by William H. Davis

The Hubble equation  $v= H_o/d$ is commonly used to estimate distances to distant objects emitting light. The observations by a number of astronomers, even before Hubble, recognized a relationship between velocity and distance of nearby objects. Velocity is estimated using spectral data from Cepheid and type 1A supernovas. The data is processed to estimate distances. Presently a linear statistical correlation to fit velocity with distance data creates a linear equation with a constant. The Hubble constant  $H_o$ is derived from $v/d$. This constant is used in a linear equation format to predict distances well beyond the distance the data was collected from.  A statistical projection even with a strong verifiable relationship has questionable accuracy. This along with the looking into the past creates higher uncertainty as the projection goes outward. If the calculation for the velocity is not corrected for relativity then a hyperbolic error will be incorporated as a function of distance. The velocity $v_e$ of the emitter corrected for SR sets the speed limit of c and corrects for the clock differences between the observer and the studied object. The actual velocity and distance compared to the observer can be achieved by correcting local velocity $v_e$ with SR to the actual proper velocity $v_o$. Create a modified Hubble equation or constant using  the actual proper velocity to Earth or have two separate correlations with $v_o$ and $v_e$:

1. The variable $v_o$ is the most important for estimating the actual distance. It produces a curved relationship that is very significant above .1c

2. Using corrected  $v_e$ will be nonlinear out to the approach to c (the speed limit) but is not accurate for determining distance.

Summary
Before modern telescopes the data without SR processing was fairly accurate within a reasonable± error of measurement. It is a different situation now because telescopes have a much greater range to view emitters at much greater distances and higher velocities. Relativity’s effect over the span of great distances and velocities is curved by γ and not linear. The Hubble tension indicates the deviation from linearity. The Hubble equation cannot exceed c. The purpose of the Hubble equation is to accurately project the relationship of velocity to distance within and beyond the established verified data from observations. We are limited at large distances to be able to observe Cepheid and type 1A supernovas. We can only derive velocity from spectral shift which should be corrected.  The Hubble equation needs updating with data corrected by relativity. Making assumptions of distance significantly beyond the data is very problematic if there is not complete confidence in a corrected and accurate as possible. What is the impact of the corrected data on the Hubble equation? Is the linear hypothesis still valid or does it need adjusting or do we require multiple Hubble models?