Further Notes on the Principle of Relativity

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Principle of Relativity

INTRODUCTION.

At the present time, different opinions are being held about the fundamental equations of Electro-dynamics for moving bodies. The Hertzian forms must be given up, for it has appeared that they are contrary to many experimental results.

In 1895 H. A. Lorentz published his theory of optical and electrical phenomena in moving bodies; this theory was based upon the atomistic conception (vorstellung) of electricity, and on account of its great success appears to have justified the bold hypotheses, by which it has been ushered into existence. In his theory, Lorentz proceeds from certain equations, which must hold at every point of “Äther”; then by forming the average values over “Physically infinitely small” regions, which however contain large numbers of electrons, the equations for electro-magnetic processes in moving bodies can be successfully built up.

In particular, Lorentz’s theory gives a good account of the non-existence of relative motion of the earth and the luminiferous “Äther”; it shows that this fact is intimately connected with the covariance of the original equation, when certain simultaneous transformations of the space and time co-ordinates are effected; these transformations have therefore obtained from H. Poincare the name of Lorentz-transformations. The covariance of these fundamental equations, when subjected to the Lorentz-transformation is a purely mathematical fact i.e. not based on any physical considerations; I will call this the Theorem of Relativity; this theorem rests essentially on the form of the differential equations for the propagation of waves with the velocity of light.

Now without recognizing any hypothesis about the connection between “Äther” and matter, we can expect these mathematically evident theorems to have their consequences so far extended—that thereby even those laws of ponderable media which are yet unknown may anyhow possess this covariance when subjected to a Lorentz-transformation; by saying this, we do not indeed express an opinion, but rather a conviction,—and this conviction I may be permitted to call the Postulate of Relativity. The position of affairs here is almost the same as when the Principle of Conservation of Energy was postulated in cases, where the corresponding forms of energy were unknown.

Now if hereafter, we succeed in maintaining this covariance as a definite connection between pure and simple observable phenomena in moving bodies, the definite connection may be styled ‘the Principle of Relativity.’

These differentiations seem to me to be necessary for enabling us to characterise the present day position of the electro-dynamics for moving bodies.

H. A. Lorentz has found out the “Relativity theorem” and has created the Relativity-postulate as a hypothesis that electrons and matter suffer contractions in consequence of their motion according to a certain law.

A. Einstein has brought out the point very clearly, that this postulate is not an artificial hypothesis but is rather a new way of comprehending the time-concept which is forced upon us by observation of natural phenomena.

The Principle of Relativity has not yet been formulated for electro-dynamics of moving bodies in the sense characterized by me. In the present essay, while formulating this principle, I shall obtain the fundamental equations for moving bodies in a sense which is uniquely determined by this principle.

But it will be shown that none of the forms hitherto assumed for these equations can exactly fit in with this principle.

We would at first expect that the fundamental equations which are assumed by Lorentz for moving bodies would correspond to the Relativity Principle. But it will be shown that this is not the case for the general equations which Lorentz has for any possible, and also for magnetic bodies; but this is approximately the case (if neglect the square of the velocity of matter in comparison to the velocity of light) for those equations which Lorentz hereafter infers for non-magnetic bodies. But this latter accordance with the Relativity Principle is due to the fact that the condition of non-magnetisation has been formulated in a way not corresponding to the Relativity Principle; therefore the accordance is due to the fortuitous compensation of two contradictions to the Relativity-Postulate. But meanwhile enunciation of the Principle in a rigid manner does not signify any contradiction to the hypotheses of Lorentz’s molecular theory, but it shall become clear that the assumption of the contraction of the electron in Lorentz’s theory must be introduced at an earlier stage than Lorentz has actually done.

In an appendix, I have gone into discussion of the position of Classical Mechanics with respect to the Relativity Postulate. Any easily perceivable modification of mechanics for satisfying the requirements of the Relativity theory would hardly afford any noticeable difference in observable processes; but would lead to very surprising consequences. By laying down the Relativity-Postulate from the outset, sufficient means have been created for deducing henceforth the complete series of Laws of Mechanics from the principle of conservation of Energy alone (the form of the Energy being given in explicit forms).

