A formula for the transformation of phase velocity between two inertial observers is given. This is a correction of the formula provided by R. A. Bachman in Am. J. Phys. 57(7), 628–630 (1989), in which the transformation of the phase velocity perpendicular to the relative velocity of the two frames was incorrectly stated.

The group velocity is the rate of change of the angular frequency with respect to the angular wave-number

$u→g=∂ω∂k→ ,$
(1)

where ω = 2πf is the signal's angular frequency, $k→=2πu→/λu$ its wave-number vector, λ its wavelength, and $u→$ its phase velocity. In the above equation and in what follows, the scalar quantity of a vector is its magnitude, i.e., $u≡‖u→‖=ux2+uy2+uz2$.

In contrast, the phase velocity is simply the ratio of the angular frequency to the angular wave-number in the direction of the wave-number vector

$u→=ωk2k→ .$
(2)

The difference between the phase velocity and the group velocity in the GPS signal is used to measure the thickness of the ionosphere.1 Given the precise requirements for the measurements and the high relative velocity between the satellite transmitting the signal and the Earth-based receiver, it is important to have a formula that accurately relates the observed phase velocity in two inertial frames.

Consider two observers $O$ and $O′$ where $O′$ is moving with a velocity $v→$ with respect to $O$. A monochromatic plane wave is described by the wave four-vector

$[−ω/ck→] ,$
(3)

where c is the speed of light in a vacuum. The wave-number vector can be separated into components parallel and perpendicular to the relative velocity $v→$

$k→=k→∥+k→⊥ ,$
(4)

from which it follows directly that:

$k→∥·v→v=k∥ and k→⊥·v→=0 .$

The Lorentz transformation for the wave four-vector is

$[−ω′/ck′∥k→⊥′]=[γβγ0βγγ000I2][−ω/ck∥k→⊥],$
(5)

where β = v/c is the normalized speed, $γ=1/1−β2$ is the Lorentz factor, and I2 is the 2 × 2 identity matrix.2

In component form, the transformation law Eq. (5) is

$ω′=γ(ω−k∥v) ,$
(6a)
$k∥′=γ(k∥−vω/c2) ,$
(6b)
$k→⊥′=k→⊥ ,$
(6c)

from which it follows that the phase velocity according to the observer $O′$ is

$u′→=ω′k′2k→′=γ(ω−k∥v)(γ(k∥−vω/c2)k̂∥+k⊥k̂⊥)k⊥2+γ2(k∥−vω/c2)2 ,$
(7)

where $k̂∥$ and $k̂⊥$ are unit vectors pointing in the direction of $k→∥$ and $k→⊥$, respectively.

While accurate, the above formula is not particularly useful. Typically, we wish to relate the phase velocity measured in one frame to the phase velocity in another. To this end, we note that it follows directly from Eq. (2) and $k2=ω2/u2$ that

(8)
$k∥=ωu∥/u2 ,$
(8a)
$k→⊥=ωu→⊥/u2 .$
(8b)
By substituting Eq. (8) into Eq. (7), and after some algebraic manipulation we determine that

$u→′=γc2(u2−u∥v)(γ(u∥c2−u2v)k̂∥+u⊥c2k̂⊥)c4u⊥2+γ2(c2u∥−u2v)2.$
(9)

Separating the phase velocity into components parallel and perpendicular to $v→$, we obtain

$u′∥=γ2c2(u2−u∥v)(u∥c2−u2v)c4u⊥2+γ2(c2u∥−u2v)2,$
(10a)
$u→⊥′=γc4(u2−u∥v)u⊥c4u⊥2+γ2(c2u∥−u2v)2k̂⊥.$
(10b)

Obvious substitutions show that Eq. (10b) is correct while Bachman's (17) is not, because it includes an extra spurious factor of u/c.3 The special cases considered in Bachman's paper still provide correct conclusions because the first one deals with parallel transformations and the second with signals traveling at the speed of light, in which case u/c = 1 and Eq. (17) in Bachman's paper and Eq. (10b) reduce to the same form. Nonetheless, it is important to provide an accurate formula for the phase velocity transformation law as it is required when high precision is needed, such as in satellite navigation.

In passing, we also note that there is one obvious typo in Bachman's paper: just above Eq. (20) it is written “Using $γ2=(c2−v2)/c2$,” clearly this should be $γ2=c2/(c2−v2)$.

1.
Elliott D.
Kaplan
.
Understanding GPS: Principles and Applications
(
Artech House
,
Boston
,
1996
).
2.
Samuel
Picton Drake
and
Alan
Purvis
, “
Everyday relativity and the Doppler effect
,”
Am. J. Phys.
82
(
1
),
52
59
(
2014
).
3.
R. A.
Bachman
, “
Relativistic phase velocity transformation
,”
Am. J. Phys.
57
(
7
),
628
630
(
1989
).