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Absolute Units

Table of Contents

1. Dimensional expressions, how to eliminate them
2. Classical Physical Basic Units
3. Other Classical Physical Units
4. Modern Basic Units
5. Other Units Modernized
6. Elimination of the Time Unit
7. Elimination of the Mass Unit
8. Elimination of the Electromagnetic Units, and the Final Unit Conversion Table
9. A Quantum Gravitational Connection

1. Dimensional expressions, how to eliminate them

Traditionally when physical values are measured in physical units, the units are included after the numerical value. Expressions of this kind are called dimensional expressions. In pure mathematics dimensional expressions are avoided in order to improve generality. The question arises, in which extent dimensions (that is, physical units) can be eliminated in physical expressions. The ideal case would eliminate them entirely. The discussion below tries to sketch up such a solution.

2. Classical Physical Basic Units

3. Other Classical Physical Units

4. Modern Basic Units

Since 1986 the international definition of one meter is the distance light travels in vacuum in one second. The background to this new definition is Einstein's theory of relativity, where time and distance both are treated as dimensions of the 4-dimensional space-time continuum. One can then take the velocity of light in vacuum to be 1 , and define the length unit accordingly, as was done in the year 1986 definition of basic units.

Thus we only have two basic units left: the time unit second and mass unit kilogram. The reference values of one second and one kg are defined using extremely stable physical phenomena, which are (for physicists) possible to determine experimentally when needed, to desired degree of accuracy. The old 'standard meter' and 'standard kilogram' probes are no longer needed.

It is now possible to convert between time and length units, and we could use time to measure distance and vice versa.

Then one meter expressed in seconds is: 1 m = 1 / 299792458.1 s = 3.3356409509·10^-9 s

And one second expressed in meters is: 1 s = 299792458.1 m

5. Other Units Modernized

Because now only second and kilogram are basic units, all the other fundamental mechanical units are to be expressed using only kg and s, velocity now becomes a dimensionless "absolute" number, expressing a fraction of the velocity of light in vacuum. Scientifically this is often an advantage, but seldom in everyday life. So the use of different units for time and length will persist, I suppose, a long time in future.

In the old classical units velocity of light in vacuum: c = 299792458.1 m / s .

But in the new scientific units c = 1 by definition.

Note that energy is measured in mass units, fully in agreement with Einstein's theory.
Energy: E = M·c² , because now c² = 1 energy in the new units gets the same value as mass.

6. Elimination of the Time Unit

In effect we have in the discussion above got rid of the length unit. Could we possibly do the trick with time and the mass units too? Yes in deed, by using certain universal constants. We need to use the quantum mechanical Planck's constant, and the universal Gravitation constant, from which we then can get the values of the Planck length, Planck time, and Planck mass. These express a presumed connection between General Relativity and Quantum Mechanics. Theories suggest such a connection, despite no generally accepted final physical theory of quantum gravitation yet exists. The more exiting it is to see a connection arise from the dimensional analysis.

The quantum mechanical Planck's constant (in old units):

Gravitational constant (in old units):

We get the "Planck length" from the formula:

In old units the value is:

Corresponding expression for "Planck time" would then be:

The value is in seconds:

One second in units of Planck time:

Now define the new time unit so that the Planck time becomes equal to one and dimensionless.
We then get for one second a dimensionless "absolute" value (see the unit conversion table below).

7. Elimination of the Mass Unit

The Planck mass is the mass which has its Compton wavelength equal to the Planck length. Generally the Compton wavelength is h / (M·c)

so we get the Planck mass from the formula: m_Pl = h / (l_Pl·c)

This is a fundamental mass constant, which can be used expressing mass in "absolute" (dimensionless) units. One kilogram expressed in Planck masses:

Setting the Planck mass to be equal to one we get for one kilogram the dimensionless "absolute" value. Then substituting the time (s) and mass (kg) unit values above we get dimensionless values. (See the unit conversion table below.)

8. Elimination of the Electromagnetic Units,
    and the Final Unit Conversion Table

The elementary charge is smallest measured charge: e = 1.60217733·10-19 C

In order to eliminate Coulomb, set e = 1.

Now we are ready to write down the final conversion table:

Physical unit conversion into absolute units
Time unit (second)	1 s = 1.8550955416·1043
Mass unit (kilogram)	1 kg = 7.3117064051·106
Force unit (Newton)	1 N = 3.3356409509·10-9 kg / s = 1.3147154289·10-45
Energy unit (Joule)	1 J = 1.1126500553·10-17 kg = 8.135370536·10-11
Power unit (Watt)	1 W = J / s = 4.3854186232·10-54
Charge unit (Coulomb)	1 C = 1 / e = 6.2415063631·1018
Current unit (Ampere)	1 A = 1 C / s = 3.3645201679·10-25
Electric potential (Volt)	1 V = 1 J / C = 1.3034306244·10-29
Magnetic strength: T (Tesla)	1 T = 1 kg / (s C) = 6.3149451121·10-56

9. A Quantum Gravitational Connection

Note that the following expression, curiously also in old units, is dimensionless and equal to one.

Setting the Planck mass to be one, as was done above, and remembering that c = 1 in the new units, we get for the universal gravitation constant an expression that connects it directly to the quantum mechanical Planck's constant:

Planck's constant in new (after year 1996) units:

In absolute units it turns out that h =1, which is nothing but an expected result. In fact the elimination of dimensions in classical and quantum mechanical expressions is a consequence of that one can set c = 1 and h = 1.

The universal gravitation constant in absolute units becomes then: G = 2p

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