Inconsistency in the Formulae for the Doppler Effect of Moving Electromagnetic Sources

1. Introduction

The Doppler effect describes the variation in the frequency of waves when the observer and the source of the wave have relative motions. Doppler effect may exist in all kinds of waves. This paper only concerns about the Doppler effect observed in the electromagnetic waves generated by the electromagnetic sources in free space.

There are mainly two types of formulae for the Doppler effect: the classical formula and the conventional relativistic formula [1,2,3,4,5]. In order to analyze the inconsistency existed between them, we are to re-derive them with transformations of the plane wave between inertial frames, as shown in Figure 1, where $K$ is the laboratory frame in which the observer is at rest, and $K^{\prime }$ is the rest inertial frame in which the source is at rest.

Assume that the electric field of the plane wave in the laboratory frame $K$ can be expressed by

$$E\left(r,t\right)=E_0\cos \left(\omega t-k\cdot r\right)$$

(1)

where $k$ is the wave vector. Denote the phase of the plane wave as:

$$\Phi =\omega t-k\cdot r$$

(2)

The classical formula for the Doppler effect can be derived using the Galilean transformations [1],

$$\left\{\begin{array}{l}{r}^{\prime }=r-vt\\ t^{\prime }=t\end{array}\right.$$

(3)

Making use of the hypothesis that the phase of the plane wave is invariant under the Galilean transformations, we can write that

$${\Phi }^{\prime }={\omega }^{\prime }{t}^{\prime }-k^{\prime }\cdot r^{\prime }=\omega t-k\cdot r$$

(4)

Inserting (3) into (4) and equating the corresponding terms, we get $k=k^{\prime }$ and the classical formula for the Doppler effect,

$$\omega ={\displaystyle \frac{{\omega }^{\prime }}{1-\widehat{k}\cdot \beta }}\triangleq {\omega }_{cl}$$

(5)

where $k=\left|k\right|$ is the wavenumber, $\widehat{k}=k/k$, and $\beta =v/c_0$. Conventionally, the angular frequency at the rest reference frame is denoted as ${\omega }_0$. In this case, ${\omega }_0={\omega }^{\prime }$. For the sake of comparison, we denote the angular frequency of the wave derived in this way as ${\omega }_{cl}$.

Einstein [4] derived the conventional relativity formula for the Doppler effect by making use of the Lorentz transformations (LTs),

$$\left\{\begin{array}{l}{r}^{\prime }=r_{\perp }+\gamma \left(r_{\parallel }-vt\right)\\ t^{\prime }=\gamma \left(t-v\cdot r/c_0^2\right)\end{array}\right.$$

(6)

where $\gamma =1/\sqrt{1-{\beta }^2}$ is the Lorentz factor. The subscript “⊥” and “||” represent the transversal and parallel component of the vector, respectively. Einstein has assumed that, in the laboratory frame $K$, the electromagnetic waves at the observation point far away from the source can be represented to a sufficient degree of approximation by a plane wave. The phase of the plane wave is expressed by (2) and is invariant under the Lorentz transformations. By inserting (6) into (4), he obtained the relation between the wave vectors in the two frames,

$$k^{\prime }=\left(\widehat{v}\times k\right)\times \widehat{v}+\gamma \left(\widehat{v}\widehat{v}\cdot k-k_{0}v\right)$$

(7)

and the relation between the angular frequencies in the two frames,

$$\omega ={\displaystyle \frac{{\omega }_0}{\gamma \left(1-\widehat{k}\cdot \beta \right)}}\triangleq {\omega }_{TSR}$$

(8)

We have denoted ${\omega }_0={\omega }^{\prime }$ in (8) to represent the angular frequency at the rest frame, and have denoted the angular frequency of the plane wave in the frame $K$ by ${\omega }_{TSR}$ to distinguish it from that in the classical formula (5). It is obvious that (8) differs from the classical formula (5) by a Lorentz factor $\gamma$. Einstein did not give explanations why the phase of plane wave should be invariant under the Lorentz transformations. Jackson [1] explained that the phase of a plane wave is an invariant quantity because the elapsed phase of a wave is proportional to the number of wave crests that have passed the observer. Since this is merely a counting operation without units, it must be independent of coordinate frame. However, this explanation also seems to lack solid theoretical foundation because it ignores the fact that the temporal period in the counting varies between inertial frames.

