Mechanics and the Relativity-Postulate

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APPENDIX

Mechanics and the Relativity-Postulate.

It would be very unsatisfactory, if the new way of looking at the time-concept, which permits a Lorentz transformation, were to be confined to a single part of Physics.

Now many authors say that classical mechanics stand in opposition to the relativity postulate, which is taken to be the basis of the new Electro-dynamics.

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.

Then the integral (7) can be denoted by

taken over all the elements of the sichel.

Now let us conceive of the space-time lines inside a space-time sichel as material curves composed of material points, and let us suppose that they are subjected to a continual change of length inside the sichel in the following manner. The entire curves are to be varied in any possible manner inside the sichel, while the end points on the lower and upper boundaries remain fixed, and the individual substantial points upon it are displaced in such a manner that they always move forward normal to the curves. The whole process may be analytically represented by means of a parameter λ, and to the value λ = 0, shall correspond the actual curves inside the sichel. Such a process may be called a virtual displacement in the sichel.

Let the point (x, y, z, t) in the sichel λ = 0 have the values x + δx, y + δy, z + δz, t + δt, when the parameter has the value λ; these magnitudes are then functions of (x, y, z, t, λ). Let us now conceive of an infinitely thin space-time filament at the point (x y z t) with the normal section of contents dJn and if dJn + δdJn be the contents of the normal section at the corresponding position of the varied filament, then according to the principle of conservation of mass—(ν + dν being the rest-mass-density at the varied position),

(8) (ν + δν) (dJn + δdJn) = νdJn = dm.

In consequence of this condition, the integral (7) taken over the whole range of the sichel, varies on account of the displacement as a definite function N + δN of λ, and we may call this function N + δN as the mass action of the virtual displacement.

If we now introduce the method of writing with indices, we shall have

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:—

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

may be called the kinetic energy of the material point.

Since dt is always greater than dτ we may call the quotient (dt - dτ)/dτ as the “Gain” (vorgehen) of the time over the proper-time of the material point and the law can then be thus expressed;—The kinetic energy of a material point is the product of its mass into the gain of the time over its proper-time.

The set of four equations (22) again shows the symmetry in (x, y, z, t), which is demanded by the relativity postulate; to the fourth equation however, a higher physical significance is to be attached, as we have already seen in the analogous case in electrodynamics. On the ground of this demand for symmetry, the triplet consisting of the first three equations are to be constructed after the model of the fourth; remembering this circumstance, we are justified in saying,—

“If the relativity-postulate be placed at the head of mechanics, then the whole set of laws of motion follows from the law of energy.”

I cannot refrain from showing that no contradiction to the assumption on the relativity-postulate can be expected from the phenomena of gravitation.

If B*(x*, y*, z*, t*) be a solid (fester) space-time point, then the region of all those space-time points B (x, y, z, t), for which

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

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