Aberration (gravitation)

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The aberration of gravitation is an effect that would occur if Newton's law of gravitation was combined with a finite gravitational velocity under certain conditions . This problem was finally solved by combining the law of gravitation with the then developing field theories . The end point of this development was Albert Einstein's general theory of relativity, which is still accepted today .

Laplace and Newtonian gravity

According to Newton's law of gravitation , changes in the gravitational field spread instantaneously, i. H. without wasting time. Around 1800, Pierre-Simon Laplace tried to combine the Newtonian model with a finite speed of propagation of gravitation by defining the gravitational field as a kind of radiation field or liquid . Changes in the movement of the central mass would have to be communicated to the environment in the form of a wave , which leads to influences on the movement of the celestial bodies of the order of magnitude v / c (where v is the speed of the body and c is the speed of the wave). So there is a similar relationship as with the aberration of starlight .

The consequences can best be explained with an example: Let us consider the earth and the sun and assume a speed of propagation of gravitation that corresponds to the speed of light . Then a force would act on the earth in the direction of the place where the sun was eight minutes ago and a force would act on the sun in the direction of the place where the earth was eight minutes ago. This delay would result in the distance between the earth and the sun increasing steadily, which means that the orbits would be unstable. Something similar would be expected for the earth and moon.

However, this contradicts the observation. B. the distance changes annually by only about 4 cm and this can be explained by the tidal friction between the earth and the moon (loss of rotational energy , loss of angular momentum ). The stability of the orbits can therefore only be achieved in the Newtonian model by assuming a higher speed of propagation of gravity. Laplace gave this as 7 · 10 6  c , where c is the speed of light. This high speed of gravitational interaction, which would be necessary with Newtonian gravity, is a point of attack that some critics in the 19th century generally opposed to all theories with finite propagation speed of gravity - e.g. B. Le Sage gravitation or gravitational explanations on an electrical basis - used.

Electrostatic fields

So aberration occurs in all fields that spread out like rays and the components of which are consequently subject to a certain direction, e.g. B. a radiation field around a light source. However, it does not occur with static fields such as B. electrostatic fields , since such a field (considered in the rest system of the source) is always stationary and does not spread once it is built up, so that the attraction is always directed exactly to the position of the field source. Deviations from this only occur when accelerations can no longer be neglected, which in turn leads to the emission of electromagnetic waves . The effects of moving charges according to Maxwell's electrodynamics are determined by the Liénard-Wiechert potential .

The first attempts to use an electrostatic field for gravitation were made at the end of the 19th century, combining the principles of Wilhelm Eduard Weber , Carl Friedrich Gauß , and Bernhard Riemann with the law of gravitation, tried and tested in the field of electrodynamics . These models were mainly used to explain the perihelion of Mercury , which could not be explained by Newton's law of gravity - but most of these models did not provide exact values. It was not until 1890 that Maurice Lévy was able to derive correct values ​​through a combination of Weber's and Riemann's basic laws. The attempt by Paul Gerber , who succeeded in 1898 in deriving the correct value for the perihelion of Mercury from a theory in which the gravitational potential spreads at finite speed, aimed in a similar direction . From the formula obtained, Gerber calculated a speed of propagation of the gravitational potential of approx. 305,000 km / s, i.e. practically the speed of light. Gerber's derivation of the corresponding equation (this equation formally corresponds to that of the general relativity theory), however, was classified as incorrect and therefore not considered by many (including Einstein) as a useful approach for a theory of gravity. From his theory it also follows that the value for the deflection of light in the gravitational field is too high by a factor of 3/2. With the replacement of Weber's electrodynamics by Maxwell's electrodynamics, however, these experiments were no longer pursued and are outdated.

The first attempt to explain gravity on the basis of Maxwell's electrodynamics was made by Hendrik Lorentz (1900). He started from the idea (like Mossotti and Zöllner before him ) that the attraction of two different electrical charges is a fraction stronger than the repulsion of two charges of the same name. The result would be nothing other than universal gravity. Lorentz was able to show that this theory is also not affected by Laplace's criticism and that only influences of the order of magnitude v² / c² occur, but he received a value that was far too low for the perihelion rotation. Lorentz summarized his efforts as follows:

“The particular form of these terms can possibly be modified. But what has been said so far is enough to show that gravity can be traced back to actions that do not spread faster than the speed of light . "

Lorentz covariant models

In 1904 Henri Poincaré stated that in order to maintain the principle of relativity it must be ensured that no signal is faster than the speed of light, otherwise the synchronization rule for light signals and thus the local time ( relativity of simultaneity ) would no longer be within the framework of the special relativity theory and Lorentz's ether theory be valid. However, in 1905 he was able to show that no orbital instabilities in the sense of Laplace can occur in a theory that takes the Lorentz transformation into account.