We should pay attention to two critical facts: one is that the above derivation is based on the hypothesis that the phase of a plane wave is invariant in all inertial frames; the other is that the Doppler effect is described by the relationship between the angular frequency $\omega$ of the fields in the frame $K$ and the angular frequency ${\omega }^{\prime }$ of the moving source in its rest frame $K^{\prime }$. A very important fact has been ignored by researchers: ${\omega }^{\prime }$ is defined in $K^{\prime }$ with ${\omega }^{\prime }=2\pi /T^{\prime }$, while $\omega$ is defined in $K$with $\omega =2\pi /T$, the time bases for their definition are not the same because of the time dilation effect. When we try to evaluate the Doppler effect in the waves far away from the source in the frame $K$, it is more natural to compare the angular frequency of the moving source and the angular frequency of the fields in the same frame on the same time base. In other words, we should express ${\omega }^{\prime }$ in terms of the period $T$ instead of the proper period $T^{\prime }$, namely,

$${\omega }^{\prime }={\displaystyle \frac{2\pi }{T^{\prime }}}={\displaystyle \frac{T}{T^{\prime }}}{\displaystyle \frac{2\pi }{T}}=\gamma {\omega }_0$$

(9)

where ${\omega }_0$ can be interpreted as the value of the angular frequency of the source when expressed in terms of the period in the frame $K$. It is important to note that we have $T=T^{\prime }$ under the Galilean transformations, so we have ${\omega }^{\prime }={\omega }_0$ in (5).

It may be more intuitive to discuss the issue using the frequency $f=1/T$ instead of the angular frequency $\omega =2\pi f$. We can treat the frequency of the source as counting the number of the period per second. Because of the time dilation under the Lorentz transformations, $T=\gamma T^{\prime }$. For the same source, when counted in $K$, the number of period per second is $f=1/T$; when counted in $K^{\prime }$, the number of period per second is $f^{\prime }=1/T^{\prime }=\gamma f$. For example, we consider a harmonic source with period of $T=0.1\mathrm{s}$. The frequency in $K$ is 10Hz, which means that we count 10 period ($T$) per second in $K$. Assume that $\gamma =2$, then the proper period is $T^{\prime }=0.05\mathrm{s}$. The frequency in $K^{\prime }$ is $f^{\prime }=1/T^{\prime }=20\mathrm{Hz}$, which means that there are 20 proper period $T^{\prime }$ in one second when we count it in $K^{\prime }$. The frequency of the source is 20Hz when it is measured in $K^{\prime }$. However, its value is converted to 10Hz when it is evaluated in the frame $K$. We should evaluate the Doppler effect by comparing the frequency of the fields we have measured in $K$ with 10Hz instead of 20Hz.

We may directly follow Einstein’s derivation and represent the fields far away from the source by a plane wave. However, instead of making use of the hypothesis of phase invariance, we derive the formula by rigorously applying the Lorentz transformations. Assume that the source is harmonically oscillating with $\cos \left({\omega }_0{t}_1\right)$ and moves uniformly with velocity $v$. The position of the source at time $t_1$ is $r_1=r_{10}+v{t}_1$. Its fields at the observation point can be approximately expressed with a plane wave with phase of $\Phi =\omega t-k\cdot r$. We are to find the relationship between $\omega$ and ${\omega }_0$ to reveal the Doppler effect. With Lorentz transformations, the position of the uniformly moving source is transformed to a fixed position ${r^{\prime }}_1$ in $K^{\prime }$. The transformation of the time is $t_1=\gamma \left({t^{\prime }}_1+v\cdot {r^{\prime }}_1/c_0^2\right)$, so the harmonically oscillating source becomes $\cos \left(\gamma {\omega }_0{t^{\prime }}_1+{\Phi }_0\right)$, where ${\Phi }_0=\gamma {\omega }_{0}v\cdot {r^{\prime }}_1/c_0^2$ is a constant because ${r^{\prime }}_1$ is a fixed position vector. Therefore, the phase of the plane wave generated by the source in $K^{\prime }$ should be

