Expressing n dimensions as n-1

John R. LaubensteinIWPD Research Center

Naperville, Illinois630-428-9842

www.iwpd.org

2009 APS March MeetingPittsburgh, Pennsylvania

March 20, 2009

IWPD Scale Metrics (ISM) DOES NOT:

Claim to identify some past error or oversight that sets the world right

Suggest that past achievements should be discarded for some new vision of reality

IWPD Scale Metrics DOES:

Suggest an alternative description of space-time

Show that ISM is equivalent to 4-Vector space-time (at least in terms of velocity)

Modify gravitation so that it can be described using ISM

Show that ISM makes predictions consistent with observation

ISM quantitatively links Scale Metrics and 4-Vector space-time through a mathematical relationship

Scale Metrics and 4-Vectors are shown to be equivalent (at least for specific conditions)

Scale Metrics adds to the body of knowledge

Approach. We will conceptually develop ISM using a two-dimensional flat manifold

Why? Because in our world we understand both 3D and 2D Euclidean geometry

Verification. You can serve as the judge and jury over the decisions made by the “Flatlanders”

Result. If successful, a model of n dimensions as n-1 will result in describing 4-Vector space-time using only three dimensions

When pondering a description for space-time this individual decides to plot time as an abstract orthogonal dimension to the two dimensions of space known in the Flatlander world

This requires three pieces of information to identify an event

(x,y) coordinates for position and a (z) coordinate for time

A series of events are depicted as a Worldline

A point tangent to the Worldline defines the 3-Velocity, which is normalized to a value of 1

The observed (2D) velocity is depicted by the blue vector that lies in the plane of the observable dimensions

The orientation of the 3-Velocity vector can be determined from its angle ( ) relative to the 2D observable plane of the Flatlander world

To describe the observed velocity of an object during a specific event will require 4 pieces of information:

x,y: position coordinates z: time coordinate for the orientation of the 3-Velocity vector

If a Worldline is due to gravitation, the challenge becomes to accurately describe the curvature of space and spacetime to accurately depict the curve of the Worldline The simplest case (a uniform spherical non rotating mass

with no charge) requires the Schwarzschild solution

22

2222221

22 2

1sin2

1 dtrc

Gmcddrdr

rc

Gmds

When pondering a description for space-time

this individual decided to plot time as an abstract orthogonal dimension to the two known dimensions of space in the Flatlander world

This individual decides to account for time within the 2 observed dimensions by plotting time – not as a point – but as a segment representing the passage of time

This approach also requires three pieces of information to identify an event

(x,y) coordinates for position

A line segment plotted on the x-y plane to designate time

Three pieces of information are required to identify an event

(x,y) coordinates for position and a (z) coordinate for time

For an object at rest, its Worldline is

orthogonal to the x-y plane

For an object at rest, the (x,y) ordered pair defines a “point” at the center of the time segment

A series of events are depicted as a

Worldline

As viewed from above, the three points may be seen “plotted” on the 2D plane

A series of events are depicted as a

Worldline

A series of events are depicted by ever-increasing time lines

A series of events are depicted as a Worldline

A series of events are depicted as points embedded in time segments

The orientation of the point relative to the timeline is denoted as (X) and is equivalent to the value

The orientation of the 3-Velocity vector can be determined from its angle ( ) relative to the 2D observable plane of the Flatlander world

The position of the timeline segment can change relative to the (x,y) position coordinates

(X) = 0.5

The position of the timeline segment can change relative to the (x,y) position coordinates

(X) = 0.75

The position of the timeline segment can change relative to the (x,y) position coordinates

(X) = 1.0

The position of the timeline segment can change relative to the (x,y) position coordinates

(X) = 0.75

The position the timeline segment can change relative to the (x,y) coordinate

(X) = 0.5

(x,y) position coordinates

segment coordinate for time

X: orientation

(x,y) position coordinates

z coordinate for time

coordinate for the orientation of the worldline

Both ( ) and (X) represent orientations

They are related by the following expression:

11sin X

ANSWER:

X has allowable values ranging from 0.5 to 1

2vm

EX

t

(X) = 0.5 (X) = 1.0

2 + 1 dimensions in the Flatlander world can be expressed in 2 dimensions with no information lost

4-Vector Space-Time may be expressed within the 3 spatial dimensions we experience

So What? Who Cares? Where is the advantage of this?

