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A cta G eod . Ge oph. Hung. , Vol . 35 (3 ) , pp . 373-378 (2000)
GAUSS AND THE THEORY OF GRAVITATION
W SCROOER1 and H-J TREOER2
[Manuscript received April 22, 1999)
A historical review is given on various aspects of Gauss' research and on hisinterpretation of the theory of gravitation.
Keywords: Gauss; geomagnetism; theory of gravitation
1. Gauss and the theory of gravitation
The most important contribution by C F Gauss (1777-1855) to the theory ofgravitation was based on his, at first very general proof deduced ind ependently fromS D Poisson (1781-1840) that the Newtonian potential
sp = _f J0 dx ~Y dz
only in vacuum fulfils Laplace's (1749-1827) homogeneous potential equ ation
In border ar eas which are filled by masses, the Lap lace-operator t::,.c.p defines themass density (!:
where f is the Newtonian gravitational constant . Based on this idea, Gauss provedin 1840 his great potential theses (among others, the Gaussian integral thesis) overthe Newtonian potential function ip which enabled him to present the Newtoniangravitational force mathematically as a potential field in spite of its remote effect character. An energy density belongs to this potential field:
Thus Gauss succeeded in making an important ste p toward a field theory of gravitation.
Gauss studied the analogies of this potential field sp with, and its deviations from ,electricity and magnetism which he studied both theoretically and experimentally
1Hechelstrasse 8, 0-28777 Bremen Roennebeck, Germany2Rosa-Luxemburg-Strasse 17a, 0- 14282 Potsdam, Germany
1217-8977/2000/ $ 5.00 © 2000 Akadem iai K iad6 , Budapest
374 W SCHRODER and H-J TREDER
together with his Gottingen physical colleague W Weber (1803-1891). In connection with this, he considered the idea of a finite propagation velocity of electricand magnetic forces, and as first, looked for a correct mathematical formulation ofsuch finite propagation velocities of physical effects. Gauss deduced from the finitevelocity the fact of a temporal retardation, the retarded appearance of the effectsof a cause due to the spatial distance between the events.
Gauss presented a mathematical initial idea in a letter from 1845 to Weber.Gauss stimulated his and Weber's student B Riemann (1826-1866) who outlined in1850 the first field theory of gravitation and electricity which are congenial in theirphysical intuition with M Faraday's (1791-1867) contemporary basic ideas on theelectromagnetic field.
Gauss' idea could also be transferred to gravitation, as shown by Riemann andby other mathematicians and astronomers influenced by Gauss. Gauss' idea led- in contrast to Laplace's opinion expressed in his famous "Mecanique celeste",1805 - to the result that a retardation (with about the velocity of light c) of thegravitational effects is acceptable in the frame of celestial mechanics of the solarsystem.
As basis for the celestial mechanical consequences of the retardation of thegravitational action according to Gauss, Riemann and C Neumann (1832-1925)developed the difference between the gravitational potential sp and the "effectiveinteraction potential of gravitation" 4> . The latter interaction potential 4> appearsin the Lagrangean action-function L (and with it, in the action-integral f Ldt, too)for the planetary motion. If T is the kinetic energy of a planet with mass m, then:
L=T+m·4> (1)
(2)
and the equations of motion are the "Euler-Lagrangean equations" to Eq. (1):
!!.- . 8L _ 8L _ 0dt 8v I 8xI - .
The gravitational potential sp ~ ¥is - with the exception of the sign - identicalwith Riemann's interaction potential 4> only for potential functions which do not
depend on the velocities vI = !kxI of the masses similarly to the Newtonian po-tential, i.e, in such cases cp = -iP-. This is, however, generally not valid, and first of
all, terms in 4> which are of the form of a total differential of the time as fitS = Shave no importance as such terms do not contribute to the equations of motion ofthe masses.
The Gaussian retardation of the gravitational action with a finite velocity ofpropagation c* = JJ of the gravitation - which has to be very large with respect
to the velocity of motion of the planets relative to the Sun vI, so as C*2 » v 2 is
valid - it yields for the Newtonian potential function 4> = I~ a series after the
radial velocity of the planet fitr = f relative to the Sun :
1M ( f f2 )=-. 1+-+-+ ...r c* C*2
Acta Geod. Geoph: Hung. 35, 2000
GAUSS AND THE GRAVITATION 375
This series was deduced by Neumann in 1868 directly from Gauss' ideas on theimportance of the finite propagation velocity of physical forces . In Eq. (2), theexpression
1M . t = 1M . i(logr) (3)c' r c' dt
is a complete differential. Thus, Neumann's series, Eq . (2), yields with2
c·2 =~ » 1'2 the effective action-function of the planetary motion:
mM ( 81'2)L' = T + m<P' ~ T +I-r
- 1 +~ . (4)
An analogous expression had been intuitively introduced into electrodynamics byWeber in 1846 on an initiative in one of Gauss' letters.
