Physical Mechanism for the Phase Invariance in Special Relativity

1 Introduction

Lorentz transformation is at the heart of special relativity. In four-dimensional (4d) space and time, physical quantities are classified as 4-d scalars, vectors, and tensors according to their behaviours under the Lorentz transformation. For example, the frequency and the 3-d wave vector of a wave forms a 4-d wave vector (ω/c, k_x, k_y, k_z). Under the Lorentz transformation, this four-vector transforms like [1, 2]

(1)ωc = γ(ω′c + βk′x)kx = γ(k′x + βω′c)ky = k′y, kz = k′z (1)

where β=υ/c and γ = 1/1 − β2.υ is the relative velocity between two frames S and S′, which are chosen such that the frame S′ moves with the constant velocity υ along the positive x-axis relative to the frame S. We also say that the 4-vector is Lorentz covariant. The transformation (1) also applies to the moving medium [3]. Considering a case that a medium is at rest in frame S′ and the light wave in the medium propagate in the negative x-axis direction, we have

(2)k′y = 0, k′z = 0, k′x = −ω′/u′ (2)

where u′ is the magnitude of the speed of the light wave in frame S′. Inserting (2) into (1), the first and second relations in (1) become

(3)ω = γω′(1 − υ/u′)kx = γk′x(1 − υu′/c2) (3)

When υ>u′, we have a negative frequency ω. However, the 3-d wave vectors in different frames keep the same sign as υu′<c². To resolve the problem of negative frequency, Huang proposed a deformed Lorentz transformation for the 4-d wave vector [2]

(4)ωc = −γ(ω′c + βk′x)kx = −γ(k′x+βω′c)ky = −k′y, kz = −k′z (4)

which is clearly not Lorentz covariant in the usual sense (1). Under the transformation (4), the scalar product of the 4-d wave vector and the 4-d space–time coordinate vector (ct, x, y, z), which is the phase of a plane wave ϕ=(ωt – xk_x – yk_y – zk_z), may be not a 4-d scalar. In another word, that the phase of a plane wave is a scalar is questionable [2]. In Ref. [4] to solve the problem of phase invariance, it is suggested to include the relativistically induced optical anisotropy. However, it is pointed out in [5] that the induced anisotropy effect is a direct consequence of the Lorentz transformation. In other words, the Lorentz transformation has already taken into account the anisotropy effect. No extra consideration is needed. In Ref. [5], there is the conclusion that the validity of phase invariance of plane wave remains questionable. We believe that why the phase invariance is often questioned is, to some extent, because the physical mechanism of phase invariance is usually not mentioned in the literature or textbook.

It is pointed out in Ref. [5] that there is a radical change of the plane wave in Ref. [4], where a harmonic wave ψ(r⇀, t) = A sin[k⇀ ⋅ (r⇀ − u⇀t)] = A sin(k⇀ ⋅ r⇀ − ωt) with ω = k⇀ ⋅ u⇀ is obtained by letting the configuration A sin(k⇀ ⋅ r⇀) move with velocity u⇀ which is defined as the wave velocity in [4]. This definition of wave velocity is different from the usual or standard definition. The usual wave velocity means the phase velocity or group velocity. For the plane wave, the phase velocity and the group velocity are the same and are in the direction of wave vector k⇀. More importantly, the energy flows along the direction of wave vector too. In another word, the wave velocity is also the one that energy flows. The magnitude of phase velocity or group velocity is ω/k. Consider a case that u⇀ is perpendicular to the wave vector k⇀, the phase velocity or the group velocity is zero, while the velocity u⇀ is non-zero, which shows the difference between the definition in [4] and the standard one.

In this article, we deal with the problem of phase invariance starting from the Lorentz transformation of the wave equation. Further, the physical mechanism for the phase invariance is given. For the mechanical wave, it is shown that the displacement and velocity of oscillation of a mass element determines the phase invariance. For the electromagnetic wave, the displacement corresponds to the field strength. The main discussions are given in the next section. The third section is the conclusion.

2 Lorentz Transformation of the Wave Equation

The wave equation in one inertial reference frame S is

(5)∇2ψ(r⇀, t) − ∂2u2∂t2ψ(r⇀, t) = 0 (5)

where ∇² is the Laplacian operator. This wave equation admits the plane wave solution

(6)ψ(r⇀, t) = A cosϕ(r⇀, t), ϕ(r⇀, t) = α⇀ ⋅ r⇀ − α0t (6)

where α⇀ and α₀ are two parameters. The time period of this wave function is T= 2π/|α₀|. According to definition, the frequency is thus ω= 2π/T= |α₀|. In another word, the quantity α₀ is only a parameter to describe the plane wave, which is named frequency by habit. The parameter α0 can be positive or negative, while the frequency is always positive. Substituting (6) into (5), we have

(7)α02 = u2α2 (7)

where α = |α⇀| = k is actually the wave number. The positive and negative values α₀=± αu corresponds to wave propagation in opposite directions.

