Chapter VI: Appendix: Mechanics and the Relativity-Postulate (1)

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_} = _r__{_ṽ_},
_r′__{_v_} = _c_(_r__{_v_} - _qt_)/√(_c²_ - _q²_),
_t′_ = (_qr__{_v_} + _c²__t_)/_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_} = _r__{_ṽ_},
_r′__{_v_} = _r__{_v_} - _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

_d_τ = √(_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 _d_J = ∫∫∫ _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 μ_d_J, where μ = mass-density at the point (_x_, _y_, _z_, _t_) of the filament (_i.e._, the principal line of the filament), _d_J = 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 _d_J_{n} of the normal cross-section of the filament which is laid through (_x_, _y_, _z_, _t_) is

(4) _d_J_{n} = (1/√(1 - _u²_))_d_J = -_i_ω₄ _d_J = (_dt_/_d_τ)_d_J.

and the function

ν = μ/-_i_ω₄ = 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 νdJ_{n} taken in the sense of (4) and (5), has got the same value. Therefore the difference of the two integrals ∫ν_dJ__{n} (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

(∂μ_u__{_x_}/∂_x_) + (∂μ_u__{_y_}/∂_y_) + (∂μ_u__{_z_}/∂_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 _d_τ 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 ν_d_J_{_n_} = _dm_ and write τ⁰, τ^l for the ‘Proper-time’ of the upper and lower boundary of the _sichel_.

Then the integral (7) can be denoted by

∫∫ ν_d_J_{_n_} _d_τ = ∫ (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_ + δ_x_, _y_ + δ_y_, _z_ + δ_z_, _t_ + δ_t_, 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 _d_J_{_n_} and if _d_J_{_n_} + δ_d_J_{_n_} 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 + _d_ν being the rest-mass-density at the varied position),

(8) (ν + δν) (_d_J_{_n_} + δ_d_J_{_n_}) = ν_d_J_{_n_} = _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_(_x__{_h_} + δ_x__{_h_}) = _dx__{_h_} + ∑_{_k_}
∂δ_x__{_h_}/∂_x__{_k_} + ∂δ_x__{_h_}/∂λ _d_λ

_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 = ∫∫∫∫ ((ν_d_(τ + δτ))/_d_τ)_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_ + _d_δ_x₁_)² - (_dx₂_ + _d_δ_x₂_)² - (_dx₃_ + _d_δ_x₃_)² -
(_dx₄_ + _d_δ_x₄_)²)

by means of (9) and

_dx₁_ = ω₁ _d_τ, _dx₂_ = ω₂ _d_τ,
_dx₃_ = ω₃ _d_τ, _dx₄_ = ω₄ _d_τ, _d_λ = 0

therefore:—

(11) (_d_(τ + δτ))/_d_τ = √( -∑(ω_{_h_} +
∑(∂δ_x__{_h_}/∂_x__{_k_})ω_{_k_})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))/_d_λ) (λ = 0)

to a transformation. Since each δ_x__{_h_} as a function of (_x_, _y_, _z_, _t_) vanishes for the zero-value of the parameter λ, so in general _d_δ_x__{_k_}/(∂_x__{_h_} = 0, for λ = 0.

Let us now put (∂δ_x__{_h_}/∂λ) = ξ_{_h_} (_h_ = 1, 2, 3, 4) (13)

λ = 0

then on the basis of (10) and (11), we have the expression (12):—

= -∫∫∫∫ ∑ ω_{_h_}((∂ξ_{_h_}/∂_x₁_)ω₁ + (∂ξ_{_h_}/∂_x₂_)ω₂
+(∂ξ_{_h_}/∂_x₃_)ω₃ + (∂ξ_{_h_}/∂_x₄_)ω₄)
_dx dy dz dt_

for the system (_x₁_ _x₂_ _x₃_ _x₄_) on the boundary of the _sichel_, (δ_x₁_ δ_x₂_ δ_x₃_ δ_x₄_) 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

∫∫∫∫ ∑ ξ_{_h_}(∂νω_{_h_}ω₁/∂_x₁_ + ∂νω_{_h_}ω₂/∂_x₂_ +
∂νω_{_h_}ω₃/∂_x₃_ + ∂νω_{_h_}ω₄/∂_x₄_)
_dx dy dz dt_

the expression within the bracket may be written as

= ω_{_h_} ∑ ∂νω_{_k_}/∂_x__{_k_} + ν∑ω_{_k_}∂ω_{_h_}/∂_x__{_k_}.

