## The Spatial and Temporal Propagation of Gravityby ## 1. The Basic Law.Gravitational phenomena are the only effects concerning separate bodies for which we cannot prove that the intervening space has a role, i.e. the existence of changes that are communicated through it from one place to another. All the more understandable therefore is the hope that we will ultimately succeed in finding the missing evidence. However we must not regard the matter as if there was no doubt that this exception was only an appearance. All known and understood observations strongly suggest the opposite. Therefore, if this is still only based on a lack of empirical information or incomplete analysis, it must first be demonstrated that there are facts that rectify and complete our previous conception in the opposite sense. Therefore it is necessary above all to exclude any hypothesis that assumes that some event occurs in the space between two gravitating masses that is responsible for gravity. Reference was made to previous similar, but inadequate treatment of the issue discussed here, in the report made by Drude on remote effects in the 69 Two gravitating masses can be recognized as such by the resistance by which they oppose an enlargement of their distance. There must therefore, while the masses themselves can be at rest or in motion, be a connection between some of the events existing in the space between them. Obviously, with the position, or with the position and the current state of motion of the masses, insofar as external influences are excluded, we mean not just the resistance from one point of mass, but also the infinite sum of resistances from all other points of mass. The necessary work to overcome the resistance is therefore the same as the single resistance of a strength characterized by that of gravity. Whether changes that propagate in space with loss of time are linked with gravity or not, it can merely, where it matters, be considered as being a parameter. For there is no sense in speaking of the concept in terms of the spatial propagation of the resistance or the attraction, because resistance and attraction exist as such only in the places where the masses are located. But if it is predicated that an event requires time to communicate from one place to another, then this means the event cannot be existing at one place while simultaneously existing at another; it must cease to exist at the first point before existing at the second. In this way the energy contained in the event would decrease with time, if it did not pass through only those points lying directly between the two places, but through other points also. The energy is equal to the work required to communicate the event, as it pertains to gravity, between two masses located in different places. This energy then depends on the position and the momentary state of motion and these cannot induce two different levels of energy. Now in order to differentiate let us call one mass the attracting and the other the attracted. Let m is being held, with reference to the likewise held attracting mass, be x, y and z. One can calculate V using the methods described in Mach’s principles of thermodynamics, by equating it with the mean of all the prevailing potentials in the immediate neighborhood of the point. V is of course not a vectored quantity and for a given location is constant with time. Let m be experiencing a force equal to f (x, y, z) and for a neighboring point equal tof(x + h, y + k, z + l).Furthermore, let represent the density of the mass Expanding
Then by setting sets, So that From this equation it follows in the well-known way, if This amounts to Newton’s law of gravity. Because is valid even at the moment when the masses are released at dT that appears, and therefore T contains at that moment, like V, an equally small change of r in time. Consequently, according to the general Lagrange equations of motion and by substituting the negative value of the force exerted by the mass m for the external force acting on it, one has for the acceleration of mNewton's law defines the potentials that reach the masses in every position if they have the necessary time to achieve their realization. This condition is always satisfied when the masses are held at a distance from each other. The condition ceases in the case of free movement directed towards each other, if the time concerned has a finite measured size. Two factors are of importance. First, the potential must indeed begin to form at a distance Apart from its speed while Because furthermore the speed with which the movements are approaching each other has the value the potential drops due to the time needed before its effect on We therefore find As long as the route whence with the aid of the Binomial theorem to the second power, we get The expression for The assumption that is small compared with ## 2. The speed of propagation.According to whether the observations provide a finite or an infinite value for the quantity We state So we then have
whence, by multiplying one equation by This is the resulting equation also in the derivation of the properties and the path of planetary motion from Newton's laws, which through integration and the introduction of polar coordinates, where We define the value Furthermore in the equations for then gives
With the constants
Because , we find from the last two equations The integrals in the denominator gradually take other values, if and to so we get
We see, by observing the immutability of , that the movement of the planet can be interpreted as if it was proceding along an ellipse whose ε and ω are changing constantly. This change only stops in the case that and the other by dividing these gives
By equating the two expressions and with and working backwards gives To gain an equation for contained in sizes observed in the middle of this value, we represent
Thus Therefore, the required equation for is or after the definition of and and after division by If we want to compare the value of the velocity calculated in this way with the observations, we must bear in mind that the calculation assumes only a single planet. Therefore, only those perihelion movements are concerned, that do not arise from interference. Such movements are only known in the case of Mercury, in an amount of about 41" in a century. This smallness excludes a priori any experiential observation of the continuous variability of . So integration must be done over a long time. In the last equation only We multiply the equation for by With suitable reordering and division, we get Dividing the numerator and denominator by we arrange in ascending powers of
so this becomes By approximating, we obtain For the perihelion movement or because
It follows that Considering that
where for the same reason 2 In this equation where
From this we find that c = 305500 km/sec.The smallest speed of light found so far was by Foucault and was equal to 298000 km/sec; the largest comes from the method of Rømer with the latest observations of 308000 km/sec; in his experiments, Hertz found the speed of electrical waves to be 320000 km/sec. Of course no one will deny that the perihelion movement of Mercury by 41" in a century could also be due to other, unknown factors, so that seeking a finite speed for gravitational potential might not be necessary. One has to remember, however, that the main deciding formula here, inducing moreover the deviation from all previous results of similar studies, for the dependence of the potential on such a completely natural speed, was progressively obtained without major assumptions. It would be a strange coincidence if the 41 arcseconds for Mercury equated precisely to the speed of light and electricity without having some connection with the spatial-temporal propagation of gravity, seeing that the medium in which this propagation and movement of light and electric waves happens, is the same as the space extending between the heavenly bodies. Not even the relatively large perihelion movement obtained with the values found for c for Venus, that is 8" in a century, can be considered a valid objection; or a review of the fluctuations of this planet would have to definitively exclude the possibility of that number. It is recalled that the calculations of the secular acceleration of the moon can fluctuate between 6" and 12". Moreover, there are nothing but imperceptibly small perihelion movements. They amount to the observed values easily found from the conventional tables of the earth in a century, 3.6", the moon 0.06", Mars 1.3", Jupiter 0.06", Saturn, 0.01", Uranus 0.002" and Neptune 0.0007". |

Original German text can be found here Die räumliche und zeitliche Ausbreitung der Gravitation |