On the gravitational nature of time

Some Prerequisite Knowledge

Special Relativity & Time Dilation

First, let's break down the difference between the generalized and specialized theories of relativity. This requires an understanding of a few concepts, but I promise, if you passed high school algebra you can follow the remainder of this note. Some calculus does make an appearance, but the concepts are what matters... not your ability to do the actual math.

Special relativity refers to a set of phenomena that occur due to relative motion and the (kind of) fact that the speed of light is constant in all reference frames. General relativity has to do with the addition of gravity to this model through a theory known as the equivalence principle. The equivalence principle simply put states that there is no measurable difference between being in a gravitational field that accelerates an observer at meters per second, and being in a completely gravity-free environment while in an elevator accelerating upwards at meters per second. If the magnitude is the same, gravity simply is this acceleration.

Next, let's break down why exactly the speed of light is presumed to be constant so we can later dissect this claim in more detail. Spoiler: It's only half true.

Here, represents the speed of light, the magnetic constant, and the electric constant. The magnetic and electric constants are often referred to as the permeability and permittivity of free space, respectively. Both of these 'constants' act as a scalar of the electric and magnetic forces, but more importantly, both have units proportional to length.

Where is a Farad, is a Henry, and is a meter. The square brackets are used only to imply the equality is demonstrating units.

Time dilation due to relative motion

Einstein was famous for quite a few incredible discoveries, and he deserves credit for even the one which I am attempting to disprove. I am not proposing that his theory was completely wrong. In fact it was far, far closer to the truth than any of his contemporaries, but his insistence on a static Universe (a Universe that is not expanding) early in his career led him to miss the obvious. Throughout the remainder of this note I will attempt to help you see the mistake that he made, but first, we must cover what he did right. The next few paragraphs will walk you through a derivation that follows the derivation provided by Einstein himself to the best of my ability without going into the details of the Maxwell equations, electromagnetism and vector calculus.

Imagine shooting a laser straight up and down from the floor of some spaceship. Thinking of this as being in a spaceship is important for it's lack of gravitational field. This will become incredibly important as we move forward and integrate gravity into this model.

At some height above that laser, place a mirror, and another mirror where the laser is placed (this isn't technically possible because the laser would be occupying that space. Pretend that the laser is like a ghost for the sake of this thought experiment). At the end of this thought experiment we should have a pair of mirrors with a laser bouncing between them, straight up and down. Imagine that each unit of time, , occurs over the period that a photon is in transit between one mirror and the other.

So that 'time clock' we set up a couple paragraphs ago? Let's set that in motion now, literally. Imagine that we are now in a different spaceship.

All of the physics majors are likely screaming right now, waiting for me to tell you how the entire basis of relativity is that all motion is relative and that there is no such thing as absolute rest or absolute motion, but I'm here to tell you that this is wrong. In fact, this is not the way we treat motion mathematically. In reality, physics carves out a unique niche for gravity, one where all motion is calculated relative to the underlying gravitational field, so that's what we should do here: The spaceship you are in is at rest relative to the local gravitational acceleration. This is equivalent to the free fall frame from the reference frame of a body in co-motion with the 'dough', but a rest frame in the coordinate system of the 'oven' as the 'dough' dilates around the observer.

Now since we're far out in outer space, far away from any stars or planets in this thought experiment we can pretend that gravitational acceleration is zero. This is a core piece of special relativity, but don't worry... we'll add gravity back in the next section.

Now imagine that we are at rest in spaceship , while spaceship (containing the light clock) zooms past us in a straight line and a constant speed at velocity . That light clock we just made is no longer a line straight up and down, but a triangle. We can use this, the 'constant' speed of light, and the Pythagorean theorem to find:

This factor of is such an important part of relativity that it gets it's own symbol, (gamma).

And that's it! You've successfully derived time dilation due to relative motion, or did you? That's what modern physics would tell us, although many pompous academics would take issue with the simplistic derivation, but let's break down what we actually discovered here.