The form of the kinetic term T (the kinetic energy of the planet) is in Eq . (4)independent from kinematics. In the Galilean and Newtonian classic al mechanicsit is valid for the motion of a planet m around the Sun in rest:
(5a)
while in Einstein's relativistic mechanics it is approximately:
(5b)
It is evident from these formulas that the retardation effects of the gravitation2
are of the order of magnitude oV2 according to Gauss, Riemann and Neumann.c
There are no correction terms of the order of magnitude v / c to the Newtonianforce laws ; all non-Newtonian correct ions in the gravitational law are proportional
to ~ . If! , if the propagation velocity of gravitation c' = ~is of the ord er ofc vo
magnitude of the velocity of light.This result implies Gauss' criticism to Laplace's thesis. Laplace supposed namely
that the series development of the retarded gravitational potential aft er the radialvelocity l' of the planet:
, 1M ( l' 1'2 )cp =-- 1+ - + - + ...r c' c· 2 (6)
should have to result in a correct ion term to the gravitational force being linear
in the velocity 1', namely as ~ 4 . I ¥ .Xl . In order to make such a corr ect ionc r
compatible with the facts of celestial mechanics, the est imat ion c" ~ ~c should haveto be valid. Thus according to Gauss the propagation velocity of the gravitationc· could be of the order of magnitude of the velocity of light c, while according toLaplace it should have to be at least of the order of magnitude
c2
C· -;- - » cv
Acta Geod. Geoph. Hung . 35, 2000
376 W SCHRODER and H-J TREDER
in order to get disturbances in the planetary motion due to retardation effects inacceptable magnitude. Laplace finally supposed that even c* 2: (f])2 c should hold.
It was a consequence of the studies made by Gauss and by his students thatLaplace's error could be corrected and the fact proved that celestial mechanicsincludes the possibility that gravitational action propagates with a velocity c* beingof the order of magnitude of the velocity of light c in accordance with the fieldphysics of gravitation. The result of such a propagation of gravitation c* = Jgyields then empirically just detectable disturbances in the Keplerian motion of twoand three-body problems of celestial mechanics, which result in a secular motion ofthe apsid-lines as described by F Tisserand's (1845-1896) formula:
(7)
(8)
Here a: is the major axis and e the eccentricity of the Keplerian ellipse. Tisserand and F Zollner (1834-1882) have already brought the motion of the perihelion as expressed by Eq. (7) in connection with the unexplained rest term foundby U Leverrier (1811-1877) in the celestial mechanical theory of the motion of Mercury.
2. Later discussions and applications
These insights into the connections between the retardation of the gravitation action in Gauss' sense and the "non-Newtonian" corrections to the classical Keplerianproblem of the motion of a planet in the gravitational field of a central body werereconsidered around 1870 by follower-mathematicians and -astronomers of Gaussand Laplace as F Tisserand in Paris, C Neumann, H Scheibner and F Zollner inLeipzig. In consequence of the re-formulation of all problems of the gravitationaltheory within Einstein's theory of relativity, these older problems were discussedagain around 1915/1920 by H v Seeliger, P Painleve and E Wiechert among others.
If the retardation velocity c* is as according to Gauss of the order of magnitudeof the velocity of light c (being the fundamental constant of electrodynamics andspecial theory of relativity), then the non-Newtonian corrections to the two- andthree-body problems of celestial mechanics are, according to Gauss', Riemann'sand Weber's potential functions, of the same order of magnitude as the relativisticeffects in celestial mechanics. Such effects were deduced from the special theory ofrelativity by H Poincare (1906) and by W de Sitter (1911). Einstein found in 1915the formula
6.'1/; = 67rfMc2a:(1 - e2 )
for the motion of the perihelion from his general theory of relativity as the only"non-Newtonian" correction of the Keplerian motion. This result was confirmed byK Schwarzschild in 1916 by the determination of the exact form of the centrallysymmetric gravitational field in the framework of the general theory of relativity.Einstein, Infeld and Hoffmann expanded this in 1938 to the two-body problem.
