General Relativity For
Tellytubbys
The Special Relativity
Section
Sir Kevin Aylward B.Sc.,
Warden of the Kings Ale
Back
to the Contents section
Overview
Special
Relativity (S.R.) has probably spawned more responses from individuals then a
naked Heather Locklear strolling through central park (sci.physics.relativity).
The main reason for this section is that it is required for an understanding of
the stress-energy tensor.
You can get a bit more of an
understanding for the rational of SR here SR
Background.
So, straight to the point.
It is noted that, independent of whether light is treated as a wave, particle
or Teletubby, and independent of ones velocity with respect to the light
source, the speed of light is invariant in vacume. That is, it is always
measured, or all experiments can only be understood, if the speed of light
always remains constant. This is of course very queer. No matter how fast one
travels into the sun, its light will always be measured to have the same speed.
Secondly,
various results of physics don’t seem to depend on what does the moving, only
relative velocities seem to matter.
These are expressed as:
1)
The
Principle Of Relativity (POR) Postulate/Axiom
The Laws Of
Physics are independent of inertial motion (non accelerated motion).
2)
The
Speed of Light (SOL) Postulate/Axiom:
The SOL, in a
vacuum, is an invariant. That is, the SOL is always measured to be the same
irrespective of the source of the light’s velocity or the observers velocity.
Postulate
1 is quite often expressed or explained quite incorrectly. The essential
bit though is that it is just as valid to swap over what object is stationary
and what object is moving. That is, there is no concept of absolute rest, all
uniform motion is equivalent. It is important to point out there is nothing new
in this postulate from Newtonian Mechanics.
The 1st Postulate fundamentally
means that the forces between co-moving
objects are independent of their joint velocity with respect to other inertial
frames.
The
Sums of the Lorentz Transformation
Everyone got to the
equations before Einstein, but somehow he managed to steal their thunder. The
Lorentz Transforms were first published in Playboy, but by someone who, for
obvious reasons, wanted to remain anonymous. So because of this Lorentz became
the fall guy, and in fact he only became aquatinted with them whilst waiting for
the number 133 bus at Dagenham East, while "reading" the news
section, or so he claimed.
Here we go. Lets pretend to
derive the co-ordinate transform equations of SR. Imagine standing in a box and
viewing another a box moving by you, left to right (the x direction) at
velocity v, that’s our co-ordinate systems described then, i.e. where the
origins are the front left corners of the boxes. The moving systems origin is
called O’, the stationary (reference) one O. If the reference frame emits a
pulse of light from its origin, the pulse’s position can be expressed by:
x
2
+
y
2
+
z
2
=
c
2
t
2
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadMhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWG6bWaaWbaaSqabeaacaaIYaaaaOGaeyypa0Jaam4yamaaCaaaleqabaGaaGOmaaaakiaadshadaahaaWcbeqaaiaaikdaaaaaaa@424E@
Since the pulse’s radial position is ct
or
x
2
+
y
2
+
z
2
−
c
2
t
2
=0
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadMhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWG6bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0Iaam4yamaaCaaaleqabaGaaGOmaaaakiaadshadaahaaWcbeqaaiaaikdaaaGccqGH9aqpcaaIWaaaaa@43FF@
For the moving box, its
equation for the light pulse would be:
x
¯
2
+
y
¯
2
+
z
¯
2
=
c
2
t
¯
2
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaaraWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIabmyEayaaraWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIabmOEayaaraWaaWbaaSqabeaacaaIYaaaaOGaeyypa0Jaam4yamaaCaaaleqabaGaaGOmaaaakiqadshagaqeamaaCaaaleqabaGaaGOmaaaaaaa@42AE@
or
x
¯
2
+
y
¯
2
+
z
¯
2
−
c
2
t
¯
2
=0
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaaraWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIabmyEayaaraWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIabmOEayaaraWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0Iaam4yamaaCaaaleqabaGaaGOmaaaakiqadshagaqeamaaCaaaleqabaGaaGOmaaaakiabg2da9iaaicdaaaa@445F@
Where, by assumption the t
bar might be different, and it is to be determined if that is indeed the case.
Note, our speed of light postulate requires that c is the same in both sets of
equations.
