Chapter IX: Appendix: Mechanics and the Relativity-Postulate (4)

and with it the velocity,

√((_dx₁_/_dx₄_)² + (_dx₂_/_dx₄_)² + (_dx₃_/_dx₄_)²) = γ,

in the sense of the Euclidean Geometry. We can easily see that, with reference to the co-ordinate system, the rays of light must appear curved in case _g__{μν}’s are not constants. If _n_ be the direction perpendicular to the direction of propagation, we have, from Huygen’s principle, that light-rays (taken in the plane (γ, _n_)] must suffer a curvature ∂λ/∂_n_.

Let us find out the curvature which a light-ray suffers when it goes by a mass M at a distance Δ from it. If we use the co-ordinate system according to the above scheme, then the total bending B of light-rays (reckoned positive when it is concave to the origin) is given as a sufficient approximation by

B = ∫_{-∞}^∞ ∂γ/∂[_x_]₁ _dx₂_

where (73) and (70) gives

γ = √(-_g₄₄_/_g₂₂_) = 1 - α/2_r_ (1 + _x₂²_/_r²_).

The calculation gives

B = 2α/Δ = KM/2πΔ.

A ray of light just grazing the sun would suffer a bending of 1·7″, whereas one coming by Jupiter would have a deviation of about ·02″.

If we calculate the gravitation-field to a greater order of approximation and with it the corresponding path of a material particle of a relatively small (infinitesimal) mass we get a deviation of the following kind from the Kepler-Newtonian Laws of Planetary motion. The Ellipse of Planetary motion suffers a slow rotation in the direction of motion, of amount

(75) _s_ = 24π³_a²_/τ²_c²_(1 - _e²_) per revolution.

In this Formula ‘_a_’ signifies the semi-major axis, _c_, the velocity of light, measured in the usual way, _e_, the eccentricity, τ, the time of revolution in seconds.

The calculation gives for the planet Mercury, a rotation of path of amount 43″ per century, corresponding sufficiently to what has been found by astronomers (Leverrier). They found a residual perihelion motion of this planet of the given magnitude which can not be explained by the perturbation of the other planets.

NOTES

Note 1.

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

curl H = 1/_c_ (∂D/∂_t_ + ρν) (1)

curl E = -1/_c_ ∂B/∂_t_ (2)

div D = ρ
B = μH
div B = 0
D = kE

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

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

(∂_R_/∂_t_) due to motion = _u_. div R + curl [_Ru_]

where _u_ is the vector velocity of the moving body and [R_u_] the vector product of R and _u_.

Hence equations (1) and (2) become

_c_ curl H = ∂D/∂_t_ + _u_ div D + curl Vect. [D_u_] + ρν (1·1)

and

-_c_ curl E = ∂B/∂_t_ + _u_ div B + curl Vect. [B_u_] (2·1)

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

∂D/∂_t_ + _u_ div D = _c_ curl (H - 1/_c_ Vect. [D_u_]) (1·2)

∂B/∂_t_ = -_c_ curl (E - 1/_c_ Vect. [_u_B]) (2·2)

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_^{_i_A(_x_ - _vt_)}, we have

∂/∂_x_ = _i_A, ∂/∂_t_ = -_i_A_v_, ∂/∂_y_ = ∂/∂_z_ = 0, _u__{_y_} =
_u__{_z_} = 0

Then ∂D__y_/∂_t_ = -_c_∂H_{_z_}/∂_x_ - _u_∂D_{_y_}/∂_z_ ... (1·21)

and ∂B_{_z_}/∂_t_ = -_c_∂E_{_y_}/∂_x_ - _u_∂B_{_z_}/∂_x_ (2·21)

Since D = KE and B = μH, we have

_i_A_v_(κE_y_) = -_ci_A(H_{_z_} + _u_KE_{_y_}) (1·22)

_i_A_v_(μH_{_z_}) = -_ci_A(E_{_y_} + _u_μH_{_z_}) (2·22)

or _v_(K - _u_)E_{_y_} = _c_H_{_z_} (1·23)

μ(_v_ - _u_)H_{_z_} = _c_E_{_y_} (2·23)

Multiplying (1·23) by (2·23)

μK(_v_ - _u_)² = _c²_

Hence (_v_ - _u_)² = _c²_/μ_k_ = _v₀_²

∴ _v_ = _v₀_ + _u_,

making Fresnelian convection co-efficient simply unity.

