Part 15
We have already mentioned several times that the special relativity theory is to be looked upon as a special case of the general, in which _g__{μν}’s have constant values (4). This signifies, according to what has been said before, a total neglect of the influence of gravitation. We get one important approximation if we consider the case when _g__{μν}’s differ from (4) only by small magnitudes (compared to 1) where we can neglect small quantities of the second and higher orders (first aspect of the approximation.)
Further it should be assumed that within the space-time region considered, _g__{μν}’s at infinite distances (using the word infinite in a spatial sense) can, by a suitable choice of co-ordinates, tend to the limiting values (4); _i.e._, we consider only those gravitational fields which can be regarded as produced by masses distributed over finite regions.
We can assume that this approximation should lead to Newton’s theory. For it however, it is necessary to treat the fundamental equations from another point of view. Let us consider the motion of a particle according to the equation (46). In the case of the special relativity theory, the components
_dx₁_/_ds_, _dx₂_/_ds_, _dx₃_/_ds_,
can take any values. This signifies that any velocity
_v_ = √((_dx₁_/_dx₄_)² + (_dx₂_/_dx₄_)² + (_dx₃_/_dx₄_)²)
can appear which is less than the velocity of light in vacuum (_v_ < 1). If we finally limit ourselves to the consideration of the case when _v_ is small compared to the velocity of light, it signifies that the components
_dx₁_/_ds_, _dx₂_/_ds_, _dx₃_/_ds_,
can be treated as small quantities, whereas _dx₄_/_ds_ is equal to 1, up to the second-order magnitudes (the second point of view for approximation).
Now we see that, according to the first view of approximation, the magnitudes γ_{μν}^τ’s are all small quantities of at least the first order. A glance at (46) will also show, that in this equation according to the second view of approximation, we are only to take into account those terms for which μ = ν = 4.
By limiting ourselves only to terms of the lowest order we get instead of (46), first, the equations:—
_d²__x__{τ}/_dt²_ = Γ₄₄^τ, where _ds_ = _dx₄_ = _dt_,
or by limiting ourselves only to those terms which according to the first stand-point are approximations of the first order,
$$ \frac{d^2 x_{\tau}}{dt^2} = \begin{bmatrix}44\\\tau\end{bmatrix} $$ (\tau = 1, 2, 3)
$$ \frac{d^2 x_{4}}{dt^2} = - \begin{bmatrix}4^4\\4\end{bmatrix] $$
If we further assume that the gravitation-field is quasi-static, _i.e._, it is limited only to the case when the matter producing the gravitation-field is moving slowly (relative to the velocity of light) we can neglect the differentiations of the positional co-ordinates on the right-hand side with respect to time, so that we get
(67) _d²__x__{τ}/_dt²_ = -½ ∂_g₄₄_/∂_x__{τ} (τ, = 1, 2, 3)
This is the equation of motion of a material point according to Newton’s theory, where _g_₄₄/₂ plays the part of gravitational potential. The remarkable thing in the result is that in the first-approximation of motion of the material point, only the component _g₄₄_ of the fundamental tensor appears.
Let us now turn to the field-equation (53). In this case, we have to remember that the energy-tensor of matter is exclusively defined in a narrow sense by the density ρ of matter, _i.e._, by the second member on the right-hand side of 58 [(58a, or 58b)]. If we make the necessary approximations, then all component vanish except
τ₄₄ = ρ = τ.
On the left-hand side of (53) the second term is an infinitesimal of the second order, so that the first leads to the following terms in the approximation, which are rather interesting for us:
$$ \frac{\partial}{\partial x_{1}} \begin{bmatrix}\mu\nu\\1\end{bmatrix} + \frac{\partial}{\partial x_{2}} \begin{bmatrix}\mu\nu\\2\end{bmatrix} + \frac{\partial}{\partial x_{3}} \begin{bmatrix}\mu\nu\\3\end{bmatrix} + \frac{\partial}{\partial x_{4}} \begin{bmatrix}\mu\nu\\4\end{bmatrix} $$
By neglecting all differentiations with regard to time, this leads, when μ = ν =4, to the expression
$$ - \frac{1}{2} ( \frac{\partial^2 g_{44}}{\partial x^2_{1}} + \frac{\partial^2 g_{44}}{\partial x^2_{2}} + \frac{\partial^2 g_{44}}{\partial x^2_{3}} ) = - \frac{1}{2} V^2 g_{44} $$
The last of the equations (53) thus leads to
(68) ▽² _g₄₄_ = κρ.
