NOTES

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NOTES

Note 1.

The fundamental electro-magnetic equations of Maxwell for stationary media are:—

According to Hertz and Heaviside, these require modification in the case of moving bodies.

Now it is known that due to motion alone there is a change in a vector R given by

(∂R/∂t) due to motion = u. div R + curl [Ru]

where u is the vector velocity of the moving body and [Ru] the vector product of R and u.

Hence equations (1) and (2) become

and

which gives finally, for ρ = 0 and div B = 0,

Since D = KE and B = μH, we have

Multiplying (1·23) by (2·23)

∴ v = v₀ + u,

making Fresnelian convection co-efficient simply unity.

Equations (1·21) and (2·21) may be obtained more simply from physical considerations.

According to Heaviside and Hertz, the real seat of both electric and magnetic polarisation is the moving medium itself. Now at a point which is fixed with respect to the ether, the rate of change of electric polarisation is δD/δt.

A short summary of the electromagnetic experiments bearing on this question, has already been given in the introduction.

According to Hertz and Heaviside the total polarisation is situated in the medium itself and is completely carried away by it. Thus the electromagnetic effect outside a moving medium should be proportional to K, the specific inductive capacity.

Rowland showed in 1876 that when a charged condenser is rapidly rotated (the dielectric remaining stationary), the magnetic effect outside is proportional to K, the Sp. Ind. Cap.

Röntgen (Annalen der Physik 1888, 1890) found that if the dielectric is rotated while the condenser remains stationary, the effect is proportional to K - 1.

Eichenwald (Annalen der Physik 1903, 1904) rotated together both condenser and dielectric and found that the magnetic effect was proportional to the potential difference and to the angular velocity, but was completely independent of K. This is of course quite consistent with Rowland and Röntgen.

H. A. Wilson (Phil. Trans. Royal Soc. 1904) repeated the experiment with a cylindrical condenser made of ebony, rotating in a magnetic field parallel to its own axis. He observed a change proportional to K — 1 and not to K.

Thus the above set of electro-magnetic experiments contradict the Hertz-Heaviside equations, and these must be abandoned.

[P. C. M.]

Note 2.
Lorentz Transformation.

Lorentz. Versuch einer theorie der elektrischen und optischen Erscheinungen im bewegten Körpern.

(Leiden—1895).

Lorentz. Theory of Electrons (English edition), pages 197-200, 230, also notes 73, 86, pages 318, 328.

Lorentz wanted to explain the Michelson-Morley null-effect. In order to do so, it was obviously necessary to explain the Fitzgerald contraction. Lorentz worked on the hypothesis that an electron itself undergoes contraction when moving. He introduced new variables for the moving system defined by the following set of equations.

and for velocities, used

With the help of the above set of equations, which is known as the Lorentz transformation, he succeeded in showing how the Fitzgerald contraction results as a consequence of “fortuitous compensation of opposing effects.”

It should be observed that the Lorentz transformation is not identical with the Einstein transformation. The Einsteinian addition of velocities is quite different as also the expression for the “relative” density of electricity.

It is true that the Maxwell-Lorentz field equations remain practically unchanged by the Lorentz transformation, but they are changed to some slight extent. One marked advantage of the Einstein transformation consists in the fact that the field equations of a moving system preserve exactly the same form as those of a stationary system.

It should also be noted that the Fresnelian convection coefficient comes out in the theory of relativity as a direct consequence of Einstein’s addition of velocities and is quite independent of any electrical theory of matter.

[P. C. M.]

 § 181, page 213.

H. Poincare, Sur la dynamique ‘electron, Rendiconti del circolo matematico di Palermo 21 (1906).

[P. C. M.]

Note 4.
Relativity Theorem and Relativity-Principle.

Lorentz showed that the Maxwell-Lorentz system of electromagnetic field-equations remained practically unchanged by the Lorentz transformation. Thus the electromagnetic laws of Maxwell and Lorentz can be definitely proved “to be independent of the manner in which they are referred to two coordinate systems which have a uniform translatory motion relative to each other.” (See “Electrodynamics of Moving Bodies,” page 5.) Thus so far as the electromagnetic laws are concerned, the principle of relativity can be proved to be true.

But it is not known whether this principle will remain true in the case of other physical laws. We can always proceed on the assumption that it does remain true. Thus it is always possible to construct physical laws in such a way that they retain their form when referred to moving coordinates. The ultimate ground for formulating physical laws in this way is merely a subjective conviction that the principle of relativity is universally true. There is no a priori logical necessity that it should be so. Hence the Principle of Relativity (so far as it is applied to phenomena other than electromagnetic) must be regarded as a postulate, which we have assumed to be true, but for which we cannot adduce any definite proof, until after the generalisation is made and its consequences tested in the light of actual experience.

[P. C. M.]

Note 5.