First, we have the emission of the photon from the 'laser'. Let's say that this event occurred at . We then have the second event, let's say at .

Time dilation is only 1 possible solution here. What we have really discovered is the Euclidean distance between these two events is elongated by an additional factor of .

So we are left with two possible solutions:

1. Time, a quantity we have never directly measured, and a quantity we aren't even sure exists in any physically meaningful manner dilates.
2. Space dilates with time. This coincides perfectly with the notion of cosmic inflation (the expansion that gives us the Big Bang model), and as you will soon see, allows us to collapse cosmic inflation, gravitational acceleration and time itself into a single process as observed from different reference frames. Some units have to change to modify quantities that are proportional to time, making them instead proportional to motion or other processes that are currently proportional to time, but all experiments validating current models of relativity are satisfied due to the equivalence that both models seek to maintain.

To be clear, what I am proposing is that time doesn't dilate with relative motion. In fact, I'm claiming that time doesn't exist at all, that time exists only as a ratio of distances. In turn, many of the attributes currently attributed to time become attributed to motion, making motion the 4th fundamental rate of change (just as is indicated in where is a function of ). If the collective motion in the coordinate system of the 'oven' for all bodies within the Universe equals zero, the passage of time ceases. Due to our co-motion across the temporal (time) axis we don't perceive this dilation of distance directly, but we do experience this dilation as gravitational acceleration. So on that note, let's add gravity back into this equation.

General Relativity & New physics

So far we've only reviewed what Einstein himself gave us. His brilliance in finding the additional factor of in the simple equivalence gives us the opportunity to derive our velocity here on Earth in two distinct forms, but first we must discuss how a single body can have two different velocities.

To dive deeper into the relationship between cosmic inflation, gravitational acceleration and time let us use some common analogies. The most common analogy you're likely to hear for cosmic inflation is that of a loaf of raisin bread or a blueberry muffin. If we imagine a piece of dough placed into an oven, each blueberry will move away from the center of the dough at a velocity that is proportional to the distance that the blueberry was from the center when the dough was placed into the oven. If the berry started further from the center, it will move away from the center faster as all of the dough in between the berry and the center of the dough expands.

This concept gives us our notion of cosmological velocity: Our velocity that takes into account this dilation of space. In this notion of velocity, it is not the motion of the berry through the dough that matters alone, but rather the cumulative motion of both the berry's motion through the dough and the motion of the dough itself.

Our kinematic velocity is different, being made of only the motion through the dough.

Cosmological Velocity

First, let's start with our cosmological velocity. The following is our standard equation for gravity

Above, is the acceleration due to gravity, is the gravitational constant, is the mass of the gravitational source, is the distance from the center of mass of the gravitational source, and just gives the acceleration a direction.

If we then take the derivative (break things into really, really small pieces) of this equation with respect to radius¹ we find

Above, indicates that a value pertains specifically to Earth, and (delta) is an arbitrary variable used for this equation elsewhere. By taking the derivative of this equation we've found another equation that when integrated (adding up all of the really, really small pieces) we get an acceleration of gravity that matches precisely with what we experience.

We're getting closer, but as is applied as a scalar, we should take the average of the integral of this equation. If you're unfamiliar with calculus, we are adding up all of the small changes in distance that would occur according to the model being proposed, and taking the average of the sum of all of these small changes to give us a scalar that we can apply to distance in a way that is symmetric with the manner in which is applied. This allows us to essentially exchange the two variables as they describe the same phenomenon in a different reference frame, and can now both be applied as scalars.

The average of this integral gives us:

This means that we can now solve for to find

And since , we can substitute:

The above value falls almost perfectly on top of probability curves found through direct observation, but we're not done! We can derive our kinematic velocity too.

