 ### Unified Theory Foundations - The ST system of units

© Engineer Xavier Borg - Blaze Labs Research

Standard units to Spacetime conversion table
Leading the way to unification Created:12/2/05

Abstract

This paper shows that all measurable quantities we learn in physics can be represented as nothing more than a number of spatial dimensions differentiated by a number of temporal dimensions or vice versa. To convert between such space-time system of units and the conventional SI system, one simply multiplies the ST numerical values by dimensionless constants in order to convert between the natural space-time units and the 'historical' SI units. Once the ST system of units presented here is applied to any set of physics parameters, one is then able to derive all laws and equations without reference to the original theory which presented said relationship. In other words, all known principles and numerical constants which took hundreds of years to be discovered, like Ohm's Law, energy mass equivalence, Newton's Laws, etc.. would simply follow naturally from the spatial and temporal dimensions themselves, and can be derived without any reference to standard theoretical background. Any relation between physical parameters one might think of, can be derived. Included is a step by step worked example showing how to derive any free space constant and quantum constant.

Dimensions and dimensional analysis

One of the most powerful mathematical tools in science is dimensional analysis. Dimensional analysis is often applied in different scientific fields to simplify a problem by reducing the number of variables to the smallest number of "essential" parameters. Systems which share these parameters are called similar and do not have to be studied separately. Most often then not, two apparently different systems are shown to obey the same laws and one of them can be considered to be analogous to the other.

Unfortunately, the term 'dimension', has two completely different meanings, both of which are going to be used in this paper, so the reader should be aware of both meanings in order to apply the correct meaning of the word according to the context in which it is being used. In mathematics the 'dimension' of a space is roughly defined as the mimimum number of coordinates needed to specify every point within it. For example the square has two dimensions since two coordinates, say x and y, can be used to specify any point within it. A cube has three dimensions since three coordinates, say x,y, and z, are enough to specify any point in space within it. In engineering and physics terminology, the term 'dimension' relates to the nature of a measurable quantity. In general, physical measurements that must be expressed in units of measurement, and quantities obtained by such measurements are dimensionful. Quantities like ratios and multiplying factors, with no physical units assigned to them are dimensionless. An example of a dimension is length, expressed in units of length, the meters, and an example of a dimensionless unit is Pi. An engineering dimension can thus be a measure of a corresponding mathematical dimension, for example, the dimension of length is a measure of a collection of small linked lines of unit length, which have a single dimension, and the dimension of area is a measure of a collection or grid of squares, which have two dimensions. Similarly the mathematical dimension of volume is three. The prefix 'hyper-' is usually used to refer to the four (and higher) dimensional analogs of three-dimensional objects, e.g. hypercube, hypersphere...

The dimension of a physical quantity is the type of unit, or relation of units, needed to express it. For instance, the dimension of speed is distance/time and the dimension of a force is mass×distance/time². Conventionally, we know that in mechanics, every physical quantity can be expressed in terms of MLT dimensions, namely mass, length and time or alternatively in terms of MLF dimensions, namely mass, length and force. Depending on the problem, it may be advantageous to choose one or the other set of fundamental units. Every unit is a product of (possibly fractional) powers of the fundamental units, and the units form a group under multiplication.

In the most primitive form, dimensional analysis is used to check the correctness of algebraic derivations: in every physically meaningful expression, only quantities of the same dimension can be added or subtracted. The two sides of any equation must have the same dimensions. Furthermore, the arguments to exponential, trigonometric and logarithmic functions must be dimensionless numbers, which is often achieved by multiplying a certain physical quantity by a suitable constant of the inverse dimension.

The Buckingham p theorem is a key theorem in dimensional analysis. The theorem states that the functional dependence between a certain number n of variables can be reduced to the number of k independent dimensions occurring in those variables to give a set of p = n - k independent, dimensionless numbers. A dimensionless number is a quantity which describes a certain physical system and which is a pure number without any physical units. Such a number is typically defined as a product or ratio of quantities which DO have units, in such a way that all units cancel. A system of fundamental units (or sometimes fundamental dimensions) is such that every other unit can be generated from them. The kilogram, metre, second, ampere, Kelvin, mole and candela are supposed to be the seven fundamental units, termed SI base units; other units such as the newton, joule, and volt can all be derived from the SI base units and are therefore termed SI derived units. The choice of dimensionless units is not unique: Buckingham's theorem only provides a way of generating sets of dimensionless parameters, and will not choose the most 'physically meaningful'.