The equations (67) and (68) together, are equivalent to Newton’s law of gravitation.
For the gravitation-potential we get from (67) and (68) the exp.
(68a.) -κ/(8π) ∫ ρ_d_τ/_r_
whereas the Newtonian theory for the chosen unit of time gives
-K/_c²_ ∫ρ_d_τ/_r_,
where K denotes usually the gravitation-constant. 6.7 x 10⁻⁸; equating them we get
(69) κ = 8πK/_c²_ = 1.87 x 10⁻²⁷.
§22. Behaviour of measuring rods and clocks in a statical gravitation-field. Curvature of light-rays. Perihelion-motion of the paths of the Planets.
In order to obtain Newton’s theory as a first approximation we had to calculate only _g₄₄_, out of the 10 components _g__{μν} of the gravitation-potential, for that is the only component which comes in the first approximate equations of motion of a material point in a gravitational field.
We see however, that the other components of _g__{μν} should also differ from the values given in (4) as required by the condition _g_ = -1.
For a heavy particle at the origin of co-ordinates and generating the gravitational field, we get as a first approximation the symmetrical solution of the equation:—
{ _g__{ρσ} = -δ_{ρσ} - α(_x__{ρ} _x__{σ})/_r³_ (ρ and σ 1, 2, 3) { (70) { _g__{ρ4} = _g__{4ρ} = 0 (ρ 1, 2, 3) { { _g₄₄_ = 1 - α/_r_.
δ_{ρσ} is 1 or 0, according as ρ = σ or not and _r_ is the quantity
+√(_x₁²_ + _x₂²_ + _x₃²_).
On account of (68a) we have
(70a) α = κM/4π
where M denotes the mass generating the field. It is easy to verify that this solution satisfies approximately the field-equation outside the mass M.
Let us now investigate the influences which the field of mass M will have upon the metrical properties of the field. Between the lengths and times measured locally on the one hand, and the differences in co-ordinates _dx__{ν} on the other, we have the relation
_ds²_ = _g__{μν} _dx__{μ} _dx__{ν}.
For a unit measuring rod, for example, placed parallel to the _x_ axis, we have to put
_ds²_ = -1, _dx₂_ = _dx₃_ = _dx₄_ = 0
then -1 = _g_₁₁_dx₁²_.
If the unit measuring rod lies on the _x_ axis, the first of the equations (70) gives
_g₁₁_ = -(1 + α/_r_).
From both these relations it follows as a first approximation that
(71) _dx_ = 1 - α/2_r_.
The unit measuring rod appears, when referred to the co-ordinate-system, shortened by the calculated magnitude through the presence of the gravitational field, when we place it radially in the field.
Similarly we can get its co-ordinate-length in a tangential position, if we put for example
_ds²_ = -1, _dx₁_ = _dx₃_ = _dx₄_ = 0, _x₁_ = _r_, _x₂_ = _x₃_ = 0
we then get
(71a) -1 = _g₂₂_ _dx₂²_ = -_dx₂²_.
The gravitational field has no influence upon the length of the rod, when we put it tangentially in the field.
Thus Euclidean geometry does not hold in the gravitational field even in the first approximation, if we conceive that one and the same rod independent of its position and its orientation can serve as the measure of the same extension. But a glance at (70a) and (69) shows that the expected difference is much too small to be noticeable in the measurement of earth’s surface.
We would further investigate the rate of going of a unit-clock which is placed in a statical gravitational field. Here we have for a period of the clock
_ds_ = 1, _dx₁_ = _dx₂_ _dx₃_ = 0;
then we have
1 = _g₄₄__dx₄²_
_dx₄_ = 1/√(_g_₄₄) = 1/√(1 + (_g_₄₄ - 1)) = 1 - (_g_₄₄ - 1)/2
or _dx₄_ = 1 + _k_/8π ∫ ρ_d_τ/_r_.
Therefore the clock goes slowly what it is placed in the neighbourhood of ponderable masses. It follows from this that the spectral lines in the light coming to us from the surfaces of big stars should appear shifted towards the red end of the spectrum.
Let us further investigate the path of light-rays in a statical gravitational field. According to the special relativity theory, the velocity of light is given by the equation
-_dx₁²_ - _dx₂²_ - _dx₃²_ + _dx₄²_ = 0;
thus also according to the generalised relativity theory it is given by the equation
(73) _ds²_ = _g__{μν} _dx__{μ} _dx__{ν} = 0.
