Download A trip to the end of the universe and the twin “paradox”

A trip to the end of the universe and the twin “paradox”
Thomas Müller,a兲 Andreas King, and Daria Adis
Institut für Astronomie und Astrophysik, Abteilung Theoretische Astrophysik, Auf der Morgenstelle 10,
72076 Tübingen, Germany
共Received 12 December 2006; accepted 8 December 2007兲
Special relativity offers the possibility of going on a trip to the center of our galaxy or even to the
end of our universe within a lifetime. On the basis of the well known twin paradox, we discuss
uniformly accelerated motion and emphasize the local perspective of each twin concerning the
interchange of light signals between both twins as well as their different views of the stellar sky. For
this purpose we developed two Java applets that students can use to explore interactively and
understand the topics presented here. © 2008 American Association of Physics Teachers.
关DOI: 10.1119/1.2830528兴
I. INTRODUCTION
The purpose of this article is to emphasize the local perspective of each twin in the well known twin paradox who
can only do measurements with respect to his/her individual
reference frame. This local perspective resembles Bondi’s
k-calculus1 or the concept of radar time as discussed by
Dolby and Gull,2 where hypersurfaces of simultaneity are
defined by local measurements of reflected light signals. The
interesting information that both twins can interchange in our
example are their individual proper times. They will not be
able to determine the time dilation from this proper time
because the information carrying the proper time of the other
twin needs some time to travel the distance between them.
This time interchange, as the motion itself, is not symmetric.
Besides the time signals, both twins also receive light from
distant stars or other astronomical sources. Depending on the
relative velocity with respect to the light source, each twin
will have a different view of the stellar sky because of aberration and the Doppler shift.
It is well known that acceleration is not crucial to explain
the twin paradox.3,4 共For a detailed discussion of the twin
paradox we refer the reader to the standard literature.5,6,8兲
The crucial idea of the twin paradox is that there is an asymmetry between both twins. Only the difference in length of
their paths makes the twins age at different rates. The traveling twin Tina leaves Earth with a rocket starting with zero
velocity. Having reached her destination the twin might return to Earth. In order for the journey to be comfortable we
consider a uniformly accelerated motion which is separated
into four phases of equal duration. Even though acceleration
is usually considered in conjunction with general relativity, it
can also be treated within special relativity.7–9
We expect the reader to be familiar with basic calculations
such as the Lorentz transformations and the description of
acceleration within special relativity. In Sec. II we review the
description of uniformly accelerated motion in special relativity. Following the example of Ruder10 we present a special
round trip consisting of four equal acceleration phases in
Sec. III. From the local perspective taken in Sec. IV only
time or light signals can be measured by each twin. As an
example, we consider a flight to Vega and examine the time
signal exchange in Sec. V. Because we stress the local perspective, we derive the frequency shift and aberration relations for the visualization of the stellar sky by means of a
local reference frame 共see Sec. VI兲. In Sec. VII we describe
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Am. J. Phys. 76 共4&5兲, April/May 2008
the visual effects due to aberration and the frequency shift. In
the last section we show how far we can travel with the help
of time dilation and length contraction.
In the appendix we give a short introduction to the Java
applets: TwinApplet and RelSkyApplet.11 The
TwinApplet illustrates how far the traveling twin can go
by means of relativistic time dilation and what time signals
they can receive from each other. To use this applet we recommend reading Secs. II–V. The visualization of the stellar
sky discussed in Sec. VII is realized in RelSkyApplet.
II. UNIFORM ACCELERATION
The earth twin Eric stays in the inertial reference frame S,
while his traveling twin sister Tina leaves the Earth with
constant acceleration ␣ with respect to her own instantaneous system S⬘. All primed quantities refer to S⬘, and
unprimed quantities to S. From the Newtonian point of view,
Tina will reach a velocity v = ␣⌬t in the time ⌬t. She will
then have covered a distance ⌬x = 21 ␣⌬t2. Here, both twins
will agree on time, velocity, and distance. In the theory of
special relativity, both twins will still agree concerning time,
velocity, and distance as long as Tina’s velocity is much less
than the speed of light, v Ⰶ c. But in general, the earth twin
Eric would measure an acceleration5,8
a = ␣共1 − ␤2兲3/2 =
␣
,
␥共␤兲3
共1兲
with ␤ = v / c and ␥ = 1 / 冑1 − ␤2, which is different to the Newtonian case. Given the initial velocity v0, we obtain Tina’s
velocity v at time t by substituting a = dv / dt in Eq. 共1兲. The
result is
v=
␣t + c␨
共2兲
冑1 + 共␣t/c + ␨兲2 ,
with ␨ = ␥共␤0兲␤0. The distance traveled x = x共t兲 follows from
Eq. 共2兲 by another integration,
x=
c2
␣
冋冑 冉 冊
1+
␣t
+␨
c
2
册
− 冑1 + ␨2 + x0 ,
共3兲
where x0 is the initial position at time t = 0.
Time in both systems will go by at different rates depending on the relative velocity ␤,
dt⬘ = ␥共␤兲−1dt.
共4兲
Synchronizing at t = t⬘ = 0 gives
http://aapt.org/ajp
© 2008 American Association of Physics Teachers
360
t⬘ =
or
t=
册
冋 冉 冊
␣t
c
+ ␨ − arcsinh ␨ ,
arcsinh
␣
c
冊 册
冋 冉
␣t⬘
c
sinh
+ arcsinh ␨ − ␨ .
␣
c
共5兲
共6兲
For ␤0 = 0 Eq. 共6兲 simplifies to
t=
␣t⬘
c
.
sinh
␣
c
共7兲
If the traveling twin starts at x0 = 0, her current position x
= x共t⬘兲 is given by
x=
冉
冊
␣t⬘
c2
−1 .
cosh
␣
c
共8兲
Tina’s velocity with respect to her proper time t⬘ follows
from Eqs. 共7兲 and 共2兲,
␣t⬘
.
v = c tanh
c
共9兲
Hence, because 兩tanh共x兲兩 艋 1, the locally measured speed is
always less than the speed of light in accordance with special
relativity.
III. A SPECIFIC ROUND TRIP
The case we will investigate is discussed in several standard textbooks on special relativity, but not in as much
detail.1,5,10
While Eric stays at home, Tina goes on a journey that is
separated into four phases each of which last time T⬘. In the
first phase she starts at space-time point ¬ and accelerates
with ␣ until she reaches the maximum velocity
vmax = c tanh
␣T⬘
c
共10兲
at point −. Next, she decelerates with −␣ until she stops at
her destination ® where she has covered the distance
xmax = 2
冉
冊
␣T⬘
c2
−1 .
cosh
␣
c
共11兲
The same procedure, but now in the opposite direction,
brings Tina back home. Thus, the whole trip takes 4T⬘ with
respect to Tina’s proper time, whereas Eric has to wait the
time
4T = 4
␣T⬘
c
sinh
␣
c
共12兲
for his sister’s return. Hence, the bigger the acceleration and
the longer the journey lasts, the bigger is the effect of time
dilation and the different aging of both twins. Tina’s worldline is shown in a spacetime diagram in Fig. 1, where Eric’s
proper time ct is plotted versus the distance x between Tina
and Eric.