Why not choose SI ? We know that measurements are the backbone of science. A lot of work has been done to get the present self-coherent SI system of physical parameters, so why not choose SI as the foundation of a unifying theory? Because if the present science is not leading to unification, it means that something in its foundations is really wrong, and where else to start searching if not in its measuring units. The present SI system of units have been laid out over the past couple of centuries while the same knowledge that generated them in the first place have changed, making the SI system more or less a database of historical units. The major fault in the SI system can be easily seen in the relation diagram shown here, officially issued by BIPM (Bureau International des Poids et Mesures). We just added the 3 green arrows for the Kelvin unit. One would expect to see the seven base units totally isolated, with arrows pointing radially outwards towards derived units, instead, what we get is a totally different picture. Here we see that the seven SI base units are not even independent, but totally interdependent like a web, and so do not even strictly qualify as fundamental dimensions. If for instance, one had to change the definition of the Kg unit, we see that the fundamental units candela, mole, Amp and Kelvin would change as well. In the original diagram issued by BIPM, the Kelvin was the only isolated unit, but as I will describe shortly, it should be well interconnected as shown by the additional green arrows. So one cannot say there are seven fundamental SI units if these units are not independent of each other. The other big fault is the obvious redundancy of units. Although not very well known to all of us, at least two of the seven base units of the SI system are officially known to be redundant, namely the mole and the candela. These two units have been dragging along, ending up in the SI system for no reason other than historic ones.
The mole is merely a certain number of atoms or molecules, in the same sense that a dozen is a number; there is no need to designate this number as a unit.
The candela is an old photometric unit which can easily be derived from radiometric units (radiated power in Watts) by multiplying it by a function to describe the optical response of the human eye. The candela unit, together with its derived units as lux and foot-candelas serve no purpose that is not served equally well by watt per steradian and its derivatives.
Temperature, is yet another base unit that can be made redundant by adopting new definitions for its unit. Temperature could be measured in energy units because, according to the equipartition theorem, temperature is proportional to the energy per degree of freedom. It is also known that for a monatomic ideal gas the temperature is related to the translational motion or average speed of the atoms. The kinetic theory of gases uses statistical mechanics to relate this motion to the average kinetic energy of atoms and molecules in the system. For this case 11605 degrees Kelvin corresponds to an average kinetic energy of one electronvolt, equivalent to 1.602E-19Joules. Hence the Kelvin could also be defined as a derived unit, equivalent to 1.3806E-23Joule per degree of freedom, having the same dimensions of energy. Every temperature T has associated with it a characteristic amount of energy kT which is present in surroundings with that temperature at the quantum and molecular levels. At any given temperature the characteristic energy E is given by kT, where k (=1.3806E-23m2kg/sec2/K) is Boltzmann constant which is nothing more than a conversion factor between characteristic energy and temperature. Temperature can be seen as an alternative scale for measuring that characteristic energy. The Joule is equivalent to Kg/m2/sec2, so for the Kelvin unit we had to add the three green arrows pointing from Kg, metres and seconds which are the SI units defining energy. Furthermore, the definitions of the supplementary units, radian and steradian, are gratuitous. These definitions properly belong in the province of mathematics and there is no need to include them in a system of physical units. So what are we left with? How many dimensions can the SI system be reduced to? Looking again at the SI relations diagram, let us see which units DO NOT depend on others, that is which are those having only outgoing arrows and no incoming arrows. We see that in the SI system, only the units Seconds and Kg are independent. So, this means that the SI system can be reduced to no more than two dimensions, without loosing any of its physical significance of all the involved units. But we know that there are a lot of other combinations that can lead to the same number of fundamental dimensions, and that Kg and Seconds might not be the most physically meaningful independent dimensions. Strictly speaking only Space and Time are fundamental dimensions .... so what are the rest? Just patches in physics covering our ignorance, our inability to accept that point particles, with the fictitious Kg dimension, do not exist.