NOTATIONS.

Let a rectangular system (x, y, z, t,) of reference be given in space and time. The unit of time shall be chosen in such a manner with reference to the unit of length that the velocity of light in space becomes unity.

Although I would prefer not to change the notations used by Lorentz, it appears important to me to use a different selection of symbols, for thereby certain homogeneity will appear from the very beginning. I shall denote the vector electric force by E, the magnetic induction by M, the electric induction by e and the magnetic force by m, so that (E, M, e, m) are used instead of Lorentz’s (E, B, D, H) respectively.

I shall further make use of complex magnitudes in a way which is not yet current in physical investigations, i.e., instead of operating with (t), I shall operate with (i t), where i denotes √(-1). If now instead of (x, y, z, i t), I use the method of writing with indices, certain essential circumstances will come into evidence; on this will be based a general use of the suffixes (1, 2, 3, 4). The advantage of this method will be, as I expressly emphasize here, that we shall have to handle symbols which have apparently a purely real appearance; we can however at any moment pass to real equations if it is understood that of the symbols with indices, such ones as have the suffix 4 only once, denote imaginary quantities, while those which have not at all the suffix 4, or have it twice denote real quantities.

An individual system of values of (x, y, z, t) i. e., of (x₁ x₂ x₃ x₄) shall be called a space-time point.

Further let u denote the velocity vector of matter, ε the dielectric constant, μ the magnetic permeability, σ the conductivity of matter, while ρ denotes the density of electricity in space, and x the vector of “Electric Current” which we shall some across in §7 and §8.

PART I

§ 2.
The Limiting Case.
The Fundamental Equations for Äther.

By using the electron theory, Lorentz in his above mentioned essay traces the Laws of Electro-dynamics of Ponderable Bodies to still simpler laws. Let us now adhere to these simpler laws, whereby we require that for the limiting case ε = 1, μ = 1, σ = 0, they should constitute the laws for ponderable bodies. In this ideal limiting case ε = 1, μ = 1, σ = 0, E will be equal to e, and M to m. At every space time point (x, y, z, t) we shall have the equations

I shall now write (x₁ x₂ x₃ x₄) for (x, y, z, t) and (ρ₁, ρ₂, ρ₃, ρ₄) for

i.e. the components of the convection current ρu, and the electric density multiplied by √ -1

Further I shall write

for

i.e., the components of m and (-i.e.) along the three axes; now if we take any two indices (h. k) out of the series

Therefore

Then the three equations comprised in (i), and the equation (ii) multiplied by i becomes

By means of this method of writing we at once notice the perfect symmetry of the 1st as well as the 2nd system of equations as regards permutation with the indices, (1, 2, 3, 4).

§ 4. Special Lorentz Transformation.

Then it follows that the equations I), II), III), IV) are transformed into the corresponding system with dashes.

The solution of the equations (10), (11), (12) leads to

§ 5. Space-time Vectors.
Of the 1st and 2nd kind.

If we take the principal result of the Lorentz transformation together with the fact that the system (A) as well as the system (B) is covariant with respect to a rotation of the coordinate-system round the null point, we obtain the general relativity theorem. In order to make the facts easily comprehensible, it may be more convenient to define a series of expressions, for the purpose of expressing the ideas in a concise form, while on the other hand I shall adhere to the practice of using complex magnitudes, in order to render certain symmetries quite evident.

Let us take a linear homogeneous transformation,

§ 6. Concept of Time.

By the Lorentz transformation, we are allowed to effect certain changes of the time parameter. In consequence of this fact, it is no longer permissible to speak of the absolute simultaneity of two events. The ordinary idea of simultaneity rather presupposes that six independent parameters, which are evidently required for defining a system of space and time axes, are somehow reduced to three. Since we are accustomed to consider that these limitations represent in a unique way the actual facts very approximately, we maintain that the simultaneity of two events exists of themselves. In fact, the following considerations will prove conclusive.

If four space-points, which do not lie in one plane, are conceived to be at the same time t₀, then it is no longer permissible to make a change of the time parameter by a Lorentz-transformation, without at the same time destroying the character of the simultaneity of these four space points.

