General
Relativity For Teletubbies
Sir Kevin Aylward B.Sc., Warden
of the King’s Ale
Galilean
Transformation
Of
Maxwell’s
Equations
Back
to the Contents section
Abstract
There are often
claims that Maxell’s Equations predict that the speed of light is invariant
irrespective of source or observer motion.
The argument pretty much, invariably, states that the EM wave equation
has a velocity given by:
c=
1
εμ
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiabg2da9maalaaabaGaaGymaaqaamaakaaabaGaeqyTduMaeqiVd0galeqaaaaaaaa@3C46@
With the claim
that, because there are no other velocity terms in the expression, the velocity
is not with respect to anything. These claims are usually the result of the
claimers simply copying from those that copied, that copied, down a long chain,
with no evaluation as to whether such a claim is even rational. The claim is
false, and it is certainly not controversial that such a claim is false. Indeed
the considerable effort in engaging in the Michaelson-Morley Experiment is
testament to this. The MMX was specifically designed to detect a change in
light’s velocity in relation to the observers velocity via way of the Earth’s
motion through the Solar system, principally because that is what Maxwell’s
Equations indicated would be the case.
However, if as
an independent assumption, Maxwell’s Equations are subject to the Lorentz
Transformations, then they do “predict” an invariant velocity of light.
The following is
a non-original summary derivation of the expected change in lights’ velocity
according to the Galilean Transformation applied to Maxwell’s Equations.
Galilean
Transformation of Maxwell’s Equation
In
order to actually determine what Maxwell’s Equations predict about observers
moving relative to EM fields, one has to actually
calculate what the wave equation will be from Maxwell’s equations under a
Galilean Transformation.
The
core of this analysis was taken from:
https://physics.stackexchange.com/questions/378861/what-does-a-galilean-transformation-of-maxwells-equations-look-like
For
two reference frames S, S’,
with one moving with relative with respect to another, the Galilean
Transformation is given by:
t
′
=t
x
′
=x−vt
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaqaaaaaaaaaWdbiaadshapaWaaWbaaeqabaWdbiabgkdiIcaacqGH9aqpcaWG0baabaGaaCiEa8aadaahaaqabeaapeGaeyOmGikaaiabg2da9iaahIhacqGHsislcaWH2bGaamiDaaaaaa@4263@
The
full Lorentz transformation of the Electromagnetic Fields and Sources, as
provided by the Wikipedia article:
https://en.wikipedia.org/wiki/Classical_electromagnetism_and_special_relativity
are:
E
′
=γ(E+v×B)−(γ−1)(E⋅
v
^
)
v
^
B
′
=γ(B−
1
c
2
v×E)−(γ−1)(B⋅
v
^
)
v
^
J
′
=J−γρv+(γ−1)(J⋅
v
^
)
v
^
ρ
′
=γ(ρ−
J.v
c
2
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@8F00@
The
Galilean Transform of the fields and sources are thus obtained by taking the
limit of c→∞
so that γ→1,
which results in:
E=
E
′
−v×
B
′
B=
B
′
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaqaaaaaaaaaWdbiaahweacqGH9aqpcaWHfbWdamaaCaaabeqaa8qacqGHYaIOaaGaeyOeI0IaaCODaiabgEna0kaahkeapaWaaWbaaeqabaWdbiabgkdiIcaaa8aabaWdbiaahkeacqGH9aqpcaWHcbWdamaaCaaabeqaa8qacqGHYaIOaaaaaaa@456A@
J=
J
′
+
ρ
′
v
ρ=
ρ
′
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaqaaaaaaaaaWdbiaahQeacqGH9aqpcaWHkbWdamaaCaaabeqaa8qacqGHYaIOaaGaey4kaSIaeqyWdi3damaaCaaabeqaa8qacqGHYaIOaaGaaCODaaWdaeaapeGaeqyWdiNaeyypa0JaeqyWdi3damaaCaaabeqaa8qacqGHYaIOaaaaaaa@4631@
After
swapping v→−v
In
the frame S, Maxwell’s Equations
are:
∇⋅E=ρ/
ϵ
0
∇⋅B=0
∇×E=−
∂B
∂t
∇×B=
μ
0
(J+
ϵ
0
∂E
∂t
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaqaaaaaaaaaWdbiabgEGirlabgwSixlaahweacqGH9aqpcqaHbpGCcaGGVaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiuGacqWF1pG8paWaaSbaaeaapeGaaGimaaWdaeqaaaqaa8qacqGHhis0cqGHflY1caWHcbGaeyypa0JaaGimaaWdaeaapeGaey4bIeTaey41aqRaaCyraiabg2da9iabgkHiTmaalaaapaqaa8qacqaHciITcaWHcbaapaqaa8qacqaHciITcaWG0baaaaWdaeaapeGaey4bIeTaey41aqRaaCOqaiabg2da9iabeY7aT9aadaWgaaqaa8qacaaIWaaapaqabaWdbiaacIcacaWHkbGaey4kaSIae8x9di=damaaBaaabaWdbiaaicdaa8aabeaapeWaaSaaa8aabaWdbiabekGi2kaahweaa8aabaWdbiabekGi2kaadshaaaGaaiykaaaaaa@70AA@
