Principle of Relativity

At the present time, different opinions are being held about the fundamental equations of Electro-dynamics for moving bodies. The Hertzian[9]forms must be given up, for it has appeared that they are contrary to many experimental results.

In 1895 H. A. Lorentz[10]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[11]the name of Lorentz-transformations. The covariance of these fundamental equations, when subjected to the Lorentz-transformation is a purely mathematical facti.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 withoutrecognizingany 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[12]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[13]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.[14]

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).

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 byeand the magnetic force bym, 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), whereidenotes √(-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 (x₁x₂x₃x₄) shall be called a space-time point.

Further letudenote 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, andxthe vector of “Electric Current” which we shall some across in§7and§8.

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 toe, and M tom. At every space time point (x,y,z,t) we shall have the equations[15]

(i) Curlm- (δe/δt) = ρu(ii) dive= ρ(iii) Curle+ δm/δt= 0(iv) div m = 0

(i) Curlm- (δe/δt) = ρu

(ii) dive= ρ

(iii) Curle+ δm/δt= 0

(iv) div m = 0

I shall now write (x₁x₂x₃x₄) for (x,y,z,t) and (ρ₁, ρ₂, ρ₃, ρ₄) for

Formula.

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

Further I shall write

f2 3,f3 1,f1 2,f1 4,f2 4,f3 4.

for

mx, my, mz, -iex, -iey, -iez.

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

3, 4),fk h= -fk h,

Therefore

f₃₂= -f₂₃,f₁₃= -f₃₁,f₂₁= -f₁₂f₄₁= -f₁₄,f₄₄= -f₂₄,f₄₃= -f₃₄

f₃₂= -f₂₃,f₁₃= -f₃₁,f₂₁= -f₁₂

f₄₁= -f₁₄,f₄₄= -f₂₄,f₄₃= -f₃₄

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

Formula."Formula A."

"Formula A."

On the other hand, the three equations comprised in (iii) and the (iv) equation multiplied by (i) becomes

Formula."Formula B."

"Formula B."

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).

It is well-known that by writing the equations i) to iv) in the symbol of vector calculus, we at once set in evidence an invariance (or rather a (covariance) of the system of equationsA)as well as ofB), when the co-ordinate system is rotated through a certain amount round the null-point. For example, if we take a rotation of the axes round the z-axis, through an amount φ, keeping e, m fixed in space, and introduce new variablesx₁′x₂′x₃′x₄′instead ofx₁x₂x₃x₄wherex′₁=x₁cos φ +x₂sin φ,x′₂= -x₁sin φ +x₂cos φ,x′₃=x₃,x′₄=x₄, and introduce magnitudes ρ′₁, ρ′₂, ρ′₃, ρ′₄, where ρ₁′ = ρ₁ cos φ + ρ₂ sin φ, ρ₂′ = - ρ₁ sin φ + ρ₂ cos φ andf′1 2, ... ...f′3 4, where

f′₂₃=f₂₃cos φ +f₃₁sin φ,f′₃₁= -f₂₃sin φ +f₃₁cos φ,f′₁₂=f₁₂,f′₁₄=f₁₄cos φ +f₂₄sin φ,f′₂₄= -f₁₄sin φ +f₂₄cos φ,f′₃₄=f₃₄3 4,f′k h= -fk h(h l k = 1, 2, 3, 4).

f′₂₃=f₂₃cos φ +f₃₁sin φ,

f′₃₁= -f₂₃sin φ +f₃₁cos φ,

f′₁₂=f₁₂,

f′₁₄=f₁₄cos φ +f₂₄sin φ,

f′₂₄= -f₁₄sin φ +f₂₄cos φ,

f′₃₄=f₃₄3 4,

f′k h= -fk h(h l k = 1, 2, 3, 4).

then out of the equations (A) would follow a corresponding system of dashed equations (A´) composed of the newly introduced dashed magnitudes.

So upon the ground of symmetry alone of the equations (A) and (B) concerning thesuffixes(1, 2, 3, 4), the theorem of Relativity, which was found out by Lorentz, follows without any calculation at all.

I will denote byiψ a purely imaginary magnitude, and consider the substitution

x₁′=x₁,x₂′=x₂,x₃′=x₃cosiψ +x₄siniψ, (1)x₄′´ = -x₃siniψ +x₄cosiψ,

x₁′=x₁,

x₂′=x₂,

x₃′=x₃cosiψ +x₄siniψ, (1)

x₄′´ = -x₃siniψ +x₄cosiψ,

Putting

Formula."(2)."

