Temporal Uniformity

[Daedalus]

On The Temporal Uniformity of Accelerating Bodies

 

The view that is to be set forth attempts to describe the universe from the point of view of a four dimensional sphere that is expanding at the speed of light. We can simplify the equations by breaking this sphere into two, three dimensional, spheres. This results in one sphere being comprised of W, X and Y axes, and the other dealing with W, Y and Z axes. The W axis is the spatial dimension of time and is shared along with the vertical axis, Y, between both spheres. If one considers that we exist on the surface of such spheres, then we can take a section of each sphere and combine them to form our three dimensional view of the universe. The spheres would be aligned such that one sphere provides the X, Y plane and the other provides the Y, Z plane. Because the curvature of a circle or sphere diminishes as the radius increases, and due to the enormous size of these spheres, our space will appear to be flat. Also, we would lose the W dimension because the four dimensional sphere is expanding at the speed of light and length contraction would have collapsed the fourth dimension to have no perceivable length.

 

For the sake of simplicity, we will only consider the W, X and Y sphere. This will allow us to derive time dilation equations across what we perceive to be a two dimensional plane. The purpose of exploring time dilation for the path of a clock as it traverses across the surface of an expanding sphere is to derive the mathematics for the Lorentz factor which is affected by such surface which is not flat. We will explore the paths for a light clock and a rotating mechanical clock and show that both approach the equation for time dilation as defined in SR as time, and subsequently the radius of the sphere, approaches infinity.

 

The equations that are to be discussed are not new and by no means considered complex as they simply deal with the basic mechanics for motion as defined in classical physics. The equations that will be discussed will include uniform acceleration and we will begin by discussing Newton's interpolation formula as it allows us to derive a single equation for simple motion that is only dependent on the locations of a body along its trajectory. This allows us to extend the equations for motion beyond that of uniform acceleration. Newton's interpolation formula can be written using the binomial theorem as follows (The locations of a body's trajectory combine in such ways that use binomial coefficients when interpolating such simple motions. We can also use the Gamma function to generate these coefficients):

 

[math]\sum_{i=0}^{n-1} \left [ \sum_{j=0}^{i} \left [ \left ( -1 \right )^{j} \binom{i}{i-j} S_j \right ] \times \binom{i-t-1}{i} \right ][/math]

 

The variable [math]n[/math] is the number of measurements or specified locations, [math]S_j[/math] is the location specified by the [math]j[/math] index, and [math]t[/math] is time. It is important to note that the locations are sampled within equal intervals of time. As a bonus, I will post the interpolation formula for locations sampled at any given time once we have completed this journey : )

 

If we let [math]n=1[/math], the formula yields the initial position [math]S_0[/math]. When [math]n=2[/math], the formula yields the equation for motion as defined by the initial position and velocity: [math]S_0 +\left (S_1 - S_0\right )t[/math]. Finally, we can derive the equation for uniform acceleration by specifying [math]n=3[/math] which yields the following:

 

[math]S_0 + \left (\left (S_1-S_0\right ) - \frac{1}{2} \left (S_2-2\, S_1+S_0\right )\right ) t + \frac{1}{2}\left (S_2-2\, S_1+S_0\right )t^2[/math]

 

We can see from the above results for [math]n=3[/math], that we can easily derive the constants for acceleration and velocity as follows:

 

[math]a=\left (S_2-2\, S_1+S_0\right )[/math]

[math]v=\left (\left (S_1-S_0\right ) - \frac{1}{2} \left (S_2-2\, S_1+S_0\right )\right )[/math]

 

However, this does not help us derive time dilation as we need to define acceleration, velocity, and the initial position in terms of the time it takes for light to travel between the reflective surfaces of a light clock that are separated by a length [math]L[/math], or the distance [math]\left \vert AB \right \vert[/math]. This results in redefining the locations, [math]S_j[/math], in terms of [math]L[/math] such that [math]S_j=\mathit{l}_j \times L[/math] where [math]\mathit{l}_j[/math] is a length scalar or multiple of [math]L[/math]. The amount of time it takes light to travel between the two reflective plates of a light clock is equal to [math]\Delta \tau = L / c[/math] such that [math]L = c \, \Delta \tau[/math]. Substituting this result back into Newton's interpolation formula allows us to completely factor out [math]c \, \Delta \tau[/math] as follows:

 

[math]c \, \Delta \tau \sum_{i=0}^{n-1} \left [ \sum_{j=0}^{i} \left [ \left ( -1 \right )^{j} \binom{i}{i-j} \mathit{l}_j \right ] \times \binom{i-t-1}{i} \right ][/math]

 

We can now reconsider the constants of acceleration, velocity, and initial position when [math]n=3[/math] as follows:

 

[math]a=c \, \Delta \tau \left (\mathit{l}_2-2\, \mathit{l}_1+\mathit{l}_0\right )[/math] or [math]a=c \, \Delta \tau \ a_l[/math] where [math]a_l=\left (\mathit{l}_2-2\, \mathit{l}_1+\mathit{l}_0\right )[/math]

 

[math]v=c \, \Delta \tau \left (\left (\mathit{l}_1-\mathit{l}_0\right ) - \frac{1}{2} \left (\mathit{l}_2-2\, \mathit{l}_1+\mathit{l}_0\right )\right )[/math] or [math]v=c \, \Delta \tau \ v_l[/math] where [math]v_l=\left (\left (\mathit{l}_1-\mathit{l}_0\right ) - \frac{1}{2} \left (\mathit{l}_2-2\, \mathit{l}_1+\mathit{l}_0\right )\right )[/math]

 

and

 

[math]S_0=c \, \Delta \tau \ \mathit{l}_0[/math]

[math]S_1=c \, \Delta \tau \ \mathit{l}_1[/math]

[math]S_2=c \, \Delta \tau \ \mathit{l}_2[/math]

 

It is important that [math]\Delta \tau[/math] is completely factored out. Otherwise, it will get trapped inside the parametric equations which describe the path that light from a light clock takes as it traverses across the expanding four dimensional sphere and we would not be able to derive time dilation accordingly when using the arc length integral to calculate the length of the path.

 

The usage of Newton's interpolation formula to describe simple motion will only be used in this post as we will only concern ourselves with uniform acceleration. The reason for discussing this equation was so that we can establish a foundation for which simple motion of any form, including that beyond uniform acceleration, can be used to derive the time dilation equation and ultimately the Lorentz factor. Next, we will apply the equation for uniform acceleration to a light clock that is traversing across an expanding sphere. But, that is for the next post.