Kinematic Velocity

Before we dive into the derivation for the kinematic velocity we first need to dig a little deeper into the muffin analogy. Imagine the muffin baking in the oven. If there was some coordinate system in the oven, and a coordinate system in the muffin, there is exactly 1 position within the muffin that does not move relative to the oven as the dough expands, assuming that the 'muffin' is not in motion with respect to the oven along any of the three spatial dimensions. If we expand that analogy to our Universe, we will call that point ... essentially

This point now would be expanded to contain the entire Universe as the Big Bang model describes, but there is one key difference in the model I am proposing: This point exists concurrently to our own, separated only by some deviation across this density axis. In other words, the 'dough' started out as the ultra-dense early Universe that gives us the Big Bang model, but that 'dough' will be the size of the Universe at which ever time is in question. The 'dough' is the Universe, but unlike our analogy, the existence of different density states is concurrent, giving us a real Euclidean distance to this point.

This concurrent nature of time should be pretty self evident, even in existing models of relativity and, I argue, has been demonstrated experimentally. Consider two observers, and . If and are initially very near one another so that they come close to sharing the same coordinates, and would exist at a shared time coordinate. If however was to be set into motion, is applied, making it so that has progressed further along the time axis than . While both existed once at a shared position along our temporal axis, they now exist concurrently at different time coordinates.

If we modify the equations above so we instead define not as proportional to time, but as proportional to motion through space we reach the cusp of the model I'm proposing: Motion occupies the 4th fundamental rate of change, not time. That is to say that for some small motion through the 'dough', the 'dough' dilates by some small quantity that is a function of that motion through the 'dough'. The faster (greater magnitude since there is no concept of time anymore) an object moves through the dough, the faster the dough dilates.

So let us slightly modify the equation for our cosmological velocity to take into account this proportionality to motion, removing the dependence on time. This in turn removes the expansion of the 'dough', giving us our kinematic velocity.

First though, we must define a 'time equivalent' quantity, since we are removing time from the equation.

Time equivalent quantities:

If the 'dough' dilates as a function of the magnitude of a body's motion through the 'dough', and this process when summed over for all bodies within the Universe gives us cosmic inflation, there is no need for some abstract quantity we naively call time. We can define time purely as a ratio of these distances... the distance a body travels through the dough, and a distance that two nearby points diverge from each other. To be more precise, our temporal coordinate is a position along this density axis. This ratio is what gives us the perceived acceleration due to gravity, cosmic inflation, and the perceived passage of time. When is greater than 1 we produce what we now consider a blackhole, however these blackholes are nothing more than a mirage in the distance that dissipates as an observer approaches that gravitational field, dilating proportionally to .²

In it's integral (adding up all of the small changes) form, this might look like:

This notation is likely confusing, so let me explain. The denominator indicates the change in displacement integrated over a single unit of "time", giving a measure of length equivalent to the velocity in units of length, while the subscript indicates an integral that continues to grow as the Universe progress temporally.

To put this in human language, there is no time anymore. There is only motion through the "dough", and motion of the "dough". When each small motion through the "dough" is added up over all of the time in question, we get . This number will be proportional to time in our standard units of measure, but there is no need for time in the physical sense. Our attribution of time is largely arbitrary, albeit proportional to the rate of cosmic inflation as we currently define cosmic inflation as being proportional to time. This relationship of course becomes inverse, but this is irrelevant as locally. The difference between local and non-local spatial dilation is what gives rise to the observable cosmic inflation and the Hubble parameter. Since the dilation of space is time, this ratio will always be 1, making the inverse of this relationship irrelevant. This is precisely why gravity lacks any apparent aberration, why the dilation of space remains hidden from us due to our co-motion along this density axis, and why Earth, being in a near equilibrium position produces an excess dilation of space at the poles due to the lack of centripetal force that precisely cancels out the excess dilation of space due to motion at the equator.