Fig. 1. Tina’s journey is separated into four phases. She starts from point ¬,
accelerates up to maximum velocity at point −, and slows down until she
reaches the turning point ®. Then she accelerates in the opposite direction
and slows down again until she comes home. Signals that were emitted by
Tina in the accelerating phase i reach the Earth twin Eric in the interval
关ti , ti+1兴.
tive. Hence, we can only use light signals exchanged by both
twins to determine the proper time in each case. To do so the
signals are coded with the proper emission time temit or temit
⬘
which will be received by the other twin at observation time
tobs
⬘ or tobs. This measured time is not the actual time of the
other twin. Because of the finite speed of light, the signals
need some time to travel the distance between both twins.
Eric’s perspective. Eric observes at his proper time tobs a
signal sent from Tina. Because of the finite speed of light, the
signal must have been emitted at Earth time temit,
tobs = temit +
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Am. J. Phys., Vol. 76, Nos. 4 & 5, April/May 2008
共13兲
The difference between tobs and temit is the time needed for
the signal to travel from Tina’s current position xemit
= x共temit兲 to the Earth. To determine the position xemit we have
to find out the interval of acceleration when Tina has emitted
the signal 共compare Fig. 1兲.
The border points ti, 共i = 2 , 3 , 4兲, of the intervals follow
from conditions such as t2 = t− + x− / c, whereas t5 follows
from Eq. 共12兲. Thus, we have
共14a兲
t1 = 0,
冉
冊
t2 =
␣T⬘
␣T⬘
c
+ cosh
−1 ,
sinh
␣
c
c
t3 =
␣T⬘
␣T⬘
c
+ 2 cosh
−2 ,
2 sinh
␣
c
c
t4 =
␣T⬘
␣T⬘
c
+ cosh
−1 ,
3 sinh
␣
c
c
IV. OBSERVATION OF TIME SIGNALS
In contrast to usual discussions of special relativity concerning length contraction or time dilation with a group of
synchronized observers, we concentrate on a local perspec-
x共temit兲
.
c
冊
冉
冉
共14b兲
冊
Müller, King, and Adis
共14c兲
共14d兲
361
t5 = 4
␣T⬘
c
.
sinh
␣
c
共14e兲
Depending on the interval 关ti , ti+1兴 Eric receives signals from
the accelerating phase i, where Tina’s current position was
xemit = x共temit兲. In the first time interval t1 − t2 we get from Eq.
共3兲 with x0 = 0,
tobs = temit +
=temit +
x¬–−共temit兲
c
c
␣
冉冑
共15a兲
冊
2
␣2temit
−1 .
c2
1+
共15b兲
We solve Eq. 共15b兲 for temit and substitute it into Eq. 共5兲 with
␤0 = 0 and find the observed signal time temit
⬘ ,
⬘ =
temit
冉
冊
␣temit c
␣tobs
c
= ln 1 +
arcsinh
.
␣
c
␣
c
x−–¯共t̃emit兲
,
tobs = T + t̃emit +
c
共17兲
where t̃emit = temit − T measures time from point −. Tina’s current position x−–¯共t̃emit兲 follows from Eq. 共3兲,
c2
␣
+
c2
␣
冉冑 冉
1+
冉冑
1+
− ␣t̃emit
+␨
c
冊
冊
2
− 冑1 + ␨2
␣ 2T 2
−1 ,
c2
冊
xobs
.
c
tobs = temit +
共23兲
In contrast to Eric’s perspective the calculations are a bit
more straightforward because we determine temit from tobs
⬘ .
While Tina is in the first accelerating phase ¬-− her proper
time tobs
⬘ transforms into tobs via Eq. 共7兲. Thus, her position is
given by
xobs =
冉
冊
␣t⬘
c2
cosh obs − 1 ,
␣
c
共24兲
and the signal from Eric which arrives at time tobs
⬘ has the
time signature
共16兲
In the accelerating time interval t2 − t4, Eric’s observation
time tobs is related to the emission time temit via
x−–¯共t̃emit兲 = −
time tobs
⬘ of the signal follows from the intersection of the
future light cone of Eric at time temit with Tina’s world line,
temit =
冉
冊
␣t⬘
␣t⬘
c
sinh obs − cosh obs + 1 .
␣
c
c
共25兲
In the accelerating phase −-¯ we obtain t̃obs = tobs − T from
Eq. 共6兲 with ␨ = sinh共␣T⬘ / c兲 for the observation time t̃obs
⬘
= tobs
⬘ − T⬘. Thus, we have
tobs = T −
冉
冊
⬘ + 2T⬘兲
␣共− tobs
␣T⬘
c
sinh
.
− sinh
␣
c
c
Tina’s position xobs when she receives the signal is given by
Eq. 共3兲 which simplifies to
共18兲
xobs = −
冉
冊
⬘ + 2T⬘兲
␣共− tobs
␣T⬘
c2
− 2 cosh
+1 .
cosh
␣
c
c
where ␨ = ␤0␥共␤0兲, and the velocity ␤0 is given by
␤0 =
␣T/c
共27兲
共19兲
冑1 + ␣2T2/c2 .
From Eq. 共17兲 we can determine t̃emit to be
1 2␰冑1 + ␨2 − ␰2␣/c
t̃emit =
,
2 冑1 + ␨2 + ␨ − ␰␣/c
with
␰ = tobs − T −
c
␣
冉冑
1+
冊
The same procedure also applies to the last accelerating
phase ¯-° for which we find
tobs = 3T +
共20兲
␣ 2T 2
−1 .
c2
共26兲
冋
⬘ − 4T⬘兲
␣共tobs
␣T⬘
c
+ sinh
sinh
␣
c
c
册
共28兲
for the observation time, and
共21兲
Equation 共5兲 gives the corresponding time t̃emit
⬘ . Hence, Eric
receives at his observation time tobs the emission time temit
⬘
= t̃emit
⬘ + T ⬘.
In the last time interval t4 − t5 Eric’s observation time tobs
follows from
tobs = t̄emit + 3T +
x¯–°共t̄emit兲
,
c
共22兲
where x¯–°共t̄emit兲 equals x−–¯共t̃emit兲 with the substitutions
␣ 哫 −␣, ␨ 哫 −␨, t̃emit 哫 t̄emit, and t̄emit = temit − 3T. To obtain
the emission time temit from tobs we follow the same procedure as in the previous accelerating time interval.
Tina’s perspective. Now we consider the opposite situation
where Eric sends his proper time temit to Tina. The arrival
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Am. J. Phys., Vol. 76, Nos. 4 & 5, April/May 2008
Fig. 2. Tina’s current velocity for her flight to Vega and return to Earth with
respect to her own proper time. The maximum speed ␤ ⬇ 0.9975 is reached
at points − and ¯.
Müller, King, and Adis
362
Fig. 3. The earth twin Eric sees/receives Tina’s proper time temit
⬘ 共ordinate兲 at
his proper time tobs 共abscissa兲.
冋
⬘ − 4T⬘兲
␣共tobs
c2
xobs =
cosh
−1
␣
c
册
共29兲
for Tina’s position. In both cases, the emission time temit
follows from Eq. 共23兲 with the corresponding time tobs and
position xobs.
V. FLIGHT TO VEGA
As an example we consider a flight to our neighboring star
Vega, which is roughly 25.3 ly away.12 To be a most comfortable journey Tina accelerates with ␣ = g ⬇ 9.81 m / s2.