See “Electrodynamics of Moving Bodies,” p. 5-8.

Note 6.
Field Equations in Minkowski’s Form.

Equations (i) and (ii) become when expanded into Cartesians:—

and ∂ex/∂x + ∂ey/∂y + ∂ez/∂z = ρ (2·1)

Substituting x₁, x₂, x₃, x₄ and x, y, z, and iτ; and ρ₁, ρ₂, ρ₃, ρ₄ for ρνx, ρνy, ρνz, iρ, where i = √(-1).

We get,

and multiplying (2·1) by i we get

Now substitute

and we get finally:—

Note 9.
On the Constancy of the Velocity of Light.

Page 12—refer also to page 6, of Einstein’s paper.

One of the two fundamental Postulates of the Principle of Relativity is that the velocity of light should remain constant whether the source is moving or stationary. It follows that even if a radiant source S move with a velocity u, it should always remain the centre of spherical waves expanding outwards with velocity c.

At first sight, it may not appear clear why the velocity should remain constant. Indeed according to the theory of Ritz, the velocity should become c + u, when the source of light moves towards the observer with the velocity u.

Prof. de Sitter has given an astronomical argument for deciding between these two divergent views. Let us suppose there is a double star of which one is revolving about the common centre of gravity in a circular orbit. Let the observer be in the plane of the orbit, at a great distance Δ.

For an experimental proof, see a paper by C. Majorana. Phil. Mag., Vol. 35, p. 163.

[M. N. S.]

Note 10.
Rest-density of Electricity.

If ρ is the volume density in a moving system then ρ√(1 - u²) is the corresponding quantity in the corresponding volume in the fixed system, that is, in the system at rest, and hence it is termed the rest-density of electricity.

[P. C. M.]

Note 11
(page 17)
Space-time vectors of the first and the second kind.

As we had already occasion to mention, Sommerfeld has, in two papers on four dimensional geometry (vide, Annalen der Physik, Bd. 32, p. 749; and Bd. 33, p. 649), translated the ideas of Minkowski into the language of four dimensional geometry. Instead of Minkowski’s space-time vector of the first kind, he uses the more expressive term ‘four-vector,’ thereby making it quite clear that it represents a directed quantity like a straight line, a force or a momentum, and has got 4 components, three in the direction of space-axes, and one in the direction of the time-axis.

The representation of the plane (defined by two straight lines) is much more difficult. In three dimensions, the plane can be represented by the vector perpendicular to itself. But that artifice is not available in four dimensions. For the perpendicular to a plane, we now have not a single line, but an infinite number of lines constituting a plane. This difficulty has been overcome by Minkowski in a very elegant manner which will become clear later on. Meanwhile we offer the following extract from the above mentioned work of Sommerfeld.

(Pp. 755, Bd. 32, Ann. d. Physik.)

“In order to have a better knowledge about the nature of the six-vector (which is the same thing as Minkowski’s space-time vector of the 2nd kind) let us take the special case of a piece of plane, having unit area (contents), and the form of a parallelogram, bounded by the four-vectors u, v, passing through the origin. Then the projection of this piece of plane on the xy plane is given by the projections ux, uy, vx, vy of the four vectors in the combination

φx y = uxvy - uyv{x}.

Let us form in a similar manner all the six components of this plane φ. Then six components are not all independent but are connected by the following relation

φy z φx l + φz x φy l + φx y φz l = 0

Further the contents | φ | of the piece of a plane is to be defined as the square root of the sum of the squares of these six quantities. In fact,

| φ |² = φy z² + φz x² + φx y² + φx l² + φy l² + φz l².

Let us now on the other hand take the case of the unit plane φ* normal to φ; we can call this plane the Complement of φ. Then we have the following relations between the components of the two plane:—

φy z* = φx l, φz x* = φy l, φx y* = φz l φz l* = φy x ...

The proof of these assertions is as follows. Let u*, v* be the four vectors defining φ*. Then we have the following relations:

ux* ux + uy* uy + uz* uz + ul* ul = 0
ux* vx + uy* vy + uz* vz + ul* vl = 0
vx* ux + vy* uy + vz* uz + vl* ul = 0
vx* vx + vy* vy + vz* vz + vl* vl = 0

from which we have

In a corresponding way we have

when the subscript (ik) denotes the component of φ in the plane contained by the lines other than (ik). Therefore the theorem is proved.

We have

We can verify that

Note 12.
Light-velocity as a maximum.

Page 23, and Electro-dynamics of Moving Bodies, p. 17.

Putting v = c - x, and w = c - λ, we get

Thus v lt; c, so long as | xλ | > 0.

Let A and B be separated by distance l, and let velocity of a “signal” in the system S be W > c. Let the (observing) system S′ have velocity +v with respect to the system S.

