Physics/Essays/Fedosin/Gravitational characteristic impedance of free space

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The Gravitational characteristic impedance of free space, \rho_{g0} \ is a physical constant that relates the magnitudes of the gravitational field strength \mathbf {\Gamma } and the gravitational torsion field \mathbf {\Omega } in covariant theory of gravitation (the gravitoelectric and gravitomagnetic fields in gravitoelectromagnetism) in gravitational radiation travelling through free space:

\rho_{g0} = \sqrt{\frac{\mu_{g0}}{\varepsilon_g}} = \frac{4\pi G}{c} = \frac {\mu_{g0} \Gamma}{\Omega}= 2.796696\cdot 10^{-18} \mathrm {m^2/(s\cdot kg)}. \

where ~\varepsilon_g = \frac{1}{4\pi G } = 1.192708\cdot 10^9 \mathrm {kg \cdot s^2 \cdot m^{-3}} is the gravitoelectric gravitational constant in the set of selfconsistent gravitational constants, ~ G is the gravitational constant, c is the speed of light, \mu_{g0} = \frac{4\pi G}{c^2} = 9.328772\cdot 10^{-27} \mathrm {m / kg} \ is the gravitomagnetic gravitational constant.

As in electromagnetism, the characteristic impedance of free space plays a central role in all radiation problems, such as in a comparison of the radiation resistance of gravity-wave antennas to the value of this impedance in order to estimate the coupling efficiency of these antennas to free space. The numerical value of this impedance is extremely small, but the impedance of all material objects must be “impedance matched” to this extremely small quantity before significant power can be transferred efficiently from gravitational waves to these detectors.

The gravitational characteristic impedance of free space may be connected with other constants:

\rho_{g0} = 2\alpha \cdot \frac{h}{m_S^2}, \

where h \ is the Planck constant, \alpha = \frac{e^2}{2\varepsilon_0 hc} is the fine structure constant for the elementary charge e \ , \varepsilon_0 is the vacuum permittivity, m_S = e\sqrt{\frac{\varepsilon_g}{\varepsilon_0}} = \frac{e}{\sqrt{4\pi G\varepsilon_0}} \ is the Stoney mass.

The gravitational characteristic impedance of free space for the atoms and nucleons level of matter is:

\rho_s = \frac{4\pi G_s }{c} = 6.346\cdot 10^{21} \mathrm {m^2/(s\cdot kg)}, \

where G_s = \frac {e^2}{4 \pi \varepsilon_0 m_p m_e} is the strong gravitational constant, m_p and m_e are the masses of proton and electron.

History

Due to McDonald [1] first who used Maxwell equations to describe gravity was Oliver Heaviside. [2] The point is that in the weak gravitational field the standard theory of gravity could be written in the form of Maxwell-like gravitational equations. [3]

In the 80-ties Maxwell-like equations were considered in the Wald book of general relativity. [4] In the 90-ties Kraus [5] first introduced the gravitational characteristic impedance of free space, which was detailed later by Kiefer [6], and now by Raymond Y. Chiao, [7] [8] [9] [10] [11] who is developing the ways of experimental determination of the gravitational waves.

Typical gravitational impedance at the megascopic scale

In the general case only planets (with their sattelites) and stars could be considered "as free, as possible" to be used as some "antenna" to the gravitational waves detection. Spherical megascopic bodies have the folloving characteristic impedance:

\rho_{gs} \approx \frac{v r}{3m}, \

where

v \ is the equatorial velocity,
r \ is the spherical body radius and
m \ is the body mass.
Solar planetary system
Object Radius, m Equator velocity, m/s Mass, kg Impedance \rho_{gs}/\rho_{g0} \
Human 2 1 100 6.67\cdot 10^{-3} 2.38\cdot 10^{15}
Sun 6.96\cdot 10^8 317.4 1.989\cdot 10^{30} 3.702\cdot 10^{-20} 1.32\cdot 10^{-2}
Mercury 2.425\cdot 10^6 2.99 3.311\cdot 10^{23} 7.3\cdot 10^{-18} 2.61\cdot 10^{0}
Venus 6.07\cdot 10^6 1.807 4.87\cdot 10^{24} 7.508\cdot 10^{-19} 2.68\cdot 10^{-1}
Earth 6.371\cdot 10^6 465 5.976\cdot 10^{24} 1.652\cdot 10^{-16} 5.9\cdot 10^{1}
Mars 3.395\cdot 10^6 240.6 6.424\cdot 10^{23} 4.239\cdot 10^{-16} 1.516\cdot 10^{2}
Jupiter 7.13\cdot 10^7 12644 1.899\cdot 10^{27} 1.582\cdot 10^{-16} 5.66\cdot 10^{1}
Saturn 6.01\cdot 10^7 10248 5.686\cdot 10^{26} 3.611\cdot 10^{-16} 1.291\cdot 10^{2}
Uranus 2.45\cdot 10^7 3952 8.689\cdot 10^{25} 3.714\cdot 10^{-16} 1.328\cdot 10^{2}
Neptune 2.51\cdot 10^7 2771 1.03\cdot 10^{26} 2.251\cdot 10^{-16} 8.05\cdot 10^{1}

Planetary data were taken from the textbook. [12] As could be seen from the Table, only the Mercury has characteristic impedance close to the free space value.