While the journey takes Ttotal
⬘ = 4T⬘ ⬇ 12.93 years with respect to her proper time, Eric has to wait Ttotal
⬇ 54.48 years for his sister to return. The worldline of Tina
with respect to Eric’s rest frame is shown in the spacetime
diagrams, Figs. 4 and 6. Even though Tina follows an accelerated motion, her worldline is almost linear. Because of the
moderate acceleration Tina’s rocket needs roughly one year
Fig. 4. Spacetime diagram for Tina’s flight to Vega and return to Earth with
respect to Eric’s frame. At point ® Tina reaches Vega and immediately
returns. At − and ¯ she changes her acceleration direction. The small
circles represent the events when Tina sends a time signal to Eric. They are
separated by a single year with respect to her proper time t⬘. Note that the
time units are not equally spaced on Tina’s worldline.
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Am. J. Phys., Vol. 76, Nos. 4 & 5, April/May 2008
Fig. 5. The rocket twin Tina sees/receives Eric’s proper time temit 共ordinate兲
at her proper time tobs
⬘ 共abscissa兲.
to reach 80% of the speed of light. At position x− she reaches
maximum speed ␤max ⬇ 0.9975 共compare with Fig. 2兲.
Consider the exchange of time signals between both twins.
The proper time sent by the one twin and received by the
other is plotted in a time-time diagram 共compare Figs. 3 and
5兲. The proper time of the twin who receives a signal is
plotted on the abscissa and the proper time, which was sent
by the other twin, is shown on the ordinate. The shapes of the
curves are difficult to understand because two effects are
mixed, time dilation and the nonlinear change of distance. To
explain the differences between these curves, we also show
the space-time diagram with Tina’s worldline where the time
signals are represented by straight lines with 45° slope 共compare Figs. 4 and 6兲. Time intervals along Tina’s worldline are
not equally spaced because of her accelerated motion.
Eric’s perspective 共Figs. 3 and 4兲. While Eric stays at
home, Tina leaves Earth with proper acceleration ␣. Because
the distance between both twins grows, it is obvious that the
signals emitted by Tina need more and more time to reach
Eric. Also, because of time dilation, Tina seems to wait a
longer time than expected before emitting the next signal.
Even though Tina is already on the way back to home when
Fig. 6. The earth twin Eric sends a time signal every year with respect to his
proper time t. The dashes on Tina’s worldline mark the years of her proper
time. Eric’s first signal does not reach Tina until she already decelerates to
her destination ®.
Müller, King, and Adis
363
she has left her destination ®, Eric will collect most of her
time signals only during the very end of her journey.
Eric observes at his proper time tobs ⬇ 20 years a time signal which was sent by Tina at her proper time temit
⬘ = 3 y when
she was 10 ly away from home. But her current position at
Eric’s observation time is x ⬇ 19 ly and her watch shows
roughly 4 y.
Tina’s perspective 共Fig. 5 and 6兲. It might be obvious that
Tina’s perspective is different. Eric’s “first year” signal
reaches Tina only when she already decelerates to her destination ®. On her way home, Tina seems to receive the signals regularly, but because of time dilation she collects most
of the signals around point ¯ when she has maximum speed
and the time dilation effect is strongest.
To investigate this situation for different proper accelerations ␣ and different durations T⬘, we have written the interactive Java applet TwinApplet.11 A short description of
the applet is given in Appendix A; a more detailed discussion
as well as two examples can be found in the help menu of the
applet.
VI. THE ACCELERATED REFERENCE FRAME
A. Aberration, Doppler shift and length contraction
Fig. 7. Wave vector k of an incoming light ray in spherical coordinates ␽
苸 共0 , ␲兲 , ␸ 苸 共−␲ , ␲兲 with respect to Eric’s rest frame. The twin Tina is
currently moving with velocity ␤ along the ex direction.
eS1⬘共t⬘兲 = sinh
␣t⬘
␣t⬘
et + cosh
e x,
c
c
eS3⬘共t⬘兲 = ez .
共35b兲
With the help of the local tetrad it is straightforward to
derive the aberration and Doppler effect formula. Consider a
wave vector k of an incoming light ray 共compare Fig. 7兲.
This wave vector can either be described with respect to
Tina’s frame 兵eSi ⬘其共i=0,1,2,3兲 or to Eric’s rest frame 兵e j其共j=t,x,z,y兲.
Thus,
In four-dimensional spacetime all observers have their
own local reference system which is given by four base vectors. Each measurement is taken with respect to this local
tetrad.
Consider the flat Minkowski spacetime which is represented by the metric
k = ␻共et − sin ␽ cos ␸ex − sin ␽ sin ␸ey − cos ␽ez兲 共36a兲
共30兲
If we transform both representations into coordinates, we can
compare each component. Thus, the time component gives
the Doppler shift
2
2
2
2
2
2
ds = − c dt + dx + dy + dz .
The local tetrad 兵ei其共i=0,. . .,3兲 of an observer moving with velocity ␤ = v / c in the direction of the x axis reads
e0 = ␥共et + ␤ex兲,
e2 = e y ,
共31a兲
e1 = ␥共␤et + ex兲,
e3 = ez ,
共31b兲
where 共et , ex , ey , ez兲 are the four base vectors of an observer
at rest with respect to the Minkowski coordinate system 共30兲.
Each base vector points in its positive coordinate direction,
where e0 is adapted to the four-velocity u of the moving
observer,
1
e0 = u.
c
dt
= c␥共␤兲,
dt⬘
ux =
dx
= ␤␥共␤兲,
dt⬘
共33兲
and uy = uz = 0. Her four-acceleration a␮ is
at =
d 2t ␣
= ␤␥共␤兲,
dt⬘2 c
ax =
d 2x
= ␣␥共␤兲,
dt⬘2
共34兲
and ay = az = 0, where ␤ = ␤共t⬘兲 is the relative velocity at Tina’s proper time t⬘. Thus, her local tetrad at time t⬘ is given
by
␣t⬘
␣t⬘
et + sinh
e x,
eS0⬘共t⬘兲 = cosh
c
c
364
− cos ␽⬘eS3⬘兲.
eS2⬘共t⬘兲 = ey ,
Am. J. Phys., Vol. 76, Nos. 4 & 5, April/May 2008
共35a兲
共36b兲
␻ = ␻⬘␥共1 − ␤ sin ␽⬘ cos ␸⬘兲,
共37兲
and from Eq. 共37兲 we obtain the redshift factor z,
z=
␻
− 1 = ␥共1 − ␤ sin ␽⬘ cos ␸⬘兲 − 1.
␻⬘
共38兲
The spatial components give the aberration formulas
cos ␽ =
cos ␽⬘
,
␥共1 − ␤ sin ␽⬘ cos ␸⬘兲
共39a兲
cos ␸ =
sin ␽⬘ cos ␸⬘ − ␤
,
sin ␽共1 − ␤ sin ␽⬘ cos ␸⬘兲
共39b兲
sin ␸ =
sin ␽⬘ sin ␸⬘
.
␥ sin ␽共1 − ␤ sin ␽⬘ cos ␸⬘兲
共39c兲
共32兲
The current four-velocity of the accelerated twin Tina at time
t⬘ is given by u = u je j , 共j = t , x , y , z兲 with
ut = c
= ␻⬘共eS0⬘ − sin ␽⬘ cos ␸⬘eS1⬘ − sin ␽⬘ sin ␸⬘eS2⬘
The inverse formulas for the Doppler shift and the aberration
follow from Eqs. 共37兲 and 共39兲 by letting 共␤ , ␻ , ␽ , ␸兲
哫 共−␤ , ␻⬘ , ␽⬘ , ␸⬘兲. Because aberration and the Doppler shift
depend only on the relative direction, we can also use the
angle ␹ between the wave vector −k and the direction of
motion ex with
cos ␹ = sin ␽ cos ␸ .