"(2)."

We shall have cosiψ = 1/√(1 -q²), siniψ =iq/√(1 -q²) where -1

Let us now writex′₁=x′,x′₂=y′,x′₃=z′,x′₄=it′(3)

then the substitution 1) takes the form

x′=x,y′=y,z′= (z-qt)/√(1 -q²),t′= (-qz+t)/√(1 -q²), (4)

the coefficients being essentially real.

If now in the above-mentioned rotation round the Z-axis, we replace 1, 2, 3, 4 throughout by 3, 4, 1, 2, and φ byiψ, we at once perceive that simultaneously, new magnitudes ρ′₁, ρ′₂, ρ′₃, ρ′₄, where

ρ′₁ = ρ₁, ρ′₂ = ρ₂, ρ′₃ = ρ₃ cosiψ + ρ₄ siniψ,ρ′₄ = - ρ₃ siniψ + ρ₄ cosiψ),

ρ′₁ = ρ₁, ρ′₂ = ρ₂, ρ′₃ = ρ₃ cosiψ + ρ₄ siniψ,

ρ′₄ = - ρ₃ siniψ + ρ₄ cosiψ),

andf′1 2...f′3 4, where

f′4 1=f4 1cosiψ +f1 3siniψ,f′1 3= -f4 1siniψ +f1 3cosiψ,f′3 4=f3 4,f′3 2=f3 2cosiψ +f4 2siniψ,f′4 2= -f3 2siniψ +f4 2cosiψ,f′1 2=f1 2,fk h= -f′k h,

f′4 1=f4 1cosiψ +f1 3siniψ,

f′1 3= -f4 1siniψ +f1 3cosiψ,

f′3 4=f3 4,

f′3 2=f3 2cosiψ +f4 2siniψ,

f′4 2= -f3 2siniψ +f4 2cosiψ,

f′1 2=f1 2,fk h= -f′k h,

must be introduced. Then the systems of equations in (A) and (B) are transformed into equations (A´), and (B´), the new equations being obtained by simply dashing the old set.

All these equations can be written in purely real figures, and we can then formulate the last result as follows.

If the real transformations 4) are taken, andx´y´z´t´be taken as a new frame of reference, then we shall have

(5) ρ´ = ρ [(-quz+ 1)/√(1 -q²)],ρ´uz´ = ρ[(uz-q)/√(1 -q²)],ρ´ux´ = ρux,ρ´uy´ = ρuy.(6)e´x´= (ex-qmy)/(√(1 -q²)),m´r´= (qex+my)/(√(1 -q²)),e´z´=ez.(7)m´x´= (mx-qey)/(√(1 -q²)),e´y´= (qmx+ey)/(√(1 -q²)),m´z´=mz.

(5) ρ´ = ρ [(-quz+ 1)/√(1 -q²)],ρ´uz´ = ρ[(uz-q)/√(1 -q²)],ρ´ux´ = ρux,ρ´uy´ = ρuy.

(5) ρ´ = ρ [(-quz+ 1)/√(1 -q²)],

ρ´uz´ = ρ[(uz-q)/√(1 -q²)],

ρ´ux´ = ρux,

ρ´uy´ = ρuy.

(6)e´x´= (ex-qmy)/(√(1 -q²)),m´r´= (qex+my)/(√(1 -q²)),e´z´=ez.

(6)e´x´= (ex-qmy)/(√(1 -q²)),

m´r´= (qex+my)/(√(1 -q²)),

e´z´=ez.

(7)m´x´= (mx-qey)/(√(1 -q²)),e´y´= (qmx+ey)/(√(1 -q²)),m´z´=mz.

(7)m´x´= (mx-qey)/(√(1 -q²)),

e´y´= (qmx+ey)/(√(1 -q²)),

m´z´=mz.

Then we have for these newly introduced vectorsu´,e´,m´(with componentsux´,uy´,uz´;ex´,ey´,ez´;mx´,my´,mz´), and the quantity ρ´ a series of equations I´), II´), III´), IV´) which are obtained from I), II), III), IV) by simply dashing the symbols.