These equations can then be
equated thus:
x
2
+
y
2
+
z
2
−
c
2
t
2
=
x
¯
2
+
y
¯
2
+
z
¯
2
−
c
2
t
¯
2
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadMhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWG6bWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0Iaam4yamaaCaaaleqabaGaaGOmaaaakiaadshadaahaaWcbeqaaiaaikdaaaGccqGH9aqpceWG4bGbaebadaahaaWcbeqaaiaaikdaaaGccqGHRaWkceWG5bGbaebadaahaaWcbeqaaiaaikdaaaGccqGHRaWkceWG6bGbaebadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGJbWaaWbaaSqabeaacaaIYaaaaOGabmiDayaaraWaaWbaaSqabeaacaaIYaaaaaaa@4FE6@
And, to cut a long story
short, a bit of rationalization will indicate that the y’s and z’s will in both
systems will be the same, thus:
x
2
−
c
2
t
2
=
x
¯
2
−
c
2
t
¯
2
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadogadaahaaWcbeqaaiaaikdaaaGccaWG0bWaaWbaaSqabeaacaaIYaaaaOGaeyypa0JabmiEayaaraWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0Iaam4yamaaCaaaleqabaGaaGOmaaaakiqadshagaqeamaaCaaaleqabaGaaGOmaaaaaaa@4468@
The bit to notice here is
that both sides of the equation are an invariant. i.e. its value don’t change
with the inertial frames motion.
So, how are the x an x bars
related. One assumes for all sorts of high-brow reasons, that the coordinates
are related by a linear transform:
x
¯
=γ(x−avt)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaaraGaeyypa0Jaeq4SdCMaaiikaiaadIhacqGHsislcaWGHbGaamODaiaadshacaGGPaaaaa@3FD3@
t
¯
=
γ
′
(t−
bv
c
2
x)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiDayaaraGaeyypa0Jafq4SdCMbauaacaGGOaGaamiDaiabgkHiTmaalaaabaGaamOyaiaadAhaaeaacaWGJbWaaWbaaSqabeaacaaIYaaaaaaakiaadIhacaGGPaaaaa@41C7@
These might look a bit
contrived, but this simply states that the x bar and t bar are linear functions
of x and t, the constants are arbitrary and to be determined, and all come out
in the wash.
Lets see what the relativity
postulate gets us first.
Well, it means that we can
simply swap over the x to x bar and t tot bar and change the sign of the
velocity, thus
x=γ(
x
¯
+av
t
¯
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg2da9iabeo7aNjaacIcaceWG4bGbaebacqGHRaWkcaWGHbGaamODaiqadshagaqeaiaacMcaaaa@3FE0@
t=
γ
′
(
t
¯
+
bv
c
2
x
¯
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabg2da9iqbeo7aNzaafaGaaiikaiqadshagaqeaiabgUcaRmaalaaabaGaamOyaiaadAhaaeaacaWGJbWaaWbaaSqabeaacaaIYaaaaaaakiqadIhagaqeaiaacMcaaaa@41D4@
Now if we substitute the
above pair of equations into the above, above x equation…
x
¯
=γ(γ(
x
¯
+av
t
¯
)−av
γ
′
(
t
¯
+
bv
x
¯
c
2
))
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaaraGaeyypa0Jaeq4SdCMaaiikaiabeo7aNjaacIcaceWG4bGbaebacqGHRaWkcaWGHbGaamODaiqadshagaqeaiaacMcacqGHsislcaWGHbGaamODaiqbeo7aNzaafaGaaiikaiqadshagaqeaiabgUcaRmaalaaabaGaamOyaiaadAhaceWG4bGbaebaaeaacaWGJbWaaWbaaSqabeaacaaIYaaaaaaakiaacMcacaGGPaaaaa@4FA7@
equating the coefficients of
t bar:
0=
γ
′
av−av
γ
′
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabg2da9iqbeo7aNzaafaGaamyyaiaadAhacqGHsislcaWGHbGaamODaiqbeo7aNzaafaaaaa@3FC9@
hence
γ
′
=γ
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4SdCMbauaacqGH9aqpcqaHZoWzaaa@3A54@
Using this result and
equating the coefficients of x bar:
1=(γ−
ab
v
2
c
2
γ)γ
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabg2da9iaacIcacqaHZoWzcqGHsisldaWcaaqaaiaadggacaWGIbGaamODamaaCaaaleqabaGaaGOmaaaaaOqaaiaadogadaahaaWcbeqaaiaaikdaaaaaaOGaeq4SdCMaaiykaiabeo7aNbaa@4496@
hence
γ=
1
1−
ab
v
2
c
2
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCMaeyypa0ZaaSaaaeaacaaIXaaabaWaaOaaaeaacaaIXaGaeyOeI0YaaSaaaeaacaWGHbGaamOyaiaadAhadaahaaWcbeqaaiaaikdaaaaakeaacaWGJbWaaWbaaSqabeaacaaIYaaaaaaaaeqaaaaaaaa@40C0@