共40兲
From Eq. 共38兲 it follows that Tina will see objects to be
blueshifted for z ⬍ 0 and redshifted for z ⬎ 0.
Müller, King, and Adis
364
Fig. 9. An object p is located at position 共r p , ␹ p兲 with respect to the initial
position x共␥ = 1兲 of Tina. After some time, Tina’s current position is x共␥兲 and
the point p would have the relative position 共r , ␹兲 with respect to an observer at rest at Tina’s current position.
Fig. 8. An object a distance d from the origin x = 0 has an apex angle ␰0. An
observer at rest at the current position x共t⬘兲 would measure an apex angle
␰共t⬘兲.
A similar consideration leads to length contraction. A fixed
distance ᐉ⬘ to some point in the direction 共␽⬘ , ␸⬘兲 with respect to Tina’s current frame would have a length
ᐉ = ᐉ⬘冑sin2 ␽⬘共␥2 cos2 ␸⬘ + sin2 ␸⬘兲 + cos2 ␽⬘
共41兲
as determined by Eric.
B. Apparent size of an object
For accelerated motion objects in the direction of motion
always seem to recede in the first moments even though one
is approaching. The reason is also based on the aberration
effect. Consider an object a distance d apart with apex angle
␰0 as seen from the origin x = 0, Fig. 8.
An observer at rest at the current position x given by Eq.
共8兲 of the accelerating twin Tina would measure a current
apex angle ␰共t⬘兲 with
tan ␰共t⬘兲 =
d tan ␰0
.
d − x共t⬘兲
共42兲
The acceleration ␣ in Eq. 共8兲 determines whether Tina approaches 共␣ ⬎ 0兲 the object or recedes 共␣ ⬍ 0兲. The apex
angle ␰⬘共t⬘兲 of the object with respect to Tina follows from
the aberration formula 共39b兲,
cos ␰⬘共t⬘兲 =
1 + ␤共t⬘兲冑1 + tan2 ␰共t⬘兲
冑1 + tan2 ␰共t⬘兲 + ␤共t⬘兲
共43兲
,
where ␤共t⬘兲 is given by Eq. 共9兲. Hence, from the derivative
of Eq. 共43兲 with respect to Tina’s proper time t⬘ it follows
that
冏 冏
d␰⬘
␣
= − sin ␰0 .
dt⬘ t⬘=0
c
共44兲
Thus, accelerating from zero velocity in the direction to the
object 共␣ ⬎ 0兲 has the effect that the object initially appears
to shrink, giving the impression of receding from it instead
of approaching. This effect holds until d␰⬘ / dt⬘ = 0,
tn⬘ =
c
ln
␣
␺1 + sign共␣兲␦冑共1 + tan2 ␰0兲␺2
2共1 + ␦兲
,
共45兲
where ␦ = ␣d / c2, ␺1 = 共1 + ␦兲2 + 1 + ␦2 tan2 ␰0, and ␺2 = 共2
+ ␦兲2 + ␦2 tan2 ␰0.
An acceleration in the opposite direction 共␣ ⬍ 0兲 results in
a magnification which has a maximum at time tn⬘. Even
though a velocity close to the speed of light has a tremen365
Am. J. Phys., Vol. 76, Nos. 4 & 5, April/May 2008
dous magnification effect, the distance to the object grows
exponentially and dominates the magnification effect.
Example. If Tina leaves home with acceleration ␣ = −g
away from the Sun,13 the Sun has an apex angle ␰0
⬇ 0.267° at a distance d ⬇ 1.5⫻ 1011 m. After tn⬘ ⬇ 500 s Tina
reaches ␤ ⬇ 1.6⫻ 10−5, and the maximum magnification is
␰⬘共tn⬘兲
1 ␣d
⬇1−
⬇ 1 + 8.2 ⫻ 10−6 .
2 c2
␰0
共46兲
The angular size as seen by an observer at rest at her current
distance d共t⬘兲 ⬇ d + 1.2⫻ 106 m would be ␰ / ␰0 ⬇ 1 − 8.2
⫻ 10−6.
C. Apparent position of an object
What’s the apparent position of an object p with respect to
the accelerating observer Tina? If we parameterize Tina’s
current position, Eq. 共8兲, by her velocity, Eq. 共9兲, we obtain
x=
c2
c2
兵cosh关arctanh共␤兲兴 − 1其 = 共␥ − 1兲.
␣
␣
共47兲
The position x p = r p cos ␹ p, y p = r p sin ␹ p of the object p,
where r p ⬎ 0 is the distance to the initial position of Tina and
0 艋 ␹ p 艋 ␲ is the initial angle with respect to Tina’s direction
of motion 共compare Fig. 9兲, transforms into Tina’s frame
according to the aberration formula,
cos ␹⬘ =
共x p − x兲/r + ␤
,
1 + ␤共x p − x兲/r
共48兲
where r2 = 共x p − x兲2 + y 2p.
Equation 共48兲 can be written for ␤ 艌 0 as
cos ␹⬘ =
关cos ␹ p − A共␥ − 1兲兴/r̃ + 冑1 − 1/␥2
1 + 冑1 − 1/␥2关cos ␹ p − A共␥ − 1兲兴/r̃
,
共49兲
with A = c2 / 共␣r p兲 and
r̃ = 冑A2共␥ − 1兲2 − 2A共␥ − 1兲cos ␹ p + 1.
共50兲
The observation angle ␹⬘ = ␹⬘共␤兲 is shown in Figs. 10 and 11
for two values of A.
From the first derivative d␹⬘ / d␥ = 0 of Eq. 共49兲 we find an
extremum at
␥e =
1
+ 1,
2A共A + cos ␹ p兲
共51兲
which is valid only if 共A + cos ␹ p兲 ⬎ 0. The associated velocity ␤e is
Müller, King, and Adis
365
Fig. 10. The observation angle ␹⬘ is plotted versus the velocity ␤ for A
= c2 / 共␣r p兲 ⬇ 9.68. In the first instance, an object with fixed distance r p depending on A and arbitrary angle ␹ p apparently approaches the center of
motion. For higher velocities, it recedes again. Because ␤ = 1 is reached only
approximately, an object at 共r p , ␹ p兲 seems to “freeze” at ␹lim
⬘ = ␹⬘共␤ → 1兲.
␤e =
冑1 + 4A cos ␹p + 4A2
1 + 2A cos ␹ p + 2A2
.
共52兲
For 共A + cos ␹ p兲 ⬎ 0 the second derivative follows after a
lengthy calculation to be
冏 冏
d 2␹ ⬘
d␥2
=
␥e
sin ␹ p
A共␥2e − 1兲2
艌 0.
共53兲
␥→⬁
sin2 ␹ p − A2
.
sin2 ␹ p + A2
共54兲
Note that two objects p1 and p2 with r p1 = r p2 and ␹ p2 = ␲
− ␹ p1 will “freeze” at the same observation angle ␹lim
⬘ .
As we have seen, the apparent position of an object depends considerably on its position 共r p , ␹ p兲 and on the accel-
Fig. 11. The observation angle ␹⬘ is plotted versus the velocity ␤ for A
= c2 / 共␣r p兲 ⬇ 0.0968. Note that even objects that are actually behind the observer 共␹ p ⬎ ␲ / 2兲 might apparently “freeze” in front of the observer.
366
eration ␣ of the observer. This dependence might be counterintuitive in some cases. From a nonrelativistic point of
view, any object is located nearly behind the observer if the
latter is infinitely apart from it. But in the relativistic case,
the object could also appear at a different position. In brief, it
is a contest between the relativistic aberration and the exponential increase of distance between star and observer.