The first sum vanishes in consequence of the continuity equation (_b_). The second may be written as

(∂ω_{_h_}/∂_x₁_)(_dx₁_/_d_τ) + (∂ω_{_h_}/∂_x₂_)(_dx₂_/_d_τ) +
(∂ω_{_h_}/∂_x₃_)(_dx₃_/_d_τ) + (∂ω_{_h_}/∂_x₄_)(_dx₄_/_d_τ)

= _d_ω_{_h_}/_d_τ = (_d_/_d_τ)(_dx__{_h_}/_d_τ)

whereby (_d_/_d_τ) 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 ξ_{_h_} is to satisfy the condition

_w₁_ξ₁ + _w₂_ξ₂ + _w₃_ξ₃ + _w₄_ξ₄ = 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 | = | X_{_x_} Y_{_x_} Z_{_x_} -_i_T_{_x_} |

| S₂₁ S₂₂ S₂₃ S₂₄ | = | X_{_y_} Y_{_y_} Z_{_y_} -_i_T_{_y_} |

| S₃₁ S₃₂ S₃₃ S₃₄ | = | X_{_z_} Y_{_z_} Z_{_z_} -_i_T_{_z_} |

| S₄₁ S₄₂ S₄₃ S₄₄ | = | -_i_X_{_t_} -_i_Y_{_t_} -_i_Z_{_t_} T_{_t_}
|

where X_{_n_} Y_{_x_} .....X_{_z_}, T_{_t_} 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 = ∫∫∫∫ (∑S_{_h k_} (∂(_x__{_k_} +
δ_x__{_k_}))/∂_x__{_h_} _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

X_{_x_} + Y_{_y_} + Z_{_z_} + T_{_t_} + X_{_x_} (∂δ_x_)/∂_x_ +
X_{_y_} (∂δ_x_)/∂_y_ + ... Z_{_z_} (∂δ_z_)/∂_z_

- X_{_t_} (∂δ_x_/∂_t_ - ... + T_{_x_} (∂δ_t_)/∂_x_ + ... T_{_t_}
(∂δ_t_)/∂_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)]/_d_λ)_{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 ∂_w__{_h_}/∂τ = K_{_h_} + χ_w__{_h_} (_h_ = 1, 2, 3, 4)

whence K_{_h_} = ∂S_{1 _h_}/∂_x₁_ + ∂S_{2 _h_}/∂_x₂_ + ∂S_{3
_h_}/∂_x₃_ + ∂S_{4 _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 _w__ẇ_ = - 1. By multiplying (19) by _w__{_h_}, and summing the four, we obtain X = K_ẇ_, and therefore clearly K + (K_ẇ_)_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, ·_i_T

Then we arrive at the following equation for the motion of matter,

(21) ν _d_/_d_τ (_dx_/_d_τ) = X, ν _d_/_d_τ (_dy_/_d_τ) = Y, ν
_d_/_d_τ (_dz_/_d_τ) = Z,

ν _d_/_d_τ (_dx_/_d_τ) = T, and we have also

(_dx_/_d_τ)² + (_dy_/_d_τ)² + (_dz_/_d_τ)² > (_dt_/_d_τ)² = -1,

and X _dx_/_d_τ + Y _dy_/_d_τ + Z _dz_/_d_τ = T _dt_/_d_τ.

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 R_{_x_} R_{_y_} R_{_z_} R_{_t_} and if _m_ be the constant mass of the filament, we obtain

(22) _m_ _d_/_d_τ _dx_/_d_τ = R_{_x_},
_m_ _d_/_d_τ _dy_/_d_τ = R_{_y_},
_m_ _d_/_d_τ _dz_/_d_τ = R_{_z_},
_m_ _d_/_d_τ _dt_/_d_τ = R_{_t_}

R is now a space-time vector of the 1st kind with the components (R_{_x_} R_{_y_} R_{_z_} R_{_t_}) which is normal to the space-time vector of the 1st kind _w_,—the velocity of the material point with the components

_dx_/_d_τ, _dy_/_d_τ, _dz_/_d_τ, _i_ _dt_/_d_τ.

