APPENDIXMechanics 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 vectorvin any direction and of any magnitudeq< 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 oft,t′,q, we shall writect,ct′, andq/c, wherecrepresents a certain positive constant, andqis

r′ṽ=rṽ,r′v=c(rv-qt)/√(c²-q²),t′= (qrv+c²t)/c√(c²-q²)

r′ṽ=rṽ,

r′v=c(rv-qt)/√(c²-q²),

t′= (qrv+c²t)/c√(c²-q²)

They denote, as we remember, thatris the space-vector (x,y,z),r′is the space-vector (x′y′z′)

If in these equations, keepingvconstant we approach the limitc= ∞, then we obtain from these

r′ṽ=rṽ,r′v=rv-qt,t′=t.

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)

whenc= ∞.

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 ofc, in another part forc= ∞.

It is evident that according to Newtonian Mechanics, this covariance holds forc= ∞ and not forc= velocity of light.

May we not then regard those traditional covariances forc= ∞ only as an approximation consistent with experience, the actual covariance of natural laws holding for a certain finite value ofc.

I may here point out that by if instead of the Newtonian Relativity-Postulate withc= ∞, we assume a relativity-postulate with a finitec, 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 treatxandtas 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 timet. The totality of the space-time points (x,y,z,t) which correspond to the positions at different timest, 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 thet′axis has the direction 0A′, 0A′ indicating the direction of the space-time line passing through P. The spacet′= const, which is to be laid through P, is the one which is perpendicular to the space-time line through P.

To the incrementdtof the time of P corresponds the increment

dτ = √(dt²-dx²-dy²) -dz²=dt√(1 -u²)

of the newly introduced time parametert′. 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 timet₀, then the region comprising all the space-time line passing through R₀ andt₀shall be called a space-time filament.

If we have an analytical expression θ(xy,z,t) so that θ(x,yzt) = 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.

-(∂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 determinedt′—axis in question here is known as the upper normal of the cross-section at the point P and the value ofdJ = ∫∫∫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 thetaxis of the filament at the pointt=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 timet, belongs a positive quantity—the mass at R at the timet. 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 thetaxis, and passing through (x,y,z,t), is constant along the whole filament.

Now the contentsdJnof the normal cross-section of the filament which is laid through (x,y,z,t) is

(4)dJn= (1/√(1 -u²))dJ = -iω₄dJ = (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 (xyzt). 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 ofton Q′ than on Q°. The finite range enclosed between Q° and Q′ shall be called a space-timesichel,[29]Q′ is the lower boundary, and Q′ is the upper boundary of thesichel.

If we decompose a filament into elementary space-time filaments, then to an entrance-point of an elementary filament through the lower boundary of thesichel, there corresponds an exit point of the same by the upper boundary, whereby for both, the product νdJntaken 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 thesichel, and (comp. (67), § 12)

lor ν[=ω] = (∂νω₁/∂x₁) + (∂νω₂/∂x₂) + (∂νω₃/∂x₃) + (∂νω₄/∂x₄).

If thesichelreduces 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 = ∫ ∫∫∫ νdx dy dz dt(7)

extending over the whole range of the space-timesichel. We shall decompose thesichelinto elementary space-time filaments, and every one of these filaments in small elementsdτ 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=dmand write τ⁰, τlfor the ‘Proper-time’ of the upper and lower boundary of thesichel.

Then the integral (7) can be denoted by

∫∫ νdJndτ = ∫ (τ′-τ⁰)dm.

taken over all the elements of the sichel.

Now let us conceive of the space-time lines inside a space-timesichelas 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 thesichel, 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 thesichel. 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 valuesx+ 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 (xyzt) with the normal section of contentsdJnand ifdJn+ δdJnbe 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 thesichel, varies on account of the displacement as a definite function N + δN of λ, and we may call this function N + δN as themass actionof 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/∂λdλk= 1, 2, 3, 4h= 1, 2, 3, 4

(9)d(xh+ δxh) =dxh+ ∑k∂δxh/∂xk+ ∂δxh/∂λdλ

k= 1, 2, 3, 4h= 1, 2, 3, 4

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)dxdydzdt,

the integration extending over the whole sicheld(τ + δτ) whered(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= ω₁dτ,dx₂= ω₂dτ,dx₃= ω₃dτ,dx₄= ω₄dτ,dλ = 0

dx₁= ω₁dτ,dx₂= ω₂dτ,

dx₃= ω₃dτ,dx₄= ω₄dτ,dλ = 0

therefore:—

(11) (d(τ + δτ))/dτ = √( -∑(ωh+ ∑(∂δxh/∂xk)ωk)²)k= 1, 2, 3, 4.h= 1, 2, 3, 4.

(11) (d(τ + δτ))/dτ = √( -∑(ωh+ ∑(∂δxh/∂xk)ωk)²)

k= 1, 2, 3, 4.h= 1, 2, 3, 4.

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 δxhas a function of (x,y,z,t) vanishes for the zero-value of the parameter λ, so in generaldδxk/(∂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

= -∫∫∫∫ ∑ ωh((∂ξh/∂x₁)ω₁ + (∂ξh/∂x₂)ω₂ +(∂ξh/∂x₃)ω₃ + (∂ξh/∂x₄)ω₄)

dx dy dz dt

for the system (x₁x₂x₃x₄) on the boundary of thesichel, (δx₁δx₂δx₃δx₄) 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

∫∫∫∫ ∑ 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)

(∂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.