If the direction, _i.e._, the ratio _dx₁_ : _dx₂_ : _dx₃_ is given, the equation (73) gives the magnitudes
_dx₁_/_dx₄_, _dx₂_/_dx₄_, _dx₃_/_dx₄_,
and with it the velocity,
√((_dx₁_/_dx₄_)² + (_dx₂_/_dx₄_)² + (_dx₃_/_dx₄_)²) = γ,
in the sense of the Euclidean Geometry. We can easily see that, with reference to the co-ordinate system, the rays of light must appear curved in case _g__{μν}’s are not constants. If _n_ be the direction perpendicular to the direction of propagation, we have, from Huygen’s principle, that light-rays (taken in the plane (γ, _n_)] must suffer a curvature ∂λ/∂_n_.
Let us find out the curvature which a light-ray suffers when it goes by a mass M at a distance Δ from it. If we use the co-ordinate system according to the above scheme, then the total bending B of light-rays (reckoned positive when it is concave to the origin) is given as a sufficient approximation by
B = ∫_{-∞}^∞ ∂γ/∂[_x_]₁ _dx₂_
where (73) and (70) gives
γ = √(-_g₄₄_/_g₂₂_) = 1 - α/2_r_ (1 + _x₂²_/_r²_).
The calculation gives
B = 2α/Δ = KM/2πΔ.
A ray of light just grazing the sun would suffer a bending of 1·7″, whereas one coming by Jupiter would have a deviation of about ·02″.
If we calculate the gravitation-field to a greater order of approximation and with it the corresponding path of a material particle of a relatively small (infinitesimal) mass we get a deviation of the following kind from the Kepler-Newtonian Laws of Planetary motion. The Ellipse of Planetary motion suffers a slow rotation in the direction of motion, of amount
(75) _s_ = 24π³_a²_/τ²_c²_(1 - _e²_) per revolution.
In this Formula ‘_a_’ signifies the semi-major axis, _c_, the velocity of light, measured in the usual way, _e_, the eccentricity, τ, the time of revolution in seconds.
The calculation gives for the planet Mercury, a rotation of path of amount 43″ per century, corresponding sufficiently to what has been found by astronomers (Leverrier). They found a residual perihelion motion of this planet of the given magnitude which can not be explained by the perturbation of the other planets.
NOTES
Note 1.
The fundamental electro-magnetic equations of Maxwell for stationary media are:—
curl H = 1/_c_ (∂D/∂_t_ + ρν) (1)
curl E = -1/_c_ ∂B/∂_t_ (2)
div D = ρ B = μH div B = 0 D = kE
According to Hertz and Heaviside, these require modification in the case of moving bodies.
Now it is known that due to motion alone there is a change in a vector _R_ given by
(∂_R_/∂_t_) due to motion = _u_. div R + curl [_Ru_]
where _u_ is the vector velocity of the moving body and [R_u_] the vector product of R and _u_.
Hence equations (1) and (2) become
_c_ curl H = ∂D/∂_t_ + _u_ div D + curl Vect. [D_u_] + ρν (1·1)
and
-_c_ curl E = ∂B/∂_t_ + _u_ div B + curl Vect. [B_u_] (2·1)
which gives finally, for ρ = 0 and div B = 0,
∂D/∂_t_ + _u_ div D = _c_ curl (H - 1/_c_ Vect. [D_u_]) (1·2)
∂B/∂_t_ = -_c_ curl (E - 1/_c_ Vect. [_u_B]) (2·2)
Let us consider a beam travelling along the _x_-axis, with apparent velocity _v_ (_i.e._, velocity with respect to the fixed ether) in medium moving with velocity _u__{_x_} = _u_ in the same direction.
Then if the electric and magnetic vectors are proportional to _e_^{_i_A(_x_ - _vt_)}, we have
∂/∂_x_ = _i_A, ∂/∂_t_ = -_i_A_v_, ∂/∂_y_ = ∂/∂_z_ = 0, _u__{_y_} = _u__{_z_} = 0
Then ∂D__y_/∂_t_ = -_c_∂H_{_z_}/∂_x_ - _u_∂D_{_y_}/∂_z_ ... (1·21)
and ∂B_{_z_}/∂_t_ = -_c_∂E_{_y_}/∂_x_ - _u_∂B_{_z_}/∂_x_ (2·21)
Since D = KE and B = μH, we have
_i_A_v_(κE_y_) = -_ci_A(H_{_z_} + _u_KE_{_y_}) (1·22)
_i_A_v_(μH_{_z_}) = -_ci_A(E_{_y_} + _u_μH_{_z_}) (2·22)
or _v_(K - _u_)E_{_y_} = _c_H_{_z_} (1·23)
μ(_v_ - _u_)H_{_z_} = _c_E_{_y_} (2·23)
Multiplying (1·23) by (2·23)
μK(_v_ - _u_)² = _c²_
Hence (_v_ - _u_)² = _c²_/μ_k_ = _v₀_²
∴ _v_ = _v₀_ + _u_,
making Fresnelian convection co-efficient simply unity.
