Part 16
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_}
In a corresponding way we have
φ_{_y_ _z_} : φ_{_x_ _y_} : φ_{_z_ _x_} = φ_{_x_ _l_}^* : φ_{_z_ _l_}^* : φ_{_y_ _l_}^*.
_i.e._ φ_{_i_ _k_}^* = λφ(_{_i_ _k_})
when the subscript (_ik_) denotes the component of φ in the plane contained by the lines other than (_ik_). Therefore the theorem is proved.
We have (φ φ*) = φ_{_y_ _z_} φ_{_y_ _z_}^* + ...
= 2 (φ_{_y_ _z_} φ_{_z_ _l_} + ...)
= 0
The general six-vector _f_ is composed from the vectors φ, φ^* in the following way:—
_f_ = ρφ + ρ^* φ^*,
ρ and ρ^* denoting the contents of the pieces of mutually perpendicular planes composing _f_. The “conjugate Vector” _f_^* (or it may be called the complement of _f_) is obtained by interchanging ρ and ρ^*.
We have
_f_^* = ρ^*φ + ρφ^*
We can verify that
_f__{_y z_}^* = _f__{_x l_} etc.
and _f²_ = ρ² + ρ^*², (_f__f_^*) = 2ρρ^*.
| _f_ |² and (_f__f_^*) may be said to be invariants of the six vectors, for their values are independent of the choice of the system of co-ordinates.
[M. N. S.]
Note 12. Light-velocity as a maximum.
Page 23, and Electro-dynamics of Moving Bodies, p. 17.
Putting _v_ = _c_ - _x_, and _w_ = _c_ - λ, we get
V = (2_c_ - (_x_ + λ))/(1 + (_c_ - _x_)(_c_ - λ)/_c²_) = (2_c_ - (_x_ + λ))/(_c²_ + _c²_ - (_x_ + λ)_c_ + _x_λ/_c²_)
= _c_ (2_c_ - (_x_ + λ))/(2_c_ - (_x_ + λ) + _x_λ/_c_)
Thus _v_ lt; _c_, so long as | _x_λ | > 0.
Thus the velocity of light is the absolute maximum velocity. We shall now see the consequences of admitting a velocity W > _c_.
Let A and B be separated by distance _l_, and let velocity of a “signal” in the system S be W > _c_. Let the (observing) system S′ have velocity +_v_ with respect to the system S.
Then velocity of signal with respect to system S′ is given by W′ = (W - _v_)/(1 - W_v_/_c²_)
Thus “time” from A to B as measured in S′, is given by _l_/W′ = _l_(1 - W_v_/_c²_)/(W - _v_) = _t′_ (1)
Now if _v_ is less than _c_, then W being greater than _c_ (by hypothesis) W is greater than _v_, _i.e._, W > _v_.
Let W = _c_ + μ and _v_ = _c_ - λ.
Then W_v_ = (_c_ + μ)(_c_ - λ) = _c²_ + (μ + λ)_c_ - μλ.
Now we can always choose _v_ in such a way that W_v_ is greater than _c²_, since W_v_ is > _c²_ if (μ + λ)_c_ - μλ is > 0, that is, if μ + λ > μλ/_c_; which can always be satisfied by a suitable choice of λ.
Thus for W > _c_ we can always choose λ in such a way as to make W_v_ > _c²_, _i.e._, λ - W_v_/_c²_ negative. But W - _v_ is always positive. Hence with W > _c_, we can always make _t′_, the time from A to B in equation (1) “negative.” That is, the signal starting from A will reach B (as observed in system S′) in less than no time. Thus the effect will be perceived before the cause commences to act, _i.e._, the future will precede the past. Which is absurd. Hence we conclude that W > _c_ is an impossibility, there can be no velocity greater than that of light.
It is _conceptually_ possible to imagine velocities greater than that of light, but such velocities cannot occur in reality. Velocities greater than _c_, will not produce any effect. Causal effect of any physical type can never travel with a velocity greater than that of light.
[P. C. M.]
Notes 13 and 14.
We have denoted the four-vector ω by the matrix | ω₁ ω₂ ω₃ ω₄ |. It is then at once seen that [=ω] denotes the reciprocal matrix
| ω₁ | | ω₂ | | ω₃ | | ω₄ |
It is now evident that while ω¹ = ωA, [=ω]¹ = A⁻¹[=ω]
[ω, _s_] The vector-product of the four-vector ω and _s_ may be represented by the combination
[ω_s_] = [=ω]_s_ - _ṡ_ω
It is now easy to verify the formula _f_¹ = A⁻¹_f_A. Supposing for the sake of simplicity that _f_ represents the vector-product of two four-vectors ω, _s_, we have
_f¹_ = [ω¹_s¹_] = [[=ω]¹_s¹_ - [=_s_]^1ω^1]
= [A⁻¹ [=ω]_s_A - A⁻¹_s_[=ω]A]
= A⁻¹[[=ω]_s_ - _s_[=ω]]A = A⁻¹_f_A.