Time Derivative Equivalent:

Consider that for as it appears in equation above, unit of pseudo-time will pass. Because the passage of pseudo-time is defined by the relationship between proportional derivatives and distances, where indicates 1 unit of pseudo-time in units applied to the traditional notion of velocity, we can easily infer the following:

This gives time derivatives per unit of time, but more importantly, a time derivative, that is proportional to if is defined by .

To put this simply, we can derive an exactly proportional time-equivalent derivative so that we can remove time, and in turn cosmic inflation from the cosmological velocity derivation above.

Applying the time equivalent quantities to find our kinematic velocity:

If is the dilation of space in it's derivative form, let's apply the time equivalent derivative to this equation. This will make this equation proportional to motion, not time, in turn removing the motion which is only secondary to the dilation of space leaving us with only the motion that is relative to the spatial system local to the observer.

Since in traditional models is proportional to time, in it's current form represents the dilation of a small quantity of space over a single unit of time, whatever the unit of time applied to may be.³ Since is also proportional to a single unit of "time", we can find a proportional value of

When we then integrate this in the same manner in which we did to find the replacement for , we will now have to integrate twice. First over a unit of pseudo-time, and then over the radius. This gives us the sum of the dilation of space for each small quantity of space, for the sum of each small quantity in time, in the coordinate system of the 'dough'.

Substituting this back in for gives:

Our observed velocity relative to the CMB (the dough in this model) is approximately , and some gravitational "turbulence" would be expected to give a reasonable margin of error.

For the math types out there, it is important to note that the integral of is treated uniquely, as the denominator of remains fixed in the same manner as when was first defined. This is consistent with the nature of this model and this quantity, as this model concludes that we've had an additional derivative in our model that is not an inherent part of the system. This maintains as a quantity that is equivalent to time, both in integral and derivative form, and allows that certain quantities remain directly hidden from us as our co-motion along this temporal axis causes our measuring apparatus to dilate along with the objects which we are attempting to measure. This is not some mathematical bandaid, but rather a remedy to an additional derivative we've arbitrarily added to the equation that is not an inherent part of the Universe: time.

Supportive Observations & Related Phenomena

Pound & Rebka

The Pound-Rebka experiment was an experiment carried out at Harvard in which researchers setup a light clock, not too dissimilar from the one we've setup in our previous thought experiment. The difference (at least the difference that matters) is that they moved the two 'mirrors' apart from each other at a velocity that would perfectly cancel out the Doppler shift induced by gravity. They found a result of

And because

If you're unfamiliar, (delta, I know, we've already used delta) just means the 'change of' , and is the height between the emitter and the receiver.

And you've likely guessed it, this coincides perfectly with the outcome we would expect in the model in which dilates according to our motion through space which in turn gives the dilation of space and the passage of time, and we did so without the need to dilate a quantity that we can't even measure!

Further, since both and would dilate proportionally to in the model being proposed, it is irrelevant which coordinate system we use to calculate as both are equivalent. In fact remains constant between reference frames much in the same manner that 4-vectors remained constant in traditional models of relativity. This ironically makes time the only quantity that doesn't dilate proportionally to , as dilates in one reference frame and in the other. This maintains a consistent energy expenditure per unit length according to the underlying spatial density, strongly suggesting a sort of quantifiable empty space with which the fundamental mechanisms giving inertia lie.

Bullet Cluster

The bullet cluster is a pair of galaxy clusters rapidly approaching each other. The extreme relative velocities accompanied by the extreme masses of each galaxy cluster give rise to a phenomenon that current models of relativity can't explain: The mass we see does not align with the gravitational lens.

This sort of 'washing' effect is precisely what would be expected in a model in which gravity, and in turn the gravitational lens is a function of not mass alone, but of mass and velocity.

Lack of Gravitational Aberration

The apparent lack of gravitational aberration seems to imply that the speed of gravity is infinite, but this can't be the case. If 'time' passes and gravity effects some distant body, it must be that either gravity does not 'move' at all, or that gravity and time are the same thing.