$${\Phi }^{\prime }=\gamma {\omega }_0{t}^{\prime }-k^{\prime }\cdot r^{\prime }+{\Phi }_0$$

(10)

from which we get ${\omega }^{\prime }=\gamma {\omega }_0$. This verifies the correctness of (9).

In order to get the plane wave solution in the frame $K$, we write the plane wave generated by the harmonic source in the frame $K^{\prime }$ as follows,

$$\left\{\begin{array}{l}{E}^{\prime }\left(r^{\prime },t^{\prime }\right)={E^{\prime }}_0\cos \left({\omega }^{\prime }{t}^{\prime }-k^{\prime }\cdot r^{\prime }\right)\\ H\left(r^{\prime },t^{\prime }\right)={\displaystyle \frac{1}{{\eta }_0}}\widehat{k^{\prime }}\times E^{\prime }\left(r^{\prime },t^{\prime }\right)\end{array}\right.$$

(11)

Note that ${\omega }^{\prime }=\gamma {\omega }_0$ in equation (11). We still use ${\eta }_0$ for the intrinsic impedance of the vacuum because the properties of the vacuum is not changed under Lorentz transformations.

Now we transform the plane wave from the frame $K^{\prime }$ to the frame $K$ with the inverse Lorentz transformations for the electromagnetic fields [9],

$$\left[\begin{array}{c}E\left(r,t\right)\\ H\left(r,t\right)\end{array}\right]=\mathcal{L}\mathcal{T}\left(-v\right)\left[\begin{array}{c}{E}^{\prime }\left(r^{\prime },t^{\prime }\right)\\ H^{\prime }\left(r^{\prime },t^{\prime }\right)\end{array}\right]$$

(12)

where $\mathcal{L}\mathcal{T}\left(-v\right)$ is the inverse Lorentz transformations,

$$\mathcal{L}\mathcal{T}\left(-v\right)=\left[\begin{array}{cc}\left[\left(1-\gamma \right)\widehat{v}\widehat{v}+\gamma \overline{I}\right]\cdot & -{\mu }_0\gamma v\times \\ {\epsilon }_0\gamma v\times & \left[\left(1-\gamma \right)\widehat{v}\widehat{v}+\gamma \overline{I}\right]\cdot \end{array}\right]$$

(13)

where $\overline{I}$ is the identity operator. The space-time coordinates are transformed with (6). With the definitions of the vectors, we can verify that the electromagnetic field in the frame $K$ is still a plane wave. Its electric field can be expressed by $E_0\cos \left(\omega t-k\cdot r\right)$, where the angular frequency is derived to be

$$\omega ={\displaystyle \frac{{\omega }^{\prime }}{\gamma \left(1-\widehat{k}\cdot \beta \right)}}={\displaystyle \frac{{\omega }_0}{1-\widehat{k}\cdot \beta }}\triangleq {\omega }_{Lo}$$

(14)

To distinguish it from that in the conventional relativistic formula (8), we have denoted the angular frequency of the plane wave obtained with LTs by ${\omega }_{Lo}$. Evidently, if derived with rigorous Lorentz transformations, the relativistic formula for Doppler effect is exactly of the same form with the classical formula (5) and not in consistent with the conventional relativistic formula (8). To distinguish them, the formula (14) is called Lorentz type relativistic formula for Doppler effect, which can be considered as the solution of moving point source obtained by solving the Maxwell’s equations with frame-hopping technique (FH) [9,11,12]. The inconsistency between the Lorentz type relativistic formula and the conventional relativistic formula for the Doppler effect is caused by the fact that the conventional relativistic formula is derived by directly applying the hypothesis of phase invariance of plane waves in all inertial frames instead of strictly applying the LTs and inverse LTs.

