Skip to main content
SpringerLink
Log in

The Frame of Fixed Stars in Relational Mechanics

  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

Relational mechanics is a gauge theory of classical mechanics whose laws do not govern the motion of individual particles but the evolution of the distances between particles. Its formulation gives a satisfactory answer to Leibniz’s and Mach’s criticisms of Newton’s mechanics: relational mechanics does not rely on the idea of an absolute space. When describing the behavior of small subsystems with respect to the so called “fixed stars”, relational mechanics basically agrees with Newtonian mechanics. However, those subsystems having huge angular momentum will deviate from the Newtonian behavior if they are described in the frame of fixed stars. Such subsystems naturally belong to the field of astronomy; they can be used to test the relational theory.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. \( \mathbf {\alpha }(t)\) is an infinitesimal vector directed along the axis of rotation (finite rotations require orthonormal matrices). Galileo transformations are included in the gauge group (2), (3); they are the elements having \({\dot{\mathbf {\xi }}}=\mathbf {V} = \) constant, and \(\dot{\mathbf {\alpha }}=0\).

  2. We call intrinsic those quantities of the form \(\sum \limits _{i<j}\frac{ m_{i}m_{j}}{2M}~f_{ij}(\mathbf {r}_{ij},\mathbf {v}_{ij})\) where \( f_{ij}=f_{ji\,}\).

  3. See Ref. [3] for the structure of constraints in the Hamiltonian formulation of the theory.

  4. It is easy to verify that the intrinsic magnitudes \(\mathbf {J}\) and \(\mathbf {I}\) are the usual angular momentum and tensor of inertia with respect to the center of mass: \( \mathbf {J}=\sum \,m_{k}~(\mathbf {r}_{k}-\mathbf {R})\times \mathbf {v}_{k}~\), \( \mathbf {I}=\sum \,m_{k}\,[|\mathbf {r}_{k}-\mathbf {R}|^{2}~\mathbf {1}-( \mathbf {r}_{k}-\mathbf {R})\otimes (\mathbf {r}_{k}-\mathbf {R})]\). In general, the intrinsic magnitudes are not additive; the intrinsic angular momentum of the universe is not the sum of the intrinsic angular momentum (spin) of its parts because of orbital contributions. However, if the system is split into several parts whose centers-of-mass are coincident, as in the case of Fig. 1, then \(\mathbf {J}\) and \(\mathbf {I}\) can be decomposed as the sum of the intrinsic quantities belonging to each part, as done in Eq. ( 14). In general, if the system is split into two parts A and B, then it follows that \(\mathbf {J}=\mathbf {J}_A+\mathbf {J}_B+(\mathbf {R}_A- \mathbf {R}_B)\times (M_B\,\mathbf {P}_A-M_A\,\mathbf {P}_B)/M\).

  5. A system of just \(N=2\) particles has only one degree of freedom (the distance between the particles). The circular motion would imply that the distance is constant. But this would only be possible in the absence of interaction. The role played by the rest of the universe as responsible of the centrifugal effect that is needed to sustain the orbital motion (and the shape of the water in the Newton’s bucket as well) is analyzed in Ref. [3].

  6. In an elliptic orbit, however, \(\tau _{Kepler}\) is the time elapsed between successive passes through the periastron. This is an observable, since the periastron is defined by the minimization of a distance. Notoriously the periastron suffers a cumulative shift in the frame of fixed stars because \(\tau _{Kepler}>\tau \).

  7. The measurement of velocities in the universe involves the Doppler shift. Ignoring general relativity effects that are beyond this framework, the Doppler shift depends on the relative radial velocity source-observer, which is a gauge invariant magnitude (it is the change of a distance per unit of time).

  8. The conservation of \(I^{-1}\ \mathbf {J}^{\prime } _{12}\) can be regarded as a consequence of the conservation of \(\mathbf {J}_{12}\) in the Newtonian frame and the way \(\mathbf {J}_{12}\) transforms under change of frame (cf. Eq. (18)).

  9. The International Celestial Reference Frame (ICRF2) is defined by the positions of about 300 extragalactic sources.

References

  1. Mach, E.: Die Mechanik in ihrer Entwicklung. Historisch-kritisch dargestellt. F.A. Brockhaus, Leipzig (1883) (The Science of Mechanics: A Critical and Historical Account of Its Development. The Open Court Publishing Co., Chicago (1893))

  2. Alexander, H.G.: The Leibniz-Clarke Correspondence Together with Extracts from Newton’s Principia and Opticks. Manchester University Press, Manchester (1956)

    MATH  Google Scholar 

  3. Ferraro, R.: Relational mechanics as a gauge theory. Gen. Relat. Gravit. 48, 23 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Einstein, A.: Gibt es eine Gravitationswirkung, die der elektrodynamischen Induktionswirkung analog ist? Vierteljahrsschrift für gerichtliche Medizin und öffentliches Sanitätswesen 44, 37–40 (1912)

    ADS  Google Scholar 

  5. Einstein, A.: Die formale Grundlage der allgemeinen Relativitätstheorie, Sber. Preuss. Akad. Wiss. Berlin, 1030–1085 (1914)