Edited September 15, 2011 by Daedalus

[michel123456]

(...)if we consider that we are always in motion through the dimension of time and that we do indeed traverse a physical distance through this fourth dimension.

 

 

Brilliant. I support this hypothesis.

 

On The Temporal Uniformity of Accelerating Bodies

 

The view that is to be set forth attempts to describe the universe from the point of view of a four dimensional sphere that is expanding at the speed of light. (...)For the sake of simplicity, we will only consider the W, X and Y sphere. This will allow us to derive time dilation equations across what we perceive to be a two dimensional plane. The purpose of exploring time dilation for the path of a clock as it traverses across the surface of an expanding sphere is (...)

 

So, if I understand correctly, you are examining a concept in which all dimensions are expanding, that is our ruler of space is expanding and our ruler of time is expanding too.

I call that a scaling theory and I support the concept.

Edited September 15, 2011 by michel123456

[Pincho Paxton]

Of course it's true, you don't need maths to apply this to every theory in the book. It is just written illogically in the way that everything is enclosed at a distance in 4D. Everything is enclosed locally and polarised.

[Daedalus]

So, if I understand correctly, you are examining a concept in which all dimensions are expanding, that is our ruler of space is expanding and our ruler of time is expanding too.

 

That is correct Michel. The concept that I have been examining is that where a four dimensional sphere is expanding through all dimensions.

 

Of course it's true, you don't need maths to apply this to every theory in the book. It is just written illogically in the way that everything is enclosed at a distance in 4D. Everything is enclosed locally and polarised.

 

Perhaps I wasn't clear in explaining how we perceive this four dimensional sphere. But to try and respond to your statement, it is true that we are enclosed locally in an expanding three dimensional sphere. The 4D distance that you are referring to, only explains why massive bodies do not speed up or fall behind us temporally. If this 4D distance was not uniform for all observable bodies, then these bodies would either appear or disappear from our view of the universe.

 

 

Continuing with the discussion, we now have to define the parametric equation that describes a path across that of a sphere. For this we must use trigonometry. Before we can define this equation, we must start by defining a line segment as an arc that traverses across the surface of the sphere. The equation for the length of an arc across a sphere with radius [math]r[/math] can be derived as follows (We will simplify this example by examining a circle):

 

The circumference of a circle is given by: [math]2\, \pi \, r[/math]. The angle, in degrees, encompassing the arc is proportional to [math]360^\circ[/math] where [math]\frac{\theta^\circ}{360^\circ}[/math] defines the percentage of the circle that the arc traverses across. This allows us to calculate the length of the arc by multiplying the circumference of the circle by this ratio:

 

ArcLength or distance, [math]d=\frac{2\, \pi \, r\, \theta^\circ}{360^\circ}=\frac{\pi \, r\, \theta^\circ}{180^\circ}[/math]

 

Now that we know how to calculate the length of an arc, we must solve for [math]\theta[/math]. This is because [math]\theta[/math] changes as a body traverses across the sphere or circle.

 

[math]\theta^\circ=\frac{180^\circ \, d}{\pi \, r}[/math]

 

We can simplify this result by converting degrees to radians (When referring to an angle from this point on we will always be using radians as it truly simplifies the mathematics):

 

[math]\theta_{rad}=\left ( \frac{\pi}{180^\circ} \right ) \frac{180^\circ \, d}{\pi \, r}=\frac{d}{r}[/math]

 

We discussed the equations for motion in the previous post and can now substitute that result for the variable, [math]d[/math], in the above equation which describes the angles that such motion produces as a body traverses across the sphere. The hypothesis states that this sphere is expanding at the speed of light. This means that the radius, [math]r[/math], of the sphere is expanding at the rate [math]c[/math]. However, we will substitute [math]V_w[/math] because it represents the temporal velocity that we have been discussing since the beginning of this thread. We will also define the motion of this expanding sphere no different than how we have defined motion in the previous post. This results in allowing temporal acceleration for completeness. However, this does not imply that we accelerate through time as it is hypothesized that this temporal acceleration, [math]A_w[/math], is now equal to zero. The reason for including it is so that we can experiment with inflation which states that the universe expanded faster than the speed of light during its initial birth as hypothesized by Alan Guth in 1980. If inflation did in fact occur, then it would be possible to have temporal acceleration. However, we will not be discussing inflation at this time.

 

It is important to understand that our measurement of time in this mathematical model is nothing more than a count of cycles as provided by some form of clock. We can convert these cycles to seconds by using a conversion factor, frequency, as previously discussed. This results in mathematical time as being calculated by taking the distance travelled and dividing it by the speed at which such distance was traversed. With that being stated, it is still proper to define time with the variable, [math]t[/math], which is equivalent to:

 

[math]time=\frac{cycles}{frequency}=\frac{meters}{meters / seconds}[/math]

 

Combining everything discussed, we can now derive the equation which describes [math]\Delta \theta[/math] for a body as it traverses across the surface of an expanding sphere. We will use the standard equation for uniform acceleration from which we will substitute our modified [math]\Delta \tau[/math] version in later.

 

The equation which describes the motion along the axis S:

 

[math]\Delta S=S_0+V_s \, \Delta t+\frac{1}{2}A_s \, \Delta t^2[/math]

 

The equation which describes the motion of the expansion of the sphere:

 

[math]\Delta W=W_0+V_w \, \Delta t+\frac{1}{2}A_w \, \Delta t^2[/math]

 

The change in angle that such motion produces as a body traverses across the expanding sphere:

 

[math]\Delta \theta_s=\frac{\Delta S}{\Delta W}=\frac{S_0+V_s \, \Delta t+\frac{1}{2}A_s \, \Delta t^2}{W_0+V_w \, \Delta t+\frac{1}{2}A_w \, \Delta t^2}[/math]

 

Now that we can calculate the [math]\Delta \theta[/math] for such motion we can define the parametric equation which describes the path of the body starting from the coordinate, {w, 0, y}:

 

[math]w'=w \, \cos(\theta_{yaw}) \, \cos(\theta_{pitch})\, -\, y \, \cos(\theta_{yaw}) \, \sin(\theta_{pitch}) \, \cos(\theta_{roll})\, +\, y \, \sin(\theta_{yaw}) \, \sin(\theta_{roll})[/math]