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

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

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/δ_t_ = 0, there will be some change in the polarisation of the body due to its motion, given by _u__{_x_}(δD/δ_x_). Hence we must add this term to a purely temporal rate of change δD/δ_t_. 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.

_x¹_ = β(_x_ - _ut_), _y¹_ = _y_, _z¹_ = _z_, _t¹_ = β(_t_ -
(_u_/_c²_)·_x_)

and for velocities, used

_v__{_x_}¹ = β²_v__{_x_} + _u_, _v__{_y_}¹ = β_v__{_y_}, _v__{_z_}¹
= β_v__{_z_} and ρ¹ = ρ/β.

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:—

∂_m__{_z_}/∂_y_ - ∂_m__{_y_}/∂_z_ - ∂_e__{_x_}/∂τ = ρν_{_x_} }
∂_m__{_x_}/∂_z_ - ∂_m__{_z_}/∂_x_ - ∂_e__{_y_}/∂τ = ρν_{_y_} } ...
(1·1)
∂_m__{_y_}/∂_x_ - ∂_m__{_x_}/∂_y_ - ∂_e__{_z_}/∂τ = ρν_{_z_} }

and ∂_e__{_x_}/∂_x_ + ∂_e__{_y_}/∂_y_ + ∂_e__{_z_}/∂_z_ = ρ (2·1)

Substituting _x₁_, _x₂_, _x₃_, _x₄_ and _x_, _y_, _z_, and _i_τ; and ρ₁, ρ₂, ρ₃, ρ₄ for ρν_{_x_}, ρν_{_y_}, ρν_{_z_}, _i_ρ, where _i_ = √(-1).

We get,

∂_m__{_z_}/∂_x₂_ - ∂_m__{_y_}/∂_x₃_ - _i_(∂_e__{_x_}/∂_x₄_) =
ρν_{_x_}{ = ρ₁ }
- ∂_m__{_z_}/∂_x₁_ + ∂_m__{_x_}/∂_x₃_ - _i_(∂_e__{_y_}/∂_x₄_) =
ρν_{_y_} = ρ₂ } ... (1·2)
∂_m__{_y_}/∂_x₁_ - ∂_m__{_x_}/∂_x₂_ - _i_(∂_e__{_z_}/∂_x₄_) =
ρν_{_z_}{ = ρ₃ }

and multiplying (2·1) by i we get

∂_ie__{_x_}/∂_x₁_ + ∂_ie__{_y_}/∂_x₂_ + ∂_ie__{_z_}/∂_x₃_ = _i_ρ =
ρ₄ ... ... (2·2)

Now substitute

_m__{_x_} = _f₂₃_ = -_f₃₂_ and _ie__{_x_} = _f_₄₁ = -_f₁₄_
_m__{_y_} = _f₃₁_ = -_f₁₃_ _ie__{_y_} = _f_₄₂ = -_f₂₄_
_m__{_z_} = _f₁₂_ = -_f₂₁_ _ie__{_z_} = _f_₄₃ = -_f₃₄_

and we get finally:—

∂_f₁₂_/∂_x₂_ + ∂_f₁₃_/∂_x₃_ + ∂_f₁₄_/∂_x₄_ = ρ₁ }

∂_f₂₁_/∂_x₁_ + ∂_f₂₃_/∂_x₃_ + ∂_f₂₄_/∂_x₄_ = ρ₂ } ... (3)

∂_f₃₁_/∂_x₁_ + ∂_f₃₂_/∂_x₂_ + ∂_f₃₄_/∂_x₄_ = ρ₃ }

∂_f₄₁_/∂_x₁_ + ∂_f₄₂_/∂_x₂_ + ∂_f₄₃_/∂_x₃_ = ρ₄ }

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 - 2Δ_u_/_c²_ and from A to B is T + 2Δ_u_/_c²_. Now if 2_u_Δ/_c²_ be comparable to T, then it is impossible that the observations should satisfy Kepler’s Law. In most of the spectroscopic binary stars, 2_u_Δ/_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 - 2_u_Δ/_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 _ku_Δ₀, _u₀_ being the maximum value of _u_, the velocity of light emitted being

_u₀_ = _c_ + _ku_,
_k_ = 0 Lorentz-Einstein
_k_ = 1 Ritz.

