Twins Paradox
Sir Kevin
Aylward B.Sc., Warden of the King’s Ale
Back to the Contents section
Overview
Yet
another true resolution of the Twins
Paradox…
The
twins paradox is the notion that if the Principle of Relativity (POR) is valid,
then if one twin jaunts off in a rocket to the star Alpha Centauri at a speed
close to the velocity of light and returns, it is concluded that there is
ambiguity in what twin shows the least age. The argument being that Special
Relativity (SR) states that the clock ticks of a clock in a frame moving
relatively to a clock in a notional stationary frame, are larger, such that time
(number of clock ticks) for the moving clock passes slower than the non-moving
clock. However, it is also stated that the
traveller can consider himself fixed in space, such that the stay at home twin
may be considered to be moving such that the stay at home twin can claim to be
the younger twin.
There
are many accounts of claims of resolving the twins paradox of Special
Relativity such as whether acceleration is required, for example trips through
“space-time” and some claiming that it is due to switching the direction of frames
for the traveller that the stay at home does not experience.
These
explanations are not correct. Fundamentally, they have lost the plot.
Neither
frame switching or acceleration form the root cause as to why the
traveller is younger.
The root cause is that the stay at home twin and the star are both in a different frame from the travelling twin.
The frames are different because the star always stays in the same frame as the
stay at home twin, whatever frame is taken to be at rest. The times in different frames are different because time
in frames is dependent on distance as well as time of other frames.
This is absolutely fundamental to the resolution of the paradox. This changes
the distances that the traveller measures from that which the stay at home twin
measures. The fact that frame times depend on distance is typically ignored.
Thus
the root cause of the asymmetry in times of the twins is:
1 The stay at home twin measures event
times at two different locations.
2 The traveller, considered at rest,
measures event times at one location.
Whether
or not there is a paradox, is whether or not a correct application of the Lorentz Transform results in the same
results for the time of the trip, independent of who is considered at rest.
Hand waving descriptions typically ignore what the true physics actually says.
Typically most alleged resolutions don’t actually show the calculations of both
viewpoints, they engage in a Strawman that notionally appears to do this, but
doesn’t.
Indeed,
whether the moving clock is outward or inward makes no difference. According to
a correct
SR calculation, both twins will agree as to the time difference between the start event
and end events of even the single way trip, and that the traveller is the
youngest, eliminating the paradox.
Key
to this calculation is:
1
Time in a frame is not simply:
t
'
=γt
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDamaaCaaaleqabaGaai4jaaaakiabg2da9iabeo7aNjaadshaaaa@3B56@
- 1
It
is:
t
'
=γ(t−
vx
c
2
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDamaaCaaaleqabaGaai4jaaaakiabg2da9iabeo7aNjaacIcacaWG0bGaeyOeI0YaaSaaaeaacaWG2bGaamiEaaqaaiaadogadaahaaWcbeqaaiaaikdaaaaaaOGaaiykaaaa@4180@
- 2
That
is, time events in inertial frames are dependent on both time events in the frames and
distance travelled in that frame.
2
Distances (lengths) in frames are not
the same, they are related by:
x
'
=γx
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCaaaleqabaGaai4jaaaakiabg2da9iabeo7aNjaadIhaaaa@3B5E@
-
3
The System
Diagram 1 Traveller Model
A
= Stay at home twin
B
= Travelling twin
S
= Star
La
= Rest frame distance of stay at home twin to the star, as measured by the stay
at home twin
Lb
= Rest frame distance of traveller twin measured for distance of stay at home
twin to the star
Event
1 = Time & space coordinates of
when B and A are at the same location
Event
2 = Time & space coordinates of
when B and S are at the same location
The
coordinates should be clear from simple inspection. It takes a time of L/V to
get to the star. The time coordinate for the case where B is considered stationary
is simply B’s own clock time.
