§12. The Differential Operator Lor.
A 4 × 4 series matrix 62) S = | S₁₁ S₁₂ S₁₃ S₁₄ | = | Skh |
| S₂₁ S₂₂ S₂₃ S₂₄ |
| S₃₁ S₃₂ S₃₃ S₃₄ |
| S₄₁ S₄₂ S₄₃ S₄₄ |
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,—
2) the product fF of two such matrices, for by a transformation A, it is replaced by (A⁻¹fA·A⁻¹FA) = A⁻¹fFA,
3) further when (ω₁, ω₂, ω₃, ω₄) and (Ω₁, Ω₂, Ω₃, Ω₄) are two space-time vectors of the 1st kind, the 4 × 4 matrix with the element Shk = ωhΩk,
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,—
| ∂/∂x, ∂/∂y, ∂/∂z, ∂/i∂t,|
or (63) | ∂/∂x₁ ∂/∂x₂ ∂/∂x₃ ∂/∂x₄ |
For this matrix I shall use the shortened from “lor.”[[25]]
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 |
where Kk = ∂S1k/∂x₁ + ∂S2k/∂x₂ + ∂S3k/∂x₃ + ∂S4h/∂x₄.
When by a Lorentz transformation A, a new reference system (x′₁ x′₂ x′₃ x₄) is introduced, we can use the operator
lor′ = | ∂/∂x₁′ ∂/∂x₂′ ∂/∂x₃′ ∂/∂x₄′ |
Then S is transformed to S′= Ā S A = | S′hk |, so by lor 'S′ is meant the 1 × 4 series matrix, whose element are
K’k = ∂S′1k/∂x₁′ + ∂S′2k/∂x₂′
+ ∂S′3k/∂x₃′ + ∂S′4k/∂x₄′.
Now for the differentiation of any function of (x y z t) we have the rule ∂/∂xk′ = ∂/∂x₁ ∂x₁/∂xk′ + ∂/∂x₂ ∂x₂/∂xk′ + ∂/∂x₃ ∂x₃/∂xk′ + ∂/∂x₄ ∂x₄/∂xk′ = ∂/∂x₁ a1k + ∂/∂x₂ a2k + ∂/∂x₃ a3k + ∂/∂x₄ a4k.
so that, we have symbolically lor′ = lor A.
Therefore it follows that
lor ′S′ = lor (A A⁻¹ SA) = (lor S)A.
i.e., lor S behaves like a space-time vector of the first kind.
If L is a multiple of the unit matrix, then by lor L will be denoted the matrix with the elements
| ∂L/∂x₁ ∂L/∂x₂ ∂L/∂x₃ ∂L/∂x₄ |
If s is a space-time vector of the 1st kind, then
lor ṡ = ∂s₁/∂x₁ + ∂s₂/∂x₂ + ∂s₃/∂x₃ + ∂s₄/∂x₄.
In case of a Lorentz transformation A, we have
lor ′ṡ′ = lor A. Ās = lor s.
i.e., lor s is an invariant in a Lorentz-transformation.
In all these operations the operator lor plays the part of a space-time vector of the first kind.
If f represents a space-time vector of the second kind,—lor f denotes a space-time vector of the first kind with the components
∂f₁₂/∂x₂ + ∂f₁₃/∂x₃ + ∂f₁₄/∂x₄,
∂f₂₁/∂x₁ + ∂f₂₃/∂x₃ + ∂f₂₄/∂x₄,
∂f₃₁/∂x₁ + ∂f₃₂/∂x₂ + ∂f₃₄/∂x₄,
∂f₄₁/∂x₁ + ∂f₄₂/∂x₂ + ∂f₄₃/∂x₃
So the system of differential equations (A) can be expressed in the concise form
{A} lor f = -s,
and the system (B) can be expressed in the form
{B} log F* = 0.
Referring back to the definition (67) for log ṡ, we find that the combinations lor ([=(lor f)=]), and lor ([=(lor F*)]) vanish identically, when f and F* are alternating matrices. Accordingly it follows out of {A}, that
(68) (∂s₁/∂x₁) + (∂s₂/∂x₂) + (∂s₃/∂x₃) + (∂s₄/∂x₄) = 0,
while the relation
(69) lor (lor F*) = 0,
signifies that of the four equations in {B}, only three represent independent conditions.
I shall now collect the results.
Let ω denote the space-time vector of the first kind
(u/√(1 - u²}), i/√(1 - u²))
(u = velocity of matter),
F the space-time vector of the second kind (M,-iE)
(M = magnetic induction, E = Electric force,
f the space-time vector of the second kind (m,-ie)
(m = magnetic force, e = Electric Induction.
s the space-time vector of the first kind (C, iρ)
(ρ = electrical space-density, C - ρu = conductivity current,
ε = dielectric constant, μ = magnetic permeability,
σ = conductivity,
then the fundamental equations for electromagnetic processes in moving bodies are[[26]]
{A} lor f = -s
{B} log F* = 0
{C} ωf = εωF
{D} ωF* = μωf*
{E} s + (ωṡ), w = - σωF.
ω ῶ = -1, and ωF, ωf, ωF*, ωf*, s + (ωs)ω which are space-time vectors of the first kind are all normal to ω, and for the system {B}, we have
lor (lor F*) = 0.