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,

Δ > 65 light-years,
_k_ is < ·002.

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

φ_{_x_ _y_} = _u__{_x_}_v__{_y_} - _u__{_y_}_v_{_x_}.

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

φ_{_y_ _z_} φ_{_x_ _l_} + φ_{_z_ _x_} φ_{_y_ _l_} + φ_{_x_ _y_}
φ_{_z_ _l_} = 0

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

| φ |² = φ_{_y_ _z_}² + φ_{_z_ _x_}² + φ_{_x_ _y_}² + φ_{_x_ _l_}² +
φ_{_y_ _l_}² + φ_{_z_ _l_}².

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:—

φ_{_y_ _z_}^* = φ_{_x_ _l_}, φ_{_z_ _x_}^* = φ_{_y_ _l_}, φ_{_x_
_y_}^* = φ_{_z_ _l_} φ_{_z_ _l_}^* = φ_{_y_ _x_} ...

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

_u__{_x_}^* _u__{_x_} + _u__{_y_}^* _u__{_y_} + _u__{_z_}^*
_u__{_z_} + _u__{_l_}^* _u__{_l_} = 0

_u__{_x_}^* _v__{_x_} + _u__{_y_}^* _v__{_y_} + _u__{_z_}^*
_v__{_z_} + _u__{_l_}^* _v__{_l_} = 0

_v__{_x_}^* _u__{_x_} + _v__{_y_}^* _u__{_y_} + _v__{_z_}^*
_u__{_z_} + _v__{_l_}^* _u__{_l_} = 0

_v__{_x_}^* _v__{_x_} + _v__{_y_}^* _v__{_y_} + _v__{_z_}^*
_v__{_z_} + _v__{_l_}^* _v__{_l_} = 0

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

_u__{_x_}^* φ_{_x_ _l_} + _u__{_y_}^* φ_{_y_ _l_} + _u__{_z_}^*
φ_{_z_ _l_} = 0

_v__{_x_}^* φ_{_z_ _l_} + _v__{_y_}^* φ_{_y_ _l_} + _v__{_z_}^*
φ_{_z_ _l_} = 0

multiplying these equations by _v__{_x_}^* . _u__{_x_}^*, or by _v__{_y_}^* . _u__{_y_}^*, we obtain

φ_{_x_ _z_}^* φ_{_x_ _l_} + φ_{_y_ _z_}^* φ_{_y_ _l_} = 0 and φ_{_x_
_y_}^* φ_{_x_ _l_} + φ_{_z_ _x_}^* φ_{_z_ _l_} = 0

from which we have

φ_{_y_ _z_}^* : φ_{_x_ _y_}^* : φ_{_z_ _x_}^* = φ_{_x_ _l_} : φ_{_z_
_l_} : φ_{_y_ _l_}

In a corresponding way we have

φ_{_y_ _z_} : φ_{_x_ _y_} : φ_{_z_ _x_} = φ_{_x_ _l_}^* : φ_{_z_
_l_}^* : φ_{_y_ _l_}^*.

_i.e._ φ_{_i_ _k_}^* = λφ(_{_i_ _k_})

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

We have (φ φ*) = φ_{_y_ _z_} φ_{_y_ _z_}^* + ...

= 2 (φ_{_y_ _z_} φ_{_z_ _l_} + ...)

= 0

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

_f_ = ρφ + ρ^* φ^*,

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

_f_^* = ρ^*φ + ρφ^*

We can verify that

_f__{_y z_}^* = _f__{_x l_} etc.

and _f²_ = ρ² + ρ^*², (_f__f_^*) = 2ρρ^*.

| _f_ |² and (_f__f_^*) 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

V = (2_c_ - (_x_ + λ))/(1 + (_c_ - _x_)(_c_ - λ)/_c²_) = (2_c_ -
(_x_ + λ))/(_c²_ + _c²_ - (_x_ + λ)_c_ + _x_λ/_c²_)

= _c_ (2_c_ - (_x_ + λ))/(2_c_ - (_x_ + λ) + _x_λ/_c_)

Thus _v_ lt; _c_, so long as | _x_λ | > 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 - W_v_/_c²_)

Thus “time” from A to B as measured in S′, is given by _l_/W′ = _l_(1 - W_v_/_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 W_v_ = (_c_ + μ)(_c_ - λ) = _c²_ + (μ + λ)_c_ - μλ.