The Calculations
The
Lorentz Transform (LT) allows the time
and space coordinates of events in one frame to be calculated from time and space coordinates of events in
another frame. That is:
x
'
=γ(x−vt)
t
'
=γ(t−
vx
c
2
)
γ=
1
1−
v
2
c
2
1
γ
2
=(1−
v
2
c
2
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5F9A@
- 4
Thus,
given the coordinates of events 1 & 2 according to A, then the coordinates
of B for events 1 & 2 can be calculated as:
x
b1
=γ(
x
a1
−v
t
a1
)
t
b1
=γ(
t
a1
−
v
x
a1
c
2
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWG4bWaaSbaaSqaaiaadkgacaaIXaaabeaakiabg2da9iabeo7aNjaacIcacaWG4bWaaSbaaSqaaiaadggacaaIXaaabeaakiabgkHiTiaadAhacaWG0bWaaSbaaSqaaiaadggacaaIXaaabeaakiaacMcaaeaacaWG0bWaaSbaaSqaaiaadkgacaaIXaaabeaakiabg2da9iabeo7aNjaacIcacaWG0bWaaSbaaSqaaiaadggacaaIXaaabeaakiabgkHiTmaalaaabaGaamODaiaadIhadaWgaaWcbaGaamyyaiaaigdaaeqaaaGcbaGaam4yamaaCaaaleqabaGaaGOmaaaaaaGccaGGPaaaaaa@5492@
- 5
x
b2
=γ(
x
a2
−v
t
a2
)
t
b2
=γ(
t
a2
−
v
x
a2
c
2
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWG4bWaaSbaaSqaaiaadkgacaaIYaaabeaakiabg2da9iabeo7aNjaacIcacaWG4bWaaSbaaSqaaiaadggacaaIYaaabeaakiabgkHiTiaadAhacaWG0bWaaSbaaSqaaiaadggacaaIYaaabeaakiaacMcaaeaacaWG0bWaaSbaaSqaaiaadkgacaaIYaaabeaakiabg2da9iabeo7aNjaacIcacaWG0bWaaSbaaSqaaiaadggacaaIYaaabeaakiabgkHiTmaalaaabaGaamODaiaadIhadaWgaaWcbaGaamyyaiaaikdaaeqaaaGcbaGaam4yamaaCaaaleqabaGaaGOmaaaaaaGccaGGPaaaaaa@5498@
- 6
The Usually Included Calculation
For
diagram 1, the time A calculates for the trip from event 1 to 2, is event 2
time
–
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=nbiaaa@37C1@
event 1 time
is:
t
a12
=
t
a2
−
t
a1
=(
L
a
v
−0)=
L
a
v
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDamaaBaaaleaacaWGHbGaaGymaiaaikdaaeqaaOGaeyypa0JaamiDamaaBaaaleaacaWGHbGaaGOmaaqabaGccqGHsislcaWG0bWaaSbaaSqaaiaadggacaaIXaaabeaakiabg2da9iaacIcadaWcaaqaaiaadYeadaWgaaWcbaGaamyyaaqabaaakeaacaWG2baaaiabgkHiTiaaicdacaGGPaGaeyypa0ZaaSaaaeaacaWGmbWaaSbaaSqaaiaadggaaeqaaaGcbaGaamODaaaaaaa@4BF1@
- 7
Also
from diagram 1
(
x
a1
,
t
a1
)=(0,0)
(
x
a2
,
t
a2
)=(
L
a
,
L
a
v
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaGGOaGaamiEamaaBaaaleaacaWGHbGaaGymaaqabaGccaGGSaGaamiDamaaBaaaleaacaWGHbGaaGymaaqabaGccaGGPaGaeyypa0JaaiikaiaaicdacaGGSaGaaGimaiaacMcaaeaacaGGOaGaamiEamaaBaaaleaacaWGHbGaaGOmaaqabaGccaGGSaGaamiDamaaBaaaleaacaWGHbGaaGOmaaqabaGccaGGPaGaeyypa0JaaiikaiaadYeadaWgaaWcbaGaamyyaaqabaGccaGGSaWaaSaaaeaacaWGmbWaaSbaaSqaaiaadggaaeqaaaGcbaGaamODaaaacaGGPaaaaaa@51AF@