VII. VISUALIZATION OF THE STELLAR SKY
A. Aberration and Doppler shift
Thus, if there is an extremum, it is always a minimum.
In the limit ␤ → 1, ␥ goes to infinity, and an object with
coordinates 共r p , ␹ p兲 seems to “freeze” at an observation angle
␹lim
⬘ with respect to Tina’s frame,14
⬘ 兲 = lim cos ␹⬘ =
cos共␹lim
Fig. 12. Stellar sky at ␤ = 0.5 in the 4␲-representation where ␸ = 共−␲ , ␲兲 is
the abscissa and ␽ = 共0 , ␲ / 2兲 is the ordinate. The center of the image 共␽
= ␲ / 2 , ␸ = 0兲 corresponds to the direction of motion. The circles of latitude
and the meridians are separated by 5°.
Am. J. Phys., Vol. 76, Nos. 4 & 5, April/May 2008
What could Tina really see if she looked through a window of her rocket? If there were a sphere fixed at infinity
with circles of latitude and meridians each separated by 5°,
Tina would see a warped lattice as in Figs. 12 and 13. In
contrast to the visualization of Scott and van Driel15 we use
the 4␲ representation where the azimuth angle ␸ and the
zenith distance ␽ are plotted like Cartesian coordinates to
show the full sky. The disadvantage of this representation is
the distortion at the nodes ␽ = 0 and ␽ = ␲. Here, the direction
of motion corresponds to the center of the representation
共␽ = ␲ / 2 , ␸ = 0兲.
Because of aberration, Eq. 共39兲 shows that the nodes of
the stellar sphere move together according to Tina’s velocity.
Their angular separation ⌬␽⬘ follows from Eq. 共39a兲,
Fig. 13. Stellar sky at ␤ = 0.9 in the 4␲ representation. Because of aberration, the nodes of the stellar sphere move together.
Müller, King, and Adis
366
Fig. 14. Lines of constant redshift z at velocity ␤ = 0.5 in the 4␲ representation. From inside to outside: z = −0.4 to z = 0.6, step 0.2; the bold line
marks z = 0.
⌬␽⬘ = ␲ − 2 arccos 冑1 − ␤2 .
Fig. 16. The observation angle ␹⬘ is plotted versus the velocity ␤. The solid
lines are lines of constant Doppler shift z according to Eq. 共59兲; the dashed
lines represent the aberration of the angle ␹ 共see Eq. 共60兲兲.
共55兲
Because angular distance becomes smaller in the direction of
motion, an object seems to be farther away in comparison to
its real distance. In contrast, objects in the opposite direction
seem to grow.
Besides the mere geometrical aspects, Figs. 14 and 15
show lines of constant Doppler shift at different velocities.
As long as the velocity ␤ = 0, there is no Doppler shift, but
for ␤ ⬎ 0 light is Doppler shifted following Eq. 共38兲,
z = ␥共1 − ␤ cos ␹⬘兲 − 1,
共56兲
where ␹⬘ is the angle between the direction of motion and
the incoming light ray.
Zero Doppler shift occurs for ␤ ⬎ 0 at an angle ␹0⬘ with
cos ␹0⬘ =
1 − 冑1 − ␤2
.
␤
共57兲
The difference ⌬z between the maximum blueshift and the
maximum redshift equals
⌬z = z共␹⬘ = ␲兲 − z共␹⬘ = 0兲 = 2␤␥ .
共58兲
Thus, the faster Tina moves, the more of the sky is redshifted. Only a small portion of the sky in the direction of
motion is blueshifted.
Another interesting detail is shown in Fig. 16, where the
observed angle ␹⬘ is plotted versus the velocity ␤ according
to the Doppler shift,
冉
1 − 共z + 1兲冑1 − ␤2
,
␤
冉
cos ␹ + ␤
.
1 + ␤ cos ␹
␹⬘ = arccos
and aberration
␹⬘ = arccos
冊
共59兲
冊
共60兲
For ␤ = 0 there is no Doppler shift and no aberration.
Objects in front of the observer, ␹ ⬍ ␲ / 2, will always be
blueshifted and seem to be in front of the observer, ␹⬘
⬍ ␲ / 2. But, for objects in back of the observer, ␹ ⬎ ␲ / 2,
whether they apparently are in front or not depends on the
velocity. An object at an angle ␹ ⬎ ␲ / 2 will turn from being
redshifted to being blueshifted when ␤ is greater than
␤red-blue = −
2 cos ␹
.
1 + cos2 ␹
共61兲
B. Temperature and brightness
For visualizing the stellar sky we use the Hipparcos star
catalogue.16 We extract the Johnson B-V color and assign a
temperature T = TB-V to each star by the empirical law17
再
B−V=
C1 log10共T兲 + C2
共log10T 艋 3.961兲
冎
C3 log10共T兲2 + C4 log10共T兲 + C5 共log10T ⬎ 3.961兲
共62兲
with constants from Table I. Equation 共62兲 is only a limited
approximation to the real temperatures of the stars, but it
simplifies the following calculations.
We also need the bolometric correction 共BC兲18 to transform from the visual M V to the bolometric magnitude M bol,
共63兲
M bol = M V + BC,
Fig. 15. Lines of constant redshift z at velocity ␤ = 0.9 in the 4␲ representation. From inside to outside: z = −0.4 to z = 2.0, step 0.2; the bold line
marks z = 0. Note that most of the sky is redshifted even for directions ␹⬘
⬍ ␲ / 2.
367
Am. J. Phys., Vol. 76, Nos. 4 & 5, April/May 2008
with
BC = C6t̃ 4 + C7t̃ 3 + C8t̃ 2 + C9t̃ + C10 ,
共64兲
where t̃ = log10共T兲 − 4.
Müller, King, and Adis
367
Table I. The coefficients Ci for Eqs. 共62兲 and 共64兲 are taken from Ref. 17.
Coefficient
C1
C2
C3
C4
C5
Value
Coefficient
Value
−3.684
14.551
0.344
−3.402
8.037
C6
C7
C8
C9
C10
−8.499
13.421
−8.131
−3.901
−0.438
Instead of the actual spectrum of each star we use a Planck
spectrum at temperature T = TB−V with spectral intensity
I␯ =
2h␯3
1
,
c2 eh␯/共kBT兲 − 1
共65兲
where h is Planck’s constant and kB is Boltzmann’s
constant.20 If the spectral intensity is expressed in terms of
wavelength, we obtain from I␯d␯ = −I␭d␭ and c = ␭␯ the expression
I␭ =
2hc2
1
.
5
hc/共␭kBT兲
␭ e
−1
共66兲
The typical shape of a Planck spectrum is shown in Fig. 17,
where the wavelength ␭max of the maximum of the intensity
follows from Wien’s displacement law
␭max T = b,
共67兲
−3
with Wien’s displacement constant b = 2.8978⫻ 10 km.
We use a Planck spectrum here, because it simply transforms with the Doppler factor according to the relativistic
Liouville theorem.21 From I␯ / ␯3 = constant together with Eqs.
共65兲 and 共38兲 the temperature transforms as
Tstar ␯
= = z + 1.
␯⬘
⬘
Tstar
共68兲
Thus, a blueshifted star with −1 ⬍ z ⬍ 0 seems to be hotter
than it really is compared to its own rest frame.