Bearing in mind this last relation, we see that we have as many independent equations at our disposal as are necessary for determining the motion of matter as well as the vector u as a function of x, y, z, t, when proper fundamental data are given.
§ 13. The Product of the Field-vectors f F.
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
f = | 0 f₁₂ f₁₃ f₁₄ |
| f₂₁ 0 f₂₃ f₂₄ |
| f₃₁ f₃₂ 0 f₃₄ |
| f₄₁ f₄₂ f₄₃ 0 |
F = | 0 F₁₂ F₁₃ F₁₄ |
| F₂₁ 0 F₂₃ F₂₄ |
| F₃₁ F₃₂ 0 F₃₄ |
| F₄₁ F₄₂ F₄₃ 0 |
are of much importance. Let us write,
(70) fF =| S₁₁ - L S₁₂ S₁₃ S₁₄ |
| S₂₁ S₂₂ - L S₂₃ S₂₄ |
| S₃₁ S₃₂ S₃₃ - L S₃₄ |
| S₄₁ S₄₂ S₄₃ S₄₄ - L |
Then (71) S₁₁ + S₂₂ + S₃₃ + S₄₄ = 0.
Let L now denote the symmetrical combination of the indices 1, 2, 3, 4, given by
(72) L = ½(f₂₃ F₂₃ + f₃₁F₃₁ + f₁₂ + F₁₂ + f₁₄ F₁₄
+ f₂₄ F₂₄ + f₃₄ F₃₄)
Then we shall have
(73) S₁₁ = ½(f₂₃ F₂₃ + f₃₄ F₃₄ + f₄₂ F₄₂ - f₁₂ F₁₂
- f₁₃ F₁₃ f₁₄ F₁₄)
S₁₂ = f₁₃ F₃₂ + f₁₄ F₄₂ etc....
In order to express in a real form, we write
(74) S = | S₁₁ S₁₂ S₁₃ S₁₄ |
| S₂₁ S₂₂ S₂₃ S₂₄ |
| S₃₁ S₃₂ S₃₃ S₃₄ |
| S₄₁ S₄₂ S₄₃ S₄₄ |
= | Xx Yx Zx -iTx |
| Xy Yy Zy -iTy |
| Xz Yz Zz -iTz |
| -iXt -iYt -iZt Tt |
Now Xx = ½[mxMx - myMy - mzMz + exEx - eyEy - ezEz]
so
(75) Xy = mxMy + eyEx, Yx = myMx + exEy etc.
Xt = eyMz - ezMy, Tx = mxEy - myEz, etc.
Tt = ½[mxMx + myMy + mzMz + exEx + eyEy + ezEz]
Lt = ½[mxMx + myMy + mzMz - exEx - eyEy - ezEz]
These quantities[[27]] are all real. In the theory for bodies at rest, the combinations (Xx, Xy, Xz, Yz, Yy, Yz, Zx, Zy, Zz) are known as “Maxwell’s Stresses,” Tx, Ty, Tz are known as the Poynting’s Vector, Tt as the electromagnetic energy-density, and L as the Langrangian function.
On the other hand, by multiplying the alternating matrices of f* and F*, we obtain
(77) F*f* =| -S₁₁ - L, -S₁₂, -S₁₃. -S₁₄ |
| -S₂₁, -S₂₂ - L, -S₂₃, -S₂₄ |
| -S₃₁ -S₃₂, -S₃₃ - L, -S₃₄ |
| -S₄₁ -S₄₂ -S₄₃ -S₄₄ - L |
and hence, we can put
(78) fF = S - L, F*f* = -S - L,
where by L, we mean L-times the unit matrix, i.e. the matrix with elements
| Lehk |, (ehh = 1, ehk = 0, h ≠ k h, k = 1, 2, 3, 4).
Since here SL = LS, we deduce that,
F*f*fF = (-S - L)(S - L) = -SS + L²,
and find, since f*f = Det½f, F*F = Det½F, we arrive at the interesting
conclusion
(79) SS = L² - Det½f Det½F
i.e. the product of the matrix S into itself can be expressed as the multiple of a unit matrix—a matrix in which all the elements except those in the principal diagonal are zero, the elements in the principal diagonal are all equal and have the value given on the right-hand side of (79). Therefore the general relations
(80) Sh1 S1k + Sh2 S2k + Sh3 S3k + Sh4 S4k = 0,
h, k being unequal indices in the series 1, 2, 3, 4, and
(81) Sh1 S1h + Sh2 S2h + Sh3 S3h + S{h4} S4h = L² -
Det½f Det½F,
for h = 1, 2, 3, 4.
Now if instead of F, and f in the combinations (72) and (73), we introduce the electrical rest-force Φ, the magnetic rest-force ψ, and the rest-ray Ω [(55), (56) and (57)], we can pass over to the expressions,—
(82) L = - ½ ε Φ [=Φ] + ½ μ ψ [=ψ],
(83) Shk = - ½ ε Φ [=Φ] ehk - ½ μ ψ [=ψ] ehk
+ ε (Φh Φk - Φ ([=Φ]) ωh Ωk
+ μ (ψh ψk - Ψ [=ψ] Ω{h} ωk) - ωh ωk - εμ ωh Ωk
(h₁ k = 1, 2, 3, 4).