[math]x'=w \, \sin(\theta_{yaw}) \, \cos(\theta_{pitch})\, -\, y \, \sin(\theta_{yaw}) \, \sin(\theta_{pitch}) \, \cos(\theta_{roll})\, -\, y \, \cos(\theta_{yaw}) \, \sin(\theta_{roll})[/math]

[math]y'=y \, \cos(\theta_{pitch}) \, \cos(\theta_{roll})\, +\, w \, \sin(\theta_{pitch})[/math]

 

The coordinate, {w, 0, y}, is the starting position of the photon that travels between the reflective surfaces of the light clock. The angle, [math]\theta_{yaw}[/math], is a rotation about the vertical Y axis and describes the angle that changes as the clock traverses what we perceive to be the X axis. We will redefine [math]\theta_{yaw}[/math] to be [math]\theta_x[/math] to avoid confusion between which angle causes horizontal translatory motion along the X axis. The angle, [math]\theta_{pitch}[/math], is a rotation about the horizontal X axis and describes the angle that changes as the clock traverses what we perceive to be the Y axis. We will also redefine [math]\theta_{pitch}[/math] to be [math]\theta_y[/math] to avoid confusion between which angle causes vertical translatory motion along the Y axis. The angle, [math]\theta_{roll}[/math] is a rotation about the W axis. We will redefine [math]\theta_{roll}[/math] to be [math]\theta_w[/math] as it defines the angle of rotation for the mechanism in a mechanical clock as will be discussed much later.

 

The above parametric equation is derived by multiplying the rotation matrix for each axis in a specific order that describes the angles as yaw, pitch and roll. The equation can also be derived by applying the following equation, which describes the rotation of a point about a specific axis, in the same manner as the matrices:

 

[math]x'=x \, \cos(\theta)\, - \, y \, \sin(\theta)[/math]

[math]y'=x \, \sin(\theta)\, + \, y \, \cos(\theta)[/math]

 

Now that we have our equations which describe the changes in angles and the path across a three dimensional sphere, we will end this post with the following summary:

 

The equations which describe motion in the form of uniform acceleration:

 

[math]\Delta W=W_0+V_w \, \Delta t+\frac{1}{2}A_w \, \Delta t^2[/math]

 

[math]\Delta X=X_0+V_x \, \Delta t+\frac{1}{2}A_x \, \Delta t^2[/math]

 

[math]\Delta Y=Y_0+V_y \, \Delta t+\frac{1}{2}A_y \, \Delta t^2[/math]

 

The equations which describe the angles as the clock traverses across the expanding sphere:

 

[math]\Delta \theta_x=\frac{\Delta X}{\Delta W}=\frac{X_0+V_x \, \Delta t+\frac{1}{2}A_x \, \Delta t^2}{W_0+V_w \, \Delta t+\frac{1}{2}A_w \, \Delta t^2}[/math]

 

[math]\Delta \theta_y=\frac{\Delta Y}{\Delta W}=\frac{Y_0+V_y \, \Delta t+\frac{1}{2}A_y \, \Delta t^2}{W_0+V_w \, \Delta t+\frac{1}{2}A_w \, \Delta t^2}[/math]

 

The parametric equation which describes a path across the sphere:

 

[math]w'=w \, \cos(\Delta \theta_{x}) \, \cos(\Delta \theta_{y})\, -\, y \, \cos(\Delta \theta_{x}) \, \sin(\Delta \theta_{y}) \, \cos(\Delta \theta_{w})\, +\, y \, \sin(\Delta \theta_{x}) \, \sin(\Delta \theta_{w})[/math]

[math]x'=w \, \sin(\Delta \theta_{x}) \, \cos(\Delta \theta_{y})\, -\, y \, \sin(\Delta \theta_{x}) \, \sin(\Delta \theta_{y}) \, \cos(\Delta \theta_{w})\, -\, y \, \cos(\Delta \theta_{x}) \, \sin(\Delta \theta_{w})[/math]

[math]y'=y \, \cos(\Delta \theta_{y}) \, \cos(\Delta \theta_{w})\, +\, w \, \sin(\Delta \theta_{y})[/math]

 

The next posting will complete the mathematics presented in this post by deriving the equations which predict the coordinates that a photon in the light clock traverses for both the up and down paths. I will also be posting images of the graphs that these equations produce for the path of the photon. We will discuss [math]\Delta \theta_w[/math] when we get to the point where we are describing the path for a mechanical clock. For now, we will set [math]\Delta \theta_w=0[/math]. Once we have completed the mathematics which describe the path that light from a light clock takes as it traverses the expanding sphere, we will substitute our equations for motion that make use of [math]\Delta \tau[/math] and derive the time dilation equation.

 

Please keep in mind that even though I have made great effort to ensure that the mathematics that I have posted is correct, I have been known to reverse and mix things up from time to time. Also, my grammar isn't the best in the world : ) Everything that I have posted has been graphed and verified using Mathematica. With that being said, please message me if you do find an error so that I can make the appropriate corrections. Thank you.

Edited September 17, 2011 by Daedalus

[Pincho Paxton]

That is correct Michel. The concept that I have been examining is that where a four dimensional sphere is expanding through all dimensions.

 

 

 

Perhaps I wasn't clear in explaining how we perceive this four dimensional sphere. But to try and respond to your statement, it is true that we are enclosed locally in an expanding three dimensional sphere. The 4D distance that you are referring to, only explains why massive bodies do not speed up or fall behind us temporally. If this 4D distance was not uniform for all observable bodies, then these bodies would either appear or disappear from our view of the universe.

 

 

Take a look at the aberration of light, and see if you find something new. Add your massive body misalignment idea to it. Maybe you get a new idea, maybe you don't. Personally, I thought about changing the word light to possibly time.

 

Edited September 17, 2011 by Pincho Paxton

[Daedalus]

Take a look at the aberration of light, and see if you find something new. Add your massive body misalignment idea to it. Maybe you get a new idea, maybe you don't. Personally, I thought about changing the word light to possibly time.