We want to know the form and solution of the wave equation in another inertial reference frame S′, which can be obtained by using the Lorentz transformation

(8)x′ = γ(x − υt)y′ = y, z′ = zt′ = γ(t − υx/c2) (8)

For a mechanical wave, ψ(r⇀, t) is the displacement of the mass element. If ψ(r⇀, t) is the transversal component relative to the motion between the two references, according to the transformation (8), we get

(9)ψ(r⇀, t) = ψ′(r⇀′, t′) (9)

Using (8) and (9), one can derive the following relations

(10a)∂ψ(r⇀, t)∂x = ∂ψ(r⇀, t)∂x′∂x′∂x + ∂ψ(r⇀, t)∂t′∂t′∂x= γ(∂ψ(r⇀, t)∂x′ − υc2∂ψ(r⇀, t)∂t′) = γ(∂ψ′(r⇀′, t′)∂x′ − υc2∂ψ′(r⇀′, t′)∂t′) (10a)

(10b)∂ψ(r⇀, t)∂t = ∂ψ(r⇀, t)∂x′∂x′∂t + ∂ψ(r⇀, t)∂t′∂t′∂t= γ(∂ψ(r⇀, t)∂t′ − υ∂ψ(r⇀, t)∂x′) = γ(∂ψ′(r⇀′, t′)∂t′ − υ∂ψ′(r⇀′, t′)∂x′) (10b)

(10c)∂ψ(r⇀, t)∂y = ∂ψ′(r⇀′, t′)∂y′, ∂ψ(r⇀, t)∂z = ∂ψ′(r⇀′, t′)∂z′ (10c)

and

(11a)∂2ψ(r⇀, t)∂x2 = γ2(∂2ψ′(r⇀′, t′)∂x′2 − 2υc2∂2ψ′(r⇀′, t′)∂x′∂t′ + υ2c4∂2ψ′(r⇀′, t′)∂t′2) (11a)

(11b)∂2ψ(r⇀, t)∂t2 = γ2(∂2ψ′(r⇀′, t′)∂t′2 − 2υ∂2ψ′(r⇀′, t′)∂x′∂t′ + ∂2ψ′(r⇀′, t′)∂x′2) (11b)

(11c)∂2ψ(r⇀, t)∂y2 = ∂2ψ′(r⇀′, t′)∂y′2, ∂2ψ(r⇀, t)∂z2 = ∂2ψ′(r⇀′, t′)∂z′2 (11c)

Substituting (11a–11c) into (5), we obtain the wave equation in the inertial reference frame S′

(12)∂2ψ′(r⇀′, t′)∂y′2 + ∂2ψ′(r⇀′, t′)∂z′2  + γ2(∂2ψ′(r⇀′, t′)∂x′2 − 2υc2∂2ψ′(r⇀′, t′)∂x′∂t′ + υ2c4∂2ψ′(r⇀′, t′)∂t′2) − γ2u2(∂2ψ′(r⇀′, t′)∂t′2 − 2υ∂2ψ′(r⇀′, t′)∂x′∂t′ + ∂2ψ′(r⇀′, t′)∂x′2) = 0 (12)

This wave equation still has the plane wave solution

(13)ψ′(r⇀′, t′) = A′cosϕ′(r⇀′, t′), ϕ′(r⇀′, t′) = α⇀′ ⋅ r⇀′ − α′0t′ (13)

Inserting (13) into (12), we obtain the relation

(14)[γ(α′0 + υα′x)]2 = u2[γ2(α′x + υα′0/c2)2 + α′y2 + α′z2] (14)

Now, we turn to the problem of phase invariance. Using (6) and (13), the equations (9), (10a), and (11a), respectively, change into

(15a)Acosϕ(r⇀, t) = A′cosϕ′(r⇀′, t′) (15a)

(15b)αxAsinϕ(r⇀, t) = γ(α′x + υα′0/c2)A′sinϕ′(r⇀′, t′) (15b)

(15c)αx2Acosϕ(r⇀, t) = [γ(α′x + υα′0/c2)]2A′cosϕ′(r⇀′, t′) (15c)

From (15a) and (15c), we know that αx2 = [γ(α′x + υα′0/c2)]2, or

(16)αx = γ(α′x + υα′0/c2) (16)

Combining this relation with equation (15b), we have

(17)Asinϕ(r⇀, t) = A′sinϕ′(r⇀′, t′) (17)

For t=t′=0 and r⇀ = r⇀′ = 0, the equation (15a) means A= A′, so that the relations (15a) and (17) become