(14) ∫∫∫∫ ν((∂ω₁/∂τ)ξ₁ + (∂ω₂/∂τ)ξ₂ + (∂ω₃/∂τ)ξ₃ + (∂ω₄/∂τ)ξ₄)

dx dy dz dt.

For a virtual displacement in thesichelwe 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 ξhis 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 | = | XxYxZx-iTx|| S₂₁ S₂₂ S₂₃ S₂₄ | = | XyYyZy-iTy|| S₃₁ S₃₂ S₃₃ S₃₄ | = | XzYzZz-iTz|| S₄₁ S₄₂ S₄₃ S₄₄ | = | -iXt-iYt-iZtTt|

(16) S =| S₁₁ S₁₂ S₁₃ S₁₄ | = | XxYxZx-iTx|

(16) S =

| S₁₁ S₁₂ S₁₃ S₁₄ | = | XxYxZx-iTx|

| S₂₁ S₂₂ S₂₃ S₂₄ | = | XyYyZy-iTy|

| S₃₁ S₃₂ S₃₃ S₃₄ | = | XzYzZz-iTz|

| S₄₁ S₄₂ S₄₃ S₄₄ | = | -iXt-iYt-iZtTt|

where XnYx.....Xz, Ttare 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))/∂xhdx dy dz dt

extended over the whole range of thesichel, 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

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 followingminimum 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= ∂S1h/∂x₁+ ∂S2h/∂x₂+ ∂S3h/∂x₃+ ∂S4h/∂x₄, (20)

(19) ν ∂wh/∂τ = Kh+ χwh(h= 1, 2, 3, 4)

whence Kh= ∂S1h/∂x₁+ ∂S2h/∂x₂+ ∂S3h/∂x₃+ ∂S4h/∂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 relationwẇ= - 1. By multiplying (19) bywh, and summing the four, we obtain X = Kẇ, and therefore clearly K + (Kẇ)wwill be a space-time vector of the 1st kind which is normal tow. 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) νd/dτ (dx/dτ) = X, νd/dτ (dy/dτ) = Y, νd/dτ (dz/dτ) = Z,νd/dτ (dx/dτ) = T, and we have also(dx/dτ)² + (dy/dτ)² + (dz/dτ)² > (dt/dτ)² = -1,and Xdx/dτ + Ydy/dτ + Zdz/dτ = Tdt/dτ.

(21) νd/dτ (dx/dτ) = X, νd/dτ (dy/dτ) = Y, νd/dτ (dz/dτ) = Z,

νd/dτ (dx/dτ) = T, and we have also

(dx/dτ)² + (dy/dτ)² + (dz/dτ)² > (dt/dτ)² = -1,

and Xdx/dτ + Ydy/dτ + Zdz/dτ = Tdt/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.

Letx,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 throughx,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 RxRyRzRtand ifmbe the constant mass of the filament, we obtain

(22)md/dτdx/dτ = Rx,md/dτdy/dτ = Ry,md/dτdz/dτ = Rz,md/dτdt/dτ = Rt

(22)md/dτdx/dτ = Rx,

md/dτdy/dτ = Ry,

md/dτdz/dτ = Rz,

md/dτdt/dτ = Rt

R is now a space-time vector of the 1st kind with the components (RxRyRzRt) which is normal to the space-time vector of the 1st kindw,—the velocity of the material point with the components

dx/dτ,dy/dτ,dz/dτ,idt/dτ.

We may call this vector Rthe 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 thetaxis, and passes through (x,y,z,t), then [See (4)] the equations (22) are obtained, but

are now multiplied bydτ/dt; in particular, the last equation comes out in the form,

md/dt(dt/dτ) =wxRxdτ/dt+wyRydτ/dt+wzRzdτ/dt.

The right side is to be looked uponas the amount of work done per unit of timeat 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+ )

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.

Sincedtis always greater thandτ 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*)²t-t* >= 0

(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 (xyzt) 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 massmmay, 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. Letx,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*)³d²z/dτ² = -m*z/(t- τ*)³,(26)d²t/dτ² = -m*/(t- τ*)²d(t- τ*)/dt

(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*)²d(t- τ*)/dt

d²z/dτ² = -m*z/(t- τ*)³,

(26)d²t/dτ² = -m*/(t- τ*)²d(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 timet, 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 (xyz) is an ellipse with the semi-major axisaand the eccentricitye. Let E denote the eccentric anomaly, Τ the increment of the proper time for a complete description of the orbit, finallynΤ = 2π, so that from a properly chosen initial point τ, we have the Kepler-equation

(29)nτ = E -esin E.

If we now change the unit of time, and denote the velocity of light byc, then from (28), we obtain

(30) (dt/dτ)² - 1= (m*/ac²) (1 +ecos E)/(1 -ecos E)

(30) (dt/dτ)² - 1

= (m*/ac²) (1 +ecos E)/(1 -ecos E)

Now neglectingc⁻⁴with regard to 1, it follows that

ndt=ndτ [ 1 + ½m*/ac²(1 +ecos E)/(1 -ecos E) ]

from which, by applying (29),

(31)nt+ const = (1 + ½m*/ac²)nτ +m*/ac²Sin E.

the factorm*/ac²is here the square of the ratio of a certain average velocity of F in its orbit to the velocity of light. If nowm* denote the mass of the sun,athe 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.

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