The Principle of Relativity · Albert Einstein
APPENDIX Mechanics and the Relativity-Postulate.
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.
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_.
(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.
“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.
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. 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.
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