Here we have
Φ [=Φ] = Φ₁² + Φ₂² + Φ₃² + Φ₄², ψ[=ψ] = ψ₁² + ψ₂² + ψ₃² + ψ₄²
ehh = 1, ehk = 0 (h ≠ k).
The right side of (82) as well as L is an invariant in a Lorentz transformation, and the 4 × 4 element on the right side of (83) as well as Sk h represent a space time vector of the second kind. Remembering this fact, it suffices, for establishing the theorems (82) and (83) generally, to prove it for the special case ω₁ = 0, ω₂ = 0, ω₃ = 0, ω₄ = i. But for this case ω = 0, we immediately arrive at the equations (82) and (83) by means (45), (51), (60) on the one hand, and e = εE, M = μm on the other hand.
The expression on the right-hand side of (81), which equals
[½ (m M - eE)²] + (em) (EM),
is >= 0, because (em = ε Φ [=ψ], (EM) = μ Φ [=ψ]; now referring back to 79), we can denote the positive square root of this expression as Det1/4 S.
Since ḟ = -f, and Ḟ = -F, we obtain for Ṡ, the transposed matrix of S, the following relations from (78),
(84) Ff = Ṡ - L, f* F* = -Ṡ - L,
Then is
Ṡ - S = | Sh k - St k |
an alternating matrix, and denotes a space-time vector of the second kind. From the expressions (83), we obtain,
(85) S - Ṡ = - (εμ - 1) [ω, Ω],
from which we deduce that [see (57), (58)].
(86) ω (S - Ṡ)* = 0,
(87) ω (S - Ṡ) = (εμ - 1) Ω
When the matter is at rest at a space-time point, ω = 0, then the equation 86) denotes the existence of the following equations
Zy = Yz, Xz = Zx, Yx = Xy,
and from 83),
Tx = Ω₁, Ty = Ω₂, Tz = Ω₃
Xt = εμΩ₁, Yt = εμΩ₂, Zt = εμΩ₃
Now by means of a rotation of the space co-ordinate system round the null-point, we can make,
Zy = Yz = 0, Xz = Zx = 0, Xx = Xy = 0,
According to 71), we have
(88) Xx + Yy + Zz + Tt = 0,
and according to 83), Tt > 0. In special cases, where ω vanishes it follows from 81) that
Xx² = Yy² = Zz² = Tt², = (Det1/4 S)²,
and if T, and one of the three magnitudes Xx, Yy, Zz are = ±Det1/4 S, the two others = -Det1/4 S. If Ω does not vanish let Ω ≠ 0, then we have in particular from 80)
Tz Xt = 0, Tz Yt = 0, Zz Tz + Tz Tt = 0,
and if Ω₁ = 0, Ω₂ = 0, Zz = -Tt It follows from (81), (see also 83) that
Xx = -Yy = ±Det1/4 S,
and -Zz = Tt = √(Det½ S + εμΩ₃²) > Det1/4S.
The space-time vector of the first kind
(89) K = lor S,
is of very great importance for which we now want to demonstrate a very important transformation
According to 78), S = L + fF, and it follows that
lor S = lor L + lor fF.
The symbol ‘lor’ denotes a differential process which in lor fF, operates on the one hand upon the components of f, on the other hand also upon the components of F. Accordingly lor fF can be expressed as the sum of two parts. The first part is the product of the matrices (lor f) F, lor f being regarded as a 1 × 4 series matrix. The second part is that part of lor fF, in which the diffentiations operate upon the components of F alone. From 78) we obtain
fF = -F*f* - 2L;
hence the second part of lor fF = -(lor F*)f* + the part of -2 lor L, in which the differentiations operate upon the components of F alone. We thus obtain
lor S = (lor f)F - (lor F*)f* + N,
where N is the vector with the components
Nh = ½(∂f₂₃/∂xh F₂₃ + ∂f₃₁/∂xh F₃₁ + ∂f₁₂/∂xh F₁₂ + ∂f₁₄/∂xh F₁₄
+ ∂f₂₄/∂xh F₂₄ + ∂f₃₄/∂xh F₃₄
- ∂F₂₃/∂xh f₂₃ - ∂F₃₁/∂xh f₃₁ - ∂F₁₂/∂xh f₁₂ - ∂F₁₄/∂xh f₁₄
- ∂F₂₄/∂xh f₂₄ - ∂F₃₄/∂xh f₃₄),
(h = 1, 2, 3, 4)
By using the fundamental relations A) and B), 90) is transformed into the fundamental relation
(91) lor S = -sF + N.
In the limitting case ε = 1, μ = 1, f = F, N vanishes identically.
Now upon the basis of the equations (55) and (56), and referring back to the expression (82) for L, and from 57) we obtain the following expressions as components of N,—
(92) Nh = - ½ Φ[=Φ]∂ε/∂xh - ½ ψ[=ψ]∂μ/∂xh
+ (εμ - 1)(Ω₁ ∂ω₁/∂xh + Ω₂ ∂ω₂/∂xh + Ω₃ ∂ω₃/∂xh + Ω₄ ∂ω₄/∂xh)
for h = 1, 2, 3, 4.