We remark here thatex-qmy,ey+qmxare components of the vectore+ [vm], wherevis a vector in the direction of the positive Z-axis, and |v| =q, and [vm] is the vector product ofvandm; similarly -qex+my,mx+qeyare the components of the vectorm- [ve].

The equations 6) and 7), as they stand in pairs, can be expressed as.

e′x′+im′x′= (ex+imx) cosiψ + (ey+imy) siniψ,e′y′+im′y′= - (ex+imx) siniψ + (ey+imy) cosiψ,e′z′+im′z′=e′z+imz.

e′x′+im′x′= (ex+imx) cosiψ + (ey+imy) siniψ,

e′y′+im′y′= - (ex+imx) siniψ + (ey+imy) cosiψ,

e′z′+im′z′=e′z+imz.

If φ denotes any other real angle, we can form the following combinations:—

(e′x′+im′x′) cos. φ + (e′y″+im′y′) sin φ= (ex+imx) cos. (φ +iψ) + (ey+imy) sin (φ +iψ),= (e′x′+im′x′) sin φ + (e′y′+im′y′) cos. φ= - (ex+imx) sin (φ +iψ) + (ey+imy) cos. (φ +iψ).

(e′x′+im′x′) cos. φ + (e′y″+im′y′) sin φ

= (ex+imx) cos. (φ +iψ) + (ey+imy) sin (φ +iψ),

= (e′x′+im′x′) sin φ + (e′y′+im′y′) cos. φ

= - (ex+imx) sin (φ +iψ) + (ey+imy) cos. (φ +iψ).

The rôle which is played by the Z-axis in the transformation (4) can easily be transferred to any other axis when the system of axes are subjected to a transformation about this last axis. So we came to a more general law:—

Letvbe a vector with the componentsvx,vy,vz, and let |v| =q< 1. Byṽwe shall denote any vector which is perpendicular tov, and byrv,rṽwe shall denote components ofrin direction ofṽandv.

Instead of (x,y,z,t), new magnetudes (x′y′z′t′) will be introduced in the following way. If for the sake of shortness,ris written for the vector with the components (x,y,z) in the first system of reference,r′for the same vector with the components (x′y′z′) in the second system of reference, then for the direction ofv, we have

(10)r′v= (rv-qt)/√(1 -q²)

and for the perpendicular directionṽ,

(11)r′ṽ=rṽ

and further (12)t′= (-qrv+t)/√(1 -q²).

The notations (r′ṽ,r′v) are to be understood in the sense that with the directionsv, and every directionṽperpendicular tovin the system (x,y,z) are always associated the directions with the same direction cosines in the system (x′y′z′).

A transformation which is accomplished by means of (10), (11), (12) with the condition 0

If further ρ′ and the vectorsu′,e′,m′, in the system (x′y′z′) are so defined that,

(13) ρ′ = ρ[(-quv+ 1)/√(1 -q²)],ρ′u′v= ρ(uv-q)/√(1 -q²),ρ′uṽ= ρ′uv,

(13) ρ′ = ρ[(-quv+ 1)/√(1 -q²)],

ρ′u′v= ρ(uv-q)/√(1 -q²),

ρ′uṽ= ρ′uv,

further

(14) (e′+im′)ṽ= ((e+im) -i[v, (e+im])']ṽ)/√(1 -q²).(15) (e′+im′)v= (e+im) -i[u, (e+im)]v.

(14) (e′+im′)ṽ= ((e+im) -i[v, (e+im])']ṽ)/√(1 -q²).

(15) (e′+im′)v= (e+im) -i[u, (e+im)]v.

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

(16)rv= (r′v+qt′)/√(1 -q²),rṽ=r′ṽ,t= (qr′v+t′)/√(1 -q²),

(16)rv= (r′v+qt′)/√(1 -q²),

rṽ=r′ṽ,

t= (qr′v+t′)/√(1 -q²),

Now we shall make a very important observation about the vectorsuandu′. We can again introduce the indices 1, 2, 3, 4, so that we write (x₁′,x₂′,x₃′,x₄′) instead of (x′,y′,z′,it′) and ρ₁′, ρ₂′, ρ₃′, ρ₄′ instead of (ρ′u′{x′}, ρ′u′{y′}, ρ′u′{z′},iρ′).