If gravity and time are the same process when observed from the coordinate system of the 'oven', , regardless of the units used or any relativistic effects, and any motion of space remains hidden from our perspective in the coordinate system of the 'dough' due to our co-motion along this axis.

Super Evolved Galaxies

With the launch of the James Webb Space Telescope in 2021 we've made quite a few incredible observations that require some additional context, if not entirely new physics. One of such discoveries is that of galaxies which appear to be too large to exist at the time at which we observe them.

Because light has to travel to us, the further away an object is the further in the past it was when the light that reaches our eyes or telescope departed that object. This problem of super-evolved galaxies at a time which appears to be too early for our current model is resolved by the concurrent nature of this density axis being proposed. If this axis is concurrent, there is no reason to believe that there should be any limit on the evolution of a cosmic process at any point in time.

Consider that for some unit of pseudo-time , some photon travels , being equivalent to one 'event'. In that same unit of pseudo-time, some body travels , making the following true and making super-evolved galaxies at early epochs not only possible, but inevitable.

Fine Tuning and the Fermi Paradox

This geometry opens an entirely new concurrent dimension that life may be spread across. It now becomes possible that life exists at some location, separated only by this density axis. If you're the super-natural type, this concurrent density axis might explain certain unexplained phenomena. Everything from ghosts to the propulsion and plausibility of extraterrestrial visitors becomes more likely if an object may exist at shared spatial coordinates, separated only by a density axis coordinate.

This model of concurrency can be extended to preclude that in order to make an object's existence linear, all that is required is a 4-dimensional coordinate for each density state. As space itself dilating gives rise to a sort of higher-order time, our motion along this density axis gives rise to time as we experience it. That is to say that we are in motion along an axis that is in motion itself. This model of concurrency does not explicitly require that an object must follow a linear path in 3-dimensional space. While the temporal axis in this model is not orthogonal, this density axis shares many properties with our 3 spatial coordinates. There is nothing to prohibit an object from traveling backwards in time, so long as the progression of (below) progresses linearly (down to the Plank scale) towards .

In other words, as the Universe expands there exists a (assumingly linear, or at least linear down to the Plank scale) density axis with possible density states. Of course if this axis is truly linear, but even a finite trends towards as the Universe expands. Each object can exist at any density state within while . This position of as it trends to gives a sort of higher-dimensional time without requiring an additional physical process. An object's existence is linear so long as it has a 4-vector for each state of , with the temporal coordinate not being necessarily equivalent to . This is to say that the coordinate system of the past, present and future exist concurrently. To borrow a phrase from Einstein, "The distinction between the past, present and future is nothing but a stubbornly persistent illusion".

I will now refer to this progression of as . is equivalent to the least dense state possible at any instant.

In regards to the super-evolved galaxies described above, there is no reason to infer that our origin at was concurrent with another distant body. A body may have existed at this more dense state for a much longer dilation of , allowing for this seemingly excess temporal progression, and this body may have existed long before our own origins at . If this axis is concurrent, there is no reason to believe that the instant we were at was concurrent to any other body. This is to say that our reliance on distance and even redshift to determine the age of a body is flawed in cases of extreme mass or velocities. There is no requirement that the distant body that emitted the light we are observing was existing at a shared temporal coordinate when it emitted the photon being received if the body's were not initially concurrent and travelling on a nearly identical trajectory at a similar velocity.

To summarize, I believe that our position along this proposed density axis exists as a sort of yet unresolved equilibrium solution. This may resolve the fine-tuning paradox while opening the door to a completely new dimension of travel with sufficient technological advancements.

• An object must have a 4-vector for each density state
• This density decreases according to (gravitational acceleration of every body in the Universe) as a function of .
• An object does not have to exist at . Instead, an object may exist at any density between the original density state of and .
  • This is what allows for super-evolved galaxies. There is nothing to exclude a body remaining at a higher density state for a greater number of events, and nothing to require that our position at was concurrent to another body's existence at .

