471010_1_En_Print.indd by 0002624

Author:0002624
Language: eng
Format: epub
Published: 2019-02-06T04:54:44+00:00

Anholonomity in Pre-and Relativistic Geodesy

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terms - lead to anholonomity as already stated by P. Teunissen [2]. Or standard spherical /ellipsoidal series expansions to the order/degree 180/180, 360/360, or 720/720

up to 7200/7200 illustrate perfectly “anholonomity”. Chapter 3 introduces “Real null frames and coframes” within General Relativity” in order to prove anholonomity.

We take reference to Blagojevic, M., Garecki, J., Hehl, F. W. and Obukhov, Y.N. [3].

More specifically, in pseudo-Riemannian spacetime we define a non-unique pseudo-

orthonormal reference frame indexed α, β, . . . , ∈ {1 , 2 , 3 , 4}. We choose a timelike 4-leg by e4 and {e1 , e2 , e3} spacelike 1 , 2 , 3-legs. Such a popular frame of reference is pseudo-orthonormal, but unfortunately anholonomic, in general. Its metric is given by g(eα, eβ) = diag( 1 , 1 , 1 , −1 ), namely locally Minkowski. With orthonormal frames of reference we have previously experienced anholonomity: Please, study the subject of “Null frames” called { l, m, n, m}.

First, surface geometry as a two-dimensional Riemann manifold we in-

tent to orthonormalize. We arrive at a non-unique orthogonal reference

frame, a special Cartan frame of reference. We have shown this when

we analyze an ellipsoidal frame of reference in E. Grafarend and F. W.

Krumm [4].

Second, in analyzing the Euler Kinematical equations as well as the

dynamical Euler equation for rigid bodies and the dynamical Euler-

Louisville equations in terms of Euler or Cartan angle we found the well

known anholonomity in terms of a Frobenius matrix. We take reference

to E. Grafarend and W. Kühnel [5].

Chapter 4 is specifying the notion Killing vectors of symmetry, namely for the sphere (3 Killing vectors) and for the ellipsoid of revolution (1 Killing vector). It was needed to understand spherical symmetry versus ellipsoidal symmetry, in particular for the geodetic Somigliana–Pizzetti reference field.

Of particular importance is Chap. 5: we study the influence of local vertical nets which cause indeed anholonomity due to the reference of the physical local vertical.

At the end we analyze the famous object of anholonomity in terrestrial networks,

namely an example of Cartan’s exterior calculus.

We conclude in Chap. 6 with special comments in the special role of anholonomity for Geodesy, namely on the irregular boundary of the planet Earth. Our final highlight is the literature list of geodetic contributions of the topic anholonomity, by no means a forgotten subject. Finally we recommend to the International Association of Geodesy to adopt the axisymmetric Kerr metric or its linear approximation Lense-Thirring for a rotating - gravitating Earth, namely replacing the axisymmetric Somigliana–Pizzetti reference field to include Relativistic Geodesy.

2

Anholonomity: Before and After Relativity

First, we study geodetic anholonomity as being established by Friedrich Robert

Helmert. He is the real founder of Physical Geodesy, for instance of gravimetry.

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E. W. Grafarend

His concept of physical heights in terms of the gravity potential W , the sum of the gravitational potential U and the centrifugal potential V , as part of the scalar-valued gravity field established holonomic coordinates.

What are holonomic heights, better: holonomic height dif-

ferences, what are holonomic coordinates in contrast to

anholonomic ones? Such questions we will answer!

It is worth to study the arguments of F.

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