Acta Geod. Geoph. Hung. 35, 2000
GAUSS AND THE GRAVITATION 377
The same rotation of the perihelion as given by Eq. (8) would be obtainedfrom Tisserand's Eq. (7) from the retardation of the Newtonian gravitational actionaccording to Gauss , if the effective gravitational potential is supposed to be
<1>' = 1M (1 +3r2
)r c2
(Gerber 1898) what yields a propagation velocity c* of the gravitational action lessthan the velocity of light c:
* c cc =y'6=J3' (9)
Thus , Eq. (9) is physically very satisfactory: the perihelion motion of Mercury isreally no consequence of a finite propagation velocity of gravitation (H v Seeliger1917/1918 , M V Laue 1917/1920) .
The perihelion motion (Eq. 8) is deduced in Einstein's general relativity theory from the geodetic motion of the planet in Schwarzschild's space-time-rnetrics,and this spherical symmetric metrics is essentially independent of time; it is the
equivalent of the Newtonian gravitational potential - If:! in the general theory ofrelativity. According to Einstein's general theory of relativity the retardation effectof the gravitational field of the Sun is purely an effect of the co-ordinates which isproduced by the choice of the reference system and which could be also eliminatedby an appropriate choice. This is the same as shown by Riemann (around 1859)that the retarded action of the gravitational potential in Neumann's approximationfulfils a time-dependent generalisation of the Laplacean potential equation. Theeffective interaction potential
<1>' = 1M (1 + ~)r c*2
is namely a solution of the bi-harmonic potential equation:
~~<I>' =o.
In this problem Einstein and his general theory of relativity closed a seriesthoughts initiated by Gauss . This conclusion led simultaneously to the new problemof gravitational waves as possible analogue of the electromagnetic waves. Physicalprocesses are looked for here which transport the energy of the gravitational fieldwith the velocity of light (as anticipated by Gauss) (Treder 1975, Schroder andTreder 1997).
References
Einstein A 1929: Die Grundlagen der allgemeinen Relativitatstheorie, 5. Aufl., LeipzigEinstein A 1969: Grundziige der Relativitatstheorie. 7. Aufl., Berlin, BraunschweigGauss C F 1889: Allgemeine Lehrsatze in Beziehung auf die im verkehrten Verhaltnisse des
Quadrates der Entfernung wirkenden Anziehungs- und Abstossungskrafte, OstwaldsKlassiker, Nr . 2, Leipzig
Acta Geod. Geoph. Hung. 35, 2000
378 W SCHROD ER and H-J T RE DER
Gauss C F 1893: Die Intensitiit der erdmagnet ischen Kraft , auf ein absolutes Mass zuriickgefiihrt.Ostwalds Klassiker, Nr . 53, Leipzig
Gauss C F 1927: Anziehung eines Ringes. Ostwalds Klassiker, Nr . 225, LeipzigGauss C F, Werke Bd V Darin 1867: Allgemeine Theorie des Erdmagnetismus. GottingenLaplace de P S 1880: Mecanique celeste. In: Oeuvres, P S de Laplace, Bd . 4, ParisLaue M v 1961: In : Gesammelte Schriften und Vortriige, M v Lau e, Bd. III, BraunschweigNeumann C 1898: Allgemeine Untersu chung en iiber die Newtonsche Theorie der Fern-
wirkungen . LeipzigRiemann B 1880: Schwere, Elektrizit iit und Magnetismus. 2. Aufl., HannoverSchroder W, Treder H-J 1997: EOS, 78, 479.Tisserand F 1896: Mecanique celeste . Bd. IV, ParisTreder H-J 1972/1975: Die Relativit iit der Triigheit . Berlin/MoskauTreder H-J 1974: Uber die Prinzipien der Dynamik bei Einstein , Hertz, Mach und Poincare,
BerlinTreder H-J 1975: Die St erne, 51, 69-81.Weber W 1890/1892: Gesammelt e Werke. Bd . III , IV , LeipzigWiechert E 1925: In : Physik in der Kultur der Gegenwart. E Lechn er (ed.) , 2. Aufl.,
LeipzigZollner F 1883: Dei Natur der Cometen . 3. Aufl. , Leipzig
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