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 _d_τ/_dt_; in particular, the last equation comes out in the form,

_m_ _d_/_dt_ (_dt_/_d_τ) = _w__{_x_} R_{_x_} _d_τ/_dt_ + _w__{_y_}
R_{_y_} _d_τ/_dt_ + _w__{_z_} R_{_z_} _d_τ/_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_/_d_τ - 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 _d_τ we may call the quotient (_dt_ - _d_τ)/_d_τ 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) _d²__x_/_d_τ² = - _m_*_x_/(_t_ - τ*)², _d²__y_/_d_τ² = -
_m_*_y_/(_t_ - τ*)³

_d²__z_/_d_τ² = -_m_*_z_/(_t_ - τ*)³,
(26) _d²__t_/_d_τ² = -_m_*/(_t_ - τ*)² _d_(_t_ - τ*)/_dt_

where (27) _x²_ + _y²_ + _z²_ = (_t_ - τ*)²

and (28) (_dx_/_d_τ)² + (_dy_/_d_τ)² + (_dz_/_d_τ)² = (_dt_/_d_τ)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 _n_Τ = 2π, so that from a properly chosen initial point τ, we have the Kepler-equation

(29) _n_τ = 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_/_d_τ)² - 1
= (_m_*/_ac²_) (1 + _e_ cos E)/(1 - _e_ cos E)

Now neglecting _c⁻⁴_ with regard to 1, it follows that

_ndt_ = _nd_τ [ 1 + ½ _m_*/_ac²_ (1 + _e_ cos E)/(1 - _e_ cos E) ]

from which, by applying (29),

(31) _nt_ + const = (1 + ½ _m_*/_ac²_) _n_τ + _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.

Footnote 29:

Sichel—a German word meaning a crescent or a scythe. The original term
is retained as there is no suitable English equivalent.

SPACE AND TIME

A Lecture delivered before the Naturforscher Versammlung (Congress of Natural Philosophers) at Cologne—(21st September, 1908).

Gentlemen,

The conceptions about time and space, which I hope to develop before you to-day, has grown on experimental physical grounds. Herein lies its strength. The tendency is radical. Henceforth, the old conception of space for itself, and time for itself shall reduce to a mere shadow, and some sort of union of the two will be found consistent with facts.

I

Now I want to show you how we can arrive at the changed concepts about time and space from mechanics, as accepted now-a-days, from purely mathematical considerations. The equations of Newtonian mechanics show a twofold invariance, (_i_) their form remains unaltered when we subject the fundamental space-coordinate system to any possible change of position, (_ii_) when we change the system in its nature of motion, _i. e._, when we impress upon it any uniform motion of translation, the null-point of time plays no part. We are accustomed to look upon the axioms of geometry as settled once for all, while we seldom have the same amount of conviction regarding the axioms of mechanics, and therefore the two invariants are seldom mentioned in the same breath. Each one of these denotes a certain group of transformations for the differential equations of mechanics. We look upon the existence of the first group as a fundamental characteristics of space. We always prefer to leave off the second group to itself, and with a light heart conclude that we can never decide from physical considerations whether the space, which is supposed to be at rest, may not finally be in uniform motion. So these two groups lead quite separate existences besides each other. Their totally heterogeneous character may scare us away from the attempt to compound them. Yet it is the whole compounded group which as a whole gives us occasion for thought.

We wish to picture to ourselves the whole relation graphically. Let (_x_, _y_, _z_) be the rectangular coordinates of space, and _t_ denote the time. Subjects of our perception are always connected with place and time. _No one has observed a place except at a particular time, or has observed a time except at a particular place._ Yet I respect the dogma that time and space have independent existences. I will call a space-point plus a time-point, _i.e._, a system of values _x_, _y_, _z_, _t_, as a _world-point_. The manifoldness of all possible values of _x_, _y_, _z_, _t_, will be the _world_. I can draw four world-axes with the chalk. Now any axis drawn consists of quickly vibrating molecules, and besides, takes part in all the journeys of the earth ; and therefore gives us occasion for reflection. The greater abstraction required for the four-axes does not cause the mathematician any trouble. In order not to allow any yawning gap to exist, we shall suppose that at every place and time, something perceptible exists. In order not to specify either matter or electricity, we shall simply style these as substances. We direct our attention to the _world-point_ _x_, _y_, _z_, _t_, and suppose that we are in a position to recognise this substantial point at any subsequent time. Let _dt_ be the time element corresponding to the changes of space coordinates of this point [_dx_, _dy_, _dz_]. Then we obtain (as a picture, so to speak, of the perennial life-career of the substantial point),—a curve in the _world_—the _world-line_, the points on which unambiguously correspond to the parameter _t_ from +∞ to -∞. The whole world appears to be resolved in such _world-lines_, and I may just deviate from my point if I say that according to my opinion the physical laws would find their fullest expression as mutual relations among these lines.

