William R. Olson

bolson@pimeson.com

The structure of the physical world has been the subject of many theories. The idea that space and time are not independent was presented in the special theory of relativity. The visible world is three-dimensional. The fact that it is changing with time may be the result of stepping in a higher dimensional physical space. There must be more than three dimensions since three dimensions are visible at each step in time. The visible world of a step in time is just the collection of all events that can emit a photon that an observer can detect in that step in time. This is the definition of a light cone. The physical world of an observer can be described as a collection of light cones for each step in time. The physical world that any one observer can detect will not be the complete physical world since in order to detect a light cone the observer’s path must cross the center of the light cone. Most of the physical world will not be observable. This does not mean that an observer cannot understand the structure of the physical world. It just means that no observer can see it all. The physical world that will be described in this paper is quantized in both space and time.

The theory of relativity deals with the measurement of space and time. It describes a way of combining the measurements of space and time into a single quantity called an interval. The interval between two events is the same for all observers. The fact that it is possible to construct a uniform lattice of identical intervals that covers all of space and time is the subject of this paper. The visible world is three-dimensional so it is difficult to visualize a higher dimensional space. One way to do this is to start with a three dimensional example. Use a two dimensional shape to describe a three dimensional space. The square is a two-dimensional shape and can be used to form a simple two-dimensional lattice.

This lattice can then be extended into a three dimensional lattice of cubes.

In two dimensions each square has four nearest neighbors corresponding to its four edges. Each point in the lattice is in common with four different squares. In a three dimensional lattice of cubes each square has more than four neighbors. If one considers the squares in the cubes above and below the plane as well as the squares in the plane then each square has twelve neighbors. However the squares in the plane are not physically as close to the original square as the ones in the cubes above or below the plane. The nearest neighbors would be the squares in the cubes above or below the two-dimensional plane. One can say that in three dimensions each square has eight nearest neighbors. If one considers nearest neighbor squares contained in one of the cubes then a new two dimensional lattice results. This is the lattice of squares in a single cube. Each point in the cube is in common with only three squares. There are still four nearest neighbors. The whole lattice consists of the six squares in a single cube. Another way to look at the three dimensional lattice of cubes is to add a new dimension called time. All squares above the plane are in the future and all squares below the plane are in the past. An object moves in time by moving through the three dimensional lattice in the up direction.

There is a similar way of describing the space-time of relativity in terms of a lattice of identical shapes. In the case of squares there were two different two-dimensional lattices. The lattice of a plane and the lattice of a cube. The most local lattice was that of a cube and could be extended to include the lattice of a plane by the fact that each square had four nearest neighbors in two different cubes. In relativity the most local region of space-time is a light cone. A light cone is a region of space-time that can be connected to a central event by a light like path. In terms of distance, the space like separation between events in the light cone and the central event is equal to the distance light would travel in the time like separation. The relativistic interval between the central event and all events in a light cone is zero. There are two different kinds of light cones that depend on whether the central event is in the past or in the future. The light cone is three-dimensional and can be filled with a lattice of identical three-dimensional shapes. In fact there are many possible shapes that can be used to construct the lattice. The simplest shape is the tetrahedron.

One can start by describing the lattice of tetrahedrons in a single light cone. Each tetrahedron has four nearest neighbors corresponding to the four faces of the tetrahedron. By constructing new tetrahedrons on each face of the old tetrahedrons one can form a lattice. The lattice is closed by the fact that exactly six tetrahedrons can be constructed with one edge in common, similar to constructing four squares in a plane with one point in common. This is not true in physical space since the angle between faces of a tetrahedron is about 70 degrees.