Now we can always choose _v_ in such a way that W_v_ is greater than _c²_, since W_v_ 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 W_v_ > _c²_, _i.e._, λ - W_v_/_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.]

Notes 13 and 14.

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

| ω₁ |
| ω₂ |
| ω₃ |
| ω₄ |

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

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

[ω_s_] = [=ω]_s_ - _ṡ_ω

It is now easy to verify the formula _f_¹ = A⁻¹_f_A. Supposing for the sake of simplicity that _f_ represents the vector-product of two four-vectors ω, _s_, we have

_f¹_ = [ω¹_s¹_] = [[=ω]¹_s¹_ - [=_s_]^1ω^1]

= [A⁻¹ [=ω]_s_A - A⁻¹_s_[=ω]A]

= A⁻¹[[=ω]_s_ - _s_[=ω]]A = A⁻¹_f_A.

Now remembering that generally

_f_ = ρφ + ρ*φ*.

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

_f_^1 = A⁻¹_f_A.

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

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

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

Let A = (A_{_x_}, A_{_y_}, A_{_z_}) denote a vector of the first type, B = (B_{_y z_}, B_{_z x_}, B_{_x y_}) denote a vector of the second type. From this last, let us form three special vectors of the first kind, namely—

B_{_x_} = (B_{_x x_}, B_{_x y_}, B_{_x z_}) }
B_{_y_} = (B_{_y x_}, B_{_y y_}, B_{_y z_}) } (B_{_i k_} = - B_{_k
i_}, B_{_i i_} = 0).
B_{_z_} = (B_{_z x_}, B_{_z y_}, B_{_z z_}) }

Since B_{_j j_} is zero, B_{_j_} is perpendicular to the _j_-axis. The _j_-component of the vector-product of A and B is equivalent to the scalar product of A and B_{_j_}, _i.e._,

(A B_{_j_},) = A_{_x_} B_{_j x_} + A_{_y_} B_{_j y_} + A_{_z_} B_{_j
z_}.

We see easily that this coincides with the usual rule for the vector-product; _e. g._, for _j_ = _x_.

(AB_{_x_}) = A_{_y_} B_{_x_ _y_} - A_{_z_} B_{_z_ _x_}.

Correspondingly let us define in the four-dimensional case the product (P_f_) of any four-vector P and the six-vector _f_. The _j_-component (_j_ = _x_, _y_, _z_, or _l_) is given by

(P_f__{_j_}) = P_{_x_}_f__{_j_ _x_} + P_{_y_}_f__{_j_ _y_} +
P_{_w_}_f__{_j_ _z_} + P_{_z_}_f__{_j_ _l_}

Each one of these components is obtained as the scalar product of P, and the vector _f__{_j_} which is perpendicular to j-axis, and is obtained from _f_ by the rule _f__{_j_} = [(_f__{_j_ _x_}, _f__{_j_ _y_}, _f__{_j_ _z_}, _f__{_j_ _l_}) _f__{_j_ _j_} = 0.]

We can also find out here the geometrical significance of vectors of the third type, when _f_ = φ, _i.e._, _f_ represents only one plane.

We replace φ by the parallelogram defined by the two four-vectors U, V, and let us pass over to the conjugate plane φ^*, which is formed by the perpendicular four-vectors U^*, V^*. The components of (Pφ) are then equal to the 4 three-rowed under-determinants D_{_x_} D_{_y_} D_{_z_} D_{_l_} of the matrix

| P_{_x_} P_{_y_} P_{_z_} P_{_l_} |
| |
| U_{_x_}^* U_{_y_}^* U_{_z_}^* U_{_l_}^* |
| |
| V_{_x_}^* V_{_y_}^* V_{_z_}^* V_{_l_}^* |

Leaving aside the first column we obtain

D_{_x_} = P_{_y_}(U_{_z_}^* V_{_l_}^* - U_{_l_}^* V_{_z_}^*) +
P_{_z_}(U_{_l_}^* V_{_y_}^* - U_{_y_}^* V_{_l_}^*)
+ P_{_l_}(U_{_y_}^* V_{_z_}^* - U_{_z_}^* V_{_y_}^*)
= P_{_y_} φ_{_z_ _y_}^* + P_{_z_}^* φ_{_l_ _y_} + P_{_l_} φ^*_{_y_
_z_}.
= P_{_y_} φ_{_x_ _y_} + P_{_z_} φ_{_x_ _z_} + _P__{_l_} φ_{_x_ _l_},

which coincides with (Pφ_{_x_}) according to our definition.