Like the rotation round the Z-axis, the transformation (4), and more generally the transformations (10), (11), (12), are also linear transformations with the determinant + 1, so that

(17)x₁²+x₂²+x₃²+x₄²i. e.x²+y²+z²-t²,

is transformed into

x₁′²+x₂′²+x₃′²+x₄′²i. e.x′²+y′²+z′²-t′².

On the basis of the equations (13), (14), we shall have (ρ₁² + ρ₂² + ρ₃² + ρ₄²) = ρ²(1 -ux², -uy², -uz²) = ρ²(1 -u²) transformed into ρ²(1 -u²) or in other words,

(18) ρ√(1 -u²)

is an invariant in a Lorentz-transformation.

If we divide (ρ₁, ρ₂, ρ₃, ρ₄) by this magnitude, we obtain the four values (ω₁, ω₂, ω₃, ω₄) = (1/√(1 -u²))(ux,uy,uz,i) so that ω₁² + ω₂² + ω₃² + ω₄² = -1.

It is apparent that these four values are determined by the vectoruand inversely the vectoruof magnitude < 1 follows from the 4 values ω₁, ω₂, ω₃, ω₄; where (ω₁, ω₂, ω₃) are real, -iω₄ real and positive and condition (19) is fulfilled.

The meaning of (ω₁, ω₂, ω₃, ω₄) here is, that they are the ratios ofdx₁,dx₂,dx₃,dx₄to

(20) √(-(dx₁²+dx₂²+dx₃²+dx₄²)) =dt√(1 -u²).

The differentials denoting the displacements of matter occupying the spacetime point (x₁,x₂,x₃,x₄) to the adjacent space-time point.

After the Lorentz-transformation is accomplished the velocity of matter in the new system of reference for the same space-time point (x′y′z′t′) is the vectoru′with the ratiosdx′/dt′,dy′/dt′,dz′/dt′,dl′/dt′, as components.

Now it is quite apparent that the system of values

x₁= ω₁,x₂= ω₂,x₃= ω₃,x₄= ω₄

is transformed into the values

x₁′= ω₁′,x₂′= ω₂′,x₃′= ω₃′,x₄′= ω₄′

in virtue of the Lorentz-transformation (10), (11), (12).

The dashed system has got the same meaning for the velocityu′after the transformation as the first system of values has got forubefore transformation.

If in particular the vectorvof the special Lorentz-transformation be equal to the velocity vectoruof matter at the space-time point (x₁,x₂,x₃,x₄) then it follows out of (10), (11), (12) that

ω₁′ = 0, ω₂′ = 0, ω₃′ = 0, ω₄′ =i

Under these circumstances therefore, the corresponding space-time point has the velocityv′= 0 after the transformation, it is as if we transform to rest. We may call the invariant ρ√(1 -u²) the rest-density of Electricity.[16]

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 generalrelativity 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,

Formula.

the Determinant of the matrix is +1, all co-efficients without the index 4 occurring once are real, whilea₄₁,a₄₂,a₄₃, are purely imaginary, buta₄₄is real and > 0, andx₁²+x₂²+x₃²+x₄²transforms intox₁′²+x₂′²+x₃′²+x₄′². The operation shall be called a general Lorentz transformation.

(This notation, which is due to Dr. C. E. Cullis of the Calcutta University, has been used throughout instead of Minkowski’s notation,x₁=a₁₁x₁′+a₁₂x₂′+a₁₃x₃′+a₁₄x₄′.)

If we putx₁′=x′,x₂′=y′,x₃′=z′,x₄′=it′, then immediately there occurs a homogeneous linear transformation of (x,y,z,t) to (x′,y′,z′,t′) with essentially real co-efficients, whereby the aggregate -x²-y²-z²+t²transforms into -x′²-y′²-z′²+t′², and to every such system of valuesx,y,z,twith a positivet, for which this aggregate > 0, there always corresponds a positivet’; this last is quite evident from the continuity of the aggregatex,y,z,t.

The last vertical column of co-efficients has to fulfil the condition 22)a₁₄²+a₂₄²+a₃₄²+a₄₄²= 1.

Ifa₁₄=a₂₄=a₃₄= 0, thena₄₄= 1, and the Lorentz transformation reduces to a simple rotation of the spatial co-ordinate system round the world-point.

Ifa₁₄,a₂₄,a₃₄are not all zero, and if we puta₁₄:a₂₄:a₃₄:a₄₄=vx:vy:vz:i

q= √(vx² +vy² +vz²) < 1.