Present maintenance and transitions in the metric SI system of units Yes, hard to believe but true! Even though such transitions are hard to implement and the inertia of the SI system of units is huge, a few transitions towards better definitions are succesfully finding their way into the present SI metric system, so all is not lost. On such idea is the transition towards definitions based solely on the unit of time, taking the atomic clock second as reference and adopted exact values of certain constants. A notable step was taken in 1983 when the meter was defined by specifying that the standard speed of light be exactly 299792458 meters per second. In 1990 the BIPM established its voltage standard by specifying that Josephson's constant be exactly 483597.9 billion cycles per second per volt. Although this standard is already in use the official definition for voltage has not yet been changed to be consistent with the method of measurement, leaving the voltage and related quantities in a state of patchwork. In 1999 the CGPM called for a redefinition of the kilogram along the lines of the 1990 standards, and the following year two leading members, Mohr and Taylor, supplied the following proposed redefinition: The kilogram is the mass of a body at rest whose equivalent energy equals the energy of a collection of photons whose frequencies sum to 135 639 274×1042 Hz. Mohr and Taylor also suggested that the larger Planck's constant be made exactly equal to 2997924582/135 639 274 × 10-42 joule second. This value follows from their suggested definition of the kilogram.

Reference: Redefinition of the kilogram: a decision whose time has come by the Institute of Physics Publishing.

Introducing the ST system of units - The Rosetta stone of a new physics Here we will go a step further over the conventional SI dimensions and its patch work and will further reduce all scientific units into the real fundamental dimensions, namely Space (metres) and Time (seconds). As shown in this diagram, all SI units have been re-mapped onto the two fundamental units. We can therefore re-map the rest of the SI-derived units onto onto our ST system as well. At first it seemed an impossible mission, but as I went through all equations currently known, I found out that we've got a lot of different branches of science that are equivalent to each other. In this paper, space takes a slightly different meaning than the conventional three dimensional property of the universe in which matter can be located, and in fact is no longer restricted to three dimensions. One starts off with the dimensions of distance as the one dimensional unit of Space S, area becomes the 2 dimensional unit of space S2, volume becomes the 3 dimensional unit of space S3, speed is distance/time which becomes S/T. To move onward to define energy related units, I make use of the knowledge presented in the standing wave EM structure of matter, which enabled me to continue the conversion work on parameters in all the other fields. Surprisingly, once you have the ST units for mass, one is able to put up a full self-coherent table of ST dimension conversions for all known phyisical quantities, while eliminating all the non-sense webbing of the conventional SI system.

Such a table sets up a much stronger foundation for a new science, and helps you visualise how scientific parameters relate to each other through space and time. Quoting John Wheeler, "There is nothing in the world except empty curved space" and "Matter, charge, electromagnetism and other fields are only manifestations of the curvature of space." Once you grasp the whole concept, you will easily understand why RC is a time constant, why mass is a volume of energy, why f=1/2pi √(LC), and how all 'mechanical' & Newton's laws are related to electrical laws. Use this table to dimensionally check all your physics equations, and compile new ones yourself! Some will look really weird, but some will definitely make a lot of sense. Note that the SI system is not less weird, for example Resistance in SI is measured in m2Kg/sec3/Amp2 (we call this Ohms), and in that in most cases, the units will look simpler when converted to ST, in this case resistance will be measured in sec2/m3, though you cannot call this Ohms since you will require a dimensionless conversion factor. You will be able for the first time to clearly see that the ratio of Energy to mass is velocity squared (E/m = c2). Using the following table, it might also be an interesting exercise to relate different parameters through integrations or differentiations of their ST parameters. You may differentiate or integrate either with respect to S or T. This is basically the Rosetta stone translating between classical theory and the new unified physics I am hereby introducing.