By this conception of time and space, the (_x_, _y_, _z_) manifoldness _t_ = 0 and its two sides _t_ < 0 and _t_ > 0 falls asunder. If for the sake of simplicity, we keep the null-point of time and space fixed, then the first named group of mechanics signifies that at _t_ = 0 we can give the _x_, _y_, and _z_-axes any possible rotation about the null-point corresponding to the homogeneous linear transformation of the expression

_x²_ + _y²_ + _z²_.

The second group denotes that without changing the expression for the mechanical laws, we can substitute (_x_ - α_t_, _y_ - β_t_, _z_ - γ_t_ for (_x_, _y_, _z_) where (α, β, γ) are any constants. According to this we can give the time-axis any possible direction in the upper half of the world _t_ > 0. Now what have the demands of orthogonality in space to do with this perfect freedom of the time-axis towards the upper half?

To establish this connection, let us take a positive parameter c, and let us consider the figure

_c²__t²_ - _x²_ - _y²_ - _z²_ = 1

According to the analogy of the hyperboloid of two sheets, this consists of two sheets separated by _t_ = 0. Let us consider the sheet, in the region of _t_ > 0, and let us now conceive the transformation of _x_, _y_, _z_, _t_ in the new system of variables; (_x’_, _y’_, _z’_, _t’_) by means of which the form of the expression will remain unaltered. Clearly the rotation of space round the null-point belongs to this group of transformations. Now we can have a full idea of the transformations which we picture to ourselves from a particular transformation in which (_y_, _z_) remain unaltered. Let us draw the cross section of the upper sheets with the plane of the _x_- and _t_-axes, _i.e._, the upper half of the hyperbola _c²__t²_ - x² = 1, with its asymptotes (_vide_ fig. 1).

Then let us draw the radius rector OA′, the tangent A′ B′ at A′, and let us complete the parallelogram OA′ B′ C′; also produce B′ C′ to meet the x-axis at D′. Let us now take Ox′, OA′ as new axes with the unit measuring rods OC′ = 1, OA′ = (1/c) ; then the hyperbola is again expressed in the form _c²__t′²_ - x′² = 1, t′ > 0 and the transition from (_x_, _y_, _z_, _t_) to (_x′_ _y′_ _z′_ _t_) is one of the transitions in question. Let us add to this characteristic transformation any possible displacement of the space and time null-points; then we get a group of transformation depending only on _c_, which we may denote by G_{_c_}.

Now let us increase _c_ to infinity. Thus (1/c) becomes zero and it appears from the figure that the hyperbola is gradually shrunk into the _x_-axis, the asymptotic angle becomes a straight one, and every special transformation in the limit changes in such a manner that the _t_-axis can have any possible direction upwards, and _x′_ more and more approximates to _x_. Remembering this point it is clear that the full group belonging to Newtonian Mechanics is simply the group G_{_c_}, with the value of _c_ = ∞. In this state of affairs, and since G_{_c_} is mathematically more intelligible than G_{∞}, a mathematician may, by a free play of imagination, hit upon the thought that natural phenomena possess an invariance not only for the group G_{∞}, but in fact also for a group G_{_c_}, where _c_ is finite, but yet exceedingly large compared to the usual measuring units. Such a preconception would be an extraordinary triumph for pure mathematics.

At the same time I shall remark for which value of _c_, this invariance can be conclusively held to be true. _For c, we shall substitute the velocity of light c in free space._ In order to avoid speaking either of space or of vacuum, we may take this quantity as the ratio between the electrostatic and electro-magnetic units of electricity.

We can form an idea of the invariant character of the expression for natural laws for the group-transformation G_{_c_} in the following manner.