2. The General Formula for Doppler Effect Derived with Moving Hertzian Dipole

We have analyzed the electromagnetic radiation properties of a uniformly moving Hertzian dipole in the vacuum [6]. The dipole is modelled with two anti-phase harmonic charges $q_{\pm }=\pm {\rho }_0\cos \left({\omega }_0{t}_1\right)$ with small spacing $l$, where ${\omega }_0$ is the oscillating angular frequency of the charge. The electric dipole moment is expressed by $p={\rho }_{0}l\cos \left({\omega }_0{t}_1\right)\widehat{p}$. To satisfy the current continuity law, there should be a very short current filament with density of $J_{dip}\left(r_1,t_1\right)=-{\omega }_0{\rho }_{0}l\sin \left({\omega }_0{t}_1\right)\widehat{p}$ between the two charges. The polarization unit vector $\widehat{p}$ points from $q_{-}$ to $q_{+}$. It is assumed that all the three sources move together with velocity $v$ in the direction perpendicular to $\widehat{p}$, as shown in Figure 2.

The center of the Hertzian dipole is denoted by $x\left(t_1\right)=v{t}_1$. The densities of the two moving charges can be expressed with the Dirac delta function as

$${\rho }_{\pm }\left(r_1,t_1\right)=\pm {\rho }_0\delta \left(r_1-v{t}_1\mp 0.5l\widehat{p}\right)\cos \left({\omega }_0{t}_1\right)$$

(15)

Their motion induces two currents with densities of

$$J_{\pm }\left(r_1,t_1\right)=\pm {\rho }_{0}v\delta \left(r_1-v{t}_1\mp 0.5l\widehat{p}\right)\cos \left({\omega }_0{t}_1\right)$$

(16)

The short current between the two charges also moves along the trajectory, and its current density becomes

$$J_{dip}\left(r_1,t_1\right)=-{\omega }_0{\rho }_{0}l\delta \left(r_1-v{t}_1\right)\sin \left({\omega }_0{t}_1\right)\widehat{p}$$

(17)

When $v\cdot \widehat{p}=0$, the Liénard-Wiechert potentials [1,2,8] of the above moving sources have been solved in [6], from which the electromagnetic fields of the uniformly moving Hertzian dipole are derived to be

$$E_{dip}\left(r,t\right)=E_0\left\{\begin{array}{l}{\displaystyle \frac{1-{\beta }^2}{R^3{\left(1-\widehat{n}\cdot \beta \right)}^3}}\left[{\displaystyle \frac{3\left(1-{\beta }^2\right)\left(\widehat{n}\cdot \widehat{p}\right)\left(\widehat{n}-\beta \right)}{{\left(1-\widehat{n}\cdot \beta \right)}^2}}-\widehat{p}\right]\cos \left({\omega }_0{t}_1\right)\\ -{\displaystyle \frac{1-{\beta }^2}{R^2{\left(1-\widehat{n}\cdot \beta \right)}^3}}\left[{\displaystyle \frac{3\left(\widehat{n}\cdot \widehat{p}\right)\left(\widehat{n}-\beta \right)}{1-\widehat{n}\cdot \beta }}-\widehat{p}\right]k_0\sin \left({\omega }_0{t}_1\right)\\ -{\displaystyle \frac{\widehat{n}\times \left(\widehat{n}\times \widehat{p}\right)+\widehat{n}\times \left(\widehat{p}\times \beta \right)}{R{\left(1-\widehat{n}\cdot \beta \right)}^3}}{k}_0^2\cos \left({\omega }_0{t}_1\right)\end{array}\right\}$$

