General Doppler Shift Equation and the Possibility of Systematic Error
  in Calculation of Z for High Redshift Type Ia Supernovae
All- Angle Doppler Shift Equation and High-Redshift Type Ia Supernova
General Doppler Shift Equation and the Possibility of Systematic 
   Error in Calculation of Z for High Redshift Type Ia Supernovae 
Steven M Taylor        
smtygc@umsl.edu 
Abstract 
Systematic error in calculation of z for high redshift type Ia supernovae could help 
explain unexpected luminosity values that indicate an accelerating rate of expansion of 
the universe. 
Introduction 
The general form of the relativistic Doppler shift equation is 
)cos1(0
' θβγνν −= ,                                                     (1)                         
where 
=  and 
=β  with u being velocity of source.    
With an emission angle the general form reduces to the familiar °= 0θ
= .        (2) 
The condition corresponds to an emission antiparallel to the source’s velocity 
vector and is typically assumed for astronomical purposes. 
°= 0θ
With the assumption  redshift parameter is defined as °= 0θ
         =z 1
.         (3) 
Evidence of Accelerating Universe and Possible Systematic Error 
The primary evidence of an accelerating rate of expansion of the Universe is that 
measurements of apparent magnitude of some high-z, type Ia supernovae are fainter than 
would be expected for non-accelerating cosmological models. [1] 
Perlmutter and Schmidt of the Cosmology Supernovae Project have noted that along with 
other possible sources of systematic errors, gravitational lensing may contribute to a 
change in luminosity of high-redshift supernovae.  Citing several authors, they note that 
mailto:smtygc@umsl.edu
as radiation traverses the large scale structure from where it is emitted and where it is 
detected, it could be lensed as it encounters fluctuations in gravitational potential.  Some 
images could be demagnified as their light passes through under-dense regions. It is also 
noted that it would also be possible for a light path to encounter denser regions magnify 
the image. It is noted that such an effect may limit the accuracy of luminosity distance 
measurements.[2] 
A lower luminosity in relation to z is to date the strongest evidence of an accelerated 
expansion rate for the Universe.  In the same sense that a change in luminosity due to 
reasons other than distance, such as gravitational lensing could produce systematic error, 
so could a false z measurement. 
Lowered luminosity would be consistent with a false z measurement if that measurement 
was less redshifted due to reasons extraneous to the expansion rate of the universe as 
presented by cosmological models. 
Whether by gravitation or other effect, any canting in angle of emission of light from a 
receding source will cause an increase in frequency as seen by an observer.  Taking the 
derivative of the general relativistic Doppler shift equation with respect to θ   yields: 
 θγβν
.                                                           (4)                        
Since  is an absolute minimum, any deviation from that angle results in a higher°= 0θ 'ν . 
Example 
According to Perivolaropoulos, the Gold supernova data set of 157 points show that 
transition from a decelerating towards and accelerating universe to be at  z= 0.46 + 0.13.  
Using a graph of apparent magnitude vs. redshift based on the Gold data, a supernova 
with an approximate 44 apparent magnitude and measured redshift parameter of 
, lies on the curve for an accelerating universe.  If it did have a redshift 
parameter of , the supernova would lie on curve consistent with a decelerating 
universe.[3] 
95.0≅z
30.1≅z
Using equation (3) a measured parameter of 95.0≅z  corresponds to 58.0=β , and 
likewise a parameter of corresponds to30.1≅z 68.0=β . 
Assuming, for the sake of argument, that the Universe is decelerating and the supernova 
does have a parameter of , we can calculate the canting in angle of emission 
from  that allows an observer to measure a parameter of 
30.1≅z
°= 0θ 95.0≅z . 
Using 0ν =1 Hz to simplify, and 68.0=β (corresponding to z =1.30), equation 2 yields 
,ν = 0.4364 Hz.  
Likewise with 0ν =1 Hz to simplify, and 58.0=β (corresponding to z =0.95), equation 2 
yields ,ν = 0.5156 Hz.   
0792.0' =Δv Hz                                                    (5) 
Inserting 68.0=β  into equation 4, yields 
)68.01(
sin68.0
 .                                              (6)  
Using 0.0792 Hz for 
 (5) into equation 6 yields an emission angle of . °≅ 90.4θ
Conclusion 
Given the large distances that light from high z supernovae travel, and the modest canting 
from in emission angle required to help explain decreased luminosity for high 
redshift supernovae, the possibility of systematic error in z measurement for high redshift 
supernova should be further investigated. 
°= 0θ
References 
1. S Perlmutter, B Schmidt, Measuring Cosmology with Supernova, arxiv:astro-    
    ph/0303428 v1 (2003) 
2. ibid 
3. L. Perivolaropoulos, Accelerating Universe: Observational Status and Theoretical  
    Implications, arXiv:astro-ph/0601014 v2 (2006)