 

I am familiar with the phenomenon, aberration of light. The mathematics set forth does not conflict with this phenomenon as our space is practically flat as supported by the mathematics. Perhaps we can discuss this phenomenon in detail after I have presented this mathematical framework to see how well it fits vs. observation or we can start a new thread regarding that discussion. Also, could you please edit your quote of my latest post to not include the entire posting as you are only referring to my repsonse to your statement. Thank you : )

Edited September 17, 2011 by Daedalus

[pantheory]

Perhaps you are correct Pantheory and time is nothing more than an interval of change. However, I ask you to consider what is change? We typically think of change as an explanation for why something is different than how it was, but this does not clearly define why something changes or how it has changed. If we dig deeper into the mechanism of change, we will find physical motion as the cause. This can be demonstrated in that a change in a body's position is caused by the motion of the body relative to our own position. The erosion of a landscape is caused by the motions of matter and energy as it applies forces upon the land that it is acting upon. Even when considering sources of change that do not seem to have motion, such as that of the wave of colors produced by electrochemical processes in a cuttlefish, there is always an underlying mechanism which is in motion that produces such change. The colors in our clothing fade due to exposure to photons and chemicals which strip the pigments from the material. I can continue to list many more examples of things that change. The point is that we will always find that it is actual physical motion that drives change. That is my reason for not accepting that time is just an interval of change because it is physical motion that drives change and not change that drives physical motion.

 

If we take the example where we can define time by dividing the distance traveled by the speed at which we traversed such distance and then place ourselves in the rest frame where our displacement and speed equals zero, then when we try to determine our time we get a division by zero which is undefined. One could state that this is a relative matter but we are still left with mathematics for which only works if we are in fact in motion relative to something else. This mathematical paradox of an undefined time for the rest frame of reference can only be resolved if we consider that we are always in motion through the dimension of time and that we do indeed traverse a physical distance through this fourth dimension.

 

I now have internet and my ISP has provided me with web space to upload the images I have created so that we can continue this discussion and explore the mathematics that I have derived as explained earlier in my posts. I understand that this mathematical framework does not assure that the theory is correct, nor does it mean that temporal uniformity will gain acceptance by my peers as Swansont has pointed out. However, a theory without a mathematical basis is forever doomed to failure as it cannot provide us with a means to check the validity of such theory. A good theory not only provides us with definitions that describe the mechanics of the universe, but provides us with the means to rule it out by checking its assertions as provided by the mathematics. I will start by breaking the mathematics into simpler components from which I will make separate posts describing the mechanics of each component. Once all of the equations have been defined I will assemble them together and post the images that they graph along with a link to the Mathematica file that I have produced so that everyone with Mathematica 5.2 or greater can reproduce the graphs and images. We will begin the discussion this week once I have set up my web space and have uploaded the images and files that I will be posting : )

 

Thanks for the extended reply to my comment Daedalus.

 

I agree. I cannot think of any kind of change that does not in some way involve motion of some kind, but I prefer using the word "change" rather than the word "motion" to define time since some motion, like particle spin, for instance, presently concerns nebulous descriptions of it in physics, in my opinion. I think your presented proposal on time can be based solely upon logic alone and needs no math. I think sometimes mathematical physics confuses more than it clarifies. As we both know in some facets of physics time has different meanings according to a particular theory/ model, but I believe all those that think that time has any other meaning aside from change or motion, are following "false gods." I think that time can best be understood and explained by definition alone using simple words such as "an interval of motion/change" and little more. Unfortunately nowadays some physicists seem to like making the simplest of concepts into post-grad college material if somebody is paying them to do it. Defining time with complicated concepts and math that seemingly reinforces the complication, I think is a good example.

[Daedalus]

Thanks for the extended reply to my comment Daedalus.

 

I agree. I cannot think of any kind of change that does not in some way involve motion of some kind, but I prefer using the word "change" rather than the word "motion" to define time since some motion, like particle spin, for instance, presently concerns nebulous descriptions of it in physics, in my opinion. I think your presented proposal on time can be based solely upon logic alone and needs no math. I think sometimes mathematical physics confuses more than it clarifies. As we both know in some facets of physics time has different meanings according to a particular theory/ model, but I believe all those that think that time has any other meaning aside from change or motion, are following "false gods." I think that time can best be understood and explained by definition alone using simple words such as "an interval of motion/change" and little more. Unfortunately nowadays some physicists seem to like making the simplest of concepts into post-grad college material if somebody is paying them to do it. Defining time with complicated concepts and math that seemingly reinforces the complication, I think is a good example.

 

You are welcome Pantheory and I also agree that time can best be understood and explained by definition alone. However, temporal uniformity is not just concerned with explaining the phenomena of time. The purpose of the mathematical framework that is being set forth is to redefine the Lorentz factor and show how we can begin to explain even more complex phenomena as a result. The mathematics and graphs that I will be posting may be of some interest to you as they do produce vortices. I understand that your latest thread, Pushing Gravity, deals with such things in the aether.

 

You might equate them with dark matter except they are much smaller and accordingly are mass-less like photons at rest, and their mechanics are strictly a physical pushing. A better description might be a physical aether with vortex motions and fluid dynamics.

Edited September 17, 2011 by Daedalus

[pantheory]

You are welcome Pantheory and I also agree that time can best be understood and explained by definition alone. However, temporal uniformity is not just concerned with explaining the phenomena of time. The purpose of mathematical framework that is being set forth is to redefine the Lorentz factor and show how we can begin to explain even more complex phenomena as a result.

 

Sounds good to me.

 

The mathematics and graphs that I will be posting may be of some interest to you as they do produce vortices. I understand that your latest thread, Pushing Gravity, deals with such things in the aether.

Although I describe gravity via the pushing forces of aether particulates in vortex and fluid dynamic motions, the math is MOND-like vector equations and not that sophisticated. I have been struggling with the realization concerning gravity formulations and galactic motions, that math alone including a particulate aether or dark matter, will never seemingly be able to predict anything at the galactic scale without direct observation of the details of the observed motions involved-- seemingly remaining an ugly step-child in physics requiring retrodiction -- even though we may be able to improve on the poor level of retrodiction that present dark-matter models are presently producing.

 

But yes, show me the math and if I understand its possible application , maybe I could somehow realize how to use it to make better gravity formulations/ predictions

Edited September 17, 2011 by pantheory

[Daedalus]

Sounds good to me.