Equations (1·21) and (2·21) may be obtained more simply from physical considerations.
According to Heaviside and Hertz, the real seat of both electric and magnetic polarisation is the moving medium itself. Now at a point which is fixed with respect to the ether, the rate of change of electric polarisation is δD/δ_t_.
Consider a slab of matter moving with velocity _u__{_x_} along the _x_-axis, then even in a stationary field of electrostatic polarisation, that is, for a field in which δD/δ_t_ = 0, there will be some change in the polarisation of the body due to its motion, given by _u__{_x_}(δD/δ_x_). Hence we must add this term to a purely temporal rate of change δD/δ_t_. Doing this we immediately arrive at equations (1·21) and (2·21) for the special case considered there.
Thus the Hertz-Heaviside form of field equations gives _unity_ as the value for the Fresnelian convection co-efficient. It has been shown in the historical introduction how this is entirely at variance with the observed optical facts. As a matter of fact, Larmor has shown (Aether and Matter) that 1 - 1/μ² is not only sufficient but is also necessary, in order to explain experiments of the Arago prism type.
A short summary of the electromagnetic experiments bearing on this question, has already been given in the introduction.
According to Hertz and Heaviside the total polarisation is situated in the medium itself and is completely carried away by it. Thus the electromagnetic effect outside a moving medium should be proportional to K, the specific inductive capacity.
_Rowland_ showed in 1876 that when a charged condenser is rapidly rotated (the dielectric remaining stationary), the magnetic effect outside is proportional to K, the Sp. Ind. Cap.
_Röntgen_ (Annalen der Physik 1888, 1890) found that if the dielectric is rotated while the condenser remains stationary, the effect is proportional to K - 1.
_Eichenwald_ (Annalen der Physik 1903, 1904) rotated together both condenser and dielectric and found that the magnetic effect was proportional to the potential difference and to the angular velocity, but was completely independent of K. This is of course quite consistent with Rowland and Röntgen.
_Blondlot_ (Comptes Rendus, 1901) passed a current of air in a steady magnetic field H_{_y_}, (H = H_{_z_} = 0). If this current of air moves with velocity _u__{_x_} along the _x_-axis, an electromotive force would be set up along the _z_-axis, due to the relative motion of matter and magnetic tubes of induction. A pair of plates at _z_ = ±_a_, will be charged up with density ρ = D_{_z_} = KE = K. _u__{_s_} H_{_y_}/c. But Blondlot failed to detect any such effect.
_H. A. Wilson_ (Phil. Trans. Royal Soc. 1904) repeated the experiment with a cylindrical condenser made of ebony, rotating in a magnetic field parallel to its own axis. He observed a change proportional to K — 1 and not to K.
Thus the above set of electro-magnetic experiments contradict the Hertz-Heaviside equations, and these must be abandoned.
[P. C. M.]
Note 2. Lorentz Transformation.
Lorentz. Versuch einer theorie der elektrischen und optischen Erscheinungen im bewegten Körpern.
(Leiden—1895).
Lorentz. Theory of Electrons (English edition), pages 197-200, 230, also notes 73, 86, pages 318, 328.
Lorentz wanted to explain the Michelson-Morley null-effect. In order to do so, it was obviously necessary to explain the Fitzgerald contraction. Lorentz worked on the hypothesis that an electron itself undergoes contraction when moving. He introduced new variables for the moving system defined by the following set of equations.
_x¹_ = β(_x_ - _ut_), _y¹_ = _y_, _z¹_ = _z_, _t¹_ = β(_t_ - (_u_/_c²_)·_x_)
and for velocities, used
_v__{_x_}¹ = β²_v__{_x_} + _u_, _v__{_y_}¹ = β_v__{_y_}, _v__{_z_}¹ = β_v__{_z_} and ρ¹ = ρ/β.