(18)cosϕ(r⇀, t) = cosϕ′(r⇀′, t′), sinϕ(r⇀, t) = sinϕ′(r⇀′, t′) (18)

The relations (18) guarantee the phase invariance or ϕ(r⇀, t) = ϕ′(r⇀′, t′). Phase invariance means the phase α⇀ ⋅ r⇀ − α0t is a 4-d scalar. Like (ct, x, y, z), the quantity (α₀/c, α_x, α_y, α_z) is then a 4 vector, which transforms as (16) and

(19)α0 = γ(α′0 + υα′x)αy = α′y, αz = α′z (19)

These transformations can also be derived from (10b) and (10c). Using (16) and (19), one may notice that the relation (14) agrees with the relation (7). The negative frequency in (3) is actually negative, α₀, not negative frequency. There does not exist negative frequency. For positive α₀, setting α₀=ω and α_i=k_i (i=x, y, z), the transformation (19) becomes (1). For negative α₀, the actual frequency is –α₀. The transformation remains in the form (19). The transformation (4) is thus unnecessary.

Now, we turn to the physical mechanism of the phase invariance of the plane wave. The first relation in (18) is from the displacement relation (9) or (15a), which gives

(20)ϕ′(r⇀′, t′) = ±ϕ(r⇀, t) (20)

The positive sign on the right hand side means phase invariance. However, the negative sign shows that the phase may be not invariant [2, 5]. The second relation in (18) guarantees the phase invariance. What’s the meaning of the second relation in (18)? This relation is associated with the oscillation velocity of the mass element

(21)∂ψ(r⇀, t)∂t = α0Asinϕ(r⇀, t) (21)

(22)∂ψ′(r⇀′, t′)∂t′=α′0A′sinϕ′(r⇀′, t′) (22)

The velocities (21) and (22) are the results in the two inertial reference frames S and S′, respectively. For positive α₀ and α′0, the two waves observed in the two frames S and S′ propagate in the same direction and the velocities (21) and (22) have the same sign, so that we still have the relation (17) and further the phase invariance

(23)ϕ′(r⇀′, t′) = ϕ(r⇀, t) (23)

For positive α₀ and negative α′0(or negative α₀ and positive α′0), the two waves in the two frames propagate in opposite directions and the velocities (21) and (22) have opposite signs. In this case, we have the relation (17) and the phase invariance (23) too. So, for the mechanical wave, the displacement, and the velocity of oscillation of a mass element together determine the phase invariance.

If ψ(r⇀, t) and ψ′(r⇀′, t′) are both the longitudinal components of the wave, we can have the same conclusion of phase invariance. In this case, ψ(r⇀, t) and ψ′(r⇀′, t′) are displacements along the relative motion and the coordinates of the particle of the medium in the two reference frames S and S′ are, respectively, x + ψ(r⇀, t) and x′ + ψ′(r⇀′, t′), which have the relation x + ψ(r⇀, t) = γ[x′ + ψ′(r⇀′, t′) + υt′] according to the Lorentz transformation (8). On the other hand, x = γ(x′ + υt′). So, we have ψ(r⇀, t) = γ ψ′(r⇀′, t′). If rewriting γA′ as A′, this relation will become (9). Further, using the Lorentz transformation (8), we can get (10) and (11) directly. Discussions between (15) and (18) give the conclusion of phase invariance.

The mechanism for the phase invariance of electromagnetic wave is similar to that of the mechanical wave. In this case, the displacement (6) or (13) represents the strength of the electric field or magnetic field. In a medium with no free electric charges and conduction currents, the Maxwell equations for the electromagnetic field are as follows:

(24)∇ × E⇀(r⇀, t) = −∂B⇀(r⇀, t)∂t, ∇ ⋅ D⇀ = 0 (24)

(25)∇ × H⇀(r⇀, t) = ∂D⇀(r⇀, t)∂t, ∇ ⋅ B⇀ = 0 (25)

where E⇀(r⇀, t) and B⇀(r⇀, t)are, respectively, the electric field and the magnetic field, D⇀(r⇀, t) = ε0E⇀(r⇀, t) + P⇀ is the electric displacement with P⇀ being the electric moment density and ε₀ the vacuum permitivity, the quantity H⇀(r⇀, t⇀) = B⇀(r⇀, t)/μ0 − M⇀ with M⇀ being the magnetic moment density and μ₀ the vacuum permeability. As pointed out in [6], the constitutive laws of media are in general not form-invariant under relativistic transformation so that the Maxwell equations of electrodynamics of media are, in general, not relativistically invariant. In the following discussion, we focus our attention on the uniform, linear, and isotropic media. On the other hand, to study the phase invariance of plane wave, there must be plane wave solution. As is known, the uniform, linear, and isotropic media permit the existence of plane wave solution.