Parameter Units SI units ST Dimensions
Distance S metres m S
Area A metres square m2 S2
Volume V metres cubed m3 S3
Time t seconds s T
Speed/ Velocity u metres/sec m/s ST-1
Acceleration a metres/sec2 m/s2 ST-2
Force/ Drag F Newtons Kgm/s2 TS-2
Damping coefficient c N sec/metre Kg/m/s T2S-3
Surface Tension g Newton per meter Kg/s2 TS-3
Spring constant/Stiffness k Newton per meter Kg/s2 TS-3
Compliance 1/k Meters per Newton s2/Kg S3T-1
Energy/ Work E Joules Kgm2/s2 TS-1
Power P Watts or J/sec m2 Kg/s3S-1
Density r kg/m3 kg/m3 T3 S-6
Mass m Kilogram Kg T3 S-3
Momentum p Kg metres/sec Kgm/s T2 S-2
Impulse J Newton Seconds Kg m/s T2 S-2
Moment m Newton metres m2 Kg/sec2 T S-1
Torque t Foot Pounds or Nm m2 Kg/sec2 T S-1
Angular Momentum L Kg m2/s Kg m2/s T2 S-1
Inertia I Kilogram m2 Kgm2 T3 S-1
Pressure/Stress P Pascal or N/m2 Kg/m/sec2 T S-4
Specific heat Capacity c J/kG/K m2/sec2/K S3 T-3
Specific Entropy J/kG/K m2/sec2/K S3 T-3
Resistance R Ohms m2Kg/sec3/Amp2 T2 S-3
Impedance Z Ohms m2Kg/sec3/Amp2 T2 S-3
Conductance S Siemens or Amp/Volts sec3 Amp2/Kg/m2 S3 T-2
Capacitance C Farads sec4Amp2/Kg/m2 S3 T-1
Inductance L Henry m2 Kg/sec2/Amp2 T3 S-3
Current I Amps Amp S T-1
Electric charge q Coulomb Amp sec S
Electric flux f Vm Volt metre T S-1
Magnetic charge qm Am Amp metre S2 T-1
Magnetic flux f Weber or Volts Sec m2 Kg/sec2/Amp T2 S-2
Magnetic flux density B Tesla /gauss/ Wb/m2 Kg/sec2/Amp T2 S-4
Magnetic reluctance R R Amp2 sec2/Kg/m2 S3 T-3
Electric flux density Jm2 Kg m4/sec2 ST
Electric field strength E N/C or V/m m Kg/sec3/Amp T S-3
Magnetic field strength H Oersted or Amp-turn/m Amp/m T-1
Poynting vector S Joule/s/m2 Kg/sec3 S-3
Frequency f Hertz sec-1 T-1
Wavelength l metres m S
Wavenumber v~ reciprocal centimetre m-1 S-1
Voltage EMF V Volts m2 Kg/sec3/Amp T S-2
Magnetic/Vector potential MMF MMF Kg/sec/Amp T2 S-3
Permittivity e Farad per metre sec4 Amp2 /Kg/m3 S2 T-1
Permeability m Henry per metre Kg m/sec2/Amp2 T3 S-4
Resistivity r Ohm metres m3Kg/sec3/Amp2 T2 S-2
Temperature T ° Kelvin K T S-1
Enthalpy H Joules Kgm2/s2 T S-1
Conductivity s Siemens per metre Sec3Amp2 /Kg/m3 S2 T-2
Thermal Conductivity W/m/° K Kg m /sec3/K S-1T-1
Thermal Resistivity ° K m/W sec3K/Kg/m ST
Thermal Conductance W/° K Kg m2 /sec3/K T-1
Thermal Resistance ° K/W sec3K/Kg/m2 T
Energy density J/m3 Kg/m/sec2 T S-4
Ion mobility m Metre2/ Volts seconds Amp sec2/Kg S4 T-2
Radioactive dose Sv Sievert or J/Kg m2/s2 S2 T -2
Dynamic Viscosity Pa sec or Poise Kg/m/s T2 S-4
Kinematic Viscosity Stoke cm2/sec S2 T-1
Fluidity 1/Pascal second m sec/Kg S4 T-2
Effective radiated power ERP Watts/m2 Kg/m/sec3 S-3
Luminance Nit Candela/m2 S-3
Luminous Intensity Candela Candela S-1
Gravitational Constant G Nm2/Kg2 m3/Kg/s2 S6 T-5
Planck Constant h Joules second Kg m2/sec T2 S-1
Coefficient of viscosity h n Kg/m/s T2 S-4
Young's Modulus of elasticity E N/m2 Kg/m/s2 T S-4
Electron Volt eV 1eV Kg m2/sec2 T S-1
Boyle's constant k Nm or Pa m3 Kg m2/sec2 T S-1
Hubble constant Ho H Km/sec/Parsec T-1
Stefan's Constant s W/m2/K4 Kg/s3/m/K4 S T-4
Strain e - - S0 T0
Refractive index h - - S0 T0
Boltzmann constant k Erg or Joule/Kelvin Kg.m2/s2/K S0 T0
Molar gas constant R J/mol/Kelvin Kg.m2/s2/K S0 T0
Mole n Mol Kg/Kg S0 T0
Fine Structure constant a - - S0 T0
Entropy S Joule/Kelvin Kg.m2/s2/K S0 T0
Reynolds Number Re - - S0 T0
Newton Power Number Np - - S0 T0