Planet's satellites
Object Radius, m Equator velocity, m/s Mass, kg Impedance \rho_{gs}/\rho_{g0} \
Moon 1.738\cdot 10^7 4.36 7.35\cdot 10^{22} 3.437\cdot 10^{-17} 1.23\cdot 10^{1}
Titan 2.44\cdot 10^7 276.8 1.37\cdot 10^{23} 1.643\cdot 10^{-15} 5.875\cdot 10^{2}
Ganymede 2.6\cdot 10^7 1116 1.54\cdot 10^{23} 6.28\cdot 10^{-15} 12.245\cdot 10^{3}
Callisto 2.36\cdot 10^6 252.7 9.5\cdot 10^{22} 2.092\cdot 10^{-17} 7.48\cdot 10^{2}

Satellites' data were taken from the textbook. [12] As could be seen from the Table, only the Moon has the closest value of characteristic impedance about 12-times greater then for free space.

Stars
Object Radius, m Equator velocity, m/s Mass, kg Impedance \rho_{gs}/\rho_{g0} \
Sun 6.96\cdot 10^8 317.4 1.989\cdot 10^{30} 3.702\cdot 10^{-20} 1.32\cdot 10^{-2}
O5 1.238\cdot 10^{10} 4\cdot 10^{5} 7.918\cdot 10^{31} 2.084\cdot 10^{-17} 7.45\cdot 10^{0}
F5 5.528\cdot 10^{10} 1\cdot 10^{5} 1.989\cdot 10^{30} 9.265\cdot 10^{-16} 3.31\cdot 10^{2}
Globular cluster 4.628\cdot 10^{17} 6\cdot 10^{4} 1.989\cdot 10^{37} 4.654\cdot 10^{-16} 1.66\cdot 10^{2}

In this table designations "O5" and "F5" define stellar types. [12] As it is seen from the Table, some stars could have characteristic impedance value about the same as the impedance of free space.

See also

References

  1. ↑ K.T. McDonald, Am. J. Phys. 65, 7 (1997) 591-2.
  2. ↑ O. Heaviside, Electromagnetic Theory (”The Electrician” Printing and Publishing Co., London, 1894) pp. 455-465.
  3. ↑ W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, MA, 1955), p. 168, 166.
  4. ↑ R. M. Wald, General Relativity (University of Chicago Press, Chicago, 1984).
  5. ↑ J. D. Kraus, IEEE Antennas and Propagation. Magazine 33, 21 (1991).
  6. ↑ C. Kiefer and C. Weber, Annalen der Physik (Leipzig) 14, 253 (2005).
  7. ↑ Raymond Y. Chiao. "Conceptual tensions between quantum mechanics and general relativity: Are there experimental consequences, e.g., superconducting transducers between electromagnetic and gravitational radiation?" arXiv:gr-qc/0208024v3 (2002). [PDF
  8. ↑ R.Y. Chiao and W.J. Fitelson. Time and matter in the interaction between gravity and quantum fluids: are there macroscopic quantum transducers between gravitational and electromagnetic waves? In Proceedings of the “Time & Matter Conference” (2002 August 11-17; Venice, Italy), ed. I. Bigi and M. Faessler (Singapore: World Scientific, 2006), p. 85. arXiv: gr-qc/0303089. PDF
  9. ↑ R.Y. Chiao. Conceptual tensions between quantum mechanics and general relativity: are there experimental consequences? In Science and Ultimate Reality, ed. J.D. Barrow, P.C.W. Davies, and C.L.Harper, Jr. (Cambridge:Cambridge University Press, 2004), p. 254. arXiv:gr-qc/0303100.
  10. ↑ Raymond Y. Chiao. "New directions for gravitational wave physics via “Millikan oil drops” arXiv:gr-qc/0610146v16 (2009). PDF
  11. ↑ Stephen Minter, Kirk Wegter-McNelly, and Raymond Chiao. Do Mirrors for Gravitational Waves Exist? arXiv:gr-qc/0903.0661v10 (2009). PDF
  12. ↑ 12.0 12.1 12.2 Allen C.W.(1973). Astrophysical quantities. 3-d edition. University of London, The Athlone Press.
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