Now if we make use of (59), and denote the space-vector which has Ω₁, Ω₂, Ω₃ as the x, y, z components by the symbol W, then the third component of 92) can be expressed in the form
(93) (εμ - 1)/√(1 - u²) (W ∂u/∂xh),
The round bracket denoting the scalar product of the vectors within it.
§ 14. The Ponderomotive Force.[[28]]
Let us now write out the relation K = lor S = -sF + N in a more practical form; we have the four equations
(94) K₁ = ∂Xx/∂x + ∂Xy/∂y + ∂Xy/∂z - ∂Xt/∂t = ρEx + syMz - szMx
- ½ Φ[=Φ] ∂ε/∂x - ½ ψ[=ψ]∂μ/∂x + (εμ - 1)/√(1 - u²) (W∂u/∂x),
(95) K₂ = ∂Yx/∂x + ∂Yy/∂y + ∂Yz/∂z - ∂Yt/∂t = ρEy + szMx - sxMy
- ½ Φ[=Φ]∂ε/∂y - ½ ψ[=ψ]∂μ/∂y + (εμ - 1)/√(1 - u²) (W∂u/∂y),
(96) K₃ = ∂Zx/∂x + ∂Zy/∂y + ∂Zz/∂z - ∂Zt/∂t = ρE₂ + sxMy - syM₄
- ½ Φ[=Φ] ∂ε/∂z - ½ ψ[=ψ] ∂μ/∂z + (εμ - 1)/√(1 - u²) (W∂u/∂z),
(97) (1/i)K₄ = ∂Ty/∂x - ∂Ty/∂y - ∂Tz/∂z - ∂Tt/∂t = sxEx + syEy + szEz
- ½ Φ[=Φ]∂ε/∂t - ½ ψ[=ψ]∂μ/∂t + (εμ - 1)/√(1 - u²) (W∂u/∂t).
It is my opinion that when we calculate the ponderomotive force which acts upon a unit volume at the space-time point x, y, z, t, it has got, x, y, z components as the first three components of the space-time vector
K + (ωK)ω,
This vector is perpendicular to ω; the law of Energy finds its expression in the fourth relation.
The establishment of this opinion is reserved for a separate tract.
In the limiting case ε = 1, μ = 1, σ = 0, the vector N = 0, S = ρω, ωK = 0, and we obtain the ordinary equations in the theory of electrons.
APPENDIX
Mechanics and the Relativity-Postulate.
It would be very unsatisfactory, if the new way of looking at the time-concept, which permits a Lorentz transformation, were to be confined to a single part of Physics.
Now many authors say that classical mechanics stand in opposition to the relativity postulate, which is taken to be the basis of the new Electro-dynamics.
In order to decide this let us fix our attention upon a special Lorentz transformation represented by (10), (11), (12), with a vector v in any direction and of any magnitude q < 1 but different from zero. For a moment we shall not suppose any special relation to hold between the unit of length and the unit of time, so that instead of t, t′, q, we shall write ct, ct′, and q/c, where c represents a certain positive constant, and q is < c. The above mentioned equations are transformed into
r′ṽ = rṽ,
r′v = c(rv - qt)/√(c² - q²),
t′ = (qrv + c²t)/c√(c² - q²)
They denote, as we remember, that r is the space-vector (x, y, z), r′ is the space-vector (x′ y′ z′)
If in these equations, keeping v constant we approach the limit c = ∞, then we obtain from these
r′ṽ = rṽ,
r′v = rv - qt,
t′ = t.
The new equations would now denote the transformation of a spatial co-ordinate system (x, y, z) to another spatial co-ordinate system (x′ y′ z′) with parallel axes, the null point of the second system moving with constant velocity in a straight line, while the time parameter remains unchanged. We can, therefore, say that classical mechanics postulates a covariance of Physical laws for the group of homogeneous linear transformations of the expression
-x² - y² - z² + c² (1)
when c = ∞.
Now it is rather confusing to find that in one branch of Physics, we shall find a covariance of the laws for the transformation of expression (1) with a finite value of c, in another part for c = ∞.
It is evident that according to Newtonian Mechanics, this covariance holds for c = ∞ and not for c = velocity of light.
May we not then regard those traditional covariances for c = ∞ only as an approximation consistent with experience, the actual covariance of natural laws holding for a certain finite value of c.
I may here point out that by if instead of the Newtonian Relativity-Postulate with c = ∞, we assume a relativity-postulate with a finite c, then the axiomatic construction of Mechanics appears to gain considerably in perfection.
The ratio of the time unit to the length unit is chosen in a manner so as to make the velocity of light equivalent to unity.
While now I want to introduce geometrical figures in the manifold of the variables (x, y, z, t), it may be convenient to leave (y, z) out of account, and to treat x and t as any possible pair of co-ordinates in a plane, referred to oblique axes.
A space time null point 0 (x, y, z, t = 0, 0, 0, 0) will be kept fixed in a Lorentz transformation.