(18)

$$B_{dip}\left(r,t\right)={\displaystyle \frac{E_0}{c_0}}\left\{\begin{array}{l}{\displaystyle \frac{\left(1-{\beta }^2\right)\beta \times }{R^3{\left(1-\widehat{n}\cdot \beta \right)}^3}}\left[{\displaystyle \frac{3\left(1-\beta \cdot \beta \right)}{{\left(1-\widehat{n}\cdot \beta \right)}^2}}\left(\widehat{n}\cdot \widehat{p}\right)\widehat{n}-\widehat{p}\right]\cos \left({\omega }_0{t}_1\right)\\ +{\displaystyle \frac{\left(1-\beta \cdot \beta \right)\widehat{n}\times }{R^2{\left(1-\widehat{n}\cdot \beta \right)}^4}}\left[3\left(\widehat{n}\cdot \widehat{p}\right)\beta +\left(1-\widehat{n}\cdot \beta \right)\widehat{p}\right]k_0\sin \left({\omega }_0{t}_1\right)\\ +{\displaystyle \frac{\widehat{n}\times }{R{\left(1-\widehat{n}\cdot \beta \right)}^2}}\left({\displaystyle \frac{\widehat{n}\cdot \widehat{p}}{1-\widehat{n}\cdot \beta }}\beta +\widehat{p}\right)k_0^2\cos \left({\omega }_0{t}_1\right)\end{array}\right\}$$

(19)

In the expressions, $R=r-x\left(t_1\right)$ is the radius vector pointing from the source point to the observation point, $R=\left|R\right|$, $\widehat{n}=R/R$, and $\beta =v/c_0=\beta \widehat{v}=v/c_0\widehat{v}$. In addition, ${\epsilon }_0$ and $k_0$ are respectively the permittivity and wavenumber in the vacuum. $E_0={\rho }_{0}l/\left(4\pi {\epsilon }_0\right)$ is a source-related constant. In the vacuum, the fields generated at $\left(r_1,t_1\right)$ propagates to the observer at $\left(r,t\right)$ with the light velocity $c_0$. The governing equation for the wave propagation is,

$$c_0\left(t-t_1\right)=R=\left|r-x\left(t_1\right)\right|$$

(20)

from which we can solve $t_1$ based on the causality principle, and obtain that [6]:

$${\omega }_0{t}_1={\displaystyle \frac{{\omega }_0\left(c_0^{2}t-r\cdot v\right)-{\omega }_0{c}_0\sqrt{\left(1-{\beta }^2{\sin }^2\theta \right)r^2-2r\cdot vt+v^2{t}^2}}{c_0^2\left(1-{\beta }^2\right)}}$$

(21)

where $\theta$ is the angle between the vector $\widehat{r}$ and $\beta$.

At places far away from the source, we have $r\gg \left|x\left(t_1\right)\right|=\left|v{t}_1\right|$. Based on (21) and making use of the causality principle, we can approximately express $t$ in terms of $t_1$ with [6]

$$t={\displaystyle \frac{c_0{t}_1\pm \sqrt{r^2-2r\cdot v{t}_1+v^2{t}_1^2}}{c_0}}\approx {\displaystyle \frac{r}{c_0}}+\left(1-\widehat{r}\cdot \beta \right)t_1$$

(22)

The field of the moving Hertzian dipole at region far away from its birthplace can be written as

$$E_{dip,1}^{far}\left(r,t\right)\approx {\displaystyle \frac{E_0{k}_0^2}{R{\left(1-\widehat{n}\cdot \beta \right)}^2}}\left[\widehat{p}-{\displaystyle \frac{\left(\widehat{n}\cdot \widehat{p}\right)\left(\widehat{n}-\beta \right)}{1-\widehat{n}\cdot \beta }}\right]\cos \left[{\omega }_1\left(t-r/c_1\right)\right]$$

(23)

where

$${\omega }_1={\displaystyle \frac{{\omega }_0}{1-\widehat{n}\cdot \beta }}={\displaystyle \frac{{\omega }_0}{1-\beta \cos \Theta }}$$

(24)

Equation (24) describes the relationship between the angular frequency of the far field and the angular frequency of the moving Hertzian dipole. It is extracted from the far fields that are obtained by solving the Maxwell’s equation in the laboratory frame without any frame transformations.