Thus,
in the transformed frame S’, Maxwell’s
Equations are:
∇⋅(
E
′
−v×
B
′
)=
ρ
′
/
ϵ
0
∇⋅
B
′
=0
∇×(
E
′
−v×
B
′
)=−
∂
B
′
∂t
∇×
B
′
=
μ
0
(
J
′
+
ρ
′
v+
ϵ
0
∂(
E
′
−v×
B
′
)
∂t
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaqaaaaaaaaaWdbiabgEGirlabgwSixlaacIcacaWHfbWdamaaCaaabeqaa8qacqGHYaIOaaGaeyOeI0IaaCODaiabgEna0kaahkeapaWaaWbaaeqabaWdbiabgkdiIcaacaGGPaGaeyypa0JaeqyWdi3damaaCaaabeqaa8qacqGHYaIOaaGaai4lamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhAG8KBLbacfiGae8x9di=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damaaBaaabaWdbiaaicdaa8aabeaapeWaaSaaa8aabaWdbiabekGi2kaacIcacaWHfbWdamaaCaaabeqaa8qacqGHYaIOaaGaeyOeI0IaaCODaiabgEna0kaahkeapaWaaWbaaeqabaWdbiabgkdiIcaacaGGPaaapaqaa8qacqaHciITcaWG0baaaiaacMcaaaaa@9BCC@
With
the derivatives also transformed by:
∇=
∇
′
∂
∂t
=
∂
∂
t
′
−v⋅∇
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaqaaaaaaaaaWdbiabgEGirlabg2da9iabgEGir=aadaahaaqabeaapeGaeyOmGikaaaWdaeaapeWaaSaaa8aabaWdbiabekGi2cWdaeaapeGaeqOaIyRaamiDaaaacqGH9aqpdaWcaaWdaeaapeGaeqOaIylapaqaa8qacqaHciITcaWG0bWdamaaCaaabeqaa8qacqGHYaIOaaaaaiabgkHiTiaahAhacqGHflY1cqGHhis0aaaa@4C8F@
And
the primes throughout, simply for convenience, now swapped to non primes,
results in the transformed Maxwell’s Equations of:
∇⋅E+v⋅(∇×B)=ρ/
ϵ
0
∇⋅B=0
∇×E=−
∂B
∂t
∇×B=
μ
0
(J+ρv+
ϵ
0
∂
∂t
(E−v×B)−
ϵ
0
v⋅∇(E−v×B))
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaqaaaaaaaaaWdbiabgEGirlabgwSixlaahweacqGHRaWkcaWH2bGaeyyXICTaaiikaiabeEGirlabgEna0kaahkeacaGGPaGaeyypa0JaeqyWdiNaai4lamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhAG8KBLbacfiGae8x9di=damaaBaaabaWdbiaaicdaa8aabeaaa8qabaGaey4bIeTaeyyXICTaaCOqaiabg2da9iaaicdaa8aabaWdbiabgEGirlabgEna0kaahweacqGH9aqpcqGHsisldaWcaaWdaeaapeGaeqOaIyRaaCOqaaWdaeaapeGaeqOaIyRaamiDaaaaa8aabaWdbiabgEGirlabgEna0kaahkeacqGH9aqpcqaH8oqBpaWaaSbaaeaapeGaaGimaaWdaeqaa8qacaGGOaGaaCOsaiabgUcaRiabeg8aYjaahAhacqGHRaWkcqWF1pG8paWaaSbaaeaapeGaaGimaaWdaeqaa8qadaWcaaWdaeaapeGaeqOaIylapaqaa8qacqaHciITcaWG0baaaiaacIcacaWHfbGaeyOeI0IaaCODaiabgEna0kaahkeacaGGPaGaeyOeI0Iae8x9di=damaaBaaabaWdbiaaicdaa8aabeaapeGaaCODaiabgwSixlabgEGirlaacIcacaWHfbGaeyOeI0IaaCODaiabgEna0kaahkeacaGGPaGaaiykaaaaaa@9495@
Taking
the curl of the last equation, and substituting for E from the 1st and 3rd equations gives the
modified wave equation for B:
c
2
∇
2
B=
∂
2
B
∂
t
2
+
(v⋅∇)
2
B−2v⋅∇(
∂B
∂t
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qacaWGJbWdamaaCaaabeqaa8qacaaIYaaaaiabgEGir=aadaahaaqabeaapeGaaGOmaaaacaWHcbGaeyypa0ZaaSaaa8aabaWdbiabekGi2+aadaahaaqabeaapeGaaGOmaaaacaWHcbaapaqaa8qacqaHciITcaWG0bWdamaaCaaabeqaa8qacaaIYaaaaaaacqGHRaWkcaGGOaGaaCODaiabgwSixlabgEGirlaacMcapaWaaWbaaeqabaWdbiaaikdaaaGaaCOqaiabgkHiTiaaikdacaWH2bGaeyyXICTaey4bIeTaaiikamaalaaapaqaa8qacqaHciITcaWHcbaapaqaa8qacqaHciITcaWG0baaaiaacMcaaaa@587F@
Whence
a solution of the form
B=
B
o
expi(k⋅x−ωt)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qacaWHcbGaeyypa0JaamOqamaaBaaabaGaam4BaaqabaGaciyzaiaacIhacaGGWbGaamyAaiaacIcacaWHRbGaeyyXICTaaCiEaiabgkHiTiabeM8a3jaadshacaGGPaaaaa@46B6@
is
obtainable with
ω=−v⋅k±c|k|
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qacqaHjpWDcqGH9aqpcqGHsislcaWH2bGaeyyXICTaaC4AaiabgglaXkaadogacaGG8bGaaC4AaiaacYhaaaa@43BC@
resulting
in a group velocity for the wave of:
∂ω
∂k
=−v±c
k
^
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qadaWcaaWdaeaapeGaeqOaIyRaeqyYdChapaqaa8qacqaHciITcaWHRbaaaiabg2da9iabgkHiTiaahAhacqGHXcqScaWGJbGabC4Aayaajaaaaa@4298@
which
results in a change in velocity :
c±v
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8qacaWGJbGaeyySaeRaaeODaaaa@39C4@
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