Out of the totality of natural phenomena, we can, by successive higher approximations, deduce a coordinate system (_x_, _y_, _z_, _t_); by means of this coordinate system, we can represent the phenomena according to definite laws. This system of reference is by no means uniquely determined by the phenomena. _We can change the system of reference in any possible manner corresponding to the above-mentioned group transformation G_{c}, but the expressions for natural laws will not be changed thereby._

For example, corresponding to the above described figure, we can call _t′_ the time, but then necessarily the space connected with it must be expressed by the manifoldness (_x′_ _y_ _z_). The physical laws are now expressed by means of _x′_, _y_, _z_, _t′_,—and the expressions are just the same as in the case of _x_, _y_, _z_, _t_. According to this, we shall have in the world, not one space, but many spaces,—quite analogous to the case that the three-dimensional space consists of an infinite number of planes. The three-dimensional geometry will be a chapter of four-dimensional physics. Now you perceive, why I said in the beginning that time and space shall reduce to mere shadows and we shall have a world complete in itself.

II

Now the question may be asked,—what circumstances lead us to these changed views about time and space, are they not in contradiction with observed phenomena, do they finally guarantee us advantages for the description of natural phenomena?

Before we enter into the discussion, a very important point must be noticed. Suppose we have individualised time and space in any manner; then a world-line parallel to the _t_-axis will correspond to a stationary point; a world-line inclined to the _t_-axis will correspond to a point moving uniformly; and a world-curve will correspond to a point moving in any manner. Let us now picture to our mind the world-line passing through any world point _x_, _y_, _z_, _t_; now if we find the world-line parallel to the radius vector OA′ of the hyperboloidal sheet, then we can introduce OA′ as a new time-axis, and then according to the new conceptions of time and space the substance will appear to be at rest in the world point concerned. We shall now introduce this fundamental axiom:—

_The substance existing at any world point can always be conceived to be at rest, if we establish our time and space suitably._ The axiom denotes that in a world-point, the expression

_c²__dt²_ - _dx²_ - _dy²_ - _dz²_

shall always be positive or what is equivalent to the same thing, every velocity V should be smaller than _c_. _c_ shall therefore be the upper limit for all substantial velocities and herein lies a deep significance for the quantity _c_. At the first impression, the axiom seems to be rather unsatisfactory. It is to be remembered that only a modified mechanics will occur, in which the square root of this differential combination takes the place of time, so that cases in which the velocity is greater than _c_ will play no part, something like imaginary coordinates in geometry.

The _impulse_ and real cause of inducement _for the assumption of the group-transformation G_{c}_ is the fact that the differential equation for the propagation of light in vacant space possesses the group-transformation G_{_c_}. On the other hand, the idea of rigid bodies has any sense only in a system mechanics with the group G_{infinity}. Now if we have an optics with G_{_c_}, and on the other hand if there are rigid bodies, it is easy to see that a _t_-direction can be defined by the two hyperboloidal shells common to the groups G_{∞}, and G_{_c_}, which has got the further consequence, that by means of suitable rigid instruments in the laboratory, we can perceive a change in natural phenomena, in case of different orientations, with regard to the direction of progressive motion of the earth. But all efforts directed towards this object, and even the celebrated interference-experiment of Michelson have given negative results. In order to supply an explanation for this result, H. A. Lorentz formed a hypothesis which practically amounts to an invariance of optics for the group G_{_c_}. According to Lorentz every substance shall suffer a contraction

1:(√(1 - v²/_c²_)) in length, in the direction of its motion

_l_/_l′_ = 1/√(1 - _v²_/_c²_) _l′_ = _l_(1 - _v²_/_c²_).

This hypothesis sounds rather phantastical. For the contraction is not to be thought of as a consequence of the resistance of ether, but purely as a gift from the skies, as a sort of condition always accompanying a state of motion.

