The Propagation of Gravity (English Translation)

Scientific Paper
Title	The Propagation of Gravity (English Translation)
Read in full	Link to paper
Author(s)	Walter Rella, Paul Gerber
Keywords	gravity, propagation velocity of gravity, Mercury perihelion, potential, action at near
Published	1917
No. of pages	17

Read the full paper here

Abstract

This paper was first published in 1898, but posthumously republished in 1917, as a challenge to Einstein's then new theory of general relativity, J. for Math. & Phys., Zeitschrift f. Mathematik und Physik, V43, pp 93-104.

If it is assumed that the hitherto unexplained advance by 41`` per century of Mercury's perihelion is caused by the delay of time spent for the spatial propagation of gravity, it follows that this value equals the velocity of light, of thermal radiation and of electric waves. Attention has to be paid to what can, on the one hand, really be proven by computation and observation and what, on the other hand, is presumed in the first place without any proof. If the gravity between two masses is transmitted from the first body to the second and back again with some lag of time, one finds that this necessarily gives rise to an advance of the planets perihelion. It is, however, impossible to prove that the actual value of the perihelions advance, although it could not be deduced from disturbances of any type, could not have another origin than the presumed time lag. If this presumed origin gave a value for the propagation velocity of gravity different from the velocity of light, this would have no further meaning. Just the coincidence of both velocities vindicates this presumption and, hence, the notion of a finite propagation velocity of gravity.

Overview

This is Walter Rella's English translation of Paul Gerber's Die Fortpflanzungsgeschwindigkeit der Gravitation, the expanded 1902 Stargard school programme reprinted posthumously in Annalen der Physik Vol. 52, pp. 415–444 (1917) at the instigation of Ernst Gehrcke, whose editorial footnote opens the piece. Gerber's shorter 1898 paper in the Zeitschrift für Mathematik und Physik Vol. 43, pp. 93–104, had already announced the result; this longer treatise supplies the reasoning that the earlier abstract had merely sketched, together with a critical history of every previous attempt to measure the speed of gravity.

The claim is a large one. Gerber assumes only that gravitational action takes time to travel, derives from that assumption alone a modified gravitational potential, and finds that it produces exactly the anomalous advance of Mercury's perihelion — provided the propagation speed is that of light. He is emphatic that this is not what he set out to find: earlier investigations had put the speed of gravity anywhere from several times to ten million times the speed of light, and Ernst Mach, in the fourth edition of his Mechanik, had drawn attention to the conflict between Gerber's answer and the older ones. Where the mainstream account of the perihelion anomaly — after 1915 — attributes it to the curvature of spacetime in General Relativity, Gerber attributes it to retardation in a medium, and treats the numerical agreement with c as the sole evidence that his assumption was the right one. His closing paragraph makes the ontological point plainly: the result "will lend support to the notion that the medium sustaining the attraction of masses is identical with the medium propagating light, thermal radiation as well as electric and magnetic influences."

The argument

The state of tension

Gerber begins with three preliminaries. The first concerns space. If two masses appeared from nothing and attracted at once, "this is all what happens — the simultaneous presence of masses," and space would be "only there as a separating distance". If instead attraction begins only some time after the masses are placed, then something must propagate: a state of tension (Zwangszustand) spreading outward from each mass independently of whether any other mass is present. Gerber is careful to deny that this is an empirical discovery — it "follows simply from the concept of a successive propagation" — and equally careful not to say what the tension is. Just as nobody knows what it is that oscillates along a light ray, so here "the essence of this alteration remains unknown; we just take notion of its existence" by the fact that a second mass within the field is attracted. He explicitly declines to give a mechanical model.

Why the potential, not the force

The second preliminary is the paper's methodological pivot. Gerber argues that with a finite propagation speed one cannot determine the accelerating force at all, only the potential. The force is a directed quantity, so its value depends on how the directed motion of the mass combines with the directed spreading of the tensed state — and to know that, one would need to know what the tensed state is, which is "out of the question". The potential is not a directed quantity: "if the distributions of the tensed state surrounding the mass are identical at rest and in motion, then this must hold also for the potentials". Hence only the potential can be carried through the argument. He adds that strictly neither "propagation of the potential" nor "propagation of the force" is a proper phrase — what spreads is the tensed state, and the potential arrives with it.

Retardation shortens the transmission time

The third preliminary supplies the mechanism. The potential is not stamped on the attracted mass instantaneously but built up "step by step", over a time proportional to the propagation speed. If the attracting mass moves toward the attracted one, its tensed state arrives faster; if the attracted mass moves to meet it, the two close at the sum of the speeds. With both in motion the relevant transmission time is set by c together with the relative radial velocity dr/dt. The potential therefore has less time to develop than it would with the masses at rest, and "a smaller value of the potential must result". Carrying this through for masses m and m′ at separation r with radial velocity r′ = dr/dt, Gerber obtains a potential in which the Newtonian μ/r is divided by the square of (1 − r′/c), and then expands it to second order in r′/c before applying Lagrange's equations to get the acceleration. He checks the construction for symmetry: the same expression must come out for m′ acting on m as for m on m′, "because the forces acting against each other need be balanced".