To the mathematician, accustomed on the one hand to the methods of treatment of the poly-dimensional manifold, and on the other hand to the conceptual figures of the so-called non-Euclidean Geometry, there can be no difficulty in adopting this concept of time to the application of the Lorentz-transformation. The paper of Einstein which has been cited in the Introduction, has succeeded to some extent in presenting the nature of the transformation from the physical standpoint.

PART II. ELECTRO-MAGNETIC PHENOMENA.
§ 7. Fundamental Equations for bodies at rest.

After these preparatory works, which have been first developed on account of the small amount of mathematics involved in the limiting case ε = 1, μ = 1, σ = 0, let us turn to the electro-magnetic phenomena in matter. We look for those relations which make it possible for us—when proper fundamental data are given—to obtain the following quantities at every place and time, and therefore at every space-time point as functions of (x, y, z, t):—the vector of the electric force E, the magnetic induction M, the electrical induction e, the magnetic force m, the electrical space-density ρ, the electric current s (whose relation hereafter to the conduction current is known by the manner in which conductivity occurs in the process), and lastly the vector v, the velocity of matter.

The relations in question can be divided into two classes.

Firstly—those equations, which,—when v, the velocity of matter is given as a function of (x, y, z, t),—lead us to a knowledge of other magnitude as functions of x, y, z, t—I shall call this first class of equations the fundamental equations—

Secondly, the expressions for the ponderomotive force, which, by the application of the Laws of Mechanics, gives us further information about the vector u as functions of (x, y, z, t).

For the case of bodies at rest, i.e. when u (x, y, z, t) = 0 the theories of Maxwell (Heaviside, Hertz) and Lorentz lead to the same fundamental equations. They are;—

(1) The Differential Equations:—which contain no constant referring to matter:—

(2) Further relations, which characterise the influence of existing matter for the most important case to which we limit ourselves i.e. for isotopic bodies;—they are comprised in the equations

By employing a modified form of writing, I shall now cause a latent symmetry in these equations to appear. I put, as in the previous work,

x₁ = x, x₂ = y, x₃ = z, x₄ = it,

§ 8. The Fundamental Equations.

We are now in a position to establish in a unique way the fundamental equations for bodies moving in any manner by means of these three axioms exclusively.

The first Axion shall be,—

When a detached region of matter is at rest at any moment, therefore the vector u is zero, for a system (x, y, z, t)—the neighbourhood may be supposed to be in motion in any possible manner, then for the space-time point x, y, z, t, the same relations (A) (B) (V) which hold in the case when all matter is at rest, shall also hold between ρ, the vectors C, e, m, M, E and their differentials with respect to x, y, z, t. The second axiom shall be:—

Every velocity of matter is < 1, smaller than the velocity of propagation of light.

The fundamental equations are of such a kind that when (x, y, z, it) are subjected to a Lorentz transformation and thereby (m - ie) and (M - iE) are transformed into space-time vectors of the second kind, (C, iρ) as a space-time vector of the 1st kind, the equations are transformed into essentially identical forms involving the transformed magnitudes.

Shortly I can signify the third axiom as:—

(m, -ie), and (M, -iE) are space-time vectors of the second kind, (C, ip) is a space-time vector of the first kind.

This axiom I call the Principle of Relativity.

In fact these three axioms lead us from the previously mentioned fundamental equations for bodies at rest to the equations for moving bodies in an unambiguous way.

According to the second axiom, the magnitude of the velocity vector | u | is < 1 at any space-time point. In consequence, we can always write, instead of the vector u, the following set of four allied quantities

§ 9. The Fundamental Equations in Lorentz’s Theory.

Let us now see how far the fundamental equations assumed by Lorentz correspond to the Relativity postulate, as defined in §8. In the article on Electron-theory (Ency., Math., Wiss., Bd. V. 2, Art 14) Lorentz has given the fundamental equations for any possible, even magnetised bodies (see there page 209, Eqn XXX′, formula (14) on page 78 of the same (part).

Lorentz’s E, D, H are here denoted by E, M, e, m while J denotes the conduction current.

and thus comes out to be in a different form than (1) here. Therefore for magnetised bodies, Lorentz’s equations do not correspond to the Relativity Principle.