Machian Inertia

This model in which cosmic inflation is derived from the sum of all gravitational acceleration gives rise to a completely Machian interpretation of inertia. As Newton's laws dictate, every force should be met with it's opposite. If this inflation gives rise to time itself due to an object's motion through the oven's coordinate system, giving an equilibrium solution in the coordinate system of the 'dough', it's opposite should resist a change from this equilibrium state. If Newton's first law is this equilibrium state, any deviation from this state should be resisted by the opposite of the dilation of space.

For body , this may be represented as:

But would be met by:

Where in the latter equation is the distance from the secondary body to and some unknown function scaling the magnitude of this interaction. It is this sum that gives rise to Machian inertia, and I believe, the source of inertial mass, as well as the proper but simplistic description of the relationship between gravity and inertia have.

Classical Simultaneity

This one is not so much an experiment, but rather an absolutely ridiculous notion that falls out of current interpretations of special relativity. Resolving this absurdity was perhaps the single greatest motivation in the development of this model. This notion, known as relativistic simultaneity, says that a single, instantaneous event can occur at two different points in time. This just falls out of the math of special relativity, and since most of the math of special relativity has been demonstrated experimentally, it is widely, albeit inexplicably accepted. This is obviously absurd, not only as a matter of physics, but as a matter of math. Through the modifications to special relativity being proposed we can remove this notion, maintaining that all instantaneous events occur at precisely one instant in time.

Discussion

Problems & Shortcomings

First, this model will, by the very nature of the symmetries it shares with existing models of relativity, satisfy all experiments supporting existing models. That is not to say that all the math has been resolved. In the narrow view we can say that all vectors have the same magnitude, some are pointing in the exact opposite direction, but if that is the case the opposite body is moving (space, not the object when it is in gravitational free fall). This gives the same kinematic velocity in both models. Cosmological velocities are a little harder to demonstrate, but by the very symmetries that this model shares with current models, all experiments must be satisfied. That is not to say that the math is easy. In fact, deriving a set of equations for a body in motion through a medium that dilates as a function of that motion, putting space itself in motion is proving to be really, really hard. I know I can do it, but with me being homeless I haven't been able to dedicate time to this model in almost a year as I've worked on releasing Fluster.

Conservation of Energy and Momentum

As the section on Machian inertia implies, inertia is zero when the body is in motion with the 'dough' and the body is in a equilibrium (inertial) state. If the body moves relative to this inertial state the dilation of space around dilates, while remains unchanged by the change of state of . This induces a gradient, giving rise to Newton's first law.

Variable speed of light

So, have you noticed the inconsistency yet? If the speed of light is constant in each reference frame, how can it possibly be a constant if either space or time dilates? If and either or dilates, it is impossible for to be both an absolute constant and equivalent in each reference frame. What is possible however is that is equivalent in each reference frame, as dilates proportional to distance as space itself dilates. This gives a measure of that both observers would agree on, but this is only due to the dilated space of the observer in motion. The kinematic velocity of is equivalent in each reference frame, not the velocity in the coordinate system of the 'oven'.

The Blackhole Mirage

One of the biggest mysteries in modern physics is how to describe the interior of a blackhole. Current models suggest that the interior of a blackhole is a point of infinite density, a place where even light can't escape. In the model I am proposing, blackholes are nothing more than a mirage in the distance that dissipates as an observer approaches that body's gravitational field, causing the speed of light to dilate proportionally. As only the speed of light through the 'dough' is constant in each reference frame, inversely proportional to the density of the 'dough', the speed of light increases proportionally to local gravitational acceleration. This fits perfectly with what is observed in the Pound & Rebka experiment once our co-motion across the temporal axis is accounted for. Even the curvature of light away from the gravitational source in the gravitational lens follows this pattern.⁴

The key to moving forward

In short, this model needs to be bound with electromagnetism. The potential outcome of such a unity would change the world on par with the wheel or the discovery of electricity. Through this new description of gravity, cosmic inflation and time we can infer that the divergence of the magnetic field is not 0, but rather equivalent to the determinant of some time dependent 3x3 matrix that describes the dilation of space at that point. In a manner that is similar to the way in which the dilation of space remains hidden apart from the apparent acceleration due to gravity, our co-motion keeps this divergence hidden from us directly. Through this relationship I am confident that gravity and electromagnetism can finally be unified.