On the other hand, with every set of values ofa₁₄,a₂₄,a₃₄,a₄₄which in this way fulfil the condition 22) with real values ofvx,vy,vz, we can construct the special Lorentz transformation (16) with (a₁₄,a₂₄,a₃₄,a₄₄) as the last vertical column,—and then every Lorentz-transformation with the same last vertical column (a₁₄,a₂₄,a₃₄,a₄₄) can be supposed to be composed of the special Lorentz-transformation, and a rotation of the spatial co-ordinate system round the null-point.

The totality of all Lorentz-Transformations forms a group. Under a space-time vector of the 1st kind shall be understood a system of four magnitudes (ρ₁, ρ₂, ρ₃, ρ₄) with the condition that in case of a Lorentz-transformation it is to be replaced by the set (ρ₁′, ρ₂′, ρ₃′, ρ₄′), where these are the values of (x₁′,x₂′,x₃′,x₄′), obtained by substituting (ρ₁, ρ₂, ρ₃, ρ₄) for (x₁,x₂,x₃,x₄) in the expression (21).

Besides the time-space vector of the 1st kind (x₁,x₂,x₃,x₄) we shall also make use of another space-time vector of the first kind (y₁,y₂,y₃,y₄), and let us form the linear combination

(23)f₂₃(x₂y₃-x₃y₂) +f₃₁(x₃y₁-x₁y₃) +f₁₂(x₁y₂-x₂y₁) +f₁₄(x₁y₄-x₄y₁) +f₂₄(x₂y₄-x₄y₂) +f₃₄(x₃y₄-x₄y₃)

(23)f₂₃(x₂y₃-x₃y₂) +f₃₁(x₃y₁-x₁y₃) +f₁₂(x₁y₂

-x₂y₁) +f₁₄(x₁y₄-x₄y₁) +f₂₄(x₂y₄-x₄y₂) +

f₃₄(x₃y₄-x₄y₃)

with six coefficientsf₂₃--f₃₄. Let us remark that in the vectorial method of writing, this can be constructed out of the four vectors.

x₁,x₂,x₃;y₁,y₂,y₃;f₂₃,f₃₁,f₁₂;f₁₄,f₂₄,f₃₄and the constantsx₄andy₄, at the same time it is symmetrical with regard the indices (1, 2, 3, 4).

If we subject (x₁,x₂,x₃,x₄) and (y₁,y₂,y₃,y₄) simultaneously to the Lorentz transformation (21), the combination (23) is changed to:

(24)f₂₃′(x₂′y₃′-x₃′y₂′) +f₃₁(x₃′y₁′-x₁′y₃′) +f₁₂(x₁′y₂′-x₂′y₁′) +f₁₄′(x₁′y₄′) -x₄′y₁′) +f₂₄′(x₂′y₄′-x₄′y₂′) +f₃₄′(x₃′y₄′-x₄′y₃′),

(24)f₂₃′(x₂′y₃′-x₃′y₂′) +f₃₁(x₃′y₁′-x₁′y₃′) +f₁₂

(x₁′y₂′-x₂′y₁′) +f₁₄′(x₁′y₄′) -x₄′y₁′) +f₂₄′(x₂′y₄′

-x₄′y₂′) +f₃₄′(x₃′y₄′-x₄′y₃′),

where the coefficientsf₂₃′,f₃₁′,f₁₂′,f₁₄′,f₂₄′,f₃₄′, depend solely on (f₂₃f₂₄) and the coefficientsa₁₁...a₄₄.

We shall define a space-time Vector of the 2nd kind as a system of six-magnitudesf₂₃,f₃₁...f₃₄, with the condition that when subjected to a Lorentz transformation, it is changed to a new systemf₂₃′... f₃₄, ... which satisfies the connection between (23) and (24).

I enunciate in the following manner the general theorem of relativity corresponding to the equations (I)-(iv),—which are the fundamental equations for Äther.

Ifx,y,z,it(space co-ordinates, and timeit) is subjected to a Lorentz transformation, and at the same time (pux,puy,puz,iρ) (convection-current, and charge density ρi) is transformed as a space time vector of the 1st kind, further (mx,my,mz, -iex, -iey, -iez) (magnetic force, and electric induction × (-i) is transformed as a space time vector of the 2nd kind, then the system of equations (I), (II), and the system of equations (III), (IV) transforms into essentially corresponding relations between the corresponding magnitudes newly introduced into the system.

