i.e., P and Q corresponding to F and G. Using the formulæ (3.3) and (2.24)–(2.26), we obtain
(4.8)
(4.9)
The last equations can be written in a more symmetrical form:
(4.8A)
(4.9A)
We are now able to solve the equations (3.3). It follows from (3.3) and (2.26) that
(3.3A)
Solving (3.3) and (3.3A) we obtain (taking into account (4.8A) and (4.9A))
(4.10)
The tensors fkl and pkl can now be treated completely symmetrically. Instead of the Lagrangian L we can use in the principle of action the Hamiltonian function H:
(4.11)
where H has to be regarded as a function of gkl and pkl. From (4.8), (4.9), and (4.10) it follows for H as a function of gkl and pkl:
(4.12)
and this can be expressed in the form
(4.12A)
The function H leads us to exactly the same equations of the field as the function L. We see that the equations
(4.13)
(4.14)
(4.15)
are entirely equivalent to the equations (3.4), (4.10), (3.2A).
The energy-impulse tensor (3.6A) can also be expressed with help of H instead of L. One has
(4.16)
The identity of this expression with (3.6) is evident,lk if we appeal to the following formula which can be deduced from (2.21), (2.22)
(4.17)
Generally, from each equation containing , φk , fkl , pkl , one obtains another correct equation changing these quantities correspond ingly into H, , , ⋅
§ 5. FIELD EQUATIONS IN SPACE-VECTOR FORM.
We now introduce the conventional units instead of the natural units. We denote by B, E and D, E, the space-vectors which characterize the electromagnetic field in the conventional units. We have in a cartesian co-ordinate system:
(5.1)
(5.2)
(5.3)
(5.4)
The quotient of the field strength expressed in the conventional units divided by the field strength in the natural units may be denoted by b. This constant of a dimension of a field strength may be called the absolute field; later we shall determine the value of b, which turns out to be very great, i.e., of the order of magnitude 1016 e.s.u.
We have
2.11)
(2.12A; 2.13A)
(3.3A)
(3.1A)
(3.2B)
(3.4A)
Our field equations (3.2B) and (3.4A) are formally identical with Maxwell’s equations for a substance which has a dielectric constance and a susceptibility, being certain functions of the field strength, but without a spatial distribution of charge and current.
For the energy-impulse tensor we find:
(3.6A)
(3.6B)
One gets another set of expressions for these quantities by changing L, B, E, H, D into H, H, D, B, E.
The conservation laws are:
(3.8A)
The function H is given by
(4.12A)
(4.6A); (4.7A)
Solving (3.3A) we obtain:
(3.10A)
§ 6. STATIC SOLUTION OF THE FIELD EQUATIONS.
We consider (in the cartesian co-ordinate system) the electrostatic case where B = H = 0 and all other field components are independent of t. Then the field equations reduce to:
(6.1)
(6.2)
We solve this equation for the case of central symmetry. Then (6.2) is simply
(6.3)
and (6.3) has the solution
(6.4)
In this case the field D is exactly the same as in Maxwell’s theory: the sources of D are point charges given by the surface integral
(6.5)
The equation (6.1) gives
(6.6)
and from (3.3A)
(6.7)
The combination of (6.4) and (6.7) gives a differential equation for φ (r) of the first order, with the solution
(6.8)
This is the elementary potential of a point charge e, which has to replace Coulomb’s law; the latter is an approximation for x >> 1, as is seen immediately, but the new potential is finite everywhere.
With help of the substitution one obtains
(6.9)
where
(6.10)
and F(k, β) is the Jacobian elliptic integral of the first kind for k = (tabulated in many books)ll
(6.11)
FIG. 1
For x = 0 one has
(6.12)
The potential has its maximum in the centre and its value is
(6.13)
The function f(x) is plotted in fig. 1. It has very similar properties to the function arc cot x. For example, one has
on the other hand
Therefore one has
(6.14)
It is sufficient, therefore, to calculate f (x) from x = 0 to x = 1 or from β = 0 to .