- 8
Thus
the coordinates of B for event 1 are:
(
x
b1
,
t
b1
)=γ(0−v×0,0−
v×
0
c
2
)=(0,0)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadIhadaWgaaWcbaGaamOyaiaaigdaaeqaaOGaaiilaiaadshadaWgaaWcbaGaamOyaiaaigdaaeqaaOGaaiykaiabg2da9iabeo7aNjaacIcacaaIWaGaeyOeI0IaamODaiabgEna0kaaicdacaGGSaGaaGimaiabgkHiTmaalaaabaGaamODaiabgEna0kaaicdadaWgaaWcbaaabeaaaOqaaiaadogadaahaaWcbeqaaiaaikdaaaaaaOGaaiykaiabg2da9iaacIcacaaIWaGaaiilaiaaicdacaGGPaaaaa@53C4@
- 9
The
coordinates of B for event 2 are:
(
x
b2
,
t
b2
)=γ(
L
a
−v×
L
a
v
,
L
a
v
−
v×
L
a
c
2
)=γ(0,
L
a
v
(1−
v
2
c
2
))=(0,
L
a
γv
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6DCF@
- 10
Thus
A concludes that the time B calculates for the trip from event 1 to 2, is event
2 time
–
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=nbiaaa@37C1@
event 1 time
is:
t
b21
=
t
b2
−
t
b1
=(
L
a
γv
−0)=
L
a
γv
=
t
a21
γ
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDamaaBaaaleaacaWGIbGaaGOmaiaaigdaaeqaaOGaeyypa0JaamiDamaaBaaaleaacaWGIbGaaGOmaaqabaGccqGHsislcaWG0bWaaSbaaSqaaiaadkgacaaIXaaabeaakiabg2da9iaacIcadaWcaaqaaiaadYeadaWgaaWcbaGaamyyaaqabaaakeaacqaHZoWzcaWG2baaaiabgkHiTiaaicdacaGGPaGaeyypa0ZaaSaaaeaacaWGmbWaaSbaaSqaaiaadggaaeqaaaGcbaGaeq4SdCMaamODaaaacqGH9aqpdaWcaaqaaiaadshadaWgaaWcbaGaamyyaiaaikdacaaIXaaabeaaaOqaaiabeo7aNbaaaaa@558B@
- 11
That
is:
t
b21
=
t
a21
γ
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDamaaBaaaleaacaWGIbGaaGOmaiaaigdaaeqaaOGaeyypa0ZaaSaaaeaacaWG0bWaaSbaaSqaaiaadggacaaIYaGaaGymaaqabaaakeaacqaHZoWzaaaaaa@3FAB@
- 12
The Usually Ignored Calculation
Now…
the bit that is pretty much always missed missed…what does B actually calculate for the time A experiences between event
1 and event 2, not what is ad-hoc claimed?
To
do this, one needs to calculate A’s coordinates, given B’s coordinates, that
is:
x
a1
=γ(
x
b1
+v
t
b1
)
t
a1
=γ(
t
b1
+
v
x
b1
c
2
)
MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWG4bWaaSbaaSqaaiaadggacaaIXaaabeaakiabg2da9iabeo7aNjaacIcacaWG4bWaaSbaaSqaaiaadkgacaaIXaaabeaakiabgUcaRiaadAhacaWG0bWaaSbaaSqaaiaadkgacaaIXaaabeaakiaacMcaaeaacaWG0bWaaSbaaSqaaiaadggacaaIXaaabeaakiabg2da9iabeo7aNjaacIcacaWG0bWaaSbaaSqaaiaadkgacaaIXaaabeaakiabgUcaRmaalaaabaGaamODaiaadIhadaWgaaWcbaGaamOyaiaaigdaaeqaaaGcbaGaam4yamaaCaaaleqabaGaaGOmaaaaaaGccaGGPaaaaaa@547E@
- 13
Noting
the change in the direction of the motion as viewed by B.
