Suggested reading: “When Gravity Balances the Lorentz Force”
In this article λ(x), the wavelength of a photon emitted from the Sun, is calculated. It is assumed that the Sun is the sole gravitating entity. x=0 corresponds to an emission point on the Sun’s surface. The Sun’s center is assumed to be at rest at x=-R. A photon propagates away from the Sun’s surface along the positive x-axis. The Sun’s gravitational mass is M.
According to gravitomagnetic theory the photon has an imaginary gravitational mass, m, with magnitude
|mgrav|=hf/c2, (1)
where f is the photon’s frequency. Or, since fλ=c for every photon,
|mgrav|=h/λc. (2)
The Sun’s gravitational field is also theoretically imaginary and, at points x>R, has the magnitude
|g|=GM/(R+x)2. (3)
Now in Gravitomagnetic Theory every particle, traveling away from the Sun, experiences a real gravitational force back toward the Sun. For a photon that force is
F=mg=-hGM/λc(R+x)2. (4)
The photon’s energy is
E=hc/λ. (5)
Therefore
F=dE/dx=(-hc/λ2) dλ/dx. (6)
Equating Eqs. 4 and 6 produces
(dλ/dx)=GMλ/c2(R+x)2 . (7)
Let us stipulate that, upon emission at x=0, a photon has the wavelength λ0=4.5e-11 (blue light). Then
λ(at x=dx)= λ0+(dλ/dx)(at x=0) dx (8)
= λ0+GMλ0dx/c2R2
λ(2 dx)= λ(dx)+(GM λ(dx)/c2(R+dx)2) dx. (9)
λ(3 dx)= λ(2 dx)+(GM λ(2 dx)/c2(R+2dx)2) dx, (10)
etc. Using tiny values of dx (say dx=1e-10 meters), we can numerically compute λ(x) for the first nine meters from the emission point. Table 1 lists the results. The data is created using the Python program specified below the data.
Table 1
x λ
0.0 4.5e-11
0.9999999999980838 4.5304743508355774e-11
2.000000000004635 4.5609487007943725e-11
3.000000000011186 4.591423049878028e-11
4.000000000017737 4.621897398084497e-11
4.999999999979879 4.652371745415758e-11
5.9999999999420215 4.682846091870374e-11
6.999999999904164 4.713320437449242e-11
7.999999999866306 4.743794782152004e-11
8.999999999828448 4.774269125978478e-11
The theoretical result, that redshift may occur (when source and receiver are at rest), has an interesting ramification in Olber’s paradox. According to Olber the sky at night should have no dark areas if there are infinite stars and galaxies homogeneously distributed. Now there may indeed be an infinite number of galaxies, homogeneously distributed. But galaxies too far away to be seen from Earth, owing to distance redshift, appear (to our eyes) as darkness between the closer, visible galaxies. Many of these galaxies should be “visible” using radio (longer wavelengths) telescopes.
Finally, pulsars, black holes, etc., viewable with radio telescopes, may emit visible light, but the visible wavelengths expand to radio wavelengths and beyond by the time the signals traverse the vast distances to Earth and/or by the time photons have “climbed” out of intense gravitational fields.
Python Program
import math
G=6.674e-11 #Gravitational Constant
M=1.989e30 #Mass of Sun
R=6.957e7 #Radius of Sun
c=3e8 #Speed of light
lamda0=4.5e-11 #Wavelength at emission point
x0=0.
dx=1e-5 #Increment of x
x=[]
lamda=[]
x.append(x0)
lamda.append(lamda0)
for index in range(1,1000000):
x.append(x[index-1]+dx)
lamda.append(lamda[index-1]+G*M*dx/(c**2*(R+x[index-1])**2))
for index in range(0,1000000,100000):
print(x[index],” “,lamda[index])
input("OK>?")