The Motion of a Photon in a
gravitostatic Field

otherwise analyzed than in Einstein's theory of gravitation

By Louis Nielsen, Senior Physics Master, M.Sc.


In the following I shall show that it is possible, by means of the Newtonian mechanics and quantum physical equations valid for a photon, to arrive at equations which are in conformity with equations deducted in Einstein's general theory of relativity. We shall consider the motion of a photon in a gravitostatic field. We find that both the velocity of the photon and the "dualistic" wavelength belonging to it depend on the gravitational field.

Motion of a photon in a gravitostatic field
Let us consider a spherical symmetric active gravitational mass m, which in the distance r from its center creates a -field given by:


where G is Newton's gravitational constant. The equation of motion for the photon is given by Newton's 2nd law:


where is the increase in momentum per time, and mf is its gravitational mass, which according to Einstein's equivalence principle is equal to its inertial mass.
The particle quantities and mf we shall express by the dualistic wavelenth of the photon.
Let us use the expression a field free vacuum; hereby we mean a domain of space absolutely free from both fields and matter.
If we consider a photon in such field free vacuum, then its velocity in a given frame of reference will be equal to the characteristic constant, c0, which a.o. is used in the equations of the special theory of relativity. If a photon is situated within a field, it is unlikely that it has the velocity c0, but another velocity c.

A photon moving in field free vacuum will have a total energy E0 given by:


where h is Planck's Constant and the wavelength of the photon. If the photon as an example moves into a gravitational field, it will have an energy E which we can write as:


where c is the velocity in the field and its wavelength in the field.
Assuming that the energy of the photon during the motion is constant, we get:


viz. the ratio between the velocity and the wavelength of the photon is constant, in spite of the fact that each of them may vary. As gives the frequency, this is seen to be constant.
The inertial mass, and thereby the gravitational mass, we can get from Einstein's energy-mass relation:


As is constant, we see that the photon retains a constant mass, even if its velocity is changing. This is not the case for particles of "matter", as their inertial mass increases with the velocity. The momentum p of the photon is given by:


By using (6) and (7) we can rewrite the equation of motion for the photon in a -field, considering a radial movement:


As   (8) can be rewritten to:


where the expression from equation (7) has been inserted.
Separating the variables and r we get:


Equation (10) can be integrated as:


where a < r1 < r2 is the distance to two points, measured from the center of the gravitating mass m, which is assumed to be spherical symmetric with radius a.   and are the wavelengths of the photon, corresponding to r1 og r2.
If we let r2 increase towards "infinity", corresponding to a field free domain with the wavelength , we can write:


where is the wavelength corresponding to the distance r.
As and c have a constant ratio, the following equation is valid for the velocity of the photon c(r) in a distance r from the gravitating mass:


If is considerably below 1, which is the case for most objects, we can with reasonably good precision use development to the first order of the series. Then we get the following expressions for and c(r) :



Equation (12) shows that a photon moving in a gravitational field is "exposed" to a change of wavelength together with a change of velocity. The velocity of a photon as function of its distance from the gravitating mass m is given by equation (13).

Gravitational shift of the wavelength
Let us calculate the relative change of wavelength corresponding to two distances r1 and r2, using the expression from equation (14). We get:


where is the wavelength at the distance r1, and is the wavelength at the distance r2.
If a photon is emitted from f.i. the surface of the Sun, where r1 ~= 7 · 108 m, and the wavelength of this photon is measured at the surface of the Earth, where r2 ~= 1,5 · 1011 m, then we get, with a solar mass m = 2 · 1030 kg and r2 >> r1:


This value is in agreement with the actually measured change of wavelength and is the same as deducted in Einstein's general theory of relativity, however with the difference that the latter theory operates with the frequency and not the wavelength. (See f.i. C. Møller: The Theory of Relativity, Oxford, 1952, p. 346).

Velocity of light in the general theory of relativity
The general theory of relativity deducts an equation for the velocity of "light" in a gravitational field (see f.i. C. Møller, p. 353). This equation is:


where we note a square root, contrary to the exponential function in equation (13).
If (18) is developed to the 1st order, it will be identical to (15). In (18) there is a critical distance rs given by:


The distance is called Schwarzschilds's radius and indicates in which distance the velocity of light becomes zero. If r < rs, crel(r) becomes imaginary, which gives non-physical conditions. The conception "black hole" is defined as an object with its mass concentrated within the distance r < rs. In equation (13) there are no critical distances, making it more physical than equation (18).

Bending of light around a gravitating mass
What made Einstein famous was an "apparent" confirmation of a phenomenon he had predicted by his general theory of relativity. Einstein calculated that a light beam, passing near to the Sun, would be bent 1.75 arc seconds. Half of this bending was caused by a change of the velocity of light, calculated according to equation (18). The second half was determined by the "curvature" of space, which was determined by the quantity of the present gravitational mass. This "curvature" of space can be calculated from Einstein's "geometrical" field equations.
At a solar eclipse in 1919 Arthur Eddington arranged an expedition to a place where this eclipse would be total. During the "totality" of the eclipse photographs were taken of the eclipse and surrounding stars, of which there were only few. Comparing the pictures with a photography of the same area of the sky, but without the Sun, a bending could be established. The analyses, however, were primitive and with great variations in the measured bendings.
A gravitational bending also follows from equation (13), which I have deducted, but using this equation we get a bending of 0.87 arc seconds caused by the Sun. A further bending is caused by the gravitational rotation field (-field), which is present around all gravitating masses in relative motion. As the Sun rotates, this will cause an -field, which will produce a force on every gravitational mass, moving within the field.   The -field shall be calculated from the gravitational field equations, as I have explained in my article: A Maxwell Analog Gravitation Theory with two gravitational fields.

Louis Nielsen

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