Similar to the lattice of squares each tetrahedron is in two different light cones. One light cone has the central event in the past and the other light cone has the central event in the future. Each tetrahedron has four nearest neighbors in the future light cone and four nearest neighbors in the past light cone. The lattice can be extended by constructing tetrahedrons in both light cones. The fact that each tetrahedron has eight nearest neighbors can be understood in terms of a fourth space dimension similar to the fact that squares have eight nearest neighbors in a three dimensional lattice of cubes. This lattice can be used to fill all of space and time. It is possible to find paths in the lattice that will move the starting tetrahedron in time without changing its position in space. Similarly paths can be found that will move the starting tetrahedron in space but not in time. The paths that move only in time or space are not independent of the starting tetrahedron.

For a more mathematical explanation of the lattice one can represent a tetrahedron by the coordinates of the four events in it and the event at the center of the light cone. The distance between any two events is the square root of the difference between the squares of the space like separation and the time like separation. The time like separation can be measured in the same units as space by defining it in terms of the distance light would travel in the time like separation. One can start with four events that have the same time coordinate but different space coordinates. One possible set of the four events and the central event are as follows.

T sqrt(3) sqrt(3) sqrt(3) sqrt(3) 0 |
X 1 1 -1 -1 0 |
Y 1 -1 1 -1 0 |
Z 1 -1 -1 1 0 |
event #1 event #2 event #3 event #4 central event |

The distance between any two events in this tetrahedron is the space like separation, since the time coordinates are the same for all events. This distance is 2*sqrt(2). The distance between any of the four events and the central event is zero since the space like separation is the same as the time like separation. The four nearest neighbors with the same central event are as follows.

T 2*sqrt(3) sqrt(3) sqrt(3) sqrt(3) 0 |
X -2 1 -1 -1 0 |
Y -2 -1 1 -1 0 |
Z -2 -1 -1 1 0 |
event #1 event #2 event #3 event #4 central event |
T sqrt(3) 2*sqrt(3) sqrt(3) sqrt(3) 0 |
X 1 -2 -1 -1 0 |
Y 1 2 1 -1 0 |
Z 1 2 -1 1 0 |
event #1 event #2 event #3 event #4 central event |

T sqrt(3) sqrt(3) 2*sqrt(3) sqrt(3) 0 |
X 1 1 2 -1 0 |
Y 1 -1 -2 -1 0 |
Z 1 -1 2 1 0 |
event #1 event #2 event #3 event #4 central event |
T sqrt(3) sqrt(3) sqrt(3) 2*sqrt(3) 0 |
X 1 1 -1 2 0 |
Y 1 -1 1 2 0 |
Z 1 -1 -1 -2 0 |
event #1 event #2 event #3 event #4 central event |

The original tetrahedron has two possible central events. The central event in the original tetrahedron has a time coordinate in the past. The second central event has the same space coordinates and a time coordinate in the future.

T sqrt(3) sqrt(3) sqrt(3) sqrt(3) 2*sqrt(3) |
X 1 1 -1 -1 0 |
Y 1 -1 1 -1 0 |
Z 1 -1 -1 1 0 |
event #1 event #2 event #3 event #4 central event |

There is a simple way to generate the nearest neighbor tetrahedrons by using matrices. The tetrahedron is represented by a four by five matrix of event coordinates. A new four by five matrix can be generated by multiplying by a five by five matrix on the left or a four by four matrix on the right. The multiplication on the right corresponds to a Lorentz transformation and is not independent of the tetrahedron. The five by five matrix on the left can be used to change only one of the events and does not depend on the original tetrahedron. The central event can also be changed by a five by five matrix. The five matrices are as follows.

(1)= | -1 0 0 0 0 |
1 1 0 0 0 |
1 0 1 0 0 |
1 0 0 1 0 |
-1 0 0 0 1 |
(2)= | 1 1 0 0 0 |
0 -1 0 0 0 |
0 1 1 0 0 |
0 1 0 1 0 |
0 -1 0 0 1 |
(3)= | 1 0 1 0 0 |
0 1 1 0 0 |
0 0 -1 0 0 |
0 0 1 1 0 |
0 0 -1 0 1 |
(4)= | 1 0 0 1 0 |
0 1 0 1 0 |
0 0 1 1 0 |
0 0 0 -1 0 |
0 0 0 -1 1 |
(5)= | 1 0 0 0 .5 |
0 1 0 0 .5 |
0 0 1 0 .5 |
0 0 0 1 .5 |
0 0 0 0 -1 |