One sees, that the D field is infinite for r = 0; E and φ, however, are always finite. One has
(6.4)
(6.15)
The components Ex, Ey , Ez are finite at the centre, but have there a discontinuity.
§ 7. SOURCES OF THE FIELD.
In the older theories, which we have called dualistic, because they considered matter and field as essentially different, the ideal would be to assume the particles to be point charges; this was impossible because of the infinite self-energy. Therefore it was necessary to assume the electron having a finite diameter and to make arbitrary assumptions about its inner structure, which lead to the difficulties pointed out in the introduction. In our theory these difficulties do not appear. We have seen that the pkl field (or D-field) has a singularity which corresponds to a point charge as the source of the field. D and E are identical only at large distances (r >> r0) from the point charge, but differ in its neighbourhood, and one can call their quotient (which is function of E) “dielectric constant” of the space. But we shall now show that another interpretation is also possible which corresponds to the old idea of a spatial distribution of charge in the electron. It consists in taking div E (instead of div D = 0) as definition of charge density ρ, which we propose to call “free charge density.”
Let us now write our set of field equations in the following form:
(3.4)
(3.2)
pkl is a given function of fkl and if we put in (3.4) for pkl the expression (3.3), in which L is not specified, we obtain:
(7.1)
We can now write the equation (7.1) in the form:
(7.2)
where
(7.3)
The equations (7.2) and (3.2) are formally identical with the equations of the Lorentz theory. But the important difference consists in this, that ρk is not a given function of the space-time co-ordinates, but is a function of the unknown field strength. If we have a solution of our set of equations, we are able to find the density of the “free charge” or the “free current” with help of (7.2) or (7.3).
We see immediately that ρk satisfies the conservation law:
(7.4)
This follows from (7.2), that is from the antisymmetrical character of fkl, and can also be checked from (7.3).
In Lorentz’s theory there exists the energy-impulse tensor of the electromagnetic field, defined by
(7.5)
but its divergence does not vanish, where the density of charge is not zero. Therefore to preservethe conservation principle in the Lorentz’s theory it was necessary to introduce an energy-impulse tensor of matter, Mkl, the meaning of which is obscure. The tensor had to fulfil the condition that the divergence of vanishes. This difficulty does not appear in our theory. We do not need to introduce the matter tensor because the conservation laws are always satisfied by our energy-impulse tensor .
We shall, however, show that it is possible by introducing the free charges to bring our conservation law
into the form used in the Lorentz theory, namely,
(7.6)
The calculations are similar to those used in § 3. The simplest way is to choose a geodetic co-ordinate system. We have then:
(7.2A)
(3.2)
Multiplying (3.2) by fkl we find:
(7.7)
and therefore
(7.8)
and taking account of (7.5):
(7.9)
One can derive the same equation directly from the conservation formula (3.8) writing it, in a geodetic co-ordinate system, in the form
(3.8)
and introducing the expression (3.6) for . The two methods are equivalent.
Let us now specialize our equations for the case in which L has the form given in (2.15). We obtain then for ρk in (7.3)
(7.10)
In the space-vector notation, where
we have
(7.10A)
We shall now apply the results here obtained to the case of the statical field. In this case E is always finite and has a non-vanishing divergence, which represents the free charge. We can, therefore, regard an electron either as a point charge, i.e., as a source of the D (pkl) field, or as a continuous distribution of the space charge which is a source of the E (fkl) field. It can easily be shown that the whole charge is in both cases the same (as is to be expected). Both∫ div D dv and ∫ div E dv
have the same value, i.e., 4πe. For the first integral it has been shown in § 6. For the second we have
everywhere else Er is finite. The discontinuity of Ex, Ey, Ez at the origin is also finite and gives no contribution to the integral. Therefore∫ div E dv = 4π∫ ρdv = 4πe.
Let us now calculate the distribution of the free charge in the statical case.
