The Principle of Relativity

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

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,

Let us consider a beam travelling along the x-axis, with apparent velocity v (i.e., velocity with respect to the fixed ether) in medium moving with velocity u_x = u in the same direction.

Then if the electric and magnetic vectors are proportional to e^{iA(x - vt)}, we have

Since D = KE and B = μH, we have

Multiplying (1·23) by (2·23)

Hence (v - u)² = c²/μk = v₀²

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.

Consider a slab of matter moving with velocity u_x along the x-axis, then even in a stationary field of electrostatic polarisation, that is, for a field in which δD/δt = 0, there will be some change in the polarisation of the body due to its motion, given by u_x(δD/δx). Hence we must add this term to a purely temporal rate of change δD/δt. Doing this we immediately arrive at equations (1·21) and (2·21) for the special case considered there.

Thus the Hertz-Heaviside form of field equations gives unity as the value for the Fresnelian convection co-efficient. It has been shown in the historical introduction how this is entirely at variance with the observed optical facts. As a matter of fact, Larmor has shown (Aether and Matter) that 1 - 1/μ² is not only sufficient but is also necessary, in order to explain experiments of the Arago prism type.

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.

Blondlot (Comptes Rendus, 1901) passed a current of air in a steady magnetic field H_y, (H = H_z = 0). If this current of air moves with velocity u_x along the x-axis, an electromotive force would be set up along the z-axis, due to the relative motion of matter and magnetic tubes of induction. A pair of plates at z = ±a, will be charged up with density ρ = D_z = KE = K. u_s H_y/c. But Blondlot failed to detect any such effect.

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.]

Note 3.

See Lorentz, Theory of Electrons (English edition), § 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 ∂e_x/∂x + ∂e_y/∂y + ∂e_z/∂z = ρ (2·1)

Substituting x₁, x₂, x₃, x₄ and x, y, z, and iτ; and ρ₁, ρ₂, ρ₃, ρ₄ for ρν_x, ρν_y, ρν_z, 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.

The light emitted by the star when at the position A will be received by the observer after a time, Δ/(c + u) while the light emitted by the star when at the position B will be received after a time Δ/(c - u). Let T be the real half-period of the star. Then the observed half-period from B to A is approximately T - 2Δu/c² and from A to B is T + 2Δu/c². Now if 2uΔ/c² be comparable to T, then it is impossible that the observations should satisfy Kepler’s Law. In most of the spectroscopic binary stars, 2uΔ/c² are not only of the same order as T, but are mostly much larger. For example, if u = 100 km/sec, T = 8 days, Δ/c = 33 years (corresponding to an annual parallax of ·1″), then T - 2uΔ/c² = 0. The existence of the Spectroscopic binaries, and the fact that they follow Kepler’s Law is therefore a proof that c is not affected by the motion of the source.

In a later memoir, replying to the criticisms of Freundlich and Günthick that an apparent eccentricity occurs in the motion proportional to kuΔ₀, u₀ being the maximum value of u, the velocity of light emitted being

Prof. de Sitter admits the validity of the criticisms. But he remarks that an upper value of k may be calculated from the observations of the double star β-Aurigae. For this star, the parallax π = ·014″, e = ·005, u₀ = 110 km/sec, T = 3·96,

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 u_x, u_y, v_x, v_y of the four vectors in the combination

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

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,

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:—

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

If we multiply these equations by v_l, u_l, v_s, and subtract the second from the first, the fourth from the third we obtain

multiplying these equations by v_x^* . u_x^*, or by v_y^* . u_y^*, we obtain

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.

The general six-vector f is composed from the vectors φ, φ^* in the following way:—

ρ and ρ^* denoting the contents of the pieces of mutually perpendicular planes composing f. The “conjugate Vector” f^* (or it may be called the complement of f) is obtained by interchanging ρ and ρ^*.

We have

We can verify that

and f² = ρ² + ρ^*², (ff^*) = 2ρρ^*.

| f |² and (ff^*) may be said to be invariants of the six vectors, for their values are independent of the choice of the system of co-ordinates.

[M. N. S.]

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.

Thus the velocity of light is the absolute maximum velocity. We shall now see the consequences of admitting a velocity W > c.

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.

Now we can always choose v in such a way that Wv is greater than c², since Wv is > c² if (μ + λ)c - μλ is > 0, that is, if μ + λ > μλ/c; which can always be satisfied by a suitable choice of λ.

Thus for W > c we can always choose λ in such a way as to make Wv > c², i.e., λ - Wv/c² negative. But W - v is always positive. Hence with W > c, we can always make t′, the time from A to B in equation (1) “negative.” That is, the signal starting from A will reach B (as observed in system S′) in less than no time. Thus the effect will be perceived before the cause commences to act, i.e., the future will precede the past. Which is absurd. Hence we conclude that W > c is an impossibility, there can be no velocity greater than that of light.

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.]