Now remembering that generally
_f_ = ρφ + ρ*φ*.
Where ρ, ρ* are scalar quantities, φ, φ* are two mutually perpendicular unit planes, there is no difficulty in seeming that
_f_^1 = A⁻¹_f_A.
Note 15. The vector product (_w__f_). (P. 36).
This represents the vector product of a four-vector and a six-vector. Now as combinations of this type are of frequent occurrence in this paper, it will be better to form an idea of their geometrical meaning. The following is taken from the above mentioned paper of Sommerfeld.
“We can also form a vectorial combination of a four-vector and a six-vector, giving us a vector of the third type. If the six-vector be of a special type, _i.e._, a piece of plane, then this vector of the third type denotes the parallelopiped formed of this four-vector and the complement of this piece of plane. In the general case, the product will be the geometric sum of two parallelopipeds, but it can always be represented by a four-vector of the 1st type. For two pieces of 3-space volumes can always be added together by the vectorial addition of their components. So by the addition of two 3-space volumes, we do not obtain a vector of a more general type, but one which can always be represented by a four-vector (loc. cit. p. 759). The state of affairs here is the same as in the ordinary vector calculus, where by the vector-multiplication of a vector of the first, and a vector of the second type (_i.e._, a polar vector), we obtain a vector of the first type (axial vector). The formal scheme of this multiplication is taken from the three-dimensional case.
Let A = (A_{_x_}, A_{_y_}, A_{_z_}) denote a vector of the first type, B = (B_{_y z_}, B_{_z x_}, B_{_x y_}) denote a vector of the second type. From this last, let us form three special vectors of the first kind, namely—
B_{_x_} = (B_{_x x_}, B_{_x y_}, B_{_x z_}) } B_{_y_} = (B_{_y x_}, B_{_y y_}, B_{_y z_}) } (B_{_i k_} = - B_{_k i_}, B_{_i i_} = 0). B_{_z_} = (B_{_z x_}, B_{_z y_}, B_{_z z_}) }
Since B_{_j j_} is zero, B_{_j_} is perpendicular to the _j_-axis. The _j_-component of the vector-product of A and B is equivalent to the scalar product of A and B_{_j_}, _i.e._,
(A B_{_j_},) = A_{_x_} B_{_j x_} + A_{_y_} B_{_j y_} + A_{_z_} B_{_j z_}.
We see easily that this coincides with the usual rule for the vector-product; _e. g._, for _j_ = _x_.
(AB_{_x_}) = A_{_y_} B_{_x_ _y_} - A_{_z_} B_{_z_ _x_}.
Correspondingly let us define in the four-dimensional case the product (P_f_) of any four-vector P and the six-vector _f_. The _j_-component (_j_ = _x_, _y_, _z_, or _l_) is given by
(P_f__{_j_}) = P_{_x_}_f__{_j_ _x_} + P_{_y_}_f__{_j_ _y_} + P_{_w_}_f__{_j_ _z_} + P_{_z_}_f__{_j_ _l_}
Each one of these components is obtained as the scalar product of P, and the vector _f__{_j_} which is perpendicular to j-axis, and is obtained from _f_ by the rule _f__{_j_} = [(_f__{_j_ _x_}, _f__{_j_ _y_}, _f__{_j_ _z_}, _f__{_j_ _l_}) _f__{_j_ _j_} = 0.]
We can also find out here the geometrical significance of vectors of the third type, when _f_ = φ, _i.e._, _f_ represents only one plane.
We replace φ by the parallelogram defined by the two four-vectors U, V, and let us pass over to the conjugate plane φ^*, which is formed by the perpendicular four-vectors U^*, V^*. The components of (Pφ) are then equal to the 4 three-rowed under-determinants D_{_x_} D_{_y_} D_{_z_} D_{_l_} of the matrix
| P_{_x_} P_{_y_} P_{_z_} P_{_l_} | | | | U_{_x_}^* U_{_y_}^* U_{_z_}^* U_{_l_}^* | | | | V_{_x_}^* V_{_y_}^* V_{_z_}^* V_{_l_}^* |
Leaving aside the first column we obtain
D_{_x_} = P_{_y_}(U_{_z_}^* V_{_l_}^* - U_{_l_}^* V_{_z_}^*) + P_{_z_}(U_{_l_}^* V_{_y_}^* - U_{_y_}^* V_{_l_}^*) + P_{_l_}(U_{_y_}^* V_{_z_}^* - U_{_z_}^* V_{_y_}^*) = P_{_y_} φ_{_z_ _y_}^* + P_{_z_}^* φ_{_l_ _y_} + P_{_l_} φ^*_{_y_ _z_}. = P_{_y_} φ_{_x_ _y_} + P_{_z_} φ_{_x_ _z_} + _P__{_l_} φ_{_x_ _l_},
which coincides with (Pφ_{_x_}) according to our definition.