This can be understood in analogy to electrostatic fields: As shown above, there is no aberration in such fields (considered in the rest system of the sun). Of course, no transformation into an inertial system can change this, in which z. B. the earth is resting and the sun is moving. In such a system the description of the sun's field would turn out to be considerably more complicated - there will be a number of speed-dependent additional terms - but the result must be the same as in the rest system of the sun and the additional terms will compensate each other. Because a transformation only changes the coordinates of the physical events; the occurrence of the events themselves is not affected. Poincaré wrote:

“Laplace showed that the propagation is either instantaneous or much faster than that of light. However, Laplace examined the hypothesis of the finite speed of propagation ceteris non mutatis; in contrast, many others are attached to this hypothesis, and a more or less complete compensation may take place between them. The application of the Lorentz transformation has already shown us many such examples. "

Similar models were subsequently designed by Hermann Minkowski (1907) and Arnold Sommerfeld (1910). However, the research in this direction was terminated by the development of general relativity.

general theory of relativity

The above hypotheses were finally replaced by the much more extensive general theory of relativity . The speed of propagation of gravity is also the speed of light here. There is no aberration in the sense of Laplace because, as in the above field theories, the effect is almost exactly canceled by parts of the gravitational field of moving bodies. The deviation of real planetary orbits from pure Kepler orbits can instead be understood in connection with the emission of gravitational waves , which causes a gradual reduction of the orbital radii. It is a direct consequence of the conservation of angular momentum and energy . These must be fulfilled because the effect is invariant under Lorentz transformations. The general theory of relativity not only explains the stability of the two-body system and the perihelion of Mercury, but also provides the correct value for the deflection of light in the gravitational field .

As far as the measurement of the gravitational speed is concerned, the detection of gravitational waves is the main indirect method. Such evidence was actually provided by Russell Hulse and Joseph Taylor through observations of the double pulsar PSR 1913 + 16 , whose orbits were reduced by a factor that corresponds to the energy loss due to the radiation of gravitational waves.

In 2002, Sergei Kopeikin and Edward Fomalont published a paper in which they claimed to have indirectly measured the speed of gravity with the help of VLBI , the result being between 0.8 and 1.2  c , thus in agreement with the general theory of relativity . However, this was rejected by other experts such as Clifford Will and Steve Carlip, who believe that in this experiment it was not the gravitational speed that was measured, but only the speed of light. An agreement has not yet been reached.

With the gravitational wave event GW170817 on August 17, 2017, an electromagnetic signal was measured at the same time (1.7 s later). From this it could be deduced that gravity spreads with the speed of light (relative uncertainty −3 × 10 −15 / + 7 × 10 −16 ).

Individual evidence

  1. ^ PS Laplace: A Treatise in Celestial Mechanics . translated by N. Bowditch. Chelsea / New York 1966 Volume IV, Book X, Chapter VII.
  2. ↑ J. Zenneck: Gravitation, Encyclopedia of the Mathematical Sciences with inclusion of their applications . Vol V. 1. Leipzig 1903–1921, pp. 25–67, facsimile
  3. ↑ Zenneck, pp. 49–51
  4. ↑ P. Gerber: The spatial and temporal spread of gravitation . In: Journal for mathematical physics , 43, 1898, pp. 93-104.
  5. ↑ Gerber's Gravity. Mathpages
  6. ^ HA Lorentz : Considerations on Gravitation . In: Proc. Acad. Amsterdam . 2, 1900, pp. 559-574.
  7. ^ "The special form of these terms may perhaps be modified. Yet, what has been said is sufficient to show that gravitation may be attributed to actions which are propagated with no greater velocity than that of light. "
  8. ^ Henri Poincaré: Sur la dynamique de l'électron . In: Rendiconti del Circolo matematico di Palermo . 21, 1906, pp. 129-176. See also German translation .
  9. ↑ Laplace a montré en effet que cette propagation est, ou bien instantanée, ou beaucoup plus rapide que celle de la lumière. Mais Laplace avait examiné l'hypothèse de la vitesse finie de propagation, ceteris non mutatis; ici, au contraire, cette hypothèse est compliquée de beaucoup d'autres, et il peut se faire qu'il y ait entre elles une compensation plus ou moins parfaite, comme celles dont leu applications de la transformation de Lorentz nous ont déjà donné tant d 'exemples.
  10. ^ Scott A. Walter: Breaking in the 4-vectors: the four-dimensional movement in gravitation, 1905-1910 . In: J. Renn, M. Schemmel (eds.): The Genesis of General Relativity , Volume 3. Springer, Berlin 2007, pp. 193-252.
  11. ↑ S. Carlip: aberration and the Speed of Gravity . In: Phys. Lett. , A 267, 1999, pp. 81-87, arxiv : gr-qc / 9909087 online
  12. ↑ The Detection of gravitational waves (PDF; 5.5 MB) January 2000. Archived from the original on March 10, 2016. Retrieved on January 7, 2009., p. 4
  13. ^ Clifford M. Will: Has the Speed ​​of Gravity Been Measured? Contains a list of works on this controversy up to 2006.
  14. ^ BP Abbott et al .: Gravitational Waves and Gamma-Rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A . In: The Astrophysical Journal Letters . 848, No. 2, 2017. arxiv : 1710.05834 . doi : 10.3847 / 2041-8213 / aa920c .