Then velocity of signal with respect to system S′ is given by W′ = (W - v)/(1 - Wv/c²)

Thus “time” from A to B as measured in S′, is given by l/W′ = l(1 - Wv/c²)/(W - v) = t′ (1)

Now if v is less than c, then W being greater than c (by hypothesis) W is greater than v, i.e., W > v.

Let W = c + μ and v = c - λ.

Then Wv = (c + μ)(c - λ) = c² + (μ + λ)c - μλ.

It is conceptually possible to imagine velocities greater than that of light, but such velocities cannot occur in reality. Velocities greater than c, will not produce any effect. Causal effect of any physical type can never travel with a velocity greater than that of light.

[P. C. M.]

Notes 13 and 14.

We have denoted the four-vector ω by the matrix | ω₁ ω₂ ω₃ ω₄ |. It is then at once seen that [=ω] denotes the reciprocal matrix

| ω₁ |
| ω₂ |
| ω₃ |
| ω₄ |

It is now evident that while ω¹ = ωA, [=ω]¹ = A⁻¹[=ω]

[ω, s] The vector-product of the four-vector ω and s may be represented by the combination

[ωs] = [=ω]s - ṡω

It is now easy to verify the formula f¹ = A⁻¹fA. Supposing for the sake of simplicity that f represents the vector-product of two four-vectors ω, s, we have

Now remembering that generally

f = ρφ + ρ*φ*.

Where ρ, ρ* are scalar quantities, φ, φ* are two mutually perpendicular unit planes, there is no difficulty in seeming that

Note 15.
The vector product (wf). (P. 36).

This represents the vector product of a four-vector and a six-vector. Now as combinations of this type are of frequent occurrence in this paper, it will be better to form an idea of their geometrical meaning. The following is taken from the above mentioned paper of Sommerfeld.

“We can also form a vectorial combination of a four-vector and a six-vector, giving us a vector of the third type. If the six-vector be of a special type, i.e., a piece of plane, then this vector of the third type denotes the parallelopiped formed of this four-vector and the complement of this piece of plane. In the general case, the product will be the geometric sum of two parallelopipeds, but it can always be represented by a four-vector of the 1st type. For two pieces of 3-space volumes can always be added together by the vectorial addition of their components. So by the addition of two 3-space volumes, we do not obtain a vector of a more general type, but one which can always be represented by a four-vector (loc. cit. p. 759). The state of affairs here is the same as in the ordinary vector calculus, where by the vector-multiplication of a vector of the first, and a vector of the second type (i.e., a polar vector), we obtain a vector of the first type (axial vector). The formal scheme of this multiplication is taken from the three-dimensional case.

We see easily that this coincides with the usual rule for the vector-product; e. g., for j = x.

Correspondingly let us define in the four-dimensional case the product (Pf) of any four-vector P and the six-vector f. The j-component (j = x, y, z, or l) is given by

Leaving aside the first column we obtain

[M. N. S.]

Note 16.
The electric-rest force. (Page 37.)

Note 17.
Operator “Lor” (§ 12, p. 41).

The operation | ∂/∂x₁ ∂/∂x₂ ∂/∂x₃ ∂/∂x₄ | which plays in four-dimensional mechanics a rôle similar to that of the operator (i∂/∂x, + j∂/∂y, + k∂/∂z = ▽) in three-dimensional geometry has been called by Minkowski ‘Lorentz-Operation’ or shortly ‘lor’ in honour of H. A. Lorentz, the discoverer of the theorem of relativity. Later writers have sometimes used the symbol □ to denote this operation. In the above-mentioned paper (Annalen der Physik, p. 649, Bd. 38) Sommerfeld has introduced the terms, Div (divergence), Rot (Rotation), Grad (gradient) as four-dimensional extensions of the corresponding three-dimensional operations in place of the general symbol lor. The physical significance of these operations will become clear when along with Minkowski’s method of treatment we also study the geometrical method of Sommerfeld. Minkowski begins here with the case of lor S, where S is a six-vector (space-time vector of the 2nd kind).

This being a complicated case, we take the simpler case of lor s,

where s is a four-vector = | s₁, s₂, s₃, s₄ |

The following geometrical method is taken from Sommerfeld.

Now if for ΔΣ we choose the four-dimensional parallelopiped with sides (dx₁, dx₂, dx₃, dx₄), we have then

Div S = ∂s₁/∂x₁ + ∂s₂/∂x₂ + ∂s₃/∂x₃ + ∂s₄/∂x₄ = lor S.

Where the integration in dσ is to be extended over the whole surface.

If now s is selected as the x-direction, Δs is then a three-dimensional parallelopiped with the sides dy, dz, dl, then we have

and generally

Hence the four-components of the four-vector lor S or Div. f is a four-vector with the components given on page 42.

∴ The vector of the third type represented by (Pf) is given by the geometrical sum of the two four-vectors of the first type PD and P*D*.

[M. N. S.]

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