Part 9
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.