The luminosity L of an isotropically radiating black body
of radius R is given by L = 4␲␴R2T4 with the Stefan–
Boltzmann constant ␴ ⬇ 5.67⫻ 10−8 Wm−2 K−4.19 Thus, the
absolute bolometric magnitude M bol of an object with a
Planck spectrum at temperature T is
M bol − M bol,䉺 = − 2.5 log10
= − 5 log10
L
L䉺
R
T
− 10 log10
,
R䉺
T䉺
共69兲
where we approximated the spectrum of the sun by a Planck
spectrum at temperature T䉺.22 Because the absolute bolometric magnitude M bol is defined as the brightness of a star at
a distance of 10 pc, the apparent bolometric magnitude mbol
is given by
mbol − M bol = 5 log10
r
,
10pc
共70兲
where r is the distance between the star and the observer.19
From Tina’s point of view, we have to transform the
Planck spectrum of a star into her current rest frame according to Eq. 共68兲 resulting in a different absolute bolometric
magnitude M bol
⬘ ,
⬘ − M bol,䉺 = − 5 log10
M bol
R⬘
T⬘
− 10 log10
.
R䉺
T䉺
共71兲
If we subtract Eq. 共69兲 from Eq. 共71兲, the transformation
between the absolute bolometric magnitudes reads
⬘ − M bol = − 5 log10
M bol
R⬘
− 10 log10共z + 1兲,
R
共72兲
where we have used Eq. 共68兲 for the transformation of the
temperatures. Because we are interested in the apparent magnitudes, we substitute the absolute magnitudes by means of
Eq. 共70兲,
Fig. 17. Planck spectrum at temperature T = 6000 K with a maximum at ␭max ⬇ 0.483 ␮m.
368
Am. J. Phys., Vol. 76, Nos. 4 & 5, April/May 2008
Müller, King, and Adis
368
Fig. 18. The stellar sky marked by some constellations as seen at rest. In the
4␲ representation the right ascension ␣ is plotted on the abscissa and the
declination ␦ is plotted on the ordinate. Abbreviations: 共Aql兲 Aquila, 共Cas兲
Cassiopeia, 共Crt兲 Crater, 共Cru兲 Crux, 共Cyg兲 Cygnus, 共Her兲 Hercules, 共Leo兲
Leo, 共Ori兲 Orion, 共Peg兲 Pegasus, 共UMi兲 Ursa Minor.
⬘ − mbol = − 5 log10
mbol
rR⬘
+ 10 log10共z + 1兲.
r ⬘R
共73兲
As long as we approximate the stars as nearly point-like
sources which are very far from Tina’s current position, we
can write ⌬␹ = 2R / r, where ⌬␹ is the angular diameter of the
star. With the apparent angular diameter ⌬␹⬘ as seen from
Tina’s reference frame, we obtain
⬘ − mbol = − 5 log10
mbol
⌬␹⬘
+ 10 log10共z + 1兲.
⌬␹
共74兲
Here, the quotient ⌬␹⬘ / ⌬␹ can be replaced by the derivative
of the aberration formula 共60兲,
1
d␹⬘
.
=
d␹ ␥共1 + ␤ cos ␹兲
共75兲
As expected, the transformation of the apparent magnitude,
Eq. 共74兲, is composed of an aberration and a Doppler shift
component.
If we cannot approximate the stars to be nearly point-like,
we have to take their expansion into account. For more information on what an expanded star would look like, we
refer the reader to Ref. 23. Because we cannot extract the
size of a star from the Hipparcos catalogue, we neglect its
expansion.
C. Constellations
We have discussed how aberration and Doppler shift determine the view of the stellar sky as seen by a relativistic
observer. Figures 18–20 show the stellar sky of an observer
from the position of the Earth but at different velocities in
the direction right ascension ␣ = 0 and declination ␦ = 0. Note
that the declination ␦ corresponds to ␲ / 2 − ␽ in comparison
to Fig. 7. The stars are connected by lines showing the
constellations.24
As explained in Sec. VI B, the aberration effect lets the
constellations apparently shrink in the direction of motion,
while the ones that are behind the observer, such as Leo,
Virgo 共Vir兲 and Crater 共Crt兲, seem to grow.
In Table II we list the stars of the constellations Orion
共Ori兲, Cassiopeia 共Cas兲, and Southern Cross 共Cru兲 with their
rest frame data. For these constellations we give in Tables III
and IV the apparent distance d⬘, the temperature T⬘, and the
369
Am. J. Phys., Vol. 76, Nos. 4 & 5, April/May 2008
Fig. 19. The stellar sky as seen by an observer passing the Earth with 50%
of the speed of light. The distortion of the constellation Southern Cross
共Cru兲 is due to the 4␲ projection and the aberration effect 共see Fig. 12兲.
bolometric magnitudes mbol for ␤ = 0.5 and ␤ = 0.9. The apparent distance d⬘ follows from the inverse form of Eq. 共41兲,
冋
d⬘ = d ␥2 − ␤2
1 − cos2 ␦ cos2 ␣
共1 + ␤ cos ␦ cos ␣兲2
册
−1/2
,
共76兲
where the distance d, measured in parsec, is related to the
trigonometric parallax ␲ via d = 1 / ␲. The redshift factor z is
given by the inverse of Eq. 共37兲,
z=
1
␻
− 1,
−1=
␻⬘
␥共1 + ␤ cos ␦ cos ␣兲
共77兲
and determines the temperature T⬘ = T / 共z + 1兲.
Because Cassiopeia and Orion are in front of the observer,
␹Cas ⬇ 1.07 and ␹Ori ⬇ 1.47, they will always be blueshifted
共see Fig. 16兲. But the Southern Cross 共Cru兲, ␹Cru ⬇ 2.14, will
be redshifted until the observer reaches the velocity
␤red-blue ⬇ 0.83, which follows from Eq. 共61兲. At ␤ = ␤red-blue
Cru has apparently already changed from the back side to the
front side of the observer. This change follows from Eq. 共60兲
and occurs at
␤␹⬘=␲/2 = − cos ␹Cru ⬇ 0.53.
共78兲
We have written the interactive Java applet
RelSkyApplet11 for visualizing the stellar sky at different
velocities as seen from the position of the Earth. The applet
clearly demonstrates the strong geometrical distortion as well
as the direction dependent Doppler shift for velocities closed
to the speed of light. A short introduction is given in
Fig. 20. The stellar sky as seen by an observer passing the Earth with 90%
of the speed of light.
Müller, King, and Adis
369
Table II. Star data of some constellations from Figs. 18–20. ␣: right ascension, ␦: declination, ␲: trigonometric
parallax 共milliarcsec兲, B-V: Johnson B-V color, T: temperature 共Kelvin兲 from Eq. 共62兲, HIP: Hipparcos number.
␣
Abbr.
␦
␲
B-V
T
HIP
␣ Ori
␤ Ori
␥ Ori
␦ Ori
⑀ Ori
␨ Ori
␬ Ori
1.5497
1.3724
1.4187
1.4487
1.4670
1.4868
1.5174
0.1293
−0.1431
0.1108
0.0052
−0.0210
−0.0339
−0.1688
7.63
4.22
13.42
3.56
2.43
3.99
4.52
1.50
−0.03
−0.22
−0.17
−0.18
−0.20
−0.17
3488
9077
19245
15279
15903
17038
14819
27989
24436
25336
25930
26311
26727
27366
␣ Cas
␤ Cas
␥ Cas
␦ Cas
⑀ Cas
0.1767
0.0400
0.2474
0.3744
0.4991
0.9868
1.0324
1.0597
1.0513
1.1113
14.27
59.89
5.32
32.81
7.38
1.17
0.38
−0.05
0.16
−0.15
4287
7025
9295
8060
13732
3179
746
4427
6686
8886
␣ Cru
␤ Cru
␥ Cru
␦ Cru
−3.0255
−2.9334
−3.0056
−3.0755
−1.1013
−1.0418
−0.9968
−1.0254
10.17
9.25
37.09
8.96
−0.24
−0.24
1.60
−0.19
21259
20696
3277
16569
60718
62434
61084
59747
Appendix A while a more detailed discussion as well as several examples can be found in the help menu of the applet.