I shall show in our figure that Lorentz’s hypothesis is fully equivalent to the new conceptions about time and space. Thereby it may appear more intelligible. Let us now, for the sake of simplicity, neglect (_y_, _z_) and fix our attention on a two dimensional world, in which let upright strips parallel to the _t_-axis represent a state of rest and another parallel strip inclined to the _t_-axis represent a state of uniform motion for a body, which has a constant spatial extension (see fig. 1). If OA′ is parallel to the second strip, we can take _t′_ as the _t_-axis and _x′_ as the _x_-axis, then the second body will appear to be at rest, and the first body in uniform motion. We shall now assume that the first body supposed to be at rest, has the length _l_, _i.e._, the cross section PP of the first strip upon the _x_-axis = _l_^. OC, where OC is the unit measuring rod upon the _x_-axis—and the second body also, when supposed to be at rest, has the same length _l_, this means that, the cross section Q′Q′ of the second strip has a cross-section _l_^· OC′, when measured parallel to the _x′_-axis. In these two bodies, we have now images of two Lorentz-electrons, one of which is at rest and the other moves uniformly. Now if we stick to our original coordinates, then the extension of the second electron is given by the cross section QQ of the strip belonging to it measured parallel to the _x_-axis. Now it is clear since Q′Q′ = _l_^· OC′, that QQ = _l_^· OD′.

If (_dc_/_dt_) = _v_, an easy calculation gives that

OD′ = OC √(1-(_v²_/_c²_)), therefore (PP/QQ) = (1/√(1-(_v²_/_c²_))

This is the sense of Lorentz’s hypothesis about the contraction of electrons in case of motion. On the other hand, if we conceive the second electron to be at rest, and therefore adopt the system (_x′_, _t′_,) then the cross-section P′P′ of the strip of the electron parallel to OC′ is to be regarded as its length and we shall find the first electron shortened with reference to the second in the same proportion, for it is,

P′P′/Q′Q′ = OD/OC′ = OD′/OC = QQ/PP

Lorentz called the combination _t′_ of (_t_ and _x_) as the _local time_ (_Ortszeit_) of the uniformly moving electron, and used a physical construction of this idea for a better comprehension of the contraction-hypothesis. But to perceive clearly that the time of an electron is as good as the time of any other electron, _i.e._ _t_, _t′_ are to be regarded as equivalent, has been the service of A. Einstein [Ann. d. Phys. 891, p. 1905, Jahrb. d. Radis. ... 4-4-11-1907.] There the concept of time was shown to be completely and unambiguously established by natural phenomena. But the concept of space was not arrived at, either by Einstein or Lorentz, probably because in the case of the above-mentioned spatial transformations, where the (_x′_, _t′_) plane coincides with the _x_-_t_ plane, the significance is possible that the _x_-axis of space some-how remains conserved in its position.

We can approach the idea of space in a corresponding manner, though some may regard the attempt as rather fantastical.

According to these ideas, the word “Relativity-Postulate” which has been coined for the demands of invariance in the group G, seems to be rather inexpressive for a true understanding of the group G_{_c_}, and for further progress. Because the sense of the postulate is that the four-dimensional world is given in space and time by phenomena only, but the projection in time and space can be handled with a certain freedom, and therefore I would rather like to give to this assertion the name “_The Postulate of the Absolute world_” [World-Postulate].

III

By the world-postulate a similar treatment of the four determining quantities _x_, _y_, _z_, _t_, of a world-point is possible. Thereby the forms under which the physical laws come forth, gain in intelligibility, as I shall presently show. Above all, the idea of acceleration becomes much more striking and clear.

I shall again use the geometrical method of expression. Let us call any world-point O as a “Space-time-null-point.” The cone

_c²__t²_ - _x²_ - _y²_ - _z²_ = O

consists of two parts with O as apex, one part having _t_ < 0, the other having _t_ > 0. The first, which we may call _the fore-cone_ consists of all those points which send light towards O, the second, which we may call _the aft-cone_, consists of all those points which receive their light from O. The region bounded by the fore-cone may be called the fore-side of O, and the region bounded by the aft-cone may be called the aft-side of O. (_Vide_ fig. 2).

On the aft-side of O we have the already considered hyperboloidal shell F = _c²__t²_ - _x²_ - _y²_ - _z²_ = 1, _t_ > 0.

The region inside the two cones will be occupied by the hyperboloid of one sheet

-F = _x²_ + _y²_ + _z²_ - _c²__t²_ = _k²_,

where _k²_ can have all possible positive values. The hyperbolas which lie upon this figure with O as centre, are important for us. For the sake of clearness the individual branches of this hyperbola will be called the “_Inter-hyperbola with centre O_.” Such a hyperbolic branch, when thought of as a world-line, would represent a motion which for _t_ = -∞ and _t_ = ∞, asymptotically approaches the velocity of light _c_.

