1. Expressing n dimensions as n-1
John R. Laubenstein
IWPD Research Center
Naperville, Illinois
2009 APS March Meeting
Pittsburgh, Pennsylvania
March 20, 2009

2. Presentation Goal 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

3. Presentation Goal 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

4. Adding to the Base of Knowlege
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

5. Flatlander 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

6. Flatlander 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

7. Flatlander
A series of events are depicted as a Worldline

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

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

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

11. Flatlander 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

12. Gravitation 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

13. Initial Alternative Scenario Scenario
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

14. Initial Alternative Scenario Scenario
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

15. Initial Alternative Scenario Scenario
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

16. Initial Alternative Scenario Scenario
As viewed from above, the
three points may be seen
“plotted” on the 2D plane
A series of events are depicted as a
Worldline

17. Initial Alternative Scenario Scenario
A series of events are
depicted by ever-increasing
time lines
A series of events are depicted as a
Worldline

18. Flatlander 3D vs. 2D
A series of events are depicted as a Worldline

19. Flatlander 3D vs. 2D

20. Flatlander 3D vs. 2D

21. Flatlander 3D vs. 2D
A series of events are depicted as points embedded
in time segments

22. Initial Alternative Scenario Scenario
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

23. The ISM Orientation (X)
The position of the timeline segment can change
relative to the (x,y) position coordinates
(X) = 0.5

24. The ISM Orientation (X)
The position of the timeline segment can change
relative to the (x,y) position coordinates
(X) = 0.75

25. The ISM Orientation (X)
The position of the timeline segment can change
relative to the (x,y) position coordinates
(X) = 1.0

26. The ISM Orientation (X)
The position of the timeline segment can change
relative to the (x,y) position coordinates
(X) = 0.75

27. The ISM Orientation (X)
The position the timeline segment can change relative to the (x,y) coordinate
(X) = 0.5

28. Initial Alternative Scenario Scenario
(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

29. What is the Relationship between θ and X ?
Both ( ) and (X) represent orientations
They are related by the following expression:

30. Does (X) Have a Physical Meaning?
ANSWER:
X has allowable values ranging from 0.5 to 1
(X) = 1.0
(X) = 0.5

31. n Dimensions as n-1 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?

32. Who c Gravitation 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

33. The Nature of ISM Gravitation
How do you determine the
directionality of the time segment?

34. The Nature of ISM Gravitation
Apply a factor of pi.
The resulting “ring” defines a fundamental entity
dubbed as the “energime”

35. The Nature of ISM Gravitation
Time emerges from everywhere within the
Initial Singularity

36. The Nature of ISM Gravitation
Time progresses as a quantized entity defining
quantized space

37. The Nature of ISM Gravitation

38. The Nature of ISM Gravitation

39. The Nature of ISM Gravitation

40. The Nature of ISM Gravitation
The collective effort results in the creation of an overall flat Background Energime Field (BEF)

41. The Nature of ISM Gravitation
Flat Background Energime Field (BEF)

42. The Nature of ISM Gravitation
Perturbation due to local effects of a gravitating mass resulting in a Local Energime Field (LEF)
Flat Background Energime Field (BEF)

43. The Nature of ISM Gravitation
Gravitation is an interaction between a local gravitating mass and the total mass-energy of the universe

44. The Nature of ISM Gravitation
The more massive the gravitating entity, the stronger the gravitation effect

45. The Nature of ISM Gravitation
The less massive the gravitating entity, the weaker the gravitational effect

46. The Nature of ISM Gravitation
As time progresses, the initial singularity increases in size as the scaling metric changes.

47. The Nature of ISM Gravitation

48. The Nature of ISM Gravitation

49. The Nature of ISM Gravitation

50. The Nature of ISM Gravitation
Fundamental Unit Time
Fundamental Unit Length

51. ISM Suggests a Linear Relationship between BEF and LEF
Velocity is typically determined by the
orthogonal relationship between 4-Velocity
and the observed 3-Velocity

52. ISM Suggests a Linear Relationship between BEF and LEF
If you attempt to subtract the 3-Velocity
from the 4-Velocity linearly, you will not get
the correct answer

53. ISM Suggests a Linear Relationship between BEF and LEF
If you attempt to subtract the 3-Velocity
from the 4-Velocity linearly, you will not get
the correct answer a b

54. ISM Suggests a Linear Relationship between BEF and LEF
However, if you apply a scaling factor, you
can achieve a linear relationship between 4-
Velocity and 3-Velocity

55. ISM Suggests a Linear Relationship between BEF and LEF
However, if you apply a scaling factor, you
can achieve a linear relationship between 4-
Velocity and 3-Velocity a b

56. ISM Suggests a Linear Relationship between BEF and LEF
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 Length (L)
Fundamental Unit Time (T)
ISM Scaling Metric (M)
Scaling Factor = M/L

57. ISM Suggests a Linear Relationship between BEF and LEF
Fundamental Unit Length (L)
Fundamental Unit Time (T)
ISM Scaling Metric (M)
Scaling Factor = M/L

58. Gravitation Gravitation 3+1 D 4 – 1 D
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:

59. Conclusions 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

60. Observational Support for ISM
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