Examples of this type of vectors will be found on page 36, Φ = wF, the electrical-rest-force, and ψ = 2wf^*, the magnetic-rest-force. The rest-ray Ω = iw[Φψ]^* also belong to the same type (page 39). It is easy to show that

Ω = -_i_ | w₁ w₂ w₃ w₄ |
| Φ₁ Φ₂ Φ₃ Φ₄ |
| ψ₁ ψ₂ ψ₃ ψ₄ |

When (Ω₁, Ω₂, Ω₃) = 0, w₄ = _i_, Ω reduces to the three-dimensional vector

| Ω₁, Ω₂, Ω₃ | = | Φ₁ Φ₂ Φ₃ |
| |
| ψ₁ ψ₂ ψ₃ |

Since in this case, Φ₁ = w₄ F₁₄ = _e__{_n_} (the electric force)
ψ₁ = -_i_w₄ f₂₃ = _m__{_x_} (the magnetic force)
we have (Ω) = | _e__{_x_} _e__{_y_} _e__{_z_} |
| _m__{_x_} _m__{_y_} _m__{_z_} |

[M. N. S.]

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

The four-vector φ = wF which is called by Minkowski the electric-rest-force (elektrische Ruh-Kraft) is very closely connected to Lorentz’s Ponderomotive force, or the force acting on a moving charge. If ρ is the density of charge, we have, when ε = 1, μ = 1, _i.e._, for free space

ρ₀φ₁ = ρ₀[w₁ F₁₁ w₂ F₁₂ + w₃ F₁₃ + w₄ F₁₄]

= ρ₀/(√(1 - V²/_c²_)) [_d__{_x_} + 1/_c_ (_v₂_ _h₃_ -
_v₃_ _h₂_)]

Now since ρ₀ = ρ√(1 - V²/_c²_)

We have ρ₀φ₁ = ρ[_d__{_x_} + 1/_c_ (_v₂_ _h₃_ - _v₃_ _h₂_)]

N. B.—We have put the components of _e_ equivalent to (_d__{_x_}, _d__{_y_}, _d__{_z_}), and the components of _m_ equivalent to _h__{_x_} _h__{_y_} _h__{_z_}), in accordance with the notation used in Lorentz’s Theory of Electrons.

We have therefore

ρ₀ (φ₁, φ₂, φ₃) = ρ (_d_ + 1/_c_ [_v_·_h_]),

_i.e._, ρ₀ (φ₁, φ₂, φ₃) represents the force acting on the electron. Compare Lorentz, Theory of Electrons, page 14.

The fourth component φ₄ when multiplied by ρ₀ represents _i_-times the rate at which work is done by the moving electron, for ρ₀ φ₄ = _i_ρ [_v__{_x_}_d__{_x_} + _v__{_y_}_d__{_y_} + _v__{_z_}_d__{_z_}] = _v__{_x_} ρ₀φ₁ + _v__{_y_} ρ₀φ₂ + _v__{_z_} ρ₀φ₃. -√(-1) times the power possessed by the electron therefore represents the fourth component, or the time component of the force-four-vector. This component was first introduced by Poincare in 1906.

The four-vector ψ = _i_ωF^* has a similar relation to the force acting on a moving magnetic pole.

[M. N. S.]

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

The operation | ∂/∂_x₁_ ∂/∂_x₂_ ∂/∂_x₃_ ∂/∂_x₄_ | which plays in four-dimensional mechanics a rôle similar to that of the operator (_i_∂/∂_x_, + _j_∂/∂_y_, + _k_∂/∂_z_ = ▽) in three-dimensional geometry has been called by Minkowski ‘Lorentz-Operation’ or shortly ‘lor’ in honour of H. A. Lorentz, the discoverer of the theorem of relativity. Later writers have sometimes used the symbol □ to denote this operation. In the above-mentioned paper (Annalen der Physik, p. 649, Bd. 38) Sommerfeld has introduced the terms, Div (divergence), Rot (Rotation), Grad (gradient) as four-dimensional extensions of the corresponding three-dimensional operations in place of the general symbol lor. The physical significance of these operations will become clear when along with Minkowski’s method of treatment we also study the geometrical method of Sommerfeld. Minkowski begins here with the case of lor S, where S is a six-vector (space-time vector of the 2nd kind).