Examples of this type of vectors will be found on page 36, Φ = wF, the electrical-rest-force, and ψ = 2wf^*, the magnetic-rest-force. The rest-ray Ω = iw[Φψ]^* also belong to the same type (page 39). It is easy to show that
Ω = -_i_ | w₁ w₂ w₃ w₄ | | Φ₁ Φ₂ Φ₃ Φ₄ | | ψ₁ ψ₂ ψ₃ ψ₄ |
When (Ω₁, Ω₂, Ω₃) = 0, w₄ = _i_, Ω reduces to the three-dimensional vector
| Ω₁, Ω₂, Ω₃ | = | Φ₁ Φ₂ Φ₃ | | | | ψ₁ ψ₂ ψ₃ |
Since in this case, Φ₁ = w₄ F₁₄ = _e__{_n_} (the electric force) ψ₁ = -_i_w₄ f₂₃ = _m__{_x_} (the magnetic force) we have (Ω) = | _e__{_x_} _e__{_y_} _e__{_z_} | | _m__{_x_} _m__{_y_} _m__{_z_} |
[M. N. S.]
Note 16. The electric-rest force. (Page 37.)
The four-vector φ = wF which is called by Minkowski the electric-rest-force (elektrische Ruh-Kraft) is very closely connected to Lorentz’s Ponderomotive force, or the force acting on a moving charge. If ρ is the density of charge, we have, when ε = 1, μ = 1, _i.e._, for free space
ρ₀φ₁ = ρ₀[w₁ F₁₁ w₂ F₁₂ + w₃ F₁₃ + w₄ F₁₄]
= ρ₀/(√(1 - V²/_c²_)) [_d__{_x_} + 1/_c_ (_v₂_ _h₃_ - _v₃_ _h₂_)]
Now since ρ₀ = ρ√(1 - V²/_c²_)
We have ρ₀φ₁ = ρ[_d__{_x_} + 1/_c_ (_v₂_ _h₃_ - _v₃_ _h₂_)]
N. B.—We have put the components of _e_ equivalent to (_d__{_x_}, _d__{_y_}, _d__{_z_}), and the components of _m_ equivalent to _h__{_x_} _h__{_y_} _h__{_z_}), in accordance with the notation used in Lorentz’s Theory of Electrons.
We have therefore
ρ₀ (φ₁, φ₂, φ₃) = ρ (_d_ + 1/_c_ [_v_·_h_]),
_i.e._, ρ₀ (φ₁, φ₂, φ₃) represents the force acting on the electron. Compare Lorentz, Theory of Electrons, page 14.
The fourth component φ₄ when multiplied by ρ₀ represents _i_-times the rate at which work is done by the moving electron, for ρ₀ φ₄ = _i_ρ [_v__{_x_}_d__{_x_} + _v__{_y_}_d__{_y_} + _v__{_z_}_d__{_z_}] = _v__{_x_} ρ₀φ₁ + _v__{_y_} ρ₀φ₂ + _v__{_z_} ρ₀φ₃. -√(-1) times the power possessed by the electron therefore represents the fourth component, or the time component of the force-four-vector. This component was first introduced by Poincare in 1906.
The four-vector ψ = _i_ωF^* has a similar relation to the force acting on a moving magnetic pole.
[M. N. S.]
Note 17. Operator “Lor” (§ 12, p. 41).
The operation | ∂/∂_x₁_ ∂/∂_x₂_ ∂/∂_x₃_ ∂/∂_x₄_ | which plays in four-dimensional mechanics a rôle similar to that of the operator (_i_∂/∂_x_, + _j_∂/∂_y_, + _k_∂/∂_z_ = ▽) in three-dimensional geometry has been called by Minkowski ‘Lorentz-Operation’ or shortly ‘lor’ in honour of H. A. Lorentz, the discoverer of the theorem of relativity. Later writers have sometimes used the symbol □ to denote this operation. In the above-mentioned paper (Annalen der Physik, p. 649, Bd. 38) Sommerfeld has introduced the terms, Div (divergence), Rot (Rotation), Grad (gradient) as four-dimensional extensions of the corresponding three-dimensional operations in place of the general symbol lor. The physical significance of these operations will become clear when along with Minkowski’s method of treatment we also study the geometrical method of Sommerfeld. Minkowski begins here with the case of lor S, where S is a six-vector (space-time vector of the 2nd kind).