now from above
x
2
−
c
2
t
2
=
x
¯
2
−
c
2
t
¯
2
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadogadaahaaWcbeqaaiaaikdaaaGccaWG0bWaaWbaaSqabeaacaaIYaaaaOGaeyypa0JabmiEayaaraWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0Iaam4yamaaCaaaleqabaGaaGOmaaaakiqadshagaqeamaaCaaaleqabaGaaGOmaaaaaaa@4468@
Substituting into the above
gets us:
x
2
−
c
2
t
2
=
γ
2
(x−avt)
2
−
c
2
γ
2
(t−
bvx
c
2
)
2
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaadogadaahaaWcbeqaaiaaikdaaaGccaWG0bWaaWbaaSqabeaacaaIYaaaaOGaeyypa0Jaeq4SdC2aaWbaaSqabeaacaaIYaaaaOGaaiikaiaadIhacqGHsislcaWGHbGaamODaiaadshacaGGPaWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0Iaam4yamaaCaaaleqabaGaaGOmaaaakiabeo7aNnaaCaaaleqabaGaaGOmaaaakiaacIcacaWG0bGaeyOeI0YaaSaaaeaacaWGIbGaamODaiaadIhaaeaacaWGJbWaaWbaaSqabeaacaaIYaaaaaaakiaacMcadaahaaWcbeqaaiaaikdaaaaaaa@559C@
x
2
−
c
2
t
2
=
γ
2
(
x
2
−2avxt+
a
2
v
2
t
2
)−
c
2
γ
2
(
t
2
−
2bvxt
c
2
+
b
2
v
2
x
2
c
4
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6830@
For this to be true for all
x and t the coefficients of each power of x and t must be equal:
xt term,
0=2av−2bv
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabg2da9iaaikdacaWGHbGaamODaiabgkHiTiaaikdacaWGIbGaamODaaaa@3DDC@
or
a=b
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabg2da9iaadkgaaaa@38C7@
x2 term
1=
γ
2
(1−
b
2
v
2
c
2
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabg2da9iabeo7aNnaaCaaaleqabaGaaGOmaaaakiaacIcacaaIXaGaeyOeI0YaaSaaaeaacaWGIbWaaWbaaSqabeaacaaIYaaaaOGaamODamaaCaaaleqabaGaaGOmaaaaaOqaaiaadogadaahaaWcbeqaaiaaikdaaaaaaOGaaiykaaaa@4303@
t2 term
1=
γ
2
(1−
a
2
v
2
c
2
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabg2da9iabeo7aNnaaCaaaleqabaGaaGOmaaaakiaacIcacaaIXaGaeyOeI0YaaSaaaeaacaWGHbWaaWbaaSqabeaacaaIYaaaaOGaamODamaaCaaaleqabaGaaGOmaaaaaOqaaiaadogadaahaaWcbeqaaiaaikdaaaaaaOGaaiykaaaa@4302@
Putting it all together
gives us.
x
¯
=γ(x−vt)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaaraGaeyypa0Jaeq4SdCMaaiikaiaadIhacqGHsislcaWG2bGaamiDaiaacMcaaaa@3EED@
t
¯
=γ(t−
vx
c
2
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiDayaaraGaeyypa0Jaeq4SdCMaaiikaiaadshacqGHsisldaWcaaqaaiaadAhacaWG4baabaGaam4yamaaCaaaleqabaGaaGOmaaaaaaGccaGGPaaaaa@40D4@
whereγ=
1
1−
v
2
c
2
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaadIgacaWGLbGaamOCaiaadwgacqaHZoWzcqGH9aqpdaWcaaqaaiaaigdaaeaadaGcaaqaaiaaigdacqGHsisldaWcaaqaaiaadAhadaahaaWcbeqaaiaaikdaaaaakeaacaWGJbWaaWbaaSqabeaacaaIYaaaaaaaaeqaaaaaaaa@43A7@
And, one must stress the
importance of the second term in the time equation. It’s the relativity of
simultaneity term that makes SR actually work. It says time is also a function
of distance separation.
Time
dilation
How does time change in the
moving system relative to the stationary, one?
d
t
¯
dt
=
d
dt
(γ(t−
vx
c
2
))
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGKbGabmiDayaaraaabaGaamizaiaadshaaaGaeyypa0ZaaSaaaeaacaWGKbaabaGaamizaiaadshaaaGaaiikaiabeo7aNjaacIcacaWG0bGaeyOeI0YaaSaaaeaacaWG2bGaamiEaaqaaiaadogadaahaaWcbeqaaiaaikdaaaaaaOGaaiykaiaacMcaaaa@47E3@
d
t
¯
dt
=γ
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGKbGabmiDayaaraaabaGaamizaiaadshaaaGaeyypa0Jaeq4SdCgaaa@3C8D@
Length
Contraction
dx
d
x
¯
=
1
γ
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGKbGaamiEaaqaaiaadsgaceWG4bGbaebaaaGaeyypa0ZaaSaaaeaacaaIXaaabaGaeq4SdCgaaaaa@3D60@
Standard
Results
The assumption here is that
the reader will get the main SR stuff from elsewhere, this is really a
refresher bit just to form a reference for the GR bits and pieces.