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

where _s_ is a four-vector = | _s₁_, _s₂_, _s₃_, _s₄_ |

and _s_ = | _s₁_ |
| _s₂_ |
| _s₃_ |
| _s₄_ |

The following geometrical method is taken from Sommerfeld.

Scalar Divergence—Let ΔΣ denote a small four-dimensional volume of any shape in the neighbourhood of the space-time point Q, _d_S denote the three-dimensional bounding surface of ΔΣ, _n_ be the outer normal to _d_S. Let S be any four-vector, P_{_n_} its normal component. Then

Div S = Lim ∫ P_{_n_}_d_S/ΔΣ.
ΔΣ = 0

Now if for ΔΣ we choose the four-dimensional parallelopiped with sides (_dx₁_, _dx₂_, _dx₃_, _dx₄_), we have then

Div S = ∂_s₁_/∂_x₁_ + ∂_s₂_/∂_x₂_ + ∂_s₃_/∂_x₃_ + ∂_s₄_/∂_x₄_ = lor
S.

If _f_ denotes a space-time vector of the second kind, lor _f_ is equivalent to a space-time vector of the first kind. The geometrical significance can be thus brought out. We have seen that the operator ‘lor’ behaves in every respect like a four-vector. The vector-product of a four-vector and a six-vector is again a four-vector. Therefore it is easy to see that lor S will be a four-vector. Let us find the component of this four-vector in any direction _s_. Let S denote the three-space which passes through the point Q (_x₁_, _x₂_, _x₃_, _x₄_) and is perpendicular to _s_, ΔS a very small part of it in the region of Q, _d_σ is an element of its two-dimensional surface. Let the perpendicular to this surface lying in the space be denoted by _n_, and let _f__{_s_ _n_} denote the component of _f_ in the plane of (_sn_) which is evidently conjugate to the plane _d_σ. Then the _s_-component of the vector divergence of _f_ because the operator lor multiplies _f_ vectorially.

= Div _f__{_s_} = Lim (∫ _f__{_s_ _n_}_d_σ)/ΔS.
Δ_s_ = 0

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

If now _s_ is selected as the _x_-direction, Δ_s_ is then a three-dimensional parallelopiped with the sides _dy_, _dz_, _dl_, then we have

$$ Div f_{x} = \frac{1}{dy dz dl} {dz. dl. \frac{\partial f_{xy}}{\partial y} dy + dl dy \frac{\partial f_{xy}}{\partial z} dz + dy dz \frac{\partial f_{xy}}{\partial l} dl} = \frac{\partial f_{xy}}{\partial y} + \frac{\partial f_{xy}}{\partial z} + \frac{\partial f_{xy}}{\partial l} $$

and generally

Div _f__{_j_} = ∂_f__{_j_ _x_}/∂_x_ + ∂_f__{_j_ _y_}/∂_y_ +
∂_f__{_j_ _z_}/∂_z_ + ∂_f__{_j_ _l_}/∂_l_ (where _f__{_j_, _j_} =
0).

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

According to the formulae of space geometry, D_{_x_} denotes a parallelopiped laid in the (_y_-_z_-_l_) space, formed out of the vectors (P_{_y_} P_{_z_} P_{_l_}), (U_{_y_}^* U_{_z_}^* U_{_l_}^*) (V_{_y_}^* V_{_z_}^* V_{_l_}^*).

D_{_x_} is therefore the projection on the _y-z-l_ space of the parallelopiped formed out of these three four-vectors (P, U^*, V^*), and could as well be denoted by Dyzl. We see directly that the four-vector of the kind represented by (D_{_x_}, D_{_y_}, D_{_z_}, D_{_l_}) is perpendicular to the parallelopiped formed by (P U^* V^*).

Generally we have

(P_f_) = PD + P^*D^*.

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

[M. N. S.]

● Transcriber’s Notes:
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symbology especially in mathematical formulas, have been retained.
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