Many Scientists believe that the proportionality constants of permittivity 'e0' and permeability 'm0' associated with free space or vacuum, depend entirely on the choice of units of the main parameters and that they do not characterize any property of free space or the vacuum. Hence it is contended that mere 'existence' of these constants should not be construed to imply the existence of any entity like 'aether' or to assign physical properties to the notion of 'vacuum'. Further, it has also been asserted that there is some sort of arbitrariness in the choice of units and physical dimensions of many universal constants like e0 and m0.
For detailed study of this problem, let us first consider a general system with inter-related parameters A1, A2, A3, A4 ... etc. Let a typical (physically observed) relationship of these parameters be written as
(A1a1).(A2a2).(A3a3)
=
constant
............(1)
where a1,a2,a3 etc. may be +ve or -ve digits or fractions. If the dimensions of parameters A1, A2, A3 are well defined, then through dimensional analysis we can ascertain whether the constant in equation (1) is a dimensional number or a dimensionless number. Only if this constant is a dimensionless number, we can declare with certainty that this constant does not characterize the system, that is, it does not represent any physical property of the system in addition to the ones already represented by A1, A2, and A3. The magnitude of such a dimensionless constant may depend on the choice of units of the parameters A1, A2, and A3. The choice of units here essentially implies the choice of scale and not a choice of dimensions of various parameters.
On the other hand, if the constant in equation (1) is a dimensional number, we can replace it with a dimensional parameter say B1, so that equation (1) can be rewritten as
(A1a1).(A2a2).(A3a3)
= B1
….....(2)
In this case the parameter B1 will certainly characterize the system, that is, it will represent a physical property of the system; even though under given observational environment this parameter may remain constant in magnitude. The magnitude of such a dimensional parameter too is governed by the choice of scale in the units of the parameters A1, A2, and A3. But the 'essence' of parameter B1 and its dimensions cannot be arbitrarily changed without simultaneously tampering with the essence and dimensions of parameters A1, A2 etc. On the whole, a unit system is highly inter-related and dimensions of any one parameter cannot be arbitrarily changed without affecting all other dependent parameters. It is, however, possible that parameter B1 may be a 'lumped' parameter, that is, it may consist of two or more physical parameters which remain constant under given observational environment.
As a typical example, let us consider the case of Boyle's Law for an ideal gas at constant temperature. The relation between pressure p and volume V is written as
p.V = Constant
........... (3)
It can be easily seen through dimensional analysis that this constant in equation (3) is a dimensional constant. Therefore, we can replace it with a dimensional parameter or with a group of dimensional parameters R.T that characterize the gas in its current state. This leads us to the perfect gas law or the equation of state.
p.V/T =
R
............ (4)
where T is the absolute temperature and R is the 'characteristic gas constant'. Here again, R is a dimensional constant, which can be further split into m(mass of the given quantity of gas in moles) and R0(the universal gas constant). Therefore, equation (4) reduces to
p.V/(m.T)=R0
......... (5)
Here too, R0 is a dimensional parameter that characterizes some most fundamental features of an ideal gas. Magnitude of R0 varies in different unit systems but it does not imply that units of R0 can be chosen arbitrarily. The essence of R0 does not change with the change in unit system. However, we cannot extract the true essence of R0 from the perfect gas law or all the associated experimental data. To fully understand as to how R0 characterizes an ideal gas system, we need a kinetic theory of gases, which tells us that R0 is a composite parameter consisting of N0 (Avogadro's number) and kB (Boltzmann's constant).
Now let us consider a slightly different case of equation (2) where the dimensions of one of the main parameters say A3 are not well established but the physical relationship represented by equation (2) is known to exist. Obviously, to fix the units of two parameters here, namely B1 and A3, we need one more relation in addition to equation (2). Let us assume the second relationship is available in the form
(A1b1).(A2b2).(A3b3)=
B2
...........(6)
where b1,b2,b3 etc. may be +ve or -ve digits or fractions. Now to fix the units of B1, B2 and A3 from equations (2) and (6), we need still another relationship between B1 and B2.
Let us take up the specific case of permittivity e0 and permeability m0. We have following three relations involving the parameters e0, m0 and the charge Q to enable us fix their units.
F = [1/(4.p.e0)].(Q2/r2)
Or e0 = [1/(4.p.F)].(Q2/r2) ............ (7)
where F is the force between two equal charges Q separated by distance r.
F/L = (m0/2p).
(I2/d)
Or m0 = 2p.(F/L ).(d/I2)
............ (8)
where F/L is the force per unit length of two parallel conductors separated by distance d and carrying current I. Since the units of time are well established, we need to establish the units of only one of the two parameters Q and I, the other will get fixed automatically.