A second multiplication by any of the five matrices results in the original tetrahedron. The first four matrices can be used to generate the lattice of tetrahedrons in a single light cone, since they do not change the central event. The fifth matrix changes the central event from one of the light cones into the other. The matrices are independent of the original tetrahedron and can be used to study the structure of the lattice. In the lattice each tetrahedron has eight nearest neighbors. Four that are in the future light cone and four in the past. If one assumes that the eight neighbors are the result of a fourth space dimension, then time could be the result of motion in this space. The lattice of tetrahedrons could represent a four dimensional physical space. The structure of this space can be studied in terms of paths through the lattice. The paths (121) and (212) both generate the same tetrahedron. This is because exactly six tetrahedrons can be constructed with a single edge in common. The paths of the form (nmnmnm) where n and m are any two of the first four matrices result in the identity matrix and generate six tetrahedrons with one edge in common. Some sequences of special interest result in translations of the original tetrahedron in one of the six rectangular coordinate directions. The sequence (12153431215343) transforms the original tetrahedron into the following tetrahedron.

T sqrt(3) sqrt(3) sqrt(3) sqrt(3) 0 |
X 13 13 11 11 12 |
Y 1 -1 1 -1 0 |
Z 1 -1 -1 1 0 |
event #1 event #2 event #3 event #4 central event |

This is the original tetrahedron translated 12 units in the X direction. The opposite sequence (34351213435121) results in the original tetrahedron translated -12 units in the X direction. A rectanglular coordinate system can be constructed in the tetrahedral lattice by using the following set of six sequences.

+X= +Y= +Z= |
(12153431215343) (13152421315242) (14152321415232) |
-X= -Y= -Z= |
(34351213435121) (24251312425131) (14152321415232) |

Each of these sequences move the original tetrahedron 12 units in space with no change in time. The sequence (5343512151215343) and all permutations of the first four matrices, transform the original tetrahedron into the following tetrahedron.

T 5*sqrt(3) 5*sqrt(3) 5*sqrt(3) 5*sqrt(3) 5*sqrt(3) |
X 1 1 -1 -1 0 |
Y 1 -1 1 -1 0 |
Z 1 -1 -1 1 0 |
event #1 event #2 event #3 event #4 central event |

This is the original tetrahedron translated 4*sqrt(3) units in time. The original tetrahedron can be translated -4*sqrt(3) units in time by moving the 5 from the left to the right end of the sequence. By adding these two time sequences to the six space sequences one can construct a rectangular coordinate system of space-time with two different types of measuring rods. Both rods are constructed from the tetrahedral lattice. The sequences that move the original tetrahedron in the plus time direction can be used as a simple clock. If the starting tetrahedron is not equilateral in space, then this sequence will move the tetrahedron in both space and time. As an example, if the starting tetrahedron is the sequence (343) times the original tetrahedron then the two tetrahedrons are as follows.

Path = (343)

T sqrt(3) sqrt(3) 3*sqrt(3) 3*sqrt(3) 0 |
X 1 1 5 5 0 |
Y 1 -1 1 -1 0 |
Z 1 -1 -1 1 0 |
event #1 event #2 event #3 event #4 central event |

path = (5343512151215343)(343)

T 9*sqrt(3) 9*sqrt(3) 11*sqrt(3) 11*sqrt(3) 8*sqrt(3) |
X 13 13 17 17 12 |
Y 1 -1 1 -1 0 |
Z 1 -1 -1 1 0 |
event #1 event #2 event #3 event #4 central event |

The two tetrahedrons are the same shape translated 12 units in the +X direction and 8*sqrt(3) units in the +T direction. This corresponds to a velocity of sqrt(3)/2 times the speed of light in the +X direction. In effect a physical object located in the tetrahedron would be moving at this velocity as it traveled through time. The twin paradox can be understood using this lattice by using the (343) sequence to accelerate one of the twins to a velocity near the speed of light. After traveling N steps through time the twin reverses direction and returns to the path of the twin that did not accelerate. The sequence required to reverse the twins direction is the path (121343). A final deceleration of (121) is required to stop the twin that moved. The complete path for the twin that moved is as follows.