With the help of the above set of equations, which is known as the Lorentz transformation, he succeeded in showing how the Fitzgerald contraction results as a consequence of “fortuitous compensation of opposing effects.”
It should be observed that the Lorentz transformation is not identical with the Einstein transformation. The Einsteinian addition of velocities is quite different as also the expression for the “relative” density of electricity.
It is true that the Maxwell-Lorentz field equations remain _practically_ unchanged by the Lorentz transformation, but they _are_ changed to some slight extent. One marked advantage of the Einstein transformation consists in the fact that the field equations of a moving system preserve _exactly_ the same form as those of a stationary system.
It should also be noted that the Fresnelian convection coefficient comes out in the theory of relativity as a direct consequence of Einstein’s addition of velocities and is quite independent of any electrical theory of matter.
[P. C. M.]
Note 3.
See Lorentz, Theory of Electrons (English edition), § 181, page 213.
H. Poincare, Sur la dynamique ‘electron, Rendiconti del circolo matematico di Palermo 21 (1906).
[P. C. M.]
Note 4. Relativity Theorem and Relativity-Principle.
Lorentz showed that the Maxwell-Lorentz system of electromagnetic field-equations remained practically unchanged by the Lorentz transformation. Thus the electromagnetic laws of Maxwell and Lorentz _can be definitely proved_ “to be independent of the manner in which they are referred to two coordinate systems which have a uniform translatory motion relative to each other.” (See “Electrodynamics of Moving Bodies,” page 5.) Thus so far as the electromagnetic laws are concerned, the principle of relativity _can be proved to be true_.
But it is not known whether this principle will remain true in the case of other physical laws. We can always proceed on the assumption that it does remain true. Thus it is always possible to construct physical laws in such a way that they retain their form when referred to moving coordinates. The ultimate ground for formulating physical laws in this way is merely a subjective conviction that the principle of relativity is universally true. There is no _a priori_ logical necessity that it should be so. Hence the Principle of Relativity (so far as it is applied to phenomena other than electromagnetic) must be regarded as a _postulate_, which we have assumed to be true, but for which we cannot adduce any definite proof, until after the generalisation is made and its consequences tested in the light of actual experience.
[P. C. M.]
Note 5.
See “Electrodynamics of Moving Bodies,” p. 5-8.
Note 6. Field Equations in Minkowski’s Form.
Equations (_i_) and (_ii_) become when expanded into Cartesians:—
∂_m__{_z_}/∂_y_ - ∂_m__{_y_}/∂_z_ - ∂_e__{_x_}/∂τ = ρν_{_x_} } ∂_m__{_x_}/∂_z_ - ∂_m__{_z_}/∂_x_ - ∂_e__{_y_}/∂τ = ρν_{_y_} } ... (1·1) ∂_m__{_y_}/∂_x_ - ∂_m__{_x_}/∂_y_ - ∂_e__{_z_}/∂τ = ρν_{_z_} }
and ∂_e__{_x_}/∂_x_ + ∂_e__{_y_}/∂_y_ + ∂_e__{_z_}/∂_z_ = ρ (2·1)
Substituting _x₁_, _x₂_, _x₃_, _x₄_ and _x_, _y_, _z_, and _i_τ; and ρ₁, ρ₂, ρ₃, ρ₄ for ρν_{_x_}, ρν_{_y_}, ρν_{_z_}, _i_ρ, where _i_ = √(-1).