VIII. A TRIP TO THE END OF THE UNIVERSE
As a first trip Tina goes on an expedition to the center of
our galaxy 共SgrA*兲 8 kpc away. From Table V we see that
her maximum speed is only 2.8 ppb 共parts per billion兲 below
the speed of light.
With this tremendous velocity even the extremely cold
microwave background radiation25 at Tcmb = 2.725 K comes
into the visual regime. But in contrast to one’s expectation,
the Doppler shifted background radiation will not fill the
whole sky. By Wien’s displacement law 共67兲 an object must
Table III. The stars of Table II have distance d⬘ 共parsec兲 and temperature T⬘
共Kelvin兲 at velocities ␤ = 0.5 and ␤ = 0.9 in the direction ␣ = ␦ = 0.
have a temperature of about T780 ⬇ 3700 K to emit its maximum radiation in the red light regime with ␭ = 780 nm. To
shift the background temperature to T780 the observer has to
move with a velocity very close to the speed of light, ␤780
⬇ 1 – 1.07⫻ 10−6. From Eq. 共57兲 it follows that the background radiation is blueshifted only in the small region of
␹0⬘ ⬍ 3°. The rest of the sky is redshifted with maximum z
⬇ 1767 at ␹⬘ = ␲.
In contrast, the redshift brings the x-ray and ␥-ray sky
down to the visual regime. For x rays with wavelengths of
␭ ⬇ 10−10 m Tina has to fly with ␤ 1 – 6.6⫻ 10−7. The much
higher velocity ␤ 1 – 6.6⫻ 10−12 is needed for ␥ rays with
␭ ⬇ 10−12 m.26 Just before the ␥-ray sky, the hydrogen 21 cm
Table IV. The apparent visual magnitude mV of the stars of Table II have
␤
at velocities ␤ = 0, ␤ = 0.5, and ␤ = 0.9 in the
bolometric magnitudes mbol
direction ␣ = ␦ = 0.
Abbr.
d␤=0
d␤⬘ =0.5
T␤⬘ =0.5
d␤⬘ =0.9
T␤⬘ =0.9
Abbr.
mV
BC
␤=0
mbol
␤=0.5
mbol
␤=0.9
mbol
␣ Ori
␤ Ori
␥ Ori
␦ Ori
⑀ Ori
␨ Ori
␬ Ori
131.06
236.97
74.52
280.90
411.52
250.63
221.24
125.61
222.56
70.35
266.08
390.64
238.46
211.27
4070
11503
23895
18717
19315
20499
17562
61.90
109.31
34.56
130.76
192.02
117.27
103.98
8153
24479
50135
38894
39886
42038
35608
␣ Ori
␤ Ori
␥ Ori
␦ Ori
⑀ Ori
␨ Ori
␬ Ori
0.45
0.18
1.64
2.25
1.69
1.74
2.07
−2.01
−0.29
−1.95
−1.36
−1.46
−1.63
−1.28
−1.56
−0.11
−0.31
0.89
0.23
0.11
0.79
−2.32
−1.27
−1.38
−0.11
−0.73
−0.81
−0.05
−6.88
−6.10
−6.14
−4.83
−5.42
−5.47
−4.66
␣ Cas
␤ Cas
␥ Cas
␦ Cas
⑀ Cas
70.08
16.70
187.97
30.48
135.50
63.34
15.13
171.11
27.78
14.49
6294
10190
13277
11457
18943
31.32
7.48
84.45
13.71
61.30
14641
23547
30424
26181
42544
␣ Cas
␤ Cas
␥ Cas
␦ Cas
⑀ Cas
2.24
2.28
2.15
2.66
3.35
−0.93
−0.08
−0.32
−0.16
−1.10
1.31
2.20
1.83
2.50
2.25
−0.57
0.37
0.07
0.78
0.67
−5.77
−4.80
−5.06
−4.35
−4.38
␣ Cru
␤ Cru
␥ Cru
␦ Cru
98.33
108.11
26.96
111.61
98.26
108.11
26.95
111.60
19032
17998
2766
14181
53.01
59.70
15.31
62.54
29046
26380
3878
20304
␣ Cru
␤ Cru
␥ Cru
␦ Cru
0.77
1.25
1.59
2.79
−2.21
−2.14
−2.45
−1.56
−1.44
−0.89
−0.86
1.23
−0.97
−0.29
−0.13
1.90
−4.14
−3.24
−2.82
−0.91
370
Am. J. Phys., Vol. 76, Nos. 4 & 5, April/May 2008
Müller, King, and Adis
370
Table V. Distance from Earth, maximum speed and proper time of both twins for several stellar destinations. In the solar system we will reach only a few
percent of the speed of light. Thus, time dilation can be neglected. However, in the neighborhood of the solar system time dilation is crucial. The “END” of
the universe represents the maximum distance of about 13.7 billion light years that astronomers are able to observe.
Object
Mars
Saturn
Pluto
␣ Cen C 共HIP 70890兲
Vega 共HIP 91262兲
SgrA*
LMC
M81
END
Distance
␤max = vmax / c
Tina’s time 2T⬘
Eric’s time 2T
0.524 AU⬇ 4.4 lm
8.53 AU⬇ 1.2 lh
38.81 AU⬇ 5.4 lh
1.29 pc
7.76 pc
8 kpc
50 kpc
2 Mpc
13.7⫻ 109 ly
0.003
0.012
0.025
0.949
0.9975
1 − 2.8⫻ 10−9
1 − 7.1⫻ 10−11
1 − 4.4⫻ 10−14
1 − 1.0⫻ 10−20
49h45min13.4sec
8d8h44min21sec
17d20h10min
3.54a
6.46a
19.74a
23.30a
30.45a
45.27a
49h45min13.7sec
8d8h44min38sec
17d20h13min
5.84a
27.17a
26ka
163ka
6.5Ma
13.7Ga
line of interstellar gas, resulting from a transition between
two hyperfine structure energy levels in the hydrogen atom,
will become visible at a velocity of ␤ ⬇ 1 – 2.7⫻ 10−11.
A wavelength ␭ will be seen at wavelength ␭⬘ for a velocity
␤=
冏
冏
␭ ⬘2 − ␭ 2
.
␭ ⬘2 + ␭ 2
共79兲
ACKNOWLEDGMENTS
Thus, the minimum and maximum wavelengths at velocity ␤
that are transformed into the visual regime follow from
冑
␭min = 380 nm
1−␤
1+␤
冑
and ␭max = 780 nm
1+␤
.
1−␤
What happens with the Milky Way in the meantime?
While Tina’s journey to the center of our galaxy lasts only
20 years with respect to her proper time, the Milky Way ages
about 26,000 years. In that time, our sun with rotational
velocity27 v ⬇ 220 km/ s ⬇ 7.34⫻ 10−4 ly/ y 共light years per
year兲 will cover a distance of about 19 ly. To fully describe
the galactic evolution, we need the position as well as the
proper motion of each star in the galaxy. The Hipparcos catalogue, which consists of about 118,000 stars, with most of
them at a distance of about 100 pc, delivers the best positional star reference 共1 milli-arc-sec兲. From the GAIA29 mission we expect an accuracy in positional astrometry of about
20 ␮as. Thus, all stars in our galaxy up to the 20th magnitude should be included.