Path =
(121)(5343512151215343)^{N}(121343)(5343512151215343)^{N}(343)

T (16N+1)*sqrt(3) (16N+1)*sqrt(3) (16N+1)*sqrt(3) (16N+1)*sqrt(3) 16N*sqrt(3) |
X 1 1 -1 -1 0 |
Y 1 -1 1 -1 0 |
Z 1 -1 -1 1 0 |
event #1 event #2 event #3 event #4 central event |

This path requires 2N steps in time. The twin that did not accelerate needs
4N steps in time to arrive at the same place in the lattice. His path is
(5343512151215343)^{4N}. The ratio predicted by the theory of relativity
is sqrt(1-(v/c)^{2}) which is 1/2 for a path with a v/c ratio of
sqrt(3)/2. The twin that did not accelerate requires twice as many steps in the
lattice to arrive at the same place as the twin that accelerated. There is no
direct refference to time, only the total number of steps required to arrive at
the same place in the lattice. The steps required to accelerate can be neglected
if N is made large.

The tetrahedron that resulted from the (343) path has a different shape than
the original tetrahedron. The events in the z direction are twice as far apart
in the new tetrahedron. This means that the volume of this tetrahedron is twice
as great. The energy contained in a physical object that is moving is
mc^{2}/sqrt(1-(v/c)^{2}). This means that a physical object
located in the (343) tetrahedron would contain twice the energy of a physical
object located in the original tetrahedron. In other words the energy and volume
ratios are the same. This relationship is true for all tetrahedrons. If a
physical object were represented by a group of tetrahedrons it would be logical
to assume that the mass of this object would be proportional to the minimum
volume this object could be transformed into using Lorentz transformations.
Consider the group of tetrahedrons consisting of the original tetrahedron and
its four nearest neighbors. Since this object is symmetrical around the original
tetrahedron its volume cannot be reduced by any transformation. The minimum
volume of this object is just the sum of the volumes of its five tetrahedrons.
The volumes of the four nearest neighbors are all the same. The ratio between
their volume and the original tetrahedron is just the ratio of the distances
from the center of a common triangle to one of the new events to that of the old
event. The ratio between these two heights is 5/4. The total volume of these
five tetrahedrons is six times the volume of the original tetrahedron. If the
object is not symmetrical around the original tetrahedron then the minimum
volume must be calculated in a different way. The volume of a tetrahedron is
proportional to the sum of the time components of the four events in the
tetrahedron if the central event is at the origin. The central event can be
moved to the origin by subtracting it from all events in the tetrahedron. The
minimum volume is the proportional to the length of the sum of the four events
in the tetrahedron. The minimum volume of a group of tetrahedrons is
proportional to the length of the sum of the sums of the four events in all the
tetrahedrons. The maticies used to define the group of tetrahedrons can be added
before multiplying by a central tetrahedron. The rows of the resulting matrix
can also be added before the multiplication. The central event can be subtracted
from the other four events by adding the first four rows and subtracting four
times the fifth row. If these sums of columns are multiplied by the original
tetrahedron and center the resulting four vector is

sqrt(3)(s1+s2+s3+s4) (s1+s2-s3-s4) (s1-s2+s3-s4) (s1-s2-s3+s4)

The square of the length of this vector is

3*(s1+s2+s3+s4)^{2}-(s1+s2-s3-s4)^{2}-(s1-s2+s3-s4)^{2}-(s1-s2-s3+s4)^{2}=

8*(s1*s2+s1*s3+s1*s4+s2*s3+s2*s4+s3*s4)