The figure -x² - y² - z² + t² = 1, t > 0 ... (2)
which represents a hyper boloidal shell, contains the space-time points A (x, y, z, t = 0, 0, 0, 1), and all points A′ which after a Lorentz-transformation enter into the newly introduced system of reference as (x′, y′, z′, t′ = 0, 0, 0, 1).
The direction of a radius vector 0A′ drawn from 0 to the point A′ of (2), and the directions of the tangents to (2) at A′ are to be called normal to each other.
Let us now follow a definite position of matter in its course through all time t. The totality of the space-time points (x, y, z, t) which correspond to the positions at different times t, shall be called a space-time line.
The task of determining the motion of matter is comprised in the following problem:—It is required to establish for every space-time point the direction of the space-time line passing through it.
To transform a space-time point P (x, y, z, t) to rest is equivalent to introducing, by means of a Lorentz transformation, a new system of reference (x′, y′, z′, t′), in which the t′ axis has the direction 0A′, 0A′ indicating the direction of the space-time line passing through P. The space t′ = const, which is to be laid through P, is the one which is perpendicular to the space-time line through P.
To the increment dt of the time of P corresponds the increment
dτ = √(dt² - dx² - dy²) - dz² = dt√(1 - u²)
of the newly introduced time parameter t′. The value of the integral
∫ dτ = ∫ √(-(dx₁² + dx₂² + dx₃² + dx₄²))
when calculated upon the space-time line from a fixed initial point P₀ to the variable point P, (both being on the space-time line), is known as the ‘Proper-time’ of the position of matter we are concerned with at the space-time point P. (It is a generalization of the idea of Positional-time which was introduced by Lorentz for uniform motion.)
If we take a body R₀ which has got extension in space at time t₀, then the region comprising all the space-time line passing through R₀ and t₀ shall be called a space-time filament.
If we have an analytical expression θ(x y, z, t) so that θ(x, y z t) = 0 is intersected by every space time line of the filament at one point,—whereby
-(∂Θ/∂x)², -(∂Θ/∂y)², -(∂Θ/∂z)²,
-(∂Θ/∂t)² > 0, ∂Θ/∂t > 0.
then the totality of the intersecting points will be called a cross section of the filament.
At any point P of such across-section, we can introduce by means of a Lorentz transformation a system of reference (x′, y, z′ t), so that according to this
∂Θ/∂x′ = 0, ∂Θ/∂y′ = 0, ∂Θ/∂z′ = 0, ∂Θ/∂t′ > 0.
The direction of the uniquely determined t′—axis in question here is known as the upper normal of the cross-section at the point P and the value of dJ = ∫∫∫ dx′ dy′ dz′ for the surrounding points of P on the cross-section is known as the elementary contents (Inhalts-element) of the cross-section. In this sense R₀ is to be regarded as the cross-section normal to the t axis of the filament at the point t = t₀, and the volume of the body R₀ is to be regarded as the contents of the cross-section.
If we allow R₀ to converge to a point, we come to the conception of an infinitely thin space-time filament. In such a case, a space-time line will be thought of as a principal line and by the term ‘Proper-time’ of the filament will be understood the ‘Proper-time’ which is laid along this principal line; under the term normal cross-section of the filament, we shall understand the cross-section upon the space which is normal to the principal line through P.
We shall now formulate the principle of conservation of mass.
To every space R at a time t, belongs a positive quantity—the mass at R at the time t. If R converges to a point (x, y, z, t), then the quotient of this mass, and the volume of R approaches a limit μ(x, y, z, t), which is known as the mass-density at the space-time point (x, y, z, t).
The principle of conservation of mass says—that for an infinitely thin space-time filament, the product μdJ, where μ = mass-density at the point (x, y, z, t) of the filament (i.e., the principal line of the filament), dJ = contents of the cross-section normal to the t axis, and passing through (x, y, z, t), is constant along the whole filament.
Now the contents dJn of the normal cross-section of the filament which is laid through (x, y, z, t) is
(4) dJn = (1/√(1 - u²))dJ = -iω₄ dJ = (dt/dτ)dJ.
and the function
ν = μ/-iω₄ = μ√(1 - u²)) = μ(∂τ/∂t. (5)
may be defined as the rest-mass density at the position (x y z t). Then the principle of conservation of mass can be formulated in this manner:—
For an infinitely thin space-time filament, the product of the rest-mass density and the contents of the normal cross-section is constant along the whole filament.
In any space-time filament, let us consider two cross-sections Q° and Q′, which have only the points on the boundary common to each other; let the space-time lines inside the filament have a larger value of t on Q′ than on Q°. The finite range enclosed between Q° and Q′ shall be called a space-time sichel,[[29]] Q′ is the lower boundary, and Q′ is the upper boundary of the sichel.
If we decompose a filament into elementary space-time filaments, then to an entrance-point of an elementary filament through the lower boundary of the sichel, there corresponds an exit point of the same by the upper boundary, whereby for both, the product νdJn taken in the sense of (4) and (5), has got the same value. Therefore the difference of the two integrals ∫νdJn (the first being extended over the upper, the second upon the lower boundary) vanishes. According to a well-known theorem of Integral Calculus the difference is equivalent to
∫∫∫∫ lor ν[=ω] dx dy dz dt,
the integration being extended over the whole range of the sichel, and (comp. (67), § 12)
lor ν[=ω] = (∂νω₁/∂x₁) + (∂νω₂/∂x₂) + (∂νω₃/∂x₃) + (∂νω₄/∂x₄).