 

 

Although I describe gravity via the pushing forces of aether particulates in vortex and fluid dynamic motions, the math is MOND-like vector equations and not that sophisticated. I have been struggling with the realization concerning gravity formulations and galactic motions, that math alone including a particulate aether or dark matter, will never seemingly be able to predict anything at the galactic scale without direct observation of the details of the observed motions involved-- seemingly remaining an ugly step-child in physics requiring retrodiction -- even though we may be able to improve on the poor level of retrodiction that present dark-matter models are presently producing.

 

But yes, show me the math and if I understand its possible application , maybe I could somehow realize how to use it to make better gravity formulations/ predictions

 

I am going to let my posting tonight stand for a day or so. But, I will be more than happy to send you an e-mail of what I have so far as I do have your e-mail address. Right now my brother is demanding that I go and jam on the guitar with him, but I will send it to you as soon as I can : )

[pantheory]

I am going to let my posting tonight stand for a day or so. But, I will be more than happy to send you an e-mail of what I have so far as I do have your e-mail address. Right now my brother is demanding that I go and jam on the guitar with him, but I will send it to you as soon as I can : )

Thanks Daedalus, looking forward to it.

 

Here is a good example of my postings concerning understandings of the concept of time. I was just researching modern concepts of time in physics and came across this book called From Eternity to Here: The Quest for the Ultimate Theory of Time. Although I haven't read it, there is a huge difference of "opinion"/ theory here. Mine is that time is one of the simplest concepts to explain and understand in physics, similar to space -- that time can simply and fully be explained and understood by a one sentence definition and no more. On the other hand the author's opinion, concerning modern physics, is that an entire book is needed just to explain the possible theoretical developments needed for such an "Ultimate Theory of Time" to someday be created. I would call the difference of these two opinions an abyss. This is what I consider a good example of the logical problems with modern physics, many of which I think stem from misguided attempts to translate theoretical physics into language.

//

Edited September 17, 2011 by pantheory

[Daedalus]

Thanks Daedalus, looking forward to it.

 

Here is a good example of my postings concerning understanding the concept of time. I was just researching modern concepts of time in physics and came across this book called From Eternity to Here: The Quest for the Ultimate Theory of Time. Although I haven't read it, there is a huge difference of "opinion"/ theory here. Mine is that time is one of the simplest concepts to explain and understand, similar to space -- that time can simply and fully be explained by a one sentence definition and no more. On the other hand the author's opinion, concerning modern physics, is that an entire book is needed just to explain the possible theoretical developments needed for such an "Ultimate Theory of Time" to someday be created. I would call the difference of these two opinions an abyss. This is what I consider a good example of the problems in modern physics.

 

Thank you for the reference Pantheory. I will try to make time ; ) to explore all of the many theories that discuss time. I have considered what you have said about time in regards to an interval of change. I must conclude that you and I are debating two sides of the same coin. I have referred to motion in many of my posts as a mathematical description of a rate of change. Mathematics is purely abstract in that any motion is described as a rate of change. I relate this concept to the reality or concreteness of our universe as pure physical motion. With that being said, one can argue that both of our concepts are balanced through opposition and are one and the same no different than up is to down. So I will concede and define that time is as you specify, "an interval of motion / change" as these are both one and the same. One being mathematically abstract, the other realized in a concrete / physical sense.

 

P.S.

 

I was browsing the forums when I seen you posted. You can expect my e-mail tomorrow when I wake up as I will not post the remainder of the mathematics for another day or so to allow everyone to view and interpret the math. This will give you some time to review what I have derived so that you can prepare your responses to what I will post and how you might be able to use it in your theories.

Edited September 17, 2011 by Daedalus

[pantheory]

 

..... I will not post the remainder of the mathematics for another day or so to allow everyone to view and interpret the math. This will give you some time to review what I have derived so that you can prepare your responses to what I will post and how you might be able to use it in your theories.

 

Thanks again, looking forward to it.

I also concur that we are in agreement concerning our understandings of time.

Edited September 17, 2011 by pantheory

[Daedalus]

We discussed the equations for describing the angles and path in the previous post, but we really didn't define the path for the photon or parameterize the equations. So we will start by discussing how we will define this path, and the mathematics needed to parameterize the equations.

 

 

The above image allows us to visualize the problem. We can see that I have positioned the arc of length [math]L[/math] so that it is centered along the horizontal W axis. This bisects the arc into two parts of equal length. We do this so that we can simplify the maximum and minimum angles (shown in blue) that the photon traverses for both, the up and down paths. Using the equation from the previous post that defines the angles, we can calculate the maximum and minimum angles as follows:

 

[math]\theta_{max}=\frac{L/2}{r}=\frac{L}{2\, r}[/math]

 

[math]\theta_{min}=-\frac{L/2}{r}=-\frac{L}{2\, r}[/math]

 

We know that the photon will traverse the length of the arc at the speed of light. The amount of time it takes for the photon to traverse this distance is [math]\Delta \tau[/math] as we have previously discussed. But instead of defining the equation for the change in angle as [math]c\, \Delta t / r[/math], we need to parameterize the equation such that the photon starts its journey when the parameter [math]u=0[/math] and has traversed the length of the arc by the time [math]u=1[/math]. We still preserve the speed of light and [math]\Delta \tau[/math] in that the parameter, [math]u[/math], only operates within the range of zero to one. This allows us to define the equations for the change in angles of the path of the photon as follows:

 

[math]\Delta \theta_{u}=\frac{u\, L}{r}[/math] where [math]\left \{u \in \mathbb{R} \ | \ 0 \le u \le 1 \right \}[/math]

 

[math]\Delta \theta_{up}=\theta_{min}+\Delta \theta_{u}=-\frac{L}{2\, r}+\frac{u\, L}{r}=\frac{L\, \left (2\, u-1\right )}{2\, r}[/math]

 

[math]\Delta \theta_{down}=\theta_{max}-\Delta \theta_{u}=\frac{L}{2\, r}-\frac{u\, L}{r}=\frac{L\, \left (1-2\, u\right )}{2\, r}[/math]

 

As expected, [math]\Delta \theta_{down}=-\Delta \theta_{up}[/math]. We can now substitute [math]c\, \Delta \tau[/math] in place of [math]L[/math] and arrive at the parameterized equation which describes the angles that the photon traverses for both paths. We will also redefine this angle as:

 

[math]\Delta \theta_{\gamma}=\frac{c\, \Delta \tau\, \left (2\, u-1\right )}{2\, r}[/math]

 

Because the photon starts at the coordinate, {w, 0, y}, we can now define the equation which describes the path of the photon. Since we only consider the horizontal W axis and the vertical Y axis in our initial coordinate, the equation is simply that which defines a circle:

 

[math]w'=r\, \cos \left (\pm \Delta \theta_{\gamma}\right )=r\, \cos \left (\pm \frac{c\, \Delta \tau\, \left (2\, u-1\right )}{2\, r}\right )[/math]

[math]y'=r\, \sin \left (\pm \Delta \theta_{\gamma}\right )=r\, \sin \left (\pm \frac{c\, \Delta \tau\, \left (2\, u-1\right )}{2\, r}\right )[/math]

 

Before we can substitute this result into the equation that we derived in the previous post for the path that traverses an expanding sphere, we must also parameterize all of the equations we have previously defined. Luckily, this step is easy to accomplish as we only have to replace [math]\Delta t[/math] in these equations with [math]\left (u+n\right )[/math]. The variable, [math]n[/math], defines integer multiples of [math]\Delta \tau[/math]. This is because we need to be able to advance the up and down paths of the photon by the amount of time it takes for the photon to traverse the length of the arc. We will also be using the modified [math]\Delta \tau[/math] versions of the equations for motion where [math]\mathit{Sl}_{0}[/math], [math]\mathit{Vl}_{S}[/math] and [math]\mathit{Al}_{S}[/math] notation will be used to represent the modified constants. We finally have all of the pieces to the puzzle (not including the mechanical clock)!!!

 

The equations which describe motion in the form of uniform acceleration ([math]\Delta W[/math] defines the radius of the sphere):

[math]\Delta W=c\, \Delta \tau\left (\mathit{Wl}_0+\mathit{Vl}_w \, \left (u+n\right )+\frac{1}{2}\mathit{Al}_w \, \left (u+n\right )^2\right )[/math]

 

[math]\Delta X=c\, \Delta \tau\left (\mathit{Xl}_0+\mathit{Vl}_x \, \left (u+n\right )+\frac{1}{2}\mathit{Al}_x \, \left (u+n\right )^2\right )[/math]

 

[math]\Delta Y=c\, \Delta \tau\left (\mathit{Yl}_0+\mathit{Vl}_y \, \left (u+n\right )+\frac{1}{2}\mathit{Al}_y \, \left (u+n\right )^2\right )[/math]

 

The equations which describe the angles as the clock traverses across the expanding sphere (take note that [math]c\, \Delta \tau[/math] is canceled out as a result of the division):

 

[math]\Delta \theta_x=\frac{\Delta X}{\Delta W}=\left (\frac{2}{2}\right ) \frac{\mathit{Xl}_0+\mathit{Vl}_x \, \left (u+n\right )+\frac{1}{2}\mathit{Al}_x \, \left (u+n\right )^2}{\mathit{Wl}_0+\mathit{Vl}_w \, \left (u+n\right )+\frac{1}{2}\mathit{Al}_w \, \left (u+n\right )^2}=\frac{2\, \mathit{Xl}_0+2\, \mathit{Vl}_x \, \left (u+n\right )+\mathit{Al}_x \, \left (u+n\right )^2}{2\, \mathit{Wl}_0+2\, \mathit{Vl}_w \, \left (u+n\right )+\mathit{Al}_w \, \left (u+n\right )^2}[/math]

 

[math]\Delta \theta_y=\frac{\Delta Y}{\Delta W}=\left (\frac{2}{2}\right ) \frac{\mathit{Yl}_0+\mathit{Vl}_y \, \left (u+n\right )+\frac{1}{2}\mathit{Al}_y \, \left (u+n\right )^2}{\mathit{Wl}_0+\mathit{Vl}_w \, \left (u+n\right )+\frac{1}{2}\mathit{Al}_w \, \left (u+n\right )^2}=\frac{2\, \mathit{Yl}_0+2\, \mathit{Vl}_y \, \left (u+n\right )+\mathit{Al}_y \, \left (u+n\right )^2}{2\, \mathit{Wl}_0+2\, \mathit{Vl}_w \, \left (u+n\right )+\mathit{Al}_w \, \left (u+n\right )^2}[/math]

 

The equations which describe the coordinates for the untransformed path of the photon (Describes what we perceive to be a stationary clock that is only moving through the temporal dimension):

The photon path angles:

 

[math]\Delta \theta_{\gamma}=\frac{c\, \Delta \tau\, \left (2\, u-1\right )}{2\, \Delta W}=\frac{2\, u-1}{2\, \mathit{Wl}_0+2\, \mathit{Vl}_w \, \left (u+n\right )+\mathit{Al}_w \, \left (u+n\right )^2}[/math]

 

The untransformed path of the photon (+ angles describe the up path and - angles describe the down path):

 

[math]W_{\gamma}=\Delta W \cos \left (\pm \Delta \theta_{\gamma} \right )[/math]

[math]Y_{\gamma}=\Delta W \sin \left (\pm \Delta \theta_{\gamma} \right )[/math]

 

or

[math]W_{\gamma}=c\, \Delta \tau\left (\mathit{Wl}_0+\mathit{Vl}_w \, \left (u+n\right )+\frac{1}{2}\mathit{Al}_w \, \left (u+n\right )^2\right ) \cos \left (\pm \frac{2\, u-1}{2\, \mathit{Wl}_0+2\, \mathit{Vl}_w \, \left (u+n\right )+\mathit{Al}_w \, \left (u+n\right )^2}\right )[/math]

 

[math]Y_{\gamma}=c\, \Delta \tau\left (\mathit{Wl}_0+\mathit{Vl}_w \, \left (u+n\right )+\frac{1}{2}\mathit{Al}_w \, \left (u+n\right )^2\right ) \sin \left (\pm \frac{2\, u-1}{2\, \mathit{Wl}_0+2\, \mathit{Vl}_w \, \left (u+n\right )+\mathit{Al}_w \, \left (u+n\right )^2}\right )[/math]

 

The equations which describe the coordinates for the transformed path of the photon:

The full non-simplified version:

 