We could calculate it from the equation (7.10A), but it is easier to do it from the equation
(7.11)
where
(6.15)
The result is
(7.12)
For r >> r0, ρ ∝ r−7, therefore diminishing very rapidly as r increases. For r < r0, ρ ∝ 1/r, therefore ρ → ∞, but r2ρ → 0 for r → 0. It is easy to verify that the space integral of ρ is equal to e. For one has, putting
Our theory combines the two possible aspects of the field; true point charges and free spatial densities are entirely equivalent. The question whether the one or the other picture of the electron is right has no meaning. This confirms the idea which has proved so fruitful in quantum mechanics, that one has to be careful in applying notions from the macroscopic world to the world of atoms: it may happen that two notions contradictory in macroscopic use are quite compatible in microphysics.
§ 8. LORENTZ’S EQUATIONS OF POINT MOTION AND MASS.
We consider once more the problem of the electron at rest. We intend to calculate the mass and to determine the absolute field constant b in terms of observable quantities. It is convenient to use the space vector notation.
The impulse-energy tensor is according to (3.6B)
(3.6c)
We calculate the space integrals of these quantities. Obviously one has with dv = dx dy dz:
(8.1)
(8.2)
Using (6.15) and (6.8) we find:
(8.3)
where
(8.4)
The integral I1 can be transformed by partial integration:
The first term vanishes; the second can be transformed by another partial integration:
The result is the so-called “theorem of Laue”lm ∫ Xx dv = ∫ Yy dv = ∫ Zz dv = 0.
In the statical case and in a co-ordinate system in which the electron is at rest the integrals of all components of the tensor vanish except the total energy
(8.5)
We find from (3.6c), (6.15), and (8.4)
(8.6)
We have obtained a finite value of the energy or the mass of the electron with a definite numerical factor. This relation enables us to complete our theory concerning the value of the absolute field b in the conventional units. For (8.6) gives the “radius” of the electron expressed in terms of its charge and mass:
(8.7)
and
(8.8)
The enormous magnitude of this field justifies the application of the Maxwell’s equations in their classical form in all cases, except those where the inner structure of the electron is concerned (field of the order b, distance or wavelength of the order r0).
It can be shown that the motion of an elementary charge, on which an external field is acting, satisfies an equation which is an obvious generalization of the classical equation of Lorentz. To find this equation we shall use here a cartesian co-ordinate system.
We assume that the strength of the external field in a region surrounding the electron is very small compared with the proper field of the point charge. We denote the proper field of the electron by
(8.9)
and the external field by
(8.10)
we do not take into consideration the sources of the external field. From the assumption, that
(8.11)
inside the sphere surrounding the electron, it follows evidently that the real solution of the field equations cannot be very different from that obtained by adding the unperturbed proper field and the external field. We construct therefore a sphere S(0) with its centre at the singularity of H and with a radius r(0), which is so small, that inside the sphere (8.11) is always satisfied. But the radius r(0) of the sphere has to be great compared with the radius of the electron, so that we can assume the validity of Maxwell’s equations on the surface of the sphere just as outside the sphere.
We make the further assumption that the acceleration (curvature of the world line) is not too large, i.e., one can choose the radius in such a way that the field inside S(0) is essentially identical with that of the charge e in uniform motion and can be derived from the formula of § 7 by a Lorentz transformation. Now we split the integral
(8.12)
into a part corresponding to the sphere S(0) and the rest of space R. In S(0) we have
(8.13)
Corresponding to (8.11) one can consider the terms f(0)kl as small of the first order (compared with f(0)kl), the terms f(e)kl as small of the second order, and these latter will be neglected. Then we have by developing (8.13) and using (4.14):
(8.14)
which holds inside the sphere S(0). We can write (8.14) in another form:
(8.15)
(8.15) differs from (8.14) only in the terms of the second order. But (8.15) holds not only inside but also outside the sphere. For in R the equation (8.15) takes, according to our assumptions about r(0), the following form:
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