On the gravitational nature of time

A novel model of relativity that describes relativistic observations through the dilation of space, not time.

Introduction

Consider an observer at relative rest . If this observer agrees with another observer, that should travel at precisely velocity between two arbitrary points in space, and , it is inconceivable that should reach point at a time other than . While current models dilate velocity in accordance with the Lorentz tensor and rely on 4 vectors being invariant, this would make it so that is not traveling at the agreed upon velocity . While current relativistic models rightly ascertain that time dependent quantities might appear dilated as an observer approaches the speed of causality, , it is unreasonable to conclude that a quantity can actually be of two different magnitudes.

Following the premise that if an observer who will remain at relative rest conspires with a second observer that the second observer should travel at precisely velocity , and that in the rest reference frame should be of an equivalent magnitude to that same velocity in the moving reference frame, , it is trivial to deduce that it must be the distance that dilates between two previously shared points in space; not time alone.

The model being proposed relies upon a relative nature of velocities that extends beyond what was proposed by Einstein, such that in many instances, velocity is not a function of time, but rather a pure magnitude to be compared proportionally to other velocities and subsequent velocity dependent forces. As itself implies that time is a function of velocity, there should exist either a rate of change which remains linear between reference frames or a physical mechanism that can be derived according to which this dilation of time occurs. While assigning time to a measurable physical process may seem uncomfortable, this leap is not unreasonable and offers a sense of completeness to cosmology.

Relative motion in the context of cosmic inflation

Recall that in On the electrodynamics of moving bodies¹ Einstein makes clear his assertion that a rod in the moving reference frame should be of the same length as the rod in the frame at relative rest. Whether this assumption was the result of Einstein's insistence on a static Universe early in his career or something he would have otherwise derived is unclear, but in the context of cosmic inflation this assumption appears to be flawed; especially when a medium we can't directly measure must then dilate to satisfy our observations.

If our Universe is expanding, it must be expanding into something. What that something is remains unclear, but there must exist a larger, encompassing space that our Universe exists within which can accommodate this expansion. Here, I will refer this larger encompassing space as space, and space within the Universe itself as space.

If our Universe is expanding within space, it must be possible for an observer to be at relative rest with respect to an reference frame, while space dilates around him; essentially placing the observer at relative rest while space itself is in motion.

Spatial dilation and the equivalence principle

Let us consider a body in motion with respect to space. As SR¹ demonstrates, this spatial dilation should be numerically equivalent to , and likewise scale with a similar proportionality to . If we consider Earth's gravitational acceleration, the equivalence principle and how they might relate to this spatial dilation, we should be able to find a symmetry if one exists as:

This gives a new which I will call .

If we then solve for , and substitute for , we find:

While this value contradicts CMB dipole observations, it does correspond with recent supernovae surveys like the ones conducted by Gordon, Land and Slosar,1 as well as Jha, Riess and Kirshner2.

Non-linear temporal progression

As the model of spatial dilation being proposed presumes that this dilation of space is what we experience as time, any integral over time should follow a similar geometric pattern, in turn giving a symmetric proportionality.²

Let us consider two forms of , the integrated form as:

and the differentiated form:

where is the derivative of with respect to . Let us then define a new peculiar velocity with each respectively.

Integrated 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 , which in turn gives

In a similar manner with which we are applying , as an averaged scalar, we should apply this proportionality to the length of time considered, .

Applying this modified to gives

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.

Differentiated non-linear time

Let us then consider the differentiated form of , . Note that in the model of relativity being proposed, acts as a pure ratio of magnitudes and is not time dependent. In turn, in order to equate to our existing measures of time, we should describe it as a velocity such that . This gives

As any measure of velocity inherently occurs over some integral of time, we should describe time not as a derivative of time, but as the elapsed period after being made proportional by the spatial derivative. This gives , such that

Applying to equation in turn gives

Relativistic Inertial Shift

Consider again the notion that may not imply a ratio of time dependent quantities, but rather a ratio of relative magnitudes, where determines a body's rate of progression along this temporal axis as described by . This gives a model of velocity and other time dependent quantities in which change does not occur over some time integral, but rather over an integral of across both space and some temporal gradient . This in turn implies a Universe in which if the sum of velocities is zero, there exists no temporal progression.

Let us review with respect to this interpretation of . As is found through , which in turn is found via a numerical equivalence to , we can infer that is in some manner bound to .

If we then consider as a derivative, such that

we can apply as a progression along this temporal gradient for Earth's center of mass. We then find that for in space, there exists a in space such that

This in turn provides a temporal derivative, such that where

electrodynamics

Note that along with a spatial density gradient implied by this model, it is straightforward to infer that should be proportional to this density gradient and not truly constant. While may be constant in each reference frame, it cannot be constant between reference frames.

Consider a single spherical massive body of a uniform density, . At any radius from there should exist a spherical surface that acts as an equi-temporal plane, . For a conserved field to have a non-zero flux within this plane, there must be a non-zero flux across this plane. A positive electric charge must behave as a faucet of sorts across this temporal plane, such that the flux of is according to our temporal progression. This is to say that for our temporal progression 'outwards', or towards a lesser spatial density, there exists a positive flux in the direction of this lesser density into the plane. Likewise, a negative electric charge, implies a positive flux in the direction of this lesser density.

Note that there is no lesser temporal density with respect to this temporal plane . Despite this spherical equi-temporal surface acting as a plane of sorts, the Universe exists as a closed surface within this postulated space, and any flux into this space cannot flow outward into some space of a lesser density.

As all magnetic components of the electromagnetic tensor act as pure space components, there should exist no difference between the electric and magnetic fields apart from the angle of flux with respect to this temporal plane. This follows closely with the lack of a magnetic monopole and the absence of a magnetic charge. Further, the radiating gradient inherent to this model provides a mechanism for the perpendicular nature of magnetic field lines and the magnetic component of the Lorentz force, as well as the tendency of magnets with a flux parallel to this temporal plane to align perpendicularly to the radial vector from a position of flux normal to this plane.

While not yet formulated, there should exist a test of this geometry which can be carried out by conducting an experiment related to this model of electromagnetism at various gravitational equi-potentials, barring any significant relative velocity.

Acknowledgements

To my aunt, Mary Beth for instilling a sense of wonder and curiosity, my grandfather for leading by example, and T.L., for demonstrating that space and time obey laws far beyond our own understanding; without whom this theory wouldn't exist.