The redshift is the phenomenon where the analysis of light spectrum of distant galaxies or stars shows characteristic features such as the absorption line of hydrogen H-$alpha$ which has been measured at 656 nm to shift to a longer wavelength. Hubble in 1929 considered objects in the range from 3.392 parsecs to 212 parsecs and found that the redshift increases linearly with distance. Subsequent studies had increased the distance ranges for the validity of this relation. Hubble’s interpretation of this was a Doppler effect from the velocity of the objects moving away from us. Fritz Zwicky had immediately reacted with the suggestion that light loses energy in their passage but this hypothesis was rejected and the recession velocity hypothesis was accepted, and with it an expansionary universe model.
In order for the S4 theory, which posits that the universe is a four dimensional sphere of fixed radius to be valid, we must find an explanation of how a redshift can occur in a stationary non-expanding universe. My initial approach to the problem was to consider that the redshift might occur as an effect of the cosmic background radiation, which has a peak at around 6 cm but has nonvanishing tails for all higher frequencies, with the idea that there could be interference with small probability for higher frequencies. This approach seems to be a failure as the cosmic background radiation in the visible frequencies in the order 400 THz produces terms like exp(-7000) which cannot be compensated by the long distances.
A more promising approach is to use the fact that gravitational fields affect light and then treat the uniform curvature of the four dimensional sphere as a uniform pseudo-gravitational field which affects all photons regardless of their frequency. The idea is that photons feel this ambient curvature as a gravitational effect that dampens their energy uniformly per unit time, and thus there is a redshift that is uniform in the distance traveled by the photons. As for conservation of energy, the natural candidate for where the energy goes is the cosmic background radiation, but the mechanism for this energy transfer remains unclear to me.
Another possible explanation of redshift in a stationary universe is from extrapolation of the fact that in transparent dense media both the speed of light as well as the wavelenghts shrink. Thus it is possible that the speed of light in the solar system, which is gravitationally denser than intersteller space has shorter wavelengths and slower speed than intersteller space. Then redshift can be explained as the expansion of wavelengths in less dense interstellar space through which light travels to reach us.
Update: Last night I was able to find an concrete explanation of the redshift that produced numbers in the order of magnitude of the redshift that is observed even in a stationary 4-sphere universe. The idea is the following. The wave equation can be solved on a four sphere and yield solutions of the type where the first factor is a spherical harmonic, ignoring some constant. So essentially for a spherical wave when the true wavelength is the frequency is not but . This error is per second, and increases with time linear in distance.
Consider the H-alpha line of 656.3 nanometers. If we take n to be c/lambda and then consider the shift over 106 parsecs, which is the scale used by Hubble, using the slope
then we find a shift of 0.023 nanometers for the wavelength which corresponds to a radial velocity of approximately 10,500 km/s in Hubble terms, which is in the right order of magnitude.
In other words, treating spherical waves on a 4-sphere as linear waves will produces an error in frequency which will produce a linear redshift of the type that Hubble interpreted as a Doppler shift. This explanation of the redshift suggests that the redshift is not even a physical phenomenon.
In order to test whether the explanation of the redshift as an artifact of mistaking spherical waves as linear waves has merit, I downloaded a modern dataset of redshift and distance from this site from table1.dat which has the ‘radial velocity’ in the second column and distance in megaparsecs in the fifth column. Then I model redshift as the linear function of distance by which the 656.3 nanometer would move as the error exacerbates. Fitting observed redshifts by this model produces an R-squared of 0.998.
Here is python code for this test.
from scipy import stats
cSpeedOfLight = 299792458.
# Estimated in nanometers for H-alpha
cExpectedRedshift = 0.2200922
fw = 
lline = list(line)
N = len(lline)
pc = lline
ilast = 0
for i in range(1,N):
c = lline[i]
if pc != ‘ ‘ and (c == ‘ ‘ or c==’\n’):
fw.append(i – ilast)
pc = c
def getFields(line, fmtString):
parse = struct.Struct(fmtString).unpack_from
file = ‘amiga1.dat’
fd = open( file, ‘rb’)
line = fd.readline()
line = fd.readline()
LineLen = len(str(line))
fieldWidths = getFieldWidths(line)
fmtString = ”.join(‘%ds’ % f for f in fieldWidths)
vel = 
dist = 
for line in fd:
s = str(line)
if len(s) < LineLen:
s = s + ‘ ‘ * (LineLen-len(s))
fields = getFields(s,fmtString)
# Field 1 is velocity and Field 4 is distance
if re.match( r’ +\d+\.\d*’, fields, re.M):
expected_redshift = 
for d in dist:
expected_redshift.append( d * cExpectedRedshift )
observed_redshift = 
for i in range(len(vel)):
observed_redshift.append( (vel[i]/cSpeedOfLight)*656.3*1000 )
grad,intercept,r_value,p_value,std_err = stats.linregress(expected_redshift,
print ‘Gradient’, grad
print “R-squared”, r_value**2
and the output is
MacBooks-MacBook:Hydrogen macbook$ python proc-amiga1.py
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