There also exists an intriguing coincidence, where if we treat as being not proportional to time, but as the fundamental rate of change mentioned above, making a pure magnitude of distance proportional only to other velocities where , we can derive a ratio of

for a set of local gravitational parameters that occurs on the surface of Earth, where is the dilation of space per the change in displacement of magnitude . Oddly enough this quantity can be found through a peculiar pairing of our equatorial and polar radii to an accuracy of .

Treating as being only proportional to other magnitudes leads to a theoretical upper limit of in units of length, making the magnitude of each vector not purely relative in the same manner that motion is currently thought to be. Instead the free-fall reference frame leaves an observer at rest relative to the 'oven', making all displacement vectors (arrows describing the change in your location) being relative to this reference frame, and the magnitude of each vector being proportional to in that coordinate system.

To The Physics Students and Professionals

First, thank you for actually reading this, despite what you may think about it. Secondly, yes... I know this is radical. It's a geometry that's hard to wrap your head around, and it leads to several uncomfortable conclusions:

• Gravity is a repulsive force
• Gravity is a function of velocity
• We've had one too many derivatives this whole time

And most significantly, nothing can be proportional to time. The implications of this are huge, but they are not unmanageable or inconsistent with observation. Every quantity we define as being proportional to time can be rewritten as being proportional to something else, even if that just is motion on a relativistic scale. This opens the door to science-fiction level possibilities... imagine every quantity being proportional to a quantity that we can directly influence. Imagine a real, quantifiable geometry to time itself.

The Lighthouse And The Clock-tower - A closing thought experiment

Now that you understand the principles of relativity, lets go through a thought experiment: Let there be two observers, and . Let these two observers agree that should travel at some agreed upon velocity between two arbitrary points in space, and . At place a time keeping device that remains visible from . As remains at and watches in motion between and , at what time will reach ?

Surely both observers must agree on the time at which reaches , which I will now call . If we were to place a camera at that captures arriving at , this image must agree with the time observes reach since there are no relativistic effects impacting the image returned from to at . We also know that must apply to this equation. Pound & Rebka, and Ives & Stillwell have both demonstrated this additional factor of experimentally.

So if must be where is the distance between and , the initial time, and is the agreed upon velocity, the only possible solution to the equivalence is that distance dilates in the coordinate system of and that and agree upon a velocity. If both and must agree on the time at which reaches and this time must be extended by an additional factor of , then the agreed upon velocity produces an elongated distance while the velocity decreases by a factor of in the coordinate system of to accommodate a that is not elongated by . It's as if there exists some sort of drag induced by the more dense coordinate system, and in a manner similar to the way in which 4-vectors remain constant in traditional models of relativity, remains constant irrespective of coordinate system.

Footnotes

1. For each small change in the radius, there is a dilation of space equivalent to this function. ↩

2. See the section on the Pound & Rebka results. dilates inversely proportional to spatial density, or in other words, proportional to in a single body approximation such as within the vicinity of a blackhole. A fundamental piece of this model is the principle that all functions limit at their Newtonian counterpart when the reference frame is shared, once this divergent geometry is accounted for. No where is this more evident than in the "constant" nature of . With being constant in each reference frame, it is impossible for it to be equivalent between reference frames when the units of measure themselves are dilating. ↩

3. in standard units. The dimensionality issue with is discussed in the section on the conservation of energy and momentum. ↩

4. The metric tensor, in the way it is currently applied, appears to describe this motion due to Einstein's reliance on the equivalence principle and the addition of vectors. ↩