  6. Einstein, A.: Die Grundlage der allgemeinen Relativitätstheorie. Annalen der Physik 354, 769–822 (1916)

    Article  ADS  MATH  Google Scholar 

  7. Einstein, A.: Prinzipielles zur allgemeinen Relativitätstheorie. Annalen der Physik 360, 241–244 (1918)

    Article  ADS  MATH  Google Scholar 

  8. Hofmann, W.: Kritische Beleuchtung der beiden Grundbegriffe der Mechanik: Bewegung und Trägheit und daraus gezogene Folgerungen betreffs der Achsendrehung der Erde und des Foucault’schen Pendelversuchs. M. Kuppitsch Wwe., Wien (1904) (partial English translation in Ref. [12])

  9. Reissner, H.: Über die Relativität der Beschleunigungen in der Mechanik, Physikalische Zeitschrift 15, 371–375 (1914) (English translation in Ref. [12])

  10. Schrödinger, E.: Die Erfüllbarkeit der Relativitätsforderung in der klassischen Mechanik, Annalen der Physik 382, 325–336 (1925) (English translation in Ref. [12])

  11. Barbour, J.B.: Forceless machian dynamics. Nuovo Cimento B 26, 16–22 (1975)

    Article  ADS  Google Scholar 

  12. Barbour, J.B., Pfister, H. (eds.): Einstein Studies, vol. 6: Mach’s Principle: From Newton’s Bucket to Quantum Gravity. Birkhäuser, Boston (1995)

  13. Hughes, V.W., Robinson, H.G., Beltran-Lopez, V.: Upper limit for the anisotropy of inertial mass from nuclear resonance experiments. Phys. Rev. Lett. 4, 342–344 (1960)

    Article  ADS  Google Scholar 

  14. Barbour, J.B., Bertotti, B.: Gravity and Inertia in a Machian Framework. Nuovo Cimento B 38, 1–27 (1977)

    Article  ADS  Google Scholar 

  15. Barbour, J.B., Bertotti, B.: Mach’s principle and the structure of dynamical theories. Proc. R. Soc. Lond. A 382, 295–306 (1982)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Barbour, J.: Scale-invariant gravity: particle dynamics. Class. Quant. Grav. 20, 1543–1570 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Gryb, S.: Implementing Mach’s principle using gauge theories. Phys. Rev. D 80, 024018 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  18. Anderson, E.: The problem of time and quantum cosmology in the relational particle mechanics arena (2011). arXiv:1111.1472v3

  19. Mercati, F.: A shape dynamics tutorial (2014). arXiv:1409.0105

  20. Lynden-Bell, D.: The relativity of acceleration. In: Warner, B. (ed.) Variable Stars and Galaxies (in honour of M.W. Feast), ASP Conference Series 30 (1992)

  21. Lynden-Bell, D.: A Relative Newtonian Mechanics. In: Barbour, J.B. and Pfister, H. (eds.) Einstein Studies, vol. 6: Mach’s Principle: From Newton’s Bucket to Quantum Gravity. Birkhäuser, Boston (1995)

  22. Lynden-Bell, D., Katz, J.: Classical mechanics without absolute space. Phys. Rev. D 52, 7322–7324 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  23. Zwicky, F.: Die Rotverschiebung von extragalaktischen Nebeln. Helv. Phys. Acta 6, 110–127 (1933)

    ADS  MATH  Google Scholar 

  24. Zwicky, F.: On the masses of nebulae and of clusters of nebulae. Astrophys. J. 86, 217–246 (1937)

    Article  ADS  MATH  Google Scholar 

  25. Rubin, V.C., Ford, W.K., Thonnard, N.: Rotational properties of 21 Sc galaxies with a large range of luminosities and radii, from NGC 4605 (R = 4 kpc) to UGC 2885 (R = 122 kpc). Astrophys. J. 238, 471–487 (1980)

    Article  ADS  Google Scholar 

  26. Persic, M., Salucci, P., Stel, F.: The universal rotation curve of spiral galaxies-I. The dark matter connection. Mon. Not. R. Astron. Soc. 281, 27–47 (1996)

    Article  ADS  Google Scholar 

Download references

Acknowledgments

This work was supported by Consejo Nacional de Investigaciones Científicas (CONICET) y Técnicas and Universidad de Buenos Aires.

Author information

Authors and Affiliations

  1. Instituto de Astronomía y Física del Espacio (IAFE, CONICET-UBA), Casilla de Correo 67, Sucursal 28, 1428, Buenos Aires, Argentina

    Rafael Ferraro

  2. Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, 1428, Buenos Aires, Argentina

    Rafael Ferraro

Authors
  1. Rafael Ferraro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rafael Ferraro.

Additional information

Member of Carrera del Investigador Científico (CONICET, Argentina).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ferraro, R. The Frame of Fixed Stars in Relational Mechanics. Found Phys 47, 71–88 (2017). https://doi.org/10.1007/s10701-016-0042-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10701-016-0042-7

Keywords

Search

Navigation