If the sichel reduces to a point, then the differential equation
lor ν[=ω] = 0, (6)
which is the condition of continuity
(∂μux/∂x) + (∂μuy/∂y) + (∂μuz/∂z) + (∂μ/∂t) = 0.
Further let us form the integral
N = ∫ ∫∫∫ ν dx dy dz dt (7)
extending over the whole range of the space-time sichel. We shall decompose the sichel into elementary space-time filaments, and every one of these filaments in small elements dτ of its proper-time, which are however large compared to the linear dimensions of the normal cross-section; let us assume that the mass of such a filament νdJn = dm and write τ⁰, τl for the ‘Proper-time’ of the upper and lower boundary of the sichel.
Then the integral (7) can be denoted by
∫∫ νdJn dτ = ∫ (τ′-τ⁰) dm.
taken over all the elements of the sichel.
Now let us conceive of the space-time lines inside a space-time sichel as material curves composed of material points, and let us suppose that they are subjected to a continual change of length inside the sichel in the following manner. The entire curves are to be varied in any possible manner inside the sichel, while the end points on the lower and upper boundaries remain fixed, and the individual substantial points upon it are displaced in such a manner that they always move forward normal to the curves. The whole process may be analytically represented by means of a parameter λ, and to the value λ = 0, shall correspond the actual curves inside the sichel. Such a process may be called a virtual displacement in the sichel.
Let the point (x, y, z, t) in the sichel λ = 0 have the values x + δx, y + δy, z + δz, t + δt, when the parameter has the value λ; these magnitudes are then functions of (x, y, z, t, λ). Let us now conceive of an infinitely thin space-time filament at the point (x y z t) with the normal section of contents dJn and if dJn + δdJn be the contents of the normal section at the corresponding position of the varied filament, then according to the principle of conservation of mass—(ν + dν being the rest-mass-density at the varied position),
(8) (ν + δν) (dJn + δdJn) = νdJn = dm.
In consequence of this condition, the integral (7) taken over the whole range of the sichel, varies on account of the displacement as a definite function N + δN of λ, and we may call this function N + δN as the mass action of the virtual displacement.
If we now introduce the method of writing with indices, we shall have
(9) d(xh + δxh) = dxh + ∑k ∂δxh/∂xk + ∂δxh/∂λ dλ
k = 1, 2, 3, 4
h = 1, 2, 3, 4
Now on the basis of the remarks already made, it is clear that the value of N + δN, when the value of the parameter is λ, will be:—
(10) N + δN = ∫∫∫∫ ((νd(τ + δτ))/dτ)dx dy dz dt,
the integration extending over the whole sichel d(τ + δτ) where d(τ + δτ) denotes the magnitude, which is deduced from
√(-(dx₁ + dδx₁)² - (dx₂ + dδx₂)² - (dx₃ + dδx₃)² - (dx₄ + dδx₄)²)
by means of (9) and
dx₁ = ω₁ dτ, dx₂ = ω₂ dτ,
dx₃ = ω₃ dτ, dx₄ = ω₄ dτ, dλ = 0
therefore:—
(11) (d(τ + δτ))/dτ = √( -∑(ωh + ∑(∂δxh/∂xk)ωk)²)
k = 1, 2, 3, 4.
h = 1, 2, 3, 4.
We shall now subject the value of the differential quotient
(12) ((d(N + δN))/dλ) (λ = 0)
to a transformation. Since each δxh as a function of (x, y, z, t) vanishes for the zero-value of the parameter λ, so in general dδxk/(∂xh = 0, for λ = 0.
Let us now put (∂δxh/∂λ) = ξh (h = 1, 2, 3, 4) (13)
λ = 0
then on the basis of (10) and (11), we have the expression (12):—
= -∫∫∫∫ ∑ ωh((∂ξh/∂x₁)ω₁ + (∂ξh/∂x₂)ω₂ +(∂ξh/∂x₃)ω₃ + (∂ξh/∂x₄)ω₄)
dx dy dz dt
for the system (x₁ x₂ x₃ x₄) on the boundary of the sichel, (δx₁ δx₂ δx₃ δx₄) shall vanish for every value of λ and therefore ξ₁, ξ₂, ξ₃, ξ₄ are nil. Then by partial integration, the integral is transformed into the form
∫∫∫∫ ∑ ξh(∂νωhω₁/∂x₁ + ∂νωhω₂/∂x₂ + ∂νωhω₃/∂x₃ + ∂νωhω₄/∂x₄)
dx dy dz dt
the expression within the bracket may be written as
= ωh ∑ ∂νωk/∂xk + ν∑ωk∂ωh/∂xk.
The first sum vanishes in consequence of the continuity equation (b). The second may be written as
(∂ωh/∂x₁)(dx₁/dτ) + (∂ωh/∂x₂)(dx₂/dτ) + (∂ωh/∂x₃)(dx₃/dτ) + (∂ωh/∂x₄)(dx₄/dτ)
= dωh/dτ = (d/dτ)(dxh/dτ)
whereby (d/dτ) is meant the differential quotient in the direction of the space-time line at any position. For the differential quotient (12), we obtain the final expression
(14) ∫∫∫∫ ν((∂ω₁/∂τ)ξ₁ + (∂ω₂/∂τ)ξ₂ + (∂ω₃/∂τ)ξ₃ + (∂ω₄/∂τ)ξ₄)
dx dy dz dt.