Sqrt(e0.m0) =
1/c
........... (9)
where c is the velocity of light with well-established units. On the basis of these three equations (7), (8) and (9), the units of three parameters e0, m0 and Q (or I) can be fixed satisfactorily. The units thus established, will be self-consistent and mutually compatible. We cannot arbitrarily change the units of any one of these parameters without affecting the units of other two. In MKSA system, units of parameters e0, m0 and I (hence Q) have been fixed this way and they satisfy all of the above mentioned three equations. We may once again highlight the fact that the proportionality constants in equations (7), (8) and (9), being dimensional parameters, play an extremely important role in characterizing the system, in characterizing the entity called 'free space, or 'vacuum' or the 'aether'.
It will not be out of place to mention here that in Gaussian CGS system of units, a bold attempt had been made to fix the proportionality constant in equation (7) to unity, apparently to give a simpler look to various equations. It is very interesting to find out how exactly it was done. Essentially the two parameters Q and 'e0' in equation (7) are lumped into one new parameter (say) Qg with units of 'statcoulomb'. Correspondingly the unit of current in this system is also named 'statamp'. Replacing (1/(4.p.e0)).Q2 with Qg2 in equation (7) and (1/(4.p.e0)).I2 with Ig2 in equation (8) we get
F = (Qg2/r2) or (1/F). (Qg2/r2)
= 1
............. (10)
and
F/L = (m0/2p).(Ig2/d).(4.p.e0) = 2.(m0.e0).(Ig2/d)
= (2/c2).(Ig2/d)
Or
(F/2L). (d/Ig2) = 1/c2
............. (11)
Units of various other parameters in Gaussian CGS system are then fixed such that they are consistent with equations (10) and (11) and mutually compatible. Here it is very important to note that the dimensions of charge Q in MKSA system are entirely different from the dimensions of charge Qg (=Q/sqrt(4.p.e0)) in Gaussian system and hence they do not represent the same physical quantity.
Finally, let us consider one more example relevant to this discussion. Take a long thin metallic rod and subject it to load test. We find stress is proportional to strain.
stress/strain = constant
................
(12)
From dimensional analysis we find that this proportionality constant is a dimensional number. We, therefore, replace it with a dimensional parameter Y known as the Young's Modulus. Obviously here Y is not just another arbitrary constant depending on arbitrary choice of units. Since the dimensions of stress and strain are well established, only magnitude of Y can vary with the variation of 'scale factor' of different units. The dimensions and hence the essence of Y remains invariant. Y represents a very important property (the elasticity) of the material of the test piece under consideration. Now let us conduct another test on the given rod. If we measure the mass and volume of any piece of this rod, we find that mass is proportional to the volume.
mass/volume = constant
................ (13)
Here again the proportionality constant is a dimensional number. We, therefore, replace it with a dimensional parameter 'r'. The density 'r' also represents a very important property (the inertial property) of the material of the test piece under consideration and is not an arbitrary constant. Finally if we measure the velocity 'c1' of longitudinal strain wave propagation in the given test piece, we find it related to Y and r as
Sqrt(Y/r) =
c1
............... (14)
This illustrates the importance of the dimensional proportionality constants, which cannot be discarded by just branding them as 'arbitrary' proportionality constants.
Now coming back to the case of permittivity e0 and permeability m0 of the so-called 'empty space' or 'vacuum' or the old 'aether' medium, let us rewrite equation (9) as
Sqrt(1/e0)/Sqrt(m0)=
c
................
(15)
Comparison of equations (14) and (15) shows a remarkable similarity of the two cases. Velocity c1 represents the velocity of strain wave propagation in the material medium and velocity c represents the velocity of electromagnetic wave propagation in the so-called 'empty space' or 'vacuum' or the 'aether' medium. Therefore, the electromagnetic wave propagation may be compared with a strain wave propagation. In this comparison (1/e0) may be seen to be identical to the elasticity property Y and (m0) may be seen to be identical to the inertial property r. Therefore it stands to reason that we must strive very hard to unravel and to comprehend the deeper significance of the proportionality constants 'e0' and 'm0' associated with the entity called 'empty space' or 'vacuum' or the 'aether' medium or identified by any other name. It is said, "Rose by any other name will smell as sweet". I personally would like to call this entity the 'Elastic Continuum'.
·
Ether, Vacuum or
the Elastic Continuum
·
What Ails the Fundamental Research in Physics
·
What If the Permittivity and Permeability of
Vacuum were Zero
· Physical Theory and Mathematical Models
· GTR is founded on a Conceptual Mistake
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