[math]w'=\left (W_{\gamma}\right ) \, \cos(\Delta \theta_{x}) \, \cos(\Delta \theta_{y})\, -\, \left (Y_{\gamma}\right ) \, \cos(\Delta \theta_{x}) \, \sin(\Delta \theta_{y}) \, \cos(\Delta \theta_{w})\, +\, \left (Y_{\gamma}\right ) \, \sin(\Delta \theta_{x}) \, \sin(\Delta \theta_{w})[/math]

[math]x'=\left (W_{\gamma}\right ) \, \sin(\Delta \theta_{x}) \, \cos(\Delta \theta_{y})\, -\, \left (Y_{\gamma}\right ) \, \sin(\Delta \theta_{x}) \, \sin(\Delta \theta_{y}) \, \cos(\Delta \theta_{w})\, -\, \left (Y_{\gamma}\right ) \, \cos(\Delta \theta_{x}) \, \sin(\Delta \theta_{w})[/math]

[math]y'=\left (Y_{\gamma}\right ) \, \cos(\Delta \theta_{y}) \, \cos(\Delta \theta_{w})\, +\, \left (W_{\gamma}\right ) \, \sin(\Delta \theta_{y})[/math]

 

The simplified version with [math]\Delta \theta_{w}=0[/math]:

 

[math]w'=\left (\Delta W\right ) \cos\left (\Delta \theta_{x}\right)\, \cos\left (\Delta \theta_{y} \pm \Delta \theta_{\gamma}\right)[/math]

[math]x'=\left (\Delta W\right ) \sin\left (\Delta \theta_{x}\right)\, \cos\left (\Delta \theta_{y} \pm \Delta \theta_{\gamma}\right)[/math]

[math]y'=\left (\Delta W\right ) \sin\left (\Delta \theta_{y} \pm \Delta \theta_{\gamma}\right)[/math]

 

We can also specify initial angles from which to start the path. This modification is very easy to do as shown below:

 

[math]w'=\left (\Delta W\right ) \cos\left (\Delta \theta_{x}+\phi_{x}\right)\, \cos\left (\Delta \theta_{y} \pm \Delta \theta_{\gamma}+\phi_{y}\right)[/math]

[math]x'=\left (\Delta W\right ) \sin\left (\Delta \theta_{x}+\phi_{x}\right)\, \cos\left (\Delta \theta_{y} \pm \Delta \theta_{\gamma}+\phi_{y}\right)[/math]

[math]y'=\left (\Delta W\right ) \sin\left (\Delta \theta_{y} \pm \Delta \theta_{\gamma}+\phi_{y}\right)[/math]

 

Images produced by graphing the equation:

 

The rings that are seen in the images below are only used for reference points. The paths of the photons are the triangle like structures. The arcs that can be seen along with the photon paths are all of length [math]L[/math]. We can see from the images that this mathematical framework models space in such a way that it will eventually appear flat as time approaches infinity.

 

This image shows two light clocks. One is moving outward with positive time and the other is moving outward with negative time (see page 2 posting #34 for an explanation of positive / negative time):

 

This image shows one clock moving outward at [math]90^\circ[/math] from the other:

 

 

This image shows acceleration in the clock that is moving through positive time (please note that I used extreme values for the acceleration so that I could produce this very cool image):

 

 

This image shows a multitude of clock paths:

 

 

This image is very interesting indeed as we get our first glimpse of the spiraling singularity:

 

 

Next, we will use the completed equation and derive time dilation. After we have derived the time dilation equation, I will describe the singularity and post detailed images that I have graphed of it. Good night everyone. I hope that you all have enjoyed reading this thread, reviewing the mathematics, and looking at the images of the graphs : )

Edited December 17, 2011 by Daedalus

[pantheory]

Wow,

 

Lots of cool equations and graphics. How do these equations and graphs relate to Temporal Uniformity?

 

best regards

[Daedalus]

Wow,

 

Lots of cool equations and graphics. How do these equations and graphs relate to Temporal Uniformity?

 

best regards

 

I'm glad you asked Pantheory as these equations and graphs are the result of five months of work. I developed my idea for Temporal Uniformity April 24th, 2011 on Easter Sunday. I opened Photoshop and began to draw the image that I had imagined would explain the phenomena of time and made a hypothesis that dark matter could be temporally displaced matter that caused gravitional effects in our view of the universe.

 

Temporally Uniformity states that regardless of a bodies motion, that body will remain temporally aligned with all other bodies. The graphs from the previous post demonstrates this concept (especially if you look at the image that demonstrates acceleration). The graphs also demonstrate that the space-time vector for all observable bodies, with the tail at the center of the expansion and the head located at the body's current position, is equal in magnitude as demonstrated by graphing two or more clocks.

 

The mathematics and graphs also show that, from our view of the universe, all three observable dimensions of space would appear to be expanding. The image showing one clock that is moving [math]90^\circ[/math] from the other illustrates this concept. A better example of this expanding space would be to plot two clocks that have a smaller angle between them.

 

The image below was created before I had derived any of the equations for the theory.

 

 

You can see that my equations and graphs fits pretty close to what I was trying to achieve by comparing the above image to the image I have graphed by combining multiple light clocks as seen below. Except, I have only considered the constancy of the speed of light.

 

 

The above image is soley produced by graphing multiple photon paths, that are radiating outward with the expansion, and by removing the restriction on the parameter, [math]u[/math], so that the path is allowed to continue beyond the boundaries of the reflective plates in the light clock. The rings are reference points as stated previously and can also be thought of as temporally displaced views of the universe. We can also see that the equations produce a spiraling singularity at the center of the expansion. I have rendered detailed graphs of this singularity and we will discuss this topic once we have concluded deriving the time dilation equations.

 

The mathematics also allows for a time before the expansion and, by working backwards from time zero, I produced the following image showing the path of photons flowing inward into the spiraling singularity.

 

 

I never intended for the mathematics to produce the singularity. My goal was to derive time dilation equations for a path across an expanding sphere so that I could make a post about how we can derive such things for spaces other than flat Euclidean space. You can only imagine my excitement when I was able to produce the mathematics and graphs, based on my original image, that seem to support my ideas.

 

I hope this explanation clarifies as to why I have developed this mathematical model and how it fits within the framework of Temporal Uniformity.

Edited December 17, 2011 by Daedalus

[pantheory]

Daedalus,

 

Again, your graphics are exceptional. Outside the framework of the Big Bang model and General Relativity, do you see how Temporal Uniformity might fit in?