The critical history

Section III is a survey and a demolition. Laplace (1805) treated gravity by analogy with stellar aberration, tilting the attractive force forward while leaving its magnitude Newtonian; Gerber objects that this amounts to a genuine action at a distance with the direction merely adjusted, "by the way the concept of a gradual propagation is eliminated," and that Laplace wrongly imagined the Sun's field at rest and ignored the planet's own field. Zöllner's 1873 proposal to time the delay of a horizontal pendulum tracking the Sun is dismissed because, as Laplace himself had noted, gravity is not shielded by intervening bodies, so no delay can appear. The nineteenth-century programme of applying electrodynamic force laws — Weber's, Riemann's, Clausius's — to planetary motion (Holzmüller 1870, Tisserand 1872, Servus 1885, Lévy 1890, Oppenheim 1894–95) fails on three counts: the constant in Weber's law is √2 times the velocity of light, not the velocity of light, so any coincidence would be misread; the spread of values gives no criterion for identifying the constant with a gravitational speed at all; and no one asked whether an electrodynamic law is applicable to gravity, which lacks the two-sided attractive–repulsive character of charge.

The closest predecessors are Lehmann-Filhés (1885) and von Hepperger (1888), who retarded the coordinates of the source in Newton's equations. Lehmann-Filhés concluded that Mercury's anomaly could not be so explained; Hepperger derived a lower bound of 500 times the speed of light. Gerber grants them real progress over Laplace but argues their scheme is inconsistent: writing the force components on Sun and planet as proportional to x₁/r₁³ and x₂/r₂³ from the respective retarded positions, the two cannot be equal and opposite unless r₁ = r₂ and the trajectories are identical. Action and reaction are therefore unbalanced in an isolated system, which he takes as a reductio. Lorentz's 1900 electrical theory of gravitation he judges "of better quality" than mechanical models but says it "must either be wrong or be corrected, because, according to this theory, the time lag associated with the propagation of gravity is unable to generate the advance of mercury's perihelion."

The computation for Mercury

Section IV runs the modified law through the standard planetary derivation: the areal-velocity integral, the substitution to the orbit equation, an ellipse whose axis ω is now slowly rotating, and finally an expression for the perihelion advance ψ per sidereal revolution in terms of the semi-major axis a, the eccentricity ε, the period τ and c. Inserting a = 0.3871 × 149 × 10⁶ km, ε = 0.2056, τ = 88 days and the observed ψ = 4.789 × 10⁻⁷ per revolution, he solves for the propagation velocity and obtains

c = 305 500 km/sec.

Section V closes with a caution and a hope. The derivation holds "as long as the speed of the masses is low compared to the speed of gravity itself"; Gerber suggests that a body raised or lowered uniformly near the Earth might in principle furnish a terrestrial test, while conceding doubt "whether, given the considerable value of the propagation velocity of gravity, the expected effects were not too small to be detectable."

Assessment

The strengths of the paper are real and are worth stating precisely. Gerber reaches the correct magnitude of Mercury's perihelion anomaly from a retarded-potential premise seventeen years before the field equations of General Relativity were published, and his formula for the advance is the same one Einstein's calculation later produced. The methodological argument — that a finite propagation speed lets you determine the potential but not the force, because the force is directional and the nature of the propagating state is unknown — is a genuinely careful piece of reasoning, and it is what distinguishes his treatment from the retarded-coordinate schemes of Lehmann-Filhés and von Hepperger. His objection to those schemes, that equal and opposite retarded force components are impossible unless the two bodies trace the same path, is a clean internal inconsistency proof. He is also unusually honest about the epistemic status of his own result: he says outright that a different value for the speed "would have no further meaning", and that only the coincidence with c vindicates the assumption. And he anticipates the modern conclusion that gravity propagates at the speed of light — a claim confirmed a century later by the near-simultaneous arrival of the Gravitational Waves and gamma-ray signals from the neutron-star merger GW170817.

The difficulties are equally real. The central one is that the retarded potential is asserted rather than derived. Gerber's step from "the potential takes time to build up" to a denominator of (1 − r′/c)² — the square, rather than the first power, or any other function — is not obtained from a field equation; it comes from a verbal argument about zones of tension being traversed at the sum of the velocities. The exponent is precisely what fixes the numerical coefficient of the perihelion advance, so the paper's one striking success rests on its least defended step. This was the substance of the objection raised against Gerber by his contemporaries and by Gehrcke's critics after the 1917 reprint, and the paper as it stands does not answer it.

The result is also not the clean confirmation it is presented as. Gerber does not predict c; he assumes the observed advance and solves for the speed, obtaining 305 500 km/sec against a then-known light speed near 300 000 km/sec — an error of roughly 2 per cent, which he does not discuss, and which is larger than the uncertainty in either the astronomical or the optical value of his day. Presenting the agreement as exact while the arithmetic is 2 per cent off is the paper's clearest internal tension.

More seriously, a velocity-dependent potential of this form is not a theory of gravitation but a single modified two-body law, and it makes no other prediction. It says nothing about the deflection of starlight, nothing about the gravitational redshift, nothing about the Time Dilation of clocks in a potential well — all of which are measured effects that any successor to Newton must now account for, and all of which follow from the same field equations that reproduce Gerber's perihelion figure. Reproducing one number is a necessary condition, not a sufficient one. Gerber's own closing appeal to a single medium carrying light, heat, electricity, magnetism and gravity — the classical Aether — was the natural conclusion in 1902, but the paper offers no account of how such a medium would behave, and Gerber explicitly and deliberately refuses to give one. Read as what it is — an argument that finite propagation alone suffices to produce the anomaly, and that the required speed turns out to be c — it is a striking and historically important piece of work whose weakness is confined to a single unjustified exponent.