This being a complicated case, we take the simpler case of lor _s_,
where _s_ is a four-vector = | _s₁_, _s₂_, _s₃_, _s₄_ |
and _s_ = | _s₁_ | | _s₂_ | | _s₃_ | | _s₄_ |
The following geometrical method is taken from Sommerfeld.
Scalar Divergence—Let ΔΣ denote a small four-dimensional volume of any shape in the neighbourhood of the space-time point Q, _d_S denote the three-dimensional bounding surface of ΔΣ, _n_ be the outer normal to _d_S. Let S be any four-vector, P_{_n_} its normal component. Then
Div S = Lim ∫ P_{_n_}_d_S/ΔΣ. ΔΣ = 0
Now if for ΔΣ we choose the four-dimensional parallelopiped with sides (_dx₁_, _dx₂_, _dx₃_, _dx₄_), we have then
Div S = ∂_s₁_/∂_x₁_ + ∂_s₂_/∂_x₂_ + ∂_s₃_/∂_x₃_ + ∂_s₄_/∂_x₄_ = lor S.
If _f_ denotes a space-time vector of the second kind, lor _f_ is equivalent to a space-time vector of the first kind. The geometrical significance can be thus brought out. We have seen that the operator ‘lor’ behaves in every respect like a four-vector. The vector-product of a four-vector and a six-vector is again a four-vector. Therefore it is easy to see that lor S will be a four-vector. Let us find the component of this four-vector in any direction _s_. Let S denote the three-space which passes through the point Q (_x₁_, _x₂_, _x₃_, _x₄_) and is perpendicular to _s_, ΔS a very small part of it in the region of Q, _d_σ is an element of its two-dimensional surface. Let the perpendicular to this surface lying in the space be denoted by _n_, and let _f__{_s_ _n_} denote the component of _f_ in the plane of (_sn_) which is evidently conjugate to the plane _d_σ. Then the _s_-component of the vector divergence of _f_ because the operator lor multiplies _f_ vectorially.
= Div _f__{_s_} = Lim (∫ _f__{_s_ _n_}_d_σ)/ΔS. Δ_s_ = 0
Where the integration in _d_σ is to be extended over the whole surface.
If now _s_ is selected as the _x_-direction, Δ_s_ is then a three-dimensional parallelopiped with the sides _dy_, _dz_, _dl_, then we have
$$ Div f_{x} = \frac{1}{dy dz dl} {dz. dl. \frac{\partial f_{xy}}{\partial y} dy + dl dy \frac{\partial f_{xy}}{\partial z} dz + dy dz \frac{\partial f_{xy}}{\partial l} dl} = \frac{\partial f_{xy}}{\partial y} + \frac{\partial f_{xy}}{\partial z} + \frac{\partial f_{xy}}{\partial l} $$
and generally
Div _f__{_j_} = ∂_f__{_j_ _x_}/∂_x_ + ∂_f__{_j_ _y_}/∂_y_ + ∂_f__{_j_ _z_}/∂_z_ + ∂_f__{_j_ _l_}/∂_l_ (where _f__{_j_, _j_} = 0).
Hence the four-components of the four-vector lor S or Div. _f_ is a four-vector with the components given on page 42.
According to the formulae of space geometry, D_{_x_} denotes a parallelopiped laid in the (_y_-_z_-_l_) space, formed out of the vectors (P_{_y_} P_{_z_} P_{_l_}), (U_{_y_}^* U_{_z_}^* U_{_l_}^*) (V_{_y_}^* V_{_z_}^* V_{_l_}^*).
D_{_x_} is therefore the projection on the _y-z-l_ space of the parallelopiped formed out of these three four-vectors (P, U^*, V^*), and could as well be denoted by Dyzl. We see directly that the four-vector of the kind represented by (D_{_x_}, D_{_y_}, D_{_z_}, D_{_l_}) is perpendicular to the parallelopiped formed by (P U^* V^*).
Generally we have
(P_f_) = PD + P^*D^*.
∴ The vector of the third type represented by (P_f_) is given by the geometrical sum of the two four-vectors of the first type PD and P^*D^*.
[M. N. S.]
● Transcriber’s Notes: ○ The book's idiosyncratic spelling, emphasis, punctuation, and symbology especially in mathematical formulas, have been retained. ○ Text that was in italics is enclosed by underscores (_italics_).
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