The Principle of Relativity

by Albert Einstein; H. Minkowski

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 iτ; and ρ₁, ρ₂, ρ₃, ρ₄ for ρν_x, ρν_y, ρν_z, iρ, 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 = iρ = ρ₄ ... ... (2·2)

Now substitute

m_x = f₂₃ = -f₃₂ and ie_x = f₄₁ = -f₁₄

m_y = f₃₁ = -f₁₃ ie_y = f₄₂ = -f₂₄

m_z = f₁₂ = -f₂₁ ie_z = f₄₃ = -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.

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 - 2Δu/c² and from A to B is T + 2Δu/c². Now if 2uΔ/c² be comparable to T, then it is impossible that the observations should satisfy Kepler’s Law. In most of the spectroscopic binary stars, 2uΔ/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 - 2uΔ/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 kuΔ₀, 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_xv_y - u_yv{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, (ff^*) = 2ρρ^*.

| f |² and (ff^*) 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 = (2c - (x + λ))/(1 + (c - x)(c - λ)/c²) = (2c - (x + λ))/(c² + c² - (x + λ)c + xλ/c²)

= c (2c - (x + λ))/(2c - (x + λ) + xλ/c)

Thus v lt; c, so long as | xλ | > 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 - Wv/c²)

Thus “time” from A to B as measured in S′, is given by l/W′ = l(1 - Wv/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 Wv = (c + μ)(c - λ) = c² + (μ + λ)c - μλ.

Now we can always choose v in such a way that Wv is greater than c², since Wv 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 Wv > c², i.e., λ - Wv/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.]