For the uniform, linear, and isotropic media, it is shown that the Maxwell equations are invariant by extending the real-valued fields to complex-valued fields and using the Fourier transform [6, 7]. Define the matrices [6, 7]

(26)Gαβ(r⇀, t) = [0−cDx(r⇀, t)−cDy(r⇀, t)−cDz(r⇀, t)cDx(r⇀, t)0−Hz(r⇀, t)Hy(r⇀, t)cDy(r⇀, t)Hz(r⇀, t)0−Hx(r⇀, t)cDz(r⇀, t)−Hy(r⇀, t)Hx(r⇀, t)0] (26)

(27)Fαβ(r⇀, t) = [0−cBx(r⇀, t)−cBy(r⇀, t)−cBz(r⇀, t)cBx(r⇀, t)0−Ez(r⇀, t)Ey(r⇀, t)cBy(r⇀, t)Ez(r⇀, t)0−Ex(r⇀, t)cBz(r⇀, t)−Ey(r⇀, t)Ex(r⇀, t)0] (27)

The homogeneous Maxwell equations (24, 25) can be written as follows:

(28)∂αTαβ(r⇀, t) = 0 (28)

where ∂_α=(∂(ct), ∇) and Tαβ(r⇀, t) can be Gαβ(r⇀, t) or Fαβ(r⇀, t). The equations (28) admit plane wave solution

(29)Tαβ(r⇀, t) = T0αβcosϕ(r⇀, t) (29)

where T0αβ are constants and the phase ϕ(r⇀, t) = α⇀ ⋅ r⇀ − α0t has the same form as that in (6). For this plane wave solution, the Maxwell equations become

(30)α⇀ × E⇀0 = α⇀0B⇀0, α⇀ ⋅ D⇀0 = 0 (30)

(31)α⇀ × H⇀0 = −α0D⇀0, α⇀ ⋅ B⇀0 = 0 (31)

where E⇀0, D⇀0, B⇀0, H⇀0 are the amplitudes of the corresponding fields. In the case of α₀>0, the wave propagates along the direction of positive α⇀ When α₀<0, the wave propagates towards the direction of negative α⇀.

In another inertial frame, the Maxwell equations have the same form as that in (28)

(32)∂′αT′αβ(r⇀′, t′) = 0 (32)

These equations have plane wave solution too

(33)T′αβ(r⇀′, t′) = T′0αβcosϕ′(r⇀′, t′) (33)

Where T′0αβ are constants and the phase ϕ′(r⇀′, t′) = α⇀′ ⋅ r⇀′ − α′0t′ has the same form as that in (13).

The transformation between physical quantities in different inertial reference frames is linear to guarantee that the Maxwell equations (28, 32) are linear, and there are the plane wave solutions (29, 33). That is, Tαβ(r⇀, t) can be written as the linear superpositions of T′αβ(r⇀′, t′)

(34)Tαβ(r⇀, t) = ∑λ,ρCλρT′λρ(r⇀′, t′) = ∑λ,ρCλρT′0λρcosϕ′(r⇀′, t′) (34)

where C_λρ are the superposition coefficients. For the usual relativistic transformations of electromagnetic fields, one can see (5, 6) and (10, 11) in [7]. For different field components, the superposition coefficients are different. We will see that to get the conclusion of phase invariance, we need not to know the specific values of the superposition coefficients. Setting

(35)ψ(r⇀, t) = Tαβ(r⇀, t) = Acosϕ(r⇀, t) (35)

(36)ψ′(r⇀′, t′) = ∑λ, ρCλρT′0λρcosϕ′(r⇀′, t′) = A′cosϕ′(r⇀′, t′) (36)

where A = T0αβ and A′ = ∑λ, ρCλρT′0λρ, the relation (34) becomes (9). Using (9) and the Lorentz transformation (8), we can still get (10) and (11). Similar to the calculations between (15) and (18), we have the phase invariance of the electromagnetic wave. For the electromagnetic wave, the fields play the role of displacements of the mechanical wave. The discussions between (20) and (23) can be applied to the electromagnetic wave or the longitudinal component of the mechanical wave. For the electromagnetic wave, (21) and (22) are the oscillation velocities of the fields.

3 Conclusions

In this article, using the Lorentz transformation for space and time coordinates, the wave equation is transformed from one inertial reference frame to another. The wave equations in both inertial reference frames have plane wave solutions. The phase invariance is obtained automatically. Further, the physical mechanism for the phase invariance is given. For the mechanical wave, the displacement and velocity of the oscillation together determine the phase invariance. For the electromagnetic wave, the field strength corresponds to the displacement and the changing speed of the field strength corresponds to the oscillation velocity of the mass element for a mechanical wave. The phase invariance is thus assured. The so-called negative frequency is actually not frequency, but a parameter describing the plane wave. According to definition, the actual frequency of oscillation cannot be negative.