If, by way of analogy to the idea of vectors in space, we call any directed length in the manifoldness _x_, _y_, _z_, _t_ a vector, then we have to distinguish between a time-vector directed from O towards the sheet ±F = 1, _t_ > 0 and a space-vector directed from O towards the sheet -F = 1. The time-axis can be parallel to any vector of the first kind. Any world-point between the _fore_ and _aft cones_ of O, may by means of the system of reference be regarded either as synchronous with O, as well as later or earlier than O. Every world-point on the fore-side of O is necessarily always earlier, every point on the aft side of O, later than O. The limit _c_ = ∞ corresponds to a complete folding up of the wedge-shaped cross-section between the fore and aft cones in the manifoldness _t_ = 0. In the figure drawn, this cross-section has been intentionally drawn with a different breadth.

Let us decompose a vector drawn from O towards (_x_, _y_, _z_, _t_) into its components. If the directions of the two vectors are respectively the directions of the radius vector OR to one of the surfaces ±F = 1, and of a tangent RS at the point R of the surface, then the vectors shall be called normal to each other. Accordingly

_c²__tt₁_ - _xx₁_ - _yy₁_ - _zz₁_ = 0,

which is the condition that the vectors with the components (_x_, _y_, _z_, _t_) and (_x₁_ _y₁_ _z₁_ _t₁_) are normal to each other.

For the _measurement_ of vectors in different directions, the unit measuring rod is to be fixed in the following manner;—a space-like vector from 0 to -F = I is always to have the measure unity, and a time-like vector from O to +F = 1, _t_ > 0 is always to have the measure 1/_c_.

Let us now fix our attention upon the world-line of a substantive point running through the world-point (_x_, _y_, _z_, _t_); then as we follow the _progress_ of the line, the quantity

_d_τ = (1/_c_) √(_c²__dt²_ - _dx²_ - _dy²_ - _dz²_),

corresponds to the time-like vector-element (_dx_, _dy_, _dz_, _dt_).

The integral τ = ∫_d_τ, taken over the world-line from any fixed initial point P₀ to any variable final point P, may be called the “Proper-time” of the substantial point at P₀ upon the _world-line_. We may regard (_x_, _y_, _z_, _t_), _i.e._, the components of the vector OP, as functions of the “proper-time” τ; let ([._x_], [._y_], [._z_], [._t_]) denote the first differential-quotients, and ([.._x_], [.._y_], [.._z_], [.._t_]) the second differential quotients of (_x_, _y_, _z_, _t_) with regard to τ, then these may respectively be called the _Velocity-vector_, and the _Acceleration-vector_ of the substantial point at P. Now we have

_c²_ [._t²_] - [._x²_] - [._y²_] - [._z²_] = _c²_

_c²_ [._t_][.._t_] - [._x_][.._x_] - [._y_][.._y_] - [._z_][.._z_] =
0

_i.e._, the ‘_Velocity-vector_’ is the time-like vector of unit measure in the direction of the world-line at P, the ‘_Acceleration-vector_’ at P is normal to the velocity-vector at P, and is in any case, a space-like vector.

Now there is, as can be easily seen, a certain hyperbola, which has three infinitely contiguous points in common with the world-line at P, and of which the asymptotes are the generators of a ‘fore-cone’ and an ‘aft-cone.’ This hyperbola may be called the “hyperbola of curvature” at P (_vide_ fig. 3). If M be the centre of this hyperbola, then we have to deal here with an ‘Inter-hyperbola’ with centre M. Let P = measure of the vector MP, then we easily perceive that the acceleration-vector at P is _a vector of magnitude_ _c²_/ρ _in the direction of_ MP.

If [.._x_], [.._y_], [.._z_], [.._t_] are nil, then the hyperbola of curvature at P reduces to the straight line touching the world-line at P, and ρ = ∞.

IV

In order to demonstrate that the assumption of the group G_{_c_} for the physical laws does not possibly lead to any contradiction, it is unnecessary to undertake a revision of the whole of physics on the basis of the assumptions underlying this group. The revision has already been successfully made in the case of “Thermodynamics and Radiation,”[30] for “Electromagnetic phenomena”,[31] and finally for “Mechanics with the maintenance of the idea of mass.”