This is proportional to the minimum volume of a group of tetrahedrons. Since the sum of each of the columns of the identity matrix is one then the ratio between the volume of a group a tetrahedrons and a single tetrahedron is

sqrt((s1*s2+s1*s3+s1*s4+s2*s3+s2*s4+s3*s4)/6)

The five matrices for the shape described above are the identity matrix and the four matrices used to form its four nearest neighbors. The sum of all five matrices is

3 1 1 1 0 |
1 3 1 1 0 |
1 1 3 1 0 |
1 1 1 3 0 |
-1 -1 -1 -1 5 |

The sum of columns is (6 6 6 6 -24)

The sum of products is 6*36 = 216

The
volume ratio is sqrt(216/6) = 6

This is the same result that was directly calculated.

One group of tetrahedrons that result in an interesting volume ratio is generated as follows. Start with a single tetrahedron. Include three of the four nearest tetrahedrons. From these three tetrahedrons step in the fourth direction to produce three more tetrahedrons. Repeat this pattern until the original tetrahedron is generated. This takes six steps and produces 74 different tetrahedrons. The matrices in this shape are as follows.

I |
(1) (2) (3) |
(41) (42) (43) |
(141) (241) (341) (142) (242) (342) (143) (243) (343) |
(4141) (4241) (4341) (4142) (4242) (4342) (4143) (4243) (4343) |
(14141)* (24141) (34141) (14241) (24241) (34241) (14341) (24341) (34341) (14142) (24142) (34142) (14242) (24242)* (34242) (14342) (24342) (34342) (14143) (24143) (34143) (14243) (34243) (34243) (14343) (24343) (34343)* |
(414141)** (424141) (434141) (414241) (424241) (434241) (414341) (424341) (434341) (414142) (424142) (434142) (414242) (424242)** (434242) (414342) (424342) (434342) (414143) (424143) (434143) (414243) (434243) (434243) (414343) (424343) (434343)** |

* all these matrices are the same as (4) so only one is counted

** all
these matrices are the identity matrix so none are counted

The sum of all the matrices used to generate this shape is

64 68 68 100 0 |
68 64 68 100 0 |
68 68 64 100 0 |
44 44 44 66 0 |
-170 -170 -170 -292 74 |

The sum of columns is (300 300 300 198 –1098)

The sum of products is
3*90000+3*59400 = 448200

The volume ratio is sqrt(448200/6) = 273.313

This volume ratio is interesting because it is close to the mass ratio of the charged pi meson to the electron (273.132). Since the electron is the smallest particle known to have a mass that is not zero one might associate it with a single tetrahedron. A projection of this shape onto a plane follows.

Another interesting group is similar in shape but centered on six tetrahedrons instead of one. The 72 matrices used to form this group are as follows.

I |
(1) (2) (3) (4) |
(21) (41) (12) (42) (43) (14) (24) (34) |
(121) (321) (421) (141) (241) (341) (312) (412) (142) (242) (342) (343) |
(4121) (4321) (1421) (2421) (3421) (2141) (3141) (1241) (3241) (4312) (1412) (2412) (3412) (2142) (3142) (1242) (3242) (1343) (2343) |
(14121) (24121) (34121) (34321) (12141) (43141) (43241) (34312) (12142) (43142) (43242) (41343) (42343) |
(214121) (314121) (124121) (324121) (134321) (234321) (134312) (234312) |
(1214121) (4314121) (4324121) (4134321) (4234321) (4134312) (4234312) |

The sum of all the matrices used to generate this shape is

0 0 0 0 0 |
0 0 0 0 0 |
189 189 207 261 0 |
108 108 117 162 0 |
-225 -225 -252 -351 72 |

The sum of columns is (0 0 846 495 –1341)

The sum of products is 846*495 =
418770

The volume ratio is sqrt(418770/6) = 264.187

This volume ratio is close to the mass ratio of the uncharged pi meson to the electron (264.143). A projection of this shape onto a plane follows.