For a virtual displacement in the sichel we have postulated the condition that the points supposed to be substantial shall advance normally to the curves giving their actual motion, which is λ = 0; this condition denotes that the ξh is to satisfy the condition
w₁ξ₁ + w₂ξ₂ + w₃ξ₃ + w₄ξ₄ = 0. (15)
Let us now turn our attention to the Maxwellian tensions in the electrodynamics of stationary bodies, and let us consider the results in § 12 and 13; then we find that Hamilton’s Principle can be reconciled to the relativity postulate for continuously extended elastic media.
At every space-time point (as in § 13), let a space time matrix of the 2nd kind be known
(16) S =
| S₁₁ S₁₂ S₁₃ S₁₄ | = | Xx Yx Zx -iTx |
| S₂₁ S₂₂ S₂₃ S₂₄ | = | Xy Yy Zy -iTy |
| S₃₁ S₃₂ S₃₃ S₃₄ | = | Xz Yz Zz -iTz |
| S₄₁ S₄₂ S₄₃ S₄₄ | = | -iXt -iYt -iZt Tt |
where Xn Yx .....Xz, Tt are real magnitudes.
For a virtual displacement in a space-time sichel (with the previously applied designation) the value of the integral
(17) W + δW = ∫∫∫∫ (∑Sh k (∂(xk + δxk))/∂xh dx dy dz dt
extended over the whole range of the sichel, may be called the tensional work of the virtual displacement.
The sum which comes forth here, written in real magnitudes, is
Xx + Yy + Zz + Tt + Xx (∂δx)/∂x + Xy (∂δx)/∂y + ... Zz (∂δz)/∂z
- Xt (∂δx/∂t - ... + Tx (∂δt)/∂x + ... Tt (∂δt)/∂t
we can now postulate the following minimum principle in mechanics.
If any space-time Sichel be bounded, then for each virtual displacement in the Sichel, the sum of the mass-works, and tension works shall always be an extremum for that process of the space-time line in the Sichel which actually occurs.
The meaning is, that for each virtual displacement,
([d(·δN + δW)]/dλ)λ = 0 = 0 (18)
By applying the methods of the Calculus of Variations, the following four differential equations at once follow from this minimal principle by means of the transformation (14), and the condition (15).
(19) ν ∂wh/∂τ = Kh + χwh (h = 1, 2, 3, 4)
whence Kh = ∂S1 h/∂x₁ + ∂S2 h/∂x₂ + ∂S3 h/∂x₃ + ∂S4 h/∂x₄, (20)
are components of the space-time vector 1st kind K = lor S, and X is a factor, which is to be determined from the relation wẇ = - 1. By multiplying (19) by wh, and summing the four, we obtain X = Kẇ, and therefore clearly K + (Kẇ)w will be a space-time vector of the 1st kind which is normal to w. Let us write out the components of this vector as
X, Y, Z, ·iT
Then we arrive at the following equation for the motion of matter,
(21) ν d/dτ (dx/dτ) = X, ν d/dτ (dy/dτ) = Y, ν d/dτ (dz/dτ) = Z,
ν d/dτ (dx/dτ) = T, and we have also
(dx/dτ)² + (dy/dτ)² + (dz/dτ)² > (dt/dτ)² = -1,
and X dx/dτ + Y dy/dτ + Z dz/dτ = T dt/dτ.
On the basis of this condition, the fourth of equations (21) is to be regarded as a direct consequence of the first three.
From (21), we can deduce the law for the motion of a material point, i.e., the law for the career of an infinitely thin space-time filament.
Let x, y, z, t, denote a point on a principal line chosen in any manner within the filament. We shall form the equations (21) for the points of the normal cross section of the filament through x, y, z, t, and integrate them, multiplying by the elementary contents of the cross section over the whole space of the normal section. If the integrals of the right side be Rx Ry Rz Rt and if m be the constant mass of the filament, we obtain
(22) m d/dτ dx/dτ = Rx,
m d/dτ dy/dτ = Ry,
m d/dτ dz/dτ = Rz,
m d/dτ dt/dτ = Rt
R is now a space-time vector of the 1st kind with the components (Rx Ry Rz Rt) which is normal to the space-time vector of the 1st kind w,—the velocity of the material point with the components
dx/dτ, dy/dτ, dz/dτ, i dt/dτ.
We may call this vector R the moving force of the material point.
If instead of integrating over the normal section, we integrate the equations over that cross section of the filament which is normal to the t axis, and passes through (x, y, z, t), then [See (4)] the equations (22) are obtained, but
are now multiplied by dτ/dt; in particular, the last equation comes out in the form,
m d/dt (dt/dτ) = wx Rx dτ/dt + wy Ry dτ/dt + wz Rz dτ/dt.