If anyone wants to add any missing parameter or knows any known equation that invalidates any of the above conversions please let me know. Here is a simple example showing you how to validate any equation into ST dimensions:

Equation to test : Casimir force F= hcA/d4

Convert each parameter to its ST dimensions from the table:

F= force= T S-2
c= speed of light= S T-1
h= Planck's constant = T2 S-1
A= Area = S2
d= 1d space = S

So the equation becomes:

T S-2 = T2 S-1 * S T-1 * S2 * S-4= T(2-1) S(-1+1+2-4)
T S-2 = T S-2... dimensionally correct.

Where does our scientific knowledge stand ?

The above conversion table makes a few things quite obvious. Since S has been defined as space in one dimension (line), S2 defines a 2D plane, S3 defines a 3D volume, and so forth, we might wonder why terms to the 6th power should exist, and what is the significance of the negative powered dimensions.
As discussed in the section Higher dimensional space, all clues point towards an ultimate fractal spacetime dimension slightly higher than 7. So, one should really expect physical parameters with space dimensions up to 7. Now, to the difficult part... time dimensions. We are normally used to talk about 3D space + 1D time, I have also introduced n*D space + 1D time in the 'Existence of higher dimensions' section. Our mind is limited to perceive everything in one 'time vector', that is one continous time line arrow, having direction from past to future. Most of the readers that went through the mentioned sections would have probably already had a hard time trying to perceive higher space dimensions a seen from such a single time vector. However, physical parameters which effect our universe, do not neccessarily exist in this single timeline, and this can be easily seen from those parameters having powers on their T dimension different than unity. The table below, shows more clearly, all known physical parameters in terms of their space & time dimensions. As expected, all known parameters fit into a 7D Spacetime. As you know, I have often referred to a self observing universe, and the negative powered dimensions are a consequence of this observation. If the observer 'lives' in a 1D timeline, then he can observe the surrounding space with respect to T, so we are able to observe space S but also to observe S with respect to T = dS/dT = velocity = S T-1. So, you see that although we write T -1, we are here differenciating S by T+1, so the + and - only indicate in which dimension (S or T) is the observer residing. Note how the inverse relation between some of the parameters is made obvious through this matrix, for example resistance vs conductance, and dynamic viscosity vs fluidity. We now use this knowledge to conclude that all physical parameters are a combination of observing space time from different dimensions of space and time, and can all fit into a table holding 7D spacetime. The table below shows the result after crossing each box of each known physical parameter from the table above. This table actually shows us where our scientific knowledge stands, counting the checked boxes one gets only about 15% of the whole table, which took humanity a few millions of years to find out. Once we will be able to fill up the complete table, we would know how to inter-relate all dimensions of our unified universe. Up to that day, no student (or lecturer) can ever think that the existing explanations of science are final and that after reading all his textbooks or finishing his course of study, he should go away satisfied with his wisdom !

 T -7 T -6 T -5 T -4 T -3 T -2 T -1 T 0 T 1 T 2 T 3 T 4 T 5 T 6 T 7 S 7 - - - - - - - - - - - - - - - S 6 - - X - - - - - - - - - - - - S 5 - - - - - - - - - - - - - - - S 4 - - - - - X - - - - - - - - - S 3 - - - - X X X X - - - - - - - S 2 - - - - - X X X - - - - - - - S 1 - - - X - X X X X - - - - - - S 0 - - - - - - X X X - - - - - - S -1 - - - - - - X X X X X - - - - S -2 - - - - - - - X X X - - - - - S -3 - - - - - - - X X X X - - - - S -4 - - - - - - - - X X X - - - - S -5 - - - - - - - - - - - - - - - S -6 - - - - - - - - - - X - - - - S -7 - - - - - - - - - - - - - - -