The fields of the uniformly moving Hertzian dipole have also been solved with the frame-hopping technique. At first, the space-time coordinates of the center of the Hertzian dipole are transformed into $K^{\prime }$ with LTs (6). The results are

$$\left\{\begin{array}{l}{r^{\prime }}_1=r_{1\perp }+\gamma \left(r_{1\parallel }-vt\right)=0\\ {t^{\prime }}_1=\gamma \left(t_1-v\cdot r_1/c_0^2\right)={\gamma }^{-1}{t}_1\end{array}\right.$$

(25)

Obviously, the center of the dipole is transformed to the origin of the frame $K^{\prime }$. Since $t_1=\gamma {t^{\prime }}_1$, we have $\left({\omega }_0{t}_1\right)=\left(\gamma {\omega }_0{t^{\prime }}_1\right)$. The moving velocity $v$ of the sources is transformed to $v^{\prime }=0$. Consequently, the three source densities can be transformed into $K^{\prime }$ as follows,

$${{\rho }^{\prime }}_{\pm }\left({r^{\prime }}_1,{t^{\prime }}_1\right)=\pm {\rho }_0\delta \left({r^{\prime }}_1\mp 0.5l{\widehat{p}}^{\prime }\right)\cos \left(\gamma {\omega }_0{t^{\prime }}_1\right)$$

(26)

$${J^{\prime }}_{\pm }\left({r^{\prime }}_1,{t^{\prime }}_1\right)=0$$

(27)

$${J^{\prime }}_{dip}\left({r^{\prime }}_1,{t^{\prime }}_1\right)=-\gamma {\omega }_0{\rho }_{0}l\delta \left({r^{\prime }}_1\right)\sin \left(\gamma {\omega }_0{t^{\prime }}_1\right){\widehat{p}}^{\prime }$$

(28)

Note that $l$ has no Lorentz contraction since $v\cdot \widehat{p}=0$. In particular, it is obvious that the angular frequency of the source is transformed from ${\omega }_0$ to $\gamma {\omega }_0$, which once again justifies the relationship of (9).

The space-time coordinate $\left(r,t\right)$ of the observation position can be transformed into the space-time coordinate $\left(r^{\prime },t^{\prime }\right)$ in the frame $K^{\prime }$ with LTs (6). The fields generated by the motionless Hertzian dipole in the frame $K^{\prime }$ are solved to be

$${E^{\prime }}_{dip}\left(r^{\prime },t^{\prime }\right)=E_0\left\{\begin{array}{l}{\displaystyle \frac{1}{{R^{\prime }}^3}}\left[3{\widehat{n}}^{\prime }\left({\widehat{n}}^{\prime }\cdot {\widehat{p}}^{\prime }\right)-{\widehat{p}}^{\prime }\right]\cos \left(\gamma {\omega }_0{t^{\prime }}_1\right)-{\displaystyle \frac{1}{{R^{\prime }}^2}}\left[3{\widehat{n}}^{\prime }\left({\widehat{n}}^{\prime }\cdot {\widehat{p}}^{\prime }\right)-{\widehat{p}}^{\prime }\right]\gamma k_0\sin \left(\gamma {\omega }_0{t^{\prime }}_1\right)\\ -{\displaystyle \frac{{\widehat{n}}^{\prime }\times \left({\widehat{n}}^{\prime }\times {\widehat{p}}^{\prime }\right)}{R^{\prime }}}{\gamma }^2{k}_0^2\cos \left(\gamma {\omega }_0{t^{\prime }}_1\right)\end{array}\right\}$$