Instead of the conventional
3D space and 1D-time descriptions, in SR objects are reformulated in one 4-D
description. For example, 3 D velocities are replaced by a 4-d velocity vector
called the 4 velocity, u.
4-position
X=[ct,x,y,z]=
x
α
e
α
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiwaiabg2da9iaacUfacaWGJbGaamiDaiaacYcacaWG4bGaaiilaiaadMhacaGGSaGaamOEaiaac2facqGH9aqpcaWG4bWaaWbaaSqabeaacqaHXoqyaaGccaWHLbWaaSbaaSqaaiabeg7aHbqabaaaaa@4718@
4
- velocity
dx
dτ
=
d
x
α
dτ
,α≠0,
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGKbGaaCiEaaqaaiaadsgacqaHepaDaaGaeyypa0ZaaSaaaeaacaWGKbGaamiEamaaCaaaleqabaGaeqySdegaaaGcbaGaamizaiabes8a0baacaGGSaGaeqySdeMaeyiyIKRaaGimaiaacYcaaaa@479C@
where the 0'th term of u is time t. Tau is the proper time.
Such that:
u
0
≡c
d
x
0
dτ
=c
dt
dτ
=γc
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaCaaaleqabaGaaGimaaaakiabggMi6kaadogadaWcaaqaaiaadsgacaWG4bWaaWbaaSqabeaacaaIWaaaaaGcbaGaamizaiabes8a0baacqGH9aqpcaWGJbWaaSaaaeaacaWGKbGaamiDaaqaaiaadsgacqaHepaDaaGaeyypa0Jaeq4SdCMaam4yaaaa@4A48@
u
α
=
d
x
α
dτ
=γ
dx
dt
=γ
v
α
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaCaaaleqabaGaeqySdegaaOGaeyypa0ZaaSaaaeaacaWGKbGaamiEamaaCaaaleqabaGaeqySdegaaaGcbaGaamizaiabes8a0baacqGH9aqpcqaHZoWzdaWcaaqaaiaadsgacaWG4baabaGaamizaiaadshaaaGaeyypa0Jaeq4SdCMaamODamaaCaaaleqabaGaeqySdegaaaaa@4C3D@
Where alpha = 1,2,3
or
u=[cγ,γ
x
˙
,γ
y
˙
,γ
z
˙
]
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCyDaiabg2da9iaacUfacaWGJbGaeq4SdCMaaiilaiabeo7aNjqadIhagaGaaiaacYcacqaHZoWzceWG5bGbaiaacaGGSaGaeq4SdCMabmOEayaacaGaaiyxaaaa@4661@
4
- Momentum
p=mu
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiCaiabg2da9iaad2gacaWH1baaaa@39E3@
where it is noted that the 0th
momentum component is:
p
0
=mcγ=
E
c
as E=mγ
c
2
where E≡KE+m
c
2
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaCaaaleqabaGaaGimaaaakiabg2da9iaad2gacaWGJbGaeq4SdCMaeyypa0ZaaSaaaeaacaWGfbaabaGaam4yaaaacaqGGaGaaeyyaiaabohacaqGGaGaaeyraiabg2da9iaab2gacqaHZoWzcaqGJbWaaWbaaSqabeaacaqGYaaaaOGaaeiiaiaabEhacaqGObGaaeyzaiaabkhacaqGLbGaaeiiaiaadweacqGHHjIUcaWGlbGaamyraiabgUcaRiaad2gacaWGJbWaaWbaaSqabeaacaaIYaaaaaaa@5642@
Or
p=[
E
c
,γm
v
x
,γm
v
y
,γm
v
z
]
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiCaiabg2da9iaacUfadaWcaaqaaiaadweaaeaacaWGJbaaaiaacYcacqaHZoWzcaWGTbGaamODamaaCaaaleqabaGaamiEaaaakiaacYcacqaHZoWzcaWGTbGaamODamaaCaaaleqabaGaamyEaaaakiaacYcacqaHZoWzcaWGTbGaamODamaaCaaaleqabaGaamOEaaaakiaac2faaaa@4BE0@
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All rights reserved
The information on the page may be
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providing that this source is acknowledged.
Website last modified 3rd December
2022
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