We get,
∂_m__{_z_}/∂_x₂_ - ∂_m__{_y_}/∂_x₃_ - _i_(∂_e__{_x_}/∂_x₄_) = ρν_{_x_}{ = ρ₁ } - ∂_m__{_z_}/∂_x₁_ + ∂_m__{_x_}/∂_x₃_ - _i_(∂_e__{_y_}/∂_x₄_) = ρν_{_y_} = ρ₂ } ... (1·2) ∂_m__{_y_}/∂_x₁_ - ∂_m__{_x_}/∂_x₂_ - _i_(∂_e__{_z_}/∂_x₄_) = ρν_{_z_}{ = ρ₃ }
and multiplying (2·1) by i we get
∂_ie__{_x_}/∂_x₁_ + ∂_ie__{_y_}/∂_x₂_ + ∂_ie__{_z_}/∂_x₃_ = _i_ρ = ρ₄ ... ... (2·2)
Now substitute
_m__{_x_} = _f₂₃_ = -_f₃₂_ and _ie__{_x_} = _f_₄₁ = -_f₁₄_ _m__{_y_} = _f₃₁_ = -_f₁₃_ _ie__{_y_} = _f_₄₂ = -_f₂₄_ _m__{_z_} = _f₁₂_ = -_f₂₁_ _ie__{_z_} = _f_₄₃ = -_f₃₄_
and we get finally:—
∂_f₁₂_/∂_x₂_ + ∂_f₁₃_/∂_x₃_ + ∂_f₁₄_/∂_x₄_ = ρ₁ }
∂_f₂₁_/∂_x₁_ + ∂_f₂₃_/∂_x₃_ + ∂_f₂₄_/∂_x₄_ = ρ₂ } ... (3)
∂_f₃₁_/∂_x₁_ + ∂_f₃₂_/∂_x₂_ + ∂_f₃₄_/∂_x₄_ = ρ₃ }
∂_f₄₁_/∂_x₁_ + ∂_f₄₂_/∂_x₂_ + ∂_f₄₃_/∂_x₃_ = ρ₄ }
Note 9. On the Constancy of the Velocity of Light.
Page 12—refer also to page 6, of Einstein’s paper.
One of the two fundamental Postulates of the Principle of Relativity is that the velocity of light should remain constant whether the source is moving or stationary. It follows that even if a radiant source S move with a velocity _u_, it should always remain the centre of spherical waves expanding outwards with velocity _c_.
At first sight, it may not appear clear why the velocity should remain constant. Indeed according to the theory of Ritz, the velocity should become _c_ + _u_, when the source of light moves towards the observer with the velocity _u_.
Prof. de Sitter has given an astronomical argument for deciding between these two divergent views. Let us suppose there is a double star of which one is revolving about the common centre of gravity in a circular orbit. Let the observer be in the plane of the orbit, at a great distance Δ.
[Illustration.]
The light emitted by the star when at the position A will be received by the observer after a time, Δ/(_c_ + _u_) while the light emitted by the star when at the position B will be received after a time Δ/(_c_ - _u_). Let T be the real half-period of the star. Then the observed half-period from B to A is approximately T - 2Δ_u_/_c²_ and from A to B is T + 2Δ_u_/_c²_. Now if 2_u_Δ/_c²_ be comparable to T, then it is impossible that the observations should satisfy Kepler’s Law. In most of the spectroscopic binary stars, 2_u_Δ/_c²_ are not only of the same order as T, but are mostly much larger. For example, if _u_ = 100 _km_/sec, T = 8 days, Δ/_c_ = 33 years (corresponding to an annual parallax of ·1″), then T - 2_u_Δ/_c²_ = 0. The existence of the Spectroscopic binaries, and the fact that they follow Kepler’s Law is therefore a proof that _c_ is not affected by the motion of the source.
In a later memoir, replying to the criticisms of Freundlich and Günthick that an apparent eccentricity occurs in the motion proportional to _ku_Δ₀, _u₀_ being the maximum value of _u_, the velocity of light emitted being
_u₀_ = _c_ + _ku_, _k_ = 0 Lorentz-Einstein _k_ = 1 Ritz.
Prof. de Sitter admits the validity of the criticisms. But he remarks that an upper value of _k_ may be calculated from the observations of the double star β-Aurigae. For this star, the parallax π = ·014″, _e_ = ·005, _u₀_ = 110 _km_/sec, T = 3·96,
Δ > 65 light-years, _k_ is < ·002.
For an experimental proof, see a paper by C. Majorana. Phil. Mag., Vol. 35, p. 163.
[M. N. S.]
Note 10. Rest-density of Electricity.
If ρ is the volume density in a moving system then ρ√(1 - _u²_) is the corresponding quantity in the corresponding volume in the fixed system, that is, in the system at rest, and hence it is termed the rest-density of electricity.
[P. C. M.]
Note 11 (page 17) Space-time vectors of the first and the second kind.
As we had already occasion to mention, Sommerfeld has, in two papers on four dimensional geometry (_vide_, Annalen der Physik, Bd. 32, p. 749; and Bd. 33, p. 649), translated the ideas of Minkowski into the language of four dimensional geometry. Instead of Minkowski’s space-time vector of the first kind, he uses the more expressive term ‘four-vector,’ thereby making it quite clear that it represents a directed quantity like a straight line, a force or a momentum, and has got 4 components, three in the direction of space-axes, and one in the direction of the time-axis.