These facts can be more concisely expressed in these words: the system of equations (I and II) as well as the system of equations (III) (IV) are covariant in all cases of Lorentz-transformation, where (ρu,iρ) is to be transformed as a space time vector of the 1st kind, (m-ie) is to be treated as a vector of the 2nd kind, or more significantly,—

(ρu,iρ) is a space time vector of the 1st kind, (m-ie)[17]is a space-time vector of the 2nd kind.

I shall add a few more remarks here in order to elucidate the conception of space-time vector of the 2nd kind. Clearly, the following are invariants for such a vector when subjected to a group of Lorentz transformation.

(i)m²-e²=f₂₃²+f₃₁²+f₁₂²+f₁₄²+f₂₄²+f₂₄²me=i(f₂₃f₁₄+f₃₁f₂₄+f₁₂f₃₄).

(i)m²-e²=f₂₃²+f₃₁²+f₁₂²+f₁₄²+f₂₄²+f₂₄²

me=i(f₂₃f₁₄+f₃₁f₂₄+f₁₂f₃₄).

A space-time vector of the second kind (m-ie), where (mande) are real magnitudes, may be called singular, when the scalar square (m-ie)² = 0,iem²-e²= 0, and at the same time (m e) = 0,iethe vectormandeare equal and perpendicular to each other; when such is the case, these two properties remain conserved for the space-time vector of the 2nd kind in every Lorentz-transformation.

If the space-time vector of the 2nd kind is not singular, we rotate the spacial co-ordinate system in such a manner that the vector-product [me] coincides with the Z-axis,i.e.mx= 0,ex= 0. Then

(mx, -i ex)² + (my, -i ey)² ≠ 0.

Therefore (ey+i my)/(ex+i ex) is different from +i, and we can therefore define a complex argument (φ +iψ) in such a manner that

tan (φ +iψ)ey+i my= ---------------ex+i mx

tan (φ +iψ)

ey+i my= ---------------ex+i mx

ey+i my

= ---------------

ex+i mx

If then, by referring back to equations (9), we carry out the transformation (1) through the angle ψ and a subsequent rotation round the Z-axis through the angle φ, we perform a Lorentz-transformation at the end of whichmy= 0,ey= 0, and thereforemandeshall both coincide with the new Z-axis. Then by means of the invariantsm²-e², (me) the final values of these vectors, whether they are of the same or of opposite directions, or whether one of them is equal to zero, would be at once settled.

By the Lorentz transformation, we are allowed to effect certainchangesof 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.[18]In fact, the following considerations will prove conclusive.

Let a reference system (x,y,z,t) for space time points (events) be somehow known. Now if a space point A (x₀,y₀,z₀) the timet₀be compared with a space point P (x,y,z) at the timet, and if the difference of timet-t₀, (lett>t₀) be less than the length A Pi.e.less than the time required for the propagation of light from A to P, and ifq= (t-t₀)/(A P) < 1, then by a special Lorentz transformation, in which A P is taken as the axis, and which has the momentq, we can introduce a time parametert′, which (see equation 11, 12, § 4) has got the same valuet′=0for both space-time points (A,t₀), and (P, t). So the two events can now be comprehended to be simultaneous.

Further, let us take at the same timet₀= 0, two different space-points A, B, or three space-points (A, B, C) which are not in the same space-line, and compare therewith a space point P, which is outside the line A B, or the plane A B C, at another timet, and let the time differencet-t₀(t >t₀) be less than the time which light requires for propagation from the line A B, or the plane (A B C) to P. Let q be the quotient of (t-t₀) by the second time. Then if a Lorentz transformation is taken in which the perpendicular from P on A B, or from P on the plane A B C is the axis, and q is the moment, then all the three (or four) events (A,t₀), (B,t₀), (C,t₀) and (P, t) are simultaneous.

If four space-points, which do not lie in one plane, are conceived to be at the same timet₀, 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.

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 inductione, the magnetic forcem, 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 vectorv, the velocity of matter.

The relations in question can be divided into two classes.

Firstly—those equations, which,—whenv, the velocity of matter is given as a function of (x,y,z,t),—lead us to a knowledge of other magnitude as functions ofx,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 vectoruas functions of (x,y,z,t).

For the case of bodies at rest,i.e.whenu(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:—

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