2. Derivation of the Planck quantities

# Comparing dimensions

Aim: From the three quantities  the Planck quantities shall be derived. The three quantities  are natural quantities, that is, they occur in physics as natural constants. For this the dimensions of  will be multiplied, such that the dimension length, mass and time result. The Planck quantities can be expressed as terms of .

2.0. Prerequisite: the notion of dimension

The dimension is a notion that is assigned to a physical quantity and describes this quantity qualitatively. The dimension is expressed in the form of a power product by means of basic quantities, which are chosen for the description of a partial field of physics. We want to designate the dimension of the quantity a as dim(a). The dimension describes the proportional correlation between the considered quantity a and the system of basic quantities bi. Here the basic dimensions Bi are assigned to these basic quantities bi. These basic quantities bi form a dimension system. For the determination of a quantity’s dimension the numerical values are left out and the symbols of the basic quantities are replaced by their dimension symbols. Besides, one or more units are assigned to each basic dimension and vice versa. The power of a quantity’s dimension is equal to the dimension of the power of this quantity with the same exponent. This means dim(A)b = dim(Ab), where A is a quantity and b a real number. It applies to the product of two quantities, that their dimension is equal to the product of the dimensions of the separate quantities. Thus, dim(A1 • A2) = dim(A1) • dim(A2) holds, where A1 and A2 are two quantities.

2.1 General Solution

For this the following equation is formed, in which x is a given dimension, e.g. the one of length:

.                                                                                           (2.1)

Here dim stands for the dimension of the respective quantity.

Now the question is which dimensions these quantities  have.

The following applies to :

c = 299,792,458 m s–1,                                                                                                                (2.2)

G = 6.673 • 10–11 m3 kg–1 s–2,                                                                                                        (2.3)

= 1.0545716 • 10–34 Js = 1.0545716 • 10–34 (kg m2 s–2) s,

= 1.0545716 • 10–34 kg m2 s–1.                                                                                                 (2.4)

For the determination of the dimensions of , first, the numerical values are left out and then the units m, kg and s are replaced by L for the dimension of length, by M for the dimension mass and by T for the dimension of time. Therefore, it follows from the equations (2.2), (2.3) and (2.4), that:

dim(c) = LM0T–1,                                                                                                                         (2.5)

dim(G) = L3M–1T–2,                                                                                                                      (2.6)

dim = L2MT–1.                                                                                                                        (2.7)

That x is a product of powers with bases L, T and M, follows from the equations (2.1), (2.5), (2.6) and (2.7). This results in:

x = L M T,                                                                                                                    (2.8)

if xL, xM and xT are the exponents of the respective basis.

Now the terms from the right sides of the equations (2.5), (2.6) and (2.7) are inserted into the equation (2.1) for dim(c), dim(G) and dim():

.

From the rules for powers it follows:

.                                                                   (2.9)

As L, M and T are independent of each other, we cannot simplify the right term of the equation (2.9) farther. That is why the exponent of the respective power of the right side of the equation (2.8) must be equal to the equation (2.9). So, it results:

(2.10)

(2.11)

(2.12)

Since x is a given dimension, xL, xM and xT are also known. Yet zc, zG and z are unknown, which are the exponents of the powers of ’s dimensions in the equation (2.1). The equations (2.10), (2.11) and (2.12) form an equation system of three equations with three unknown quantities, which can be determined unambiguously then, if the determinant of the coefficients of these three equations is unequal to zero.

The determinant is calculated as follows:

First, the terms of the equation (2.10) are added to the ones of the equation (2.12):

(2.13)

Now the terms of equation (2.11) are added to the terms of equation (2.13):

| :2

 (2.14)

Further, z is replaced by the term of equation (2.14)’s right side in equation (2.13):

 . (2.15)

Finally, in equation (2.10) for z the term of the right side of equation (2.14) is inserted and for zG the one of equation (2.15).

 (2.16)

The quantities zc, zG and z can be calculated at a given x from the equations (2.14), (2.15) and (2.16).

2.2. Calculation of the Planck length

The values of zc, zG and z will be calculated for x = L.

Hence, xL = 1, xM = 0 and xT = 0 must be set.

Having inserted these values for xL, xM and xT in equations (2.14), (2.15) and (2.16), it results:

From equation (2.1) it follows after replacing x by L and zc, zG and z by these worked out values:

(2.17)

As the Planck length can be expressed as a expression of c, G and , it is true, that:

Here al is a unit-less quantity that is caused by the fact that for the determination of a quantity’s dimension the numerical value of this quantity is left out. In order to obtain the Planck length, we make al equal 1, because the Planck length is the simplest expression of c, G and  and 1 the simplest factor.

2.3. Calculation of the Planck mass

Now the values of zc, zG and z will be calculated for x = M.

Consequently, it holds: xL = 0, xM = 1 and xT = 0.

Having inserted these values of xL, xM and xT in equations (2.14), (2.15) and (2.16), it yields:

From equation (2.1) it follows after replacing x by M and zc, zG and z by the calculated values:

(2.18)

Since the Planck mass can be expressed as a expression of c, G and  and there is only one possible such expression, it is true, that:

Here am is a unit-less quantity that is caused by the fact that for the determination of a quantity’s dimension the numerical value of this quantity is left out. The Planck mass is also the simplest expression of c, G and . Thus we set the factor am to 1, in order to get the Planck mass, for 1 is the simplest factor:

2.4. Calculation of the Planck time

Eventually, the values of zc, zG and z will be calculated for x = T.

Therefore it must be: xL = 0, xM = 0 and xT = 1.

After replacing xL, xM and xT by these values in equations (2.14), (2.15) and (2.16), it results:

From equation (2.1) it follows after inserting T for x and the worked out values for zc, zG and z, that:

(2.19)

That results:

Here at is again a quantity of dimension 1 that arises from the fact that for the determination of a quantity’s dimension the numerical value of this quantity is left out. at is set to 1 because of a reasoning analogous to the previous ones.

2.5. On the pre-factor a at the derived the Planck quantities

All here derived Planck quantities are products of a pre-factor a of dimension 1 and a term of G,  and c. But this factor a equals 1, because 1 is the simplest factor and the Planck quantities are the simplest expression of the three natural constants G,  and c. Giving a the value 1 is useful, as then this factor is dropped. Though, there are no standard Planck constants. However, there is also a definition of the Planck quantities at which h is used instead of . The dimensions of h and  are equal, because of

so that they both differ only by a factor of dimension 1. At the here derived quantities we could also assign the value of the pre-factor a to (2p)0.5. This would be equivalent to the use of h instead of . But  is more often needed.

2.6. Summary of the outcomes

The calculation and considerations results: