Clocks and Spacetime Metrology (Bern, Switzerland)

11. Post-Newtonian reference spheroid
Uniformly rotating perfect fluid of homogeneous density 𝜌 takes the form of a
Maclaurin biaxial ellipsoid
𝑥 2 + 𝑦 2 𝑎 𝑧 2 𝑏 2 =1
Post-Newtonian equations tell us that the shape of a such uniformly rotating fluid body
has a shape of spheroid
𝜎 2 𝑎 𝑧 2 𝑏 2 =1+𝜅 𝐾 1 𝜎 2 𝑎 2 + 𝐾 2 𝑧 2 𝑏 2 + 𝐵 1 𝜎 4 𝑎 4 + 𝐵 2 𝑧 4 𝑏 4 + 𝐵 3 𝜎 2 𝑎 2 𝑧 2 𝑏 2
where 𝜅= 𝜋𝐺𝜌 𝑎 2 𝑐 2 ≃5.2× 10 − This equation can be recast to the following form
𝜎 2 𝑟 𝑒 𝑧 2 𝑟 𝑝 2 =1+𝜅 𝐵 1 + 𝐵 2 − 𝐵 3 𝜎 2 𝑎 2 𝑧 2 𝑏 2
where 𝑟 𝑒 =𝑎 𝜅( 𝐾 1 + 𝐵 1 ) - equatorial radius, and 𝑟 𝑝 =𝑏 𝜅( 𝐾 2 + 𝐵 2 ) -
polar radius.
March 19-23, 2018
ISSI Workshop on Clocks and Spacetime Metrology (Bern, Switzerland)

12. Gauge freedom and the choice of coordinates
The residual coordinate freedom is described by a PN coordinate transformation
𝑥 ′𝛼 = 𝑥 𝛼 +𝜅 𝜉 𝛼 𝑥
where the gauge functions 𝜉 𝛼 obey the Laplace equation
∆ 𝜉 𝛼 =0
Solution of this equation are harmonic polynomials. In the case of PN spheroid they are
𝜉 1 =ℎ𝑥+𝑝 𝑥 𝑎 2 𝑥 2 + 𝑦 2 −4 𝑧 2
𝜉 2 =ℎ𝑦+𝑝 𝑦 𝑎 2 𝑥 2 + 𝑦 2 −4 𝑧 2
𝜉 1 =𝑘𝑧+𝑞 𝑧 𝑏 𝑥 2 + 3𝑦 2 −2 𝑧 2
Equation of PN spheroid is form-invariant under the coordinate transformations. It
means that the parametric equation of PN spheroid has the following coordinate
freedom:
𝐾 1 → 𝐾 1 +2ℎ 𝐾 2 → 𝐾 2 +2𝑘
𝐵 1 → 𝐵 1 +2𝑝 𝐵 2 → 𝐵 2 −4ℎ
𝐵 3 → 𝐵 3 −8𝑝 𝑏 2 𝑎 2 +6𝑞 𝑎 2 𝑏 2
March 19-23, 2018
ISSI Workshop on Clocks and Spacetime Metrology (Bern, Switzerland)

13. Gauge freedom and the choice of coordinates
Arbitrary 𝐾 1 , 𝐾 2 ,𝐵 1 , 𝐵 2 , 𝐵 3
𝐾 1 = 𝐾 2 = 𝐵 1 = 𝐵 2 =0, 𝐵 3 ≠0
∼𝑎 𝑓𝑒𝑤 𝑚𝑒𝑡𝑒𝑟𝑠
𝒓 𝒑 𝒃 𝒂
𝒓 𝒆
~ 1 𝑐𝑚
𝑖𝑓 𝐵 3 ≃ 10 −10
The problem has four degrees of freedom corresponding to three infinitesimal coordinate transformations having four free parameters. Hence, we can arbitrary fix four out of the five constants in the mathematical equation of the PN spheroid. The remaining constant is determined from Einstein’s field equations along with a relationship between 𝜔 and the geometric parameters of the spheroid.
March 19-23, 2018
ISSI Workshop on Clocks and Spacetime Metrology (Bern, Switzerland)

14. ISSI Workshop on Clocks and Spacetime Metrology (Bern, Switzerland)
Solving PN equations
March 19-23, 2018
ISSI Workshop on Clocks and Spacetime Metrology (Bern, Switzerland)

15. External Scalar Potential
where
𝑚 𝑁 = 𝑀 𝑁 /𝛼
Notice that
Therefore,
March 19-23, 2018
ISSI Workshop on Clocks and Spacetime Metrology (Bern, Switzerland)

16. Mass Multipole Moments of Scalar Potential→
H. Moritz
PN correction
March 19-23, 2018
ISSI Workshop on Clocks and Spacetime Metrology (Bern, Switzerland)

17. ISSI Workshop on Clocks and Spacetime Metrology (Bern, Switzerland)
Comparison with GRS80
Inhomogeneous Ellipsoid
Homogeneous Spheroid
March 19-23, 2018
ISSI Workshop on Clocks and Spacetime Metrology (Bern, Switzerland)

18. Spin Multipole Moments
Total angular momentum
March 19-23, 2018
ISSI Workshop on Clocks and Spacetime Metrology (Bern, Switzerland)

19. Spin Multipole Moments
PN Gravitomagnetic Field
Soffel & Frutos, 2016 – error fixed
March 19-23, 2018
ISSI Workshop on Clocks and Spacetime Metrology (Bern, Switzerland)

20. Relativistic Potential of Normal Gravity Field→ →
𝑊 𝑃𝑁 − 𝑊 𝑁 =
2.5× 10 −3 𝑚 2 / 𝑠 2
on the reference
surface or about
2 mm in height
March 19-23, 2018
ISSI Workshop on Clocks and Spacetime Metrology (Bern, Switzerland)

21. The Somigliana Formula
Newtonian part
March 19-23, 2018
ISSI Workshop on Clocks and Spacetime Metrology (Bern, Switzerland)

22. Normal Gravity Field of the Kerr Metric
Scalar potential:
Vector potential:
March 19-23, 2018
ISSI Workshop on Clocks and Spacetime Metrology (Bern, Switzerland)

23. Multipole Moments of the Kerr Metric
ℓ>1
Multipole Moments of
PN Normal Gravity Field
March 19-23, 2018
ISSI Workshop on Clocks and Spacetime Metrology (Bern, Switzerland)

24. ISSI Workshop on Clocks and Spacetime Metrology (Bern, Switzerland)
Conclusions:
Mathematical technique for relativistic geodesy
Relativistic effects in geodesy can reach several meters but they are mostly coordinate-dependent effects. Physical effects reach the level not exceeding 1 cm.
At this level there is a large gauge freedom in choosing the reference body generating normal gravity field
Relativistic potential W of the normal gravity field differs from that 𝑾 𝟎 of the Newtonian gravity by 2.5×𝟏 𝟎 −𝟑 𝒎 𝟐 / 𝒔 𝟐 on the level surface of reference ellipsoid or about 2 mm in height.
Relativistic corrections to the multipole moments of normal gravity field are of the order of 𝟏𝟎 −𝟏𝟐 .
Normal gravity field generated by exact solutions of Einstein’s equations are mathematically exact but impractical for they cannot be matched with any realistic distribution of matter inside the reference body (the disturbing potential cannot be evaluated).
March 19-23, 2018
ISSI Workshop on Clocks and Spacetime Metrology (Bern, Switzerland)