(29)

$${B^{\prime }}_{dip}\left(r^{\prime },t^{\prime }\right)={\mu }_0{H^{\prime }}_{dip}\left(r^{\prime },t^{\prime }\right)={\displaystyle \frac{E_0}{c_0}}\left\{{\displaystyle \frac{{\widehat{n}}^{\prime }\times {\widehat{p}}^{\prime }}{{R^{\prime }}^2}}\gamma k_0\sin \left(\gamma {\omega }_0{t^{\prime }}_1\right)+{\displaystyle \frac{{\widehat{n}}^{\prime }\times {\widehat{p}}^{\prime }}{R^{\prime }}}{\gamma }^2{k}_0^2\cos \left(\gamma {\omega }_0{t^{\prime }}_1\right)\right\}$$

(30)

In the expressions, $R^{\prime }=r^{\prime }-{r^{\prime }}_1=\left|r^{\prime }\right|$, $R^{\prime }=\left|r^{\prime }-{r^{\prime }}_1\right|=r^{\prime }$,${\widehat{p}}^{\prime }=\widehat{p}$. The electromagnetic pulse generated at $\left({r^{\prime }}_1,{t^{\prime }}_1\right)$ propagates to $\left(r^{\prime },t^{\prime }\right)$ with velocity $c_0$, hence, $R^{\prime }=\left|r^{\prime }-{r^{\prime }}_1\right|=r^{\prime }=c_0\left(t^{\prime }-{t^{\prime }}_1\right)$, from which we can obtain that

$${t^{\prime }}_1=t^{\prime }-{\displaystyle \frac{\left|r^{\prime }\right|}{c_0}}=\gamma \left(t-v\cdot r/c_0^2\right)-{\displaystyle \frac{\left|r_{\perp }+\gamma \left(r_{\parallel }-vt\right)\right|}{c_0}}$$

(31)

We can solve $t^{\prime }$ from (31) and check that

$$\gamma {\omega }_0{t^{\prime }}_1={\displaystyle \frac{{\omega }_0\left(c_0^{2}t-v\cdot r\right)-{\omega }_0{c}_0\sqrt{r^2\left(1-{\beta }^2{\sin }^2\theta \right)-2r\cdot vt+V^2{t}^2}}{c_0^2{\gamma }^{-2}}}$$

(32)

Substituting ${\gamma }^{-2}=1-{\beta }^2$ into (32), we can verify that the result is exactly the same as (21).

The fields $E_{dip}\left(r,t\right)$ and $H_{dip}\left(r,t\right)$ of the Hertzian dipole in the frame $K$can be obtained with the inverse Lorentz transformations (12) for the electromagnetic fields. It has been proved that the fields obtained with Lorentz Transformations are exactly the same as those obtained with the Lienard-Wiechert potentials [8,10]. In addition, we have recently analyzed the scattered fields of uniformly moving PEC objects with the two methods. Our numerical results have demonstrated that the scattered fields obtained with the frame-hopping techniques and that with the Lienard-Wiechert potentials are in agreement with high accuracy even in situations when the PEC object moves as fast as $0.8c_0$[11,12]. These results suggest that the formula (24) derived from the far fields of the moving Hertzian dipole is a general relativistic formula for the Doppler effect. The classical formula and the Lorentz type relativistic formula are the approximate forms of (24) by letting $\widehat{n}=\widehat{k}$, that is, representing the far field by a plane wave.

3. Discussions

The proposed Lorentz type relativistic formula for the Doppler effect is exactly of the same form as the classical formula, where the angular frequency $\omega$ of the fields and the angular frequency ${\omega }_0$ of the moving source are measured in the same frame $K$. The formula is obtained by approximately representing the fields far away from the source with a plane wave, i.e., it is an approximate form of the formula (24) with $\widehat{n}=\widehat{k}$.