§11. Typical Representations of the Fundamental Equations.

In the statement of the fundamental equations, our leading idea had been that they should retain a covariance of form, when subjected to a group of Lorentz-transformations. Now we have to deal with ponderomotive reactions and energy in the electro-magnetic field. Here from the very first there can be no doubt that the settlement of this question is in some way connected with the simplest forms which can be given to the fundamental equations, satisfying the conditions of covariance. In order to arrive at such forms, I shall first of all put the fundamental equations in a typical form which brings out clearly their covariance in case of a Lorentz-transformation. Here I am using a method of calculation, which enables us to deal in a simple manner with the space-time vectors of the 1st, and 2nd kind, and of which the rules, as far as required are given below.

these elements being formed by combination of the horizontal rows of A with the vertical columns of B. For such a point, the associative law (AB)S = A(BS) holds, where S is a third matrix which has got as many horizontal rows as B (or AB) has got vertical columns.

For the transposed matrix of C = BA, we have Ċ = ḂĀ

3⁰. We shall have principally to deal with matrices with at most four vertical columns and for horizontal rows.

As a unit matrix (in equations they will be known for the sake of shortness as the matrix 1) will be denoted the following matrix (4 × 4 series) with the elements.

i.e. We shall have a 4 × 4 series matrix in which all the elements except those on the diagonal from left up to right down are zero, and the elements in this diagonal agree with each other, and are each equal to the above mentioned combination in (36).

4⁰. A linear transformation

xh = αh1 x₁′ + αh2 x₂′ + αh3 x₃′ + αh4 x₄′ (h = 1,2,3,

which is accomplished by the matrix

By the transformation A, the expression

x²₁ + x²₂ + x²₃ + x²₄ is changed into the quadratic for m ∑ αhk xh′ xk′,

A has to correspond to the following relation, if transformation (38) is to be a Lorentz-transformation. For the determinant of A) it follows out of (39) that (Det A)² = 1, or Det A = ± 1.

A⁻¹ = Ā,

i.e. the reciprocal matrix of A is equivalent to the transposed matrix of A.

For A as Lorentz transformation, we have further Det A = +1, the quantities involving the index 4 once in the subscript are purely imaginary, the other co-efficients are real, and a₄₄ > 0.

5⁰. A space time vector of the first kind which s represented by the 1 × 4 series matrix,

is to be replaced by sA in case of a Lorentz transformation

After these preparatory works let us engage ourselves with the equations (C,) (D,) (E) by means which the constants ε μ, σ will be introduced.

Instead of the space vector u, the velocity of matter, we shall introduce the space-time vector of the first kind ω with the components.

I shall call the space-time vector Φ of the first kind as the Electric Rest Force.

The vector ψ is perpendicular to ω; we can call it the Magnetic rest-force.

I shall call Ω, which is a space-time vector 1st kind the Rest-Ray.

This formula contains four equations, of which the fourth follows from the first three, since this is a space-time vector which is perpendicular to ω.

Lastly, we shall transform the differential equations (A) and (B) into a typical form.

§12. The Differential Operator Lor.

with the condition that in case of a Lorentz transformation it is to be replaced by ĀSA, may be called a space-time matrix of the II kind. We have examples of this in:—

1) the alternating matrix f, which corresponds to the space-time vector of the II kind,—

lastly in a multiple L of the unit matrix of 4 × 4 series in which all the elements in the principal diagonal are equal to L, and the rest are zero.

We shall have to do constantly with functions of the space-time point (x, y, z, it), and we may with advantage

employ the 1 × 4 series matrix, formed of differential symbols,—

For this matrix I shall use the shortened from “lor.”

Then if S is, as in (62), a space-time matrix of the II kind, by lor S′ will be understood the 1 × 4 series matrix

| K₁ K₂ K₃ K₄ |

Finally let us enquire about the laws which lead to the determination of the vector ω as a function of (x, y, z, t.) In these investigations, the expressions which are obtained by the multiplication of two alternating matrices

§ 14. The Ponderomotive Force.

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