For this last mentioned province of physics, the question may be asked: if there is a force with the components X, Y, Z (in the direction of the space-axes) at a world-point (_x_, _y_, _z_, _t_), where the velocity-vector is ([._x_], [._y_], [._z_], [._t_]), then how are we to regard this force when the system of reference is changed in any possible manner? Now it is known that there are certain well-tested theorems about the ponderomotive force in electromagnetic fields, where the group G_{_c_} is undoubtedly permissible. These theorems lead us to the following simple rule; _if the system of reference be changed in any way, then the supposed force is to be put as a force in the new space-coordinates in such a manner, that the corresponding vector with the components_

[._t_]X, [._t_]Y, [._t_]Z, [._t_]T,

_where_ T = 1/_c²_ ([._x_]/[._t_] X + [._y_]/[._t_] Y +
[._z_]/[._t_] Z) = 1/_c²_
(_the rate of
which work is done at the world-point_), _remains unaltered_.

This vector is always normal to the velocity-vector at P. Such a force-vector, representing a force at P, may be called a _moving force-vector at_ P.

Now the world-line passing through P will be described by a substantial point with the constant _mechanical mass m_. Let us call _m-times_ the velocity-vector at P as the _impulse-vector_, and _m-times_ the acceleration-vector at P as the _force-vector of motion_, at P. According to these definitions, the following law tells us how the motion of a point-mass takes place under any moving force-vector[32]:

_The force-vector of motion is equal to the moving force-vector._

This enunciation comprises four equations for the components in the four directions, of which the fourth can be deduced from the first three, because both of the above-mentioned vectors are perpendicular to the velocity-vector. From the definition of T, we see that the fourth simply expresses the “Energy-law.” Accordingly _c²_-_times the component of the impulse-vector in the direction of the t-axis is_ to be defined as _the kinetic-energy_ of the point-mass. The expression for this is

_mc²_ _dt_/_d_τ = _mc²_ /√(1 - _v²_/_c²_)

_i.e._, if we deduct from this the additive constant _mc²_, we obtain the expression ½ _mv²_ of Newtonian-mechanics up to magnitudes of _the order of_ 1/_c²_. Hence it appears that _the energy_ depends _upon the system of reference_. But since the _t_-axis can be laid in the direction of any time-like axis, therefore the energy-law comprises, for any possible system of reference, the whole system of equations of motion. This fact retains its significance even in the limiting case c = ∞, for the axiomatic construction of Newtonian mechanics, as has already been pointed out by T. R. Schütz.[33]

From the very beginning, we can establish the ratio between the units of time and space in such a manner, that the velocity of light becomes unity. If we now write √-1 _t_ = _l_, in the place of _l_, then the differential expression

_d_τ² = -(_dx²_ + _dy²_ + _dz²_ + _dl²_),

becomes symmetrical in (_x_, _y_, _r_, _l_); this symmetry then enters into each law, which does not contradict the _world-postulate_. We can clothe the “essential nature of this postulate in the mystical, but mathematically significant formula

3·10⁵ _km_ = √-1 Sec.

V

The advantages arising from the formulation of the world-postulate are illustrated by nothing so strikingly as by the expressions which tell us about the reactions exerted by a point-charge moving in any manner according to the Maxwell-Lorentz theory.

Let us conceive of the world-line of such an electron with the charge (_e_), and let us introduce upon it the “Proper-time” τ reckoned from any possible initial point. In order to obtain the field caused by the electron at any world-point P₁ let us construct the fore-cone belonging to P₁ (_vide_ fig. 4). Clearly this cuts the unlimited world-line of the electron at a single point P, because these directions are all time-like vectors. At P, let us draw the tangent to the world-line, and let us draw from P₁ the normal to this tangent. Let _r_ be the measure of P₁Q. According to the definition of a fore-cone, _r_/_e_ is to be reckoned as the measure of PQ. Now at the world-point P₁, the vector-potential of the field excited by _e_ is represented by the vector in direction PQ, having the magnitude _e_/_cr_, in its three space components along the _x_-, _y_-, _z_-axes; the scalar-potential is represented by the component along the _t_-axis. This is the elementary law found out by A. Lienard, and E. Wiechert.[34]

If the field caused by the electron be described in the above-mentioned way, then it will appear that the division of the field into electric and magnetic forces is a relative one, and depends upon the time-axis assumed; the two forces considered together bears some analogy to the force-screw in mechanics; the analogy is, however, imperfect.