The right side is to be looked upon as the amount of work done per unit of time at the material point. In this equation, we obtain the energy-law for the motion of the material point and the expression
m (dt/dτ - 1) = m [1/√(1 - w²) - 1]
= m (½ |w₁² + 3/8 |w₁⁴ + )
may be called the kinetic energy of the material point.
Since dt is always greater than dτ we may call the quotient (dt - dτ)/dτ as the “Gain” (vorgehen) of the time over the proper-time of the material point and the law can then be thus expressed;—The kinetic energy of a material point is the product of its mass into the gain of the time over its proper-time.
The set of four equations (22) again shows the symmetry in (x, y, z, t), which is demanded by the relativity postulate; to the fourth equation however, a higher physical significance is to be attached, as we have already seen in the analogous case in electrodynamics. On the ground of this demand for symmetry, the triplet consisting of the first three equations are to be constructed after the model of the fourth; remembering this circumstance, we are justified in saying,—
“If the relativity-postulate be placed at the head of mechanics, then the whole set of laws of motion follows from the law of energy.”
I cannot refrain from showing that no contradiction to the assumption on the relativity-postulate can be expected from the phenomena of gravitation.
If B*(x*, y*, z*, t*) be a solid (fester) space-time point, then the region of all those space-time points B (x, y, z, t), for which
(23) (x - x*)² + (y - y*)² + (z - z*)² = (t - t*)²
t - t* >= 0
may be called a “Ray-figure” (Strahl-gebilde) of the space time point B*.
A space-time line taken in any manner can be cut by this figure only at one particular point; this easily follows from the convexity of the figure on the one hand, and on the other hand from the fact that all directions of the space-time lines are only directions from B* towards to the concave side of the figure. Then B* may be called the light-point of B.
If in (23), the point (x y z t) be supposed to be fixed, the point (x* y* z* t*) be supposed to be variable, then the relation (23) would represent the locus of all the space-time points B*, which are light-points of B.
Let us conceive that a material point F of mass m may, owing to the presence of another material point F*, experience a moving force according to the following law. Let us picture to ourselves the space-time filaments of F and F* along with the principal lines of the filaments. Let BC be an infinitely small element of the principal line of F; further let B* be the light point of B, C* be the light point of C on the principal line of F*; so that OA′ is the radius vector of the hyperboloidal fundamental figure (23) parallel to B*C*, finally D* is the point of intersection of line B*C* with the space normal to itself and passing through B. The moving force of the mass-point F in the space-time point B is now the space-time vector of the first kind which is normal to BC, and which is composed of the vectors
(24) mm*(OA′/B*D*)³ BD* in the direction of BD*, and another vector of suitable value in direction of B*C*.
Now by (OA′/B*D*) is to be understood the ratio of the two vectors in question. It is clear that this proposition at once shows the covariant character with respect to a Lorentz-group.
Let us now ask how the space-time filament of F behaves when the material point F* has a uniform translatory motion, i.e., the principal line of the filament of F* is a line. Let us take the space time null-point in this, and by means of a Lorentz-transformation, we can take this axis as the t-axis. Let x, y, z, t, denote the point B, let τ* denote the proper time of B*, reckoned from O. Our proposition leads to the equations
(25) d²x/dτ² = - m*x/(t - τ*)², d²y/dτ² = - m*y/(t - τ*)³
d²z/dτ² = -m*z/(t - τ*)³,
(26) d²t/dτ² = -m*/(t - τ*)² d(t - τ*)/dt
where (27) x² + y² + z² = (t - τ*)²
and (28) (dx/dτ)² + (dy/dτ)² + (dz/dτ)² = (dt/dτ)² - 1.
In consideration of (27), the three equations (25) are of the same form as the equations for the motion of a material point subjected to attraction from a fixed centre according to the Newtonian Law, only that instead of the time t, the proper time τ of the material point occurs. The fourth equation (26) gives then the connection between proper time and the time for the material point.
Now for different values of τ′, the orbit of the space-point (x y z) is an ellipse with the semi-major axis a and the eccentricity e. Let E denote the eccentric anomaly, Τ the increment of the proper time for a complete description of the orbit, finally nΤ = 2π, so that from a properly chosen initial point τ, we have the Kepler-equation
(29) nτ = E - e sin E.
If we now change the unit of time, and denote the velocity of light by c, then from (28), we obtain
(30) (dt/dτ)² - 1
= (m*/ac²) (1 + e cos E)/(1 - e cos E)
Now neglecting c⁻⁴ with regard to 1, it follows that
ndt = ndτ [ 1 + ½ m*/ac² (1 + e cos E)/(1 - e cos E) ]
from which, by applying (29),
(31) nt + const = (1 + ½ m*/ac²) nτ + m*/ac² Sin E.
the factor m*/ac² is here the square of the ratio of a certain average velocity of F in its orbit to the velocity of light. If now m* denote the mass of the sun, a the semi major axis of the earth’s orbit, then this factor amounts to 10⁻⁸.
The law of mass attraction which has been just described and which is formulated in accordance with the relativity postulate would signify that gravitation is propagated with the velocity of light. In view of the fact that the periodic terms in (31) are very small, it is not possible to decide out of astronomical observations between such a law (with the modified mechanics proposed above) and the Newtonian law of attraction with Newtonian mechanics.