If we define according to the model of relative spatial dilation as described in On the gravitational nature of time, an interesting question arises. If time in this model is the relative spatial dilation, how do we then describe a velocity?

While this model proposes a nature of time that is velocity dependent, such that

there must exist a transformation that satisfies our mathematics as they currently exist.

Application of non-linear time

Let us again consider ¹ as follows

which is equivalent to

While current models of space and time indicate that is simply , we must acknowledge that any measurement of time exists as an integral between two temporal positions.

If exists as a spatial scalar, as

then this dilation of space is clearly non-linear. If time itself is this dilation of space, then time should follow a proportional non-linearity.

Non-linear time applied to & velocity relative to the CMB dipole

In the truest sense, exists in two forms:

• The spatially integrated form as

• the differential form:

While both give different numerical results, once this non-linear time transformation is applied, both yield a peculiar velocity within 1% of recent calculations deriving such a value from the CMB dipole.¹

Modifying both forms of gives us the appearance of acceleration in free fall, although instead of integrating over time itself, we are modifying the integral of to apply in a manner which is non-linear. That is to say, that for , is not necessarily of the same magnitude as without applying the proper transformation.

Integration of non-linear time

Let us first consider the integrated form of , . As no relativistic corrections were applied to this derivation of , we should apply time as part of , where

Applying time in this manner inherently applies the proportionality scalar as proportional to . This gives

Note that is scaled proportionally to , as is integrated over .

By modifying in this manner, we also modify to be

This value falls within of the value found by Amendola, Catena, Masina, and colleagues, and falls within the error margin of many recent observational surveys.²

Differentiated non-linear time

Let us now consider the differentiated form of , . If we consider the way velocity and other time dependent quantities are described in modern physics and mathematics, should equate to in some form. Let us describe that relationship as

As is in a differential form here, yet any observation of time integrates between two temporal positions, we must find a proportional manner with which to scale this derivative so that it aligns with observations. We should therefore not describe time in a differential form, and instead describe time in it's integrated form where

The above must be undertaken to compensate for the manner with which we currently integrate over time, where even in a so called curvy-linear system, the passage of time is treated as if it's linearly proportionate.

Since we measure velocity not as , but rather of some larger approximation , we should scale this according to the average of the proportionality scalar over that integral and it's derivative. As our observations of already measure time in the integrated form, we should find the non-linear time derivative as:

Revisiting , this gives

This value again falls within of Amendola, Catena, Masina, and their colleagues, and fits probability curves derived from multiple CMB surveys even more closely than the result of the revision to .