_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.
The representation of the plane (defined by two straight lines) is much
more difficult. In three dimensions, the plane can be represented by the
vector perpendicular to itself. But that artifice is not available in
four dimensions. For the perpendicular to a plane, we now have not a
single line, but an infinite number of lines constituting a plane. This
difficulty has been overcome by Minkowski in a very elegant manner which
will become clear later on. Meanwhile we offer the following extract
from the above mentioned work of Sommerfeld.
(Pp. 755, Bd. 32, Ann. d. Physik.)
“In order to have a better knowledge about the nature of the six-vector
(which is the same thing as Minkowski’s space-time vector of the _2nd_
kind) let us take the special case of a piece of plane, having unit area
(contents), and the form of a parallelogram, bounded by the four-vectors
_u_, _v_, passing through the origin. Then the projection of this piece
of plane on the _xy_ plane is given by the projections _u__{_x_},
_u__{_y_}, _v__{_x_}, _v__{_y_} of the four vectors in the combination
φ_{_x_ _y_} = _u__{_x_}_v__{_y_} - _u__{_y_}_v_{_x_}.
Let us form in a similar manner all the six components of this plane φ.
Then six components are not all independent but are connected by the
following relation
φ_{_y_ _z_} φ_{_x_ _l_} + φ_{_z_ _x_} φ_{_y_ _l_} + φ_{_x_ _y_}
φ_{_z_ _l_} = 0
Further the contents | φ | of the piece of a plane is to be defined as
the square root of the sum of the squares of these six quantities. In
fact,
| φ |² = φ_{_y_ _z_}² + φ_{_z_ _x_}² + φ_{_x_ _y_}² + φ_{_x_ _l_}² +
φ_{_y_ _l_}² + φ_{_z_ _l_}².
Let us now on the other hand take the case of the unit plane φ^* normal
to φ; we can call this plane the Complement of φ. Then we have the
following relations between the components of the two plane:—
φ_{_y_ _z_}^* = φ_{_x_ _l_}, φ_{_z_ _x_}^* = φ_{_y_ _l_}, φ_{_x_
_y_}^* = φ_{_z_ _l_} φ_{_z_ _l_}^* = φ_{_y_ _x_} ...
The proof of these assertions is as follows. Let _u_^*, _v_^* be the
four vectors defining φ^*. Then we have the following relations:—
_u__{_x_}^* _u__{_x_} + _u__{_y_}^* _u__{_y_} + _u__{_z_}^*
_u__{_z_} + _u__{_l_}^* _u__{_l_} = 0
_u__{_x_}^* _v__{_x_} + _u__{_y_}^* _v__{_y_} + _u__{_z_}^*
_v__{_z_} + _u__{_l_}^* _v__{_l_} = 0
_v__{_x_}^* _u__{_x_} + _v__{_y_}^* _u__{_y_} + _v__{_z_}^*
_u__{_z_} + _v__{_l_}^* _u__{_l_} = 0
_v__{_x_}^* _v__{_x_} + _v__{_y_}^* _v__{_y_} + _v__{_z_}^*
_v__{_z_} + _v__{_l_}^* _v__{_l_} = 0
If we multiply these equations by _v__{_l_}, _u__{_l_}, _v__{_s_}, and
subtract the second from the first, the fourth from the third we obtain
_u__{_x_}^* φ_{_x_ _l_} + _u__{_y_}^* φ_{_y_ _l_} + _u__{_z_}^*
φ_{_z_ _l_} = 0
_v__{_x_}^* φ_{_z_ _l_} + _v__{_y_}^* φ_{_y_ _l_} + _v__{_z_}^*
φ_{_z_ _l_} = 0
multiplying these equations by _v__{_x_}^* . _u__{_x_}^*, or by
_v__{_y_}^* . _u__{_y_}^*, we obtain
φ_{_x_ _z_}^* φ_{_x_ _l_} + φ_{_y_ _z_}^* φ_{_y_ _l_} = 0 and φ_{_x_
_y_}^* φ_{_x_ _l_} + φ_{_z_ _x_}^* φ_{_z_ _l_} = 0
from which we have
φ_{_y_ _z_}^* : φ_{_x_ _y_}^* : φ_{_z_ _x_}^* = φ_{_x_ _l_} : φ_{_z_
_l_} : φ_{_y_ _l_}