[Daedalus]

Daedalus,

 

Again, your graphics are exceptional. Outside the framework of the Big Bang model and General Relativity, do you see how Temporal Uniformity might fit in?

 

I appreciate the comment Pantheory. As you can probably tell, I managed to pick up a few things in graphic design while developing class 2 casino gaming devices.

 

To answer your question, Temporal Uniformity by itself is currently just an extension to SR. I am researching QM to see if there is a common ground between my spiraling singularity in comparison to that of ring singularities. I found some interesting material by Don J. Stevens referring to Alexander Burinskii's model which describes electrons as naked ring singularites. Read up on what Don J. Stevens has to say about the electron as a ring singularity:

 

Electron - Ring Singularity

 

Granted, they may be correct or they could be wrong. But, I must consider all sources of information if I am going to find a possible link between the mathematics that I have currently produced with those of QM. To achieve this, I must consider other forces and physical constants within my mathematical model and explore many other theories in Physics.

Edited September 20, 2011 by Daedalus

[pantheory]

Pardon me. I should have said both Special and General Relativity, concerning my last question -- considering all the LT's involved in the math The General Relativity comment related to GR being the mathematical basis for the Big Bang model

 

Quantum Theory, I believe, is where all the complicated theories concerning time, stem from

 

regards

Edited September 20, 2011 by pantheory

[Daedalus]

Pardon me. I should have said both Special and General Relativity, concerning my last question -- considering all the LT's involved in the math The General Relativity comment related to GR being the mathematical basis for the Big Bang model

 

You are most definitely pardoned Pantheory : ) I made that reference to SR because I have not yet considered gravity except in stating a hypothesis about temporally displaced mass-energy that could explain dark matter and galactic rotation curves. But as you and I have discussed, Temporal Uniformity does have something to say concerning the Big Bang. The next step is to either tackle gravity / GR or attempt to relate Temporal Uniformity to QM. I personally feel that integration with QM is the next logical step because I believe the solution regarding gravity will emerge as a result. However, I do not have illusions of grandeur in that I will achieve this step or that the work I have done thus far will be considered "Nobel Prize" material. But like many of us who pose theories, I do hope that the work I am doing will, in some way, prove to have merit.

 

It is on that note that I would like to extend an invitation to everyone who has the knowledge and is willing to be a part of developing this theory to join in on the fun : ) I would also like to thank AJB, Michel123456, and you Pantheory for all of the support you guys have given me as well as Klaynos and Swansont for debating the topic.

 

 

Quantum Theory, I believe, is where all the complicated theories concerning time, stem from

 

I absolutely agree. The main reason which sparked the idea of Temporal Uniformity was due to some of these theories regarding time and QM. My friends and I were watching "The Universe" the Saturday before Easter Sunday. It was the episode that discussed different theories of time. They were talking about the possibility of time travel when I looked over at my friend, Jim, and said "That mechanism for time travel it preposterous!". I don't quite remember which theory they were discussing, but I proceeded to explain to him why I felt that way. The next day, I began developing Temporal Uniformity and have been working on the theory in my spare time ever since.

Edited September 21, 2011 by Daedalus

[pantheory]

Daedalus,

 

I think that most of the preposterous proposals stemming from Quantum Theory involve the fact that QM does not recognize the existence of background field particles that make up the ZPF. If there were such particles which influence quantum behavior, I believe that by their discovery or recognition, many if not most of the preposterous aspects of Quantum Theory would slowly disappear. On the other hand such hypothetical proposals of background field particles are common and many, such as dark matter, Higg's particles, gravitons, quantum sand, field strings, etc. etc.

 

I believe there are Planck size dark matter particles that are so small that they rightfully could not be called matter (like photons), which I think will lead back to rational models of time which will relate to change/ motion, like a relatively simple definition

[michel123456]

I don't understand where is acceleration in your graphs. I only see uniform expansion, so I guess I am missing something.

[Daedalus]

Daedalus,

 

I think that most of the preposterous proposals stemming from Quantum Theory involve the fact that QM does not recognize the existence of background field particles that make up the ZPF. If there were such particles which influence quantum behavior, I believe that by their discovery or recognition, many if not most of the preposterous aspects of Quantum Theory would slowly disappear. On the other hand such hypothetical proposals of background field particles are common and many, such as dark matter, Higg's particles, gravitons, quantum sand, field strings, etc. etc.

 

I believe there are Planck size dark matter particles that are so small that they rightfully could not be called matter (like photons), which I think will lead back to rational models of time which will relate to change/ motion, like a relatively simple definition

 

That definitely gives me something to think about.

 

I don't understand where is acceleration in your graphs. I only see uniform expansion, so I guess I am missing something.

 

Hi Michel, the following image shows the clock that is moving outward with positive time, accelerating along what we perceive to be the X axis:

 

 

That is why I labeled this section, "On The Temporal Uniformity of Accelerating Bodies". We clearly see that all bodies will remain temporally aligned regardless of thier motion. The point I am trying to make is that time dilation is not some form of temporal displacement that allows us to disappear from our view of the universe and reappear at a point in the future. Time dilation only slows down or speeds up clocks. As stated before, all clocks work by some form of oscillation. The effect that we perceive to be a dilation of time, is nothing more than our atomic processes slowing down or speeding up which does effect our motions and aging process. That is how time dilation is viewed in Temporal Uniformity. I will post more images tonight that demonstrate accelerating bodies : )

 

Tommorow we will begin to discuss time dilation. We will only include velocity and not acceleration as defined in SR. After we have covered time dilation, I will post detailed images of the spiraling singularity and we can discuss its properties.

Edited December 17, 2011 by Daedalus

[Daedalus]

As promised Michel, here are some more images that show how bodies remain temporally aligned.

 

This image shows one clock moving away from the other at a constant velocity:

 

 

This image shows one clock accelerating away from the other:

 

 

This image shows both clocks accelerating along what we perceive to be the X axis:

 

 

This image shows both clocks having a ridiculously extreme acceleration:

 

Edited December 17, 2011 by Daedalus

[michel123456]

I don't understand the graph.

1. It looks like a cube seen in perspective.

2. The center (the origin) coincides with a corner, It is a bit confusing.

3. If there is expansion, how does it come that the clocks are represented as these blue stripes? I would expect to see the blue part like a triangle (a stripe getting wider & wider) radially. Or I understand nothing.