The
coordinates of B, from the diagram are:
(
x
b1
,
t
b1
)=(0,0)
(
x
b2
,
t
b2
)=(0,
t
b2
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaGGOaGaamiEamaaBaaaleaacaWGIbGaaGymaaqabaGccaGGSaGaamiDamaaBaaaleaacaWGIbGaaGymaaqabaGccaGGPaGaeyypa0JaaiikaiaaicdacaGGSaGaaGimaiaacMcaaeaacaGGOaGaamiEamaaBaaaleaacaWGIbGaaGOmaaqabaGccaGGSaGaamiDamaaBaaaleaacaWGIbGaaGOmaaqabaGccaGGPaGaeyypa0JaaiikaiaaicdacaGGSaGaamiDamaaBaaaleaacaWGIbGaaGOmaaqabaGccaGGPaaaaaa@505A@
-
14
Thus
the coordinates of A for event 1 are:
(
x
a1
,
t
a1
)=γ(0+v×0,0+
v×
0
c
2
)=(0,0)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadIhadaWgaaWcbaGaamyyaiaaigdaaeqaaOGaaiilaiaadshadaWgaaWcbaGaamyyaiaaigdaaeqaaOGaaiykaiabg2da9iabeo7aNjaacIcacaaIWaGaey4kaSIaamODaiabgEna0kaaicdacaGGSaGaaGimaiabgUcaRmaalaaabaGaamODaiabgEna0kaaicdadaWgaaWcbaaabeaaaOqaaiaadogadaahaaWcbeqaaiaaikdaaaaaaOGaaiykaiabg2da9iaacIcacaaIWaGaaiilaiaaicdacaGGPaaaaa@53AC@
- 15
The
coordinates of A for event 2 are:
(
x
a2
,
t
a2
)=γ(0+v×
t
b2
,
t
b2
+0)=γ(
L
b
,
t
b2
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadIhadaWgaaWcbaGaamyyaiaaikdaaeqaaOGaaiilaiaadshadaWgaaWcbaGaamyyaiaaikdaaeqaaOGaaiykaiabg2da9iabeo7aNjaacIcacaaIWaGaey4kaSIaamODaiabgEna0kaadshadaWgaaWcbaGaamOyaiaaikdaaeqaaOGaaiilaiaadshadaWgaaWcbaGaamOyaiaaikdaaeqaaOGaey4kaSIaaGimaiaacMcacqGH9aqpcqaHZoWzcaGGOaGaamitamaaBaaaleaacaWGIbaabeaakiaacYcacaWG0bWaaSbaaSqaaiaadkgacaaIYaaabeaakiaacMcaaaa@579E@
- 16
Thus
B concludes that A’s distance to the star, and its time difference between
events is:
(
x
a21
,
t
a21
)=γ(
L
b
,
t
b2
)−(0,0)=γ(
L
b
,
t
b2
)
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadIhadaWgaaWcbaGaamyyaiaaikdacaaIXaaabeaakiaacYcacaWG0bWaaSbaaSqaaiaadggacaaIYaGaaGymaaqabaGccaGGPaGaeyypa0Jaeq4SdCMaaiikaiaadYeadaWgaaWcbaGaamOyaaqabaGccaGGSaGaamiDamaaBaaaleaacaWGIbGaaGOmaaqabaGccaGGPaGaeyOeI0IaaiikaiaaicdacaGGSaGaaGimaiaacMcacqGH9aqpcqaHZoWzcaGGOaGaamitamaaBaaaleaacaWGIbaabeaakiaacYcacaWG0bWaaSbaaSqaaiaadkgacaaIYaaabeaakiaacMcaaaa@5650@
- 17
That
is, B concludes that A’s length and time is:
x
a21
=γ
L
b
t
a21
=γ
t
b21
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWG4bWaaSbaaSqaaiaadggacaaIYaGaaGymaaqabaGccqGH9aqpcqaHZoWzcaWGmbWaaSbaaSqaaiaadkgaaeqaaaGcbaGaamiDamaaBaaaleaacaWGHbGaaGOmaiaaigdaaeqaaOGaeyypa0Jaeq4SdCMaamiDamaaBaaaleaacaWGIbGaaGOmaiaaigdaaeqaaaaaaa@47C3@
- 18
That
is, B concludes, by simple algebra, that:
L
b
=
x
a21
γ
=
L
a
γ
t
b21
=
t
a21
γ
MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGGipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGmbWaaSbaaSqaaiaadkgaaeqaaOGaeyypa0ZaaSaaaeaacaWG4bWaaSbaaSqaaiaadggacaaIYaGaaGymaaqabaaakeaacqaHZoWzaaGaeyypa0ZaaSaaaeaacaWGmbWaaSbaaSqaaiaadggaaeqaaaGcbaGaeq4SdCgaaaqaaiaadshadaWgaaWcbaGaamOyaiaaikdacaaIXaaabeaakiabg2da9maalaaabaGaamiDamaaBaaaleaacaWGHbGaaGOmaiaaigdaaeqaaaGcbaGaeq4SdCgaaaaaaa@4C97@
- 19
Thus
the time that B concludes A measures for B’s events, is exactly the same as
that A (eq. 12) concludes B measures for B’s events, both agree that B reads
less time, thus there is no
paradox.
That
is, both A & B conclude that making a one way trip to the star results in
less time for B. Returning just
results in doubling up the time, as can be easily calculated simply by resetting
t=0 at the star and performing, the same calculation with velocities swapped.
The
crucial point is that B views the distance from A to the star as shorter, thus
B views events that are synchronised by that length, take less time. Thus
despite a notional symmetrical γ in the transform equations when inverting
viewpoints, γ is not the sole determinator of the time between events.
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