When using ISM, time is not defined as orthogonal to the spatial dimensions

A time segment with a defined point is equivalent to the 4-Vector Worldline

The orientation of the point (X) is related to the velocity of an object just as the slope of the Worldline is related to velocity

Just as gravity influences the 4-Vector Worldline, gravity must also be shown to influence the value of X in ISM

Who c

How do you determine the directionality of the time segment?

Apply a factor of pi.

The resulting “ring” defines a fundamental entity dubbed as the “energime”

Time emerges from everywhere within the Initial Singularity

Time progresses as a quantized entity defining quantized space

The collective effort results in the creation of an overall flat Background Energime Field (BEF)

Flat Background Energime Field (BEF)

Flat Background Energime Field (BEF)

Perturbation due to local effects of a gravitating mass resulting in a Local Energime Field (LEF)

Gravitation is an interaction between a local gravitating mass and the total mass-energy of the universe

The more massive the gravitating entity, the stronger the gravitation effect

The less massive the gravitating entity, the weaker the gravitational effect

As time progresses, the initial singularity increases in size as the scaling metric changes.

Fundamental Unit Time

Fundamental Unit Length

Velocity is typically determined by the orthogonal relationship between 4-Velocity and the observed 3-Velocity

If you attempt to subtract the 3-Velocity from the 4-Velocity linearly, you will not get the correct answer

If you attempt to subtract the 3-Velocity from the 4-Velocity linearly, you will not get the correct answer

a

b ba

However, if you apply a scaling factor, you can achieve a linear relationship between 4- Velocity and 3-Velocity

However, if you apply a scaling factor, you can achieve a linear relationship between 4- Velocity and 3-Velocity

a

b

ba

ANSWER: The ISM Scaling Metric (M), relative to the Fundamental Unit Length (L), defines the magnitude of the Scaling Factor required to make a = b.

Fundamental Unit Time (T)

Fundamental Unit Length (L)

Scaling Factor = M/L

ISM Scaling Metric (M)

Fundamental Unit Time (T)

Fundamental Unit Length (L)

Scaling Factor = M/L

ISM Scaling Metric (M)

L

MLEFBEFv

If a Worldline is due to gravitation, the challenge becomes to accurately describe the curvature of space and space-time to accurately depict the curve of the Worldline

The simplest case (a uniform spherical non rotating mass with no charge) requires the Schwarzschild solution

In the case of ISM, an object under the influence of gravitation must have a specific value of X

The value of X and therefore the geometry of ISM space-time is defined by:

11sin X 22

2222221

22 2

1sin2

1 dtrc

Gmcddrdr

rc

Gmds

All of the information in 4-Vector space-time can be captured in 3 spatial dimensions by incorporating:

a quantized time segment (ring) with an orientation value (X)

The relationship between time and (X) defines velocity ISM coordinates are consistent with a new formalism for gravitation ISM is supported by observational data

A quantum theory of gravity Physical explanation of the fine structure constant

A university that is 14.2 billion years old A new interpretation of objectivity and local causality

An accelerating rate of expansion Absolute definition of mass, distance and time

Inflationary epoch falling naturally out of expansion A link between gravitation and electrostatic force

A clear definition of the initial singularity A link between gravitation and strong nuclear force

A physical definition of space Defined relationship between energy and momentum

A physical definition of Cold Dark Matter Explanation of the effects of Special Relativity

A physical explanation of Dark Energy 4-Vectors expressed in a 3D ISM coordinate system