As a matter of fact, some researchers have suggested that the conventional relativistic formula for the Doppler effect based on the phase invariance hypothesis might need to be rechecked [13,14]. However, their opinions have not attracted much attention because most of them failed to provide convincing theoretical solutions, and are simply criticized by being not in consistent with the Theory of Special Relativity (TSR). We would like to mention that those critics might have ignored the basic fact that the phase invariance of plane waves among inertial frames is not an axiom but a hypothesis that has not been proved with rigorous theory or verified with solid experimental results. We think that it is of significance to show that the it is possible to derive the relativistic formula for the Doppler effect with Lorentz transformations instead of making use of the hypothesis.

In order to give an intuitive illustration, we have schematically compared the relationships between the sources and waves in Figure 3. To Derive the Lorentz type formula, we have compared the angular frequency of the electromagnetic fields and that of electromagnetic source in the same frame, as shown in Figure 3(a). The electromagnetic fields are solved either through Lienard-Wiechert potentials, or with the frame-hopping methods. The two solutions are completely in agreement with each other. However, to derive the conventional relativistic formula, we have to compare the angular frequency of the electromagnetic fields in the laboratory frame $K$ and the angular frequency of the electromagnetic source in the frame $K^{\prime }$ based on hypothesis that the phase of a plane wave is invariant in all inertial frames.

When $\widehat{k}\cdot \beta =0$, there is no Doppler shift if we use the classical formula and the Lorentz type relativistic formula. However, if we use the conventional relativistic formula, there is a transversal Doppler shift due to the Lorentz factor $\gamma$. Denote $\lambda =2\pi c_0/\omega$ and ${\lambda }_0=2\pi c_0/{\omega }_0$. When $\beta \ll 1$, we can derive from the conventional relativistic formula (8) that

$\Delta \lambda /{\lambda }_0=\lambda /{\lambda }_0-1\approx -\beta \cos \theta +0.5{\beta }^2$The transversal Doppler occurs when $\theta =\pi /2$, with a coefficient of 0.5 for the second order term of $\beta$.

Historically, the transversal Doppler shift is regarded as a pure relativistic effect, and was expected to provide a critical experimental support for the TSR [15]. As a matter of fact, several experiments have been conducted to detect the relativistic Doppler effect [16,17,18,19,20]. As summarized in [20], there are mainly two types of experiments for measuring the relativistic Doppler shift. The first type is to take the arithmetic mean of the longitudinal Doppler shift at $\theta =0,\pi$[18,19]. The experiments have reported a coefficient of $0.498\pm 0.025$[18] and $0.491\pm 0.017$[19]. The other type is to directly detect the relativistic Doppler shift at $\theta =\pi /2$[20]. The experiment reported a coefficient of $0.52\pm 0.03$. These experiments seem to have confirmed the transversal Doppler shift predicted by TSR. However, we would like to mention that it is extremely difficult to align the observation angle accurately. Although the researchers had tried their best to improve the accuracy, the experimental results are still not accurately enough to reach a convincing verification. Including the relativistic correction in the GPS signals does not necessarily prove that the conventional relativistic formula is absolutely correct.

4. Conclusions

We have shown that the inconsistency between the Lorentz type relativistic formula and the conventional relativistic formula is caused by the fact that the reference angular frequency for the comparison is different. For the Lorentz type relativistic formula, the frequency of the electromagnetic fields observed in $K$ is compared with the frequency of the moving source in the same frame. On the contrary, the conventional relativistic formula is derived by comparing the frequency of the fields in the frame $K$with the frequency of the moving source in the different frame $K^{\prime }$. We have clearly demonstrated that the classical formula, which is exactly the same as the Lorentz type formula, is by no means the low-speed approximation of the conventional relativistic formula. In this paper, we just want to reveal the cause of the inconsistency between the Lorentz type relativistic formula and the conventional relativistic formula for the Doppler effect. We think that the inconsistency between the two kinds of formulas for the Doppler effect is one of the several inconsistencies between the Maxwell’s theory and TSR.