As in the title of this article, Tina could also go on a trip to
the “end of the universe” within her lifetime. Here, the “end”
means the maximum distance of about 13.7 billion light
years that astronomers are able to observe. For these speculations we neglect the expansion of the universe, but do the
calculations in the flat Minkowskian spacetime instead of a
Robertson–Walker5 spacetime.
When Tina reaches her maximum velocity ␤ = 1 − ⑀ with
⑀ = 1.0⫻ 10−20 in the middle of her journey, the relativistic
effects are extremely dramatic. A ␥ factor of about 7.07
⫻ 109 results in a different time rate: ⌬t⬘ = 1 s corresponds to
⌬t = 224a! Thus, in roughly 12 days with respect to Tina’s
proper time our Sun would have finished one revolution
around the center of our galaxy. Zero Doppler shift occurs at
an angle ␹0⬘ ⬇ 冑4 8⑀ ⬇ 3.5 as. The maximum blueshift in the
direction of motion is z共␹⬘ = 0兲 ⬇ 冑⑀ / 2 ⬇ −1 + 7.07⫻ 10−11,
whereas the maximum redshift is z共␹⬘ = ␲兲 ⬇ 冑2 / ⑀ ⬇ 1.4
⫻ 1010. Because the transition from redshift to blueshift is
371
quite strong, Tina would see only a very small, bright dot in
the direction of motion. In contrast, most of the sky would be
very cold and very dark. Navigation by stars would be completely impossible.
One might ask if she would be able to see the evolution of
the universe.37
Am. J. Phys., Vol. 76, Nos. 4 & 5, April/May 2008
The authors thank Professor Hanns Ruder for the idea of
this work and Professor Jörg Frauendiener for many discussions and for carefully reading the manuscript. Thanks also
to Professor Jeff Rabin for the suggestion of the book Tau
Zero 共see Ref. 30兲. This work was supported by the Deutsche
Forschungsgesellschaft 共DFG兲, SFB 382, Teilprojekt D4.
APPENDIX A: JAVA APPLETS
The Java applets used to generate the diagrams in this
article are available from the Electronic Physics Auxiliary
Publication Service 共EPAPS兲.11 The Java applet TwinApplet generates the diagrams of Sec. V. There are three input
parameters, two of them are obligatory: the acceleration of
the traveling twin Tina in terms of Earth’s gravity g and
either her travel time or the maximum reachable distance.
With these parameters four plots are generated. They are
grouped in two panels: one showing the distance and the
velocity of Tina and the other showing the time signals
which the twins receive of each other.
The acceleration of Tina can be modified between 10−6 g
and 103 g. The travel time of one acceleration phase can be
entered either into a text field or with a slider. While moving
the time slider, we can observe the immediate change of the
plots and the maximum reachable distance. The maximum
time is limited to the time that the twin needs to reach the
end of the universe. Alternatively we can choose the travel
distance and hence calculate the time needed. It is possible to
save the plots as PNG images or the calculated data points in
a text file.
Because we have to deal with velocities very close to the
speed of light, the usual double-precision floating-point numbers will not suffice, compare Appendix B. Thus, we use a
series expansion for the equations depending on the velocity
␤.
The Java applet RelSkyApplet can visualize either the
stellar sky, the star constellations, or the cosmic microwave
background, each with different velocities as seen by an obMüller, King, and Adis
371
server passing the Earth. The main input parameters for all
the views are the velocity and the line of sight 共right ascension and declination兲. For the stellar sky there are some additional parameters such as the maximum displayed magnitude and the magnitude scale. For the stellar sky and the star
constellations it is possible to increase the velocity stepwise
共without changing position兲 to observe the increase of aberration and Doppler shift. By varying the velocity, the user
obtains a better understanding of how relativistic aberration
and the Doppler shift affects the visualization.
It is possible to choose between two views for the visualizations: a 4␲-representation or an hemispherical representation. In both cases the line of sight corresponds to the center
of the representation and can be set manually. The stellar sky
can also be seen from any desired point in our galaxy 共either
in equatorial coordinates or in galactic coordinates兲. By moving the mouse over the sky view, the sky coordinates of the
actual mouse position are displayed in the lower left corner
of the applet. The longitude and latitude of the sky can also
be displayed as a grid, which deforms according to the increasing velocity.
The stars are realized as small disks whose colors are calculated according to Appendix C and the star temperatures. It
is possible to change the color mode in order to examine a
particular temperature range. In manual mode the colors that
are displayed correspond to an arbitrary color scale. The size
of the disk represents the star’s apparent bolometric magnitude mbol.
By clicking on a star its Hipparcos identifier and its common name 共if available兲 are displayed. A click on Get Info
provides further information about the star: the complete
Hipparcos data set, and some of its real values compared to
its apparent values.
The star constellations view is suitable for observing the
distortion due to the aberration effect. Remember that the
distortion of some constellations in the border area of the
4␲-representation might be caused by the representation itself.
The basic data of the cosmic microwave background was
obtained from the Wilkinson Microwave Anisotropy Probe
共WMAP兲.31 To speed up the applets’ execution, the original
data was downgraded to a 4␲ representation with an image
resolution of 720⫻ 360 pixels. By selecting the microwave
mode the color scale is automatically set to a temperature
range of 2.5– 2.9 K 共which corresponds to microwave wavelengths兲. Due to the Doppler effect, it is also possible to
observe the microwave background even in the visual regime
共visual color mode兲 starting at a velocity of about 0.999c.
APPENDIX B: NUMERICAL CALCULATIONS
As we can see from Table V, we have to consider velocities ␤ that are very close to the speed of light. But computers
can handle only a limited number of digits. In general,
double-precision floating-point numbers with a 52 bit mantissa are used for precise calculations.32 For clarity we will
describe here the velocity ␤ = 1 − 10−5 as a single-precision
floating-point number with a 23 bit mantissa
␤ˆ = 1.111111111111111010110002 ⫻ 2−1
⬇ 0.999989986410 ,
共B1兲
where ␤ˆ is expressed either in normalized binary form 共subscript 2兲 or in decimal form 共subscript 10兲. This machine
372
Am. J. Phys., Vol. 76, Nos. 4 & 5, April/May 2008
Fig. 21. CIE 1931 2-deg color matching functions x̄␭ , ȳ ␭ , z̄␭.
impreciseness propagates. Hence, a straightforward numerical calculation of the ␥-factor results in
␥ˆ = 1.101111101110100100000112 ⫻ 27
⬇ 223.45512310 .
共B2兲
The series expansion of the ␥ factor is
␥=
1
1
冑1 − 共1 − ␧兲2 ⬇ 冑2␧ + O共␧
1/2
兲,
共B3兲
and the evaluation of ␧ = 10−5 gives
␥ˆ = 1.101111110011011010101112 ⫻ 27
⬇ 223.60679610 .
共B4兲
rel
From the relative errors E for the straightforward calculation and the series expansion,
rel
Estraight
⬇ 6.8 ⫻ 10−4
rel
and Eseries
⬇ 2.5 ⫻ 10−6 ,
共B5兲
we conclude that it is important to replace an equation by its
series expansion for high velocities.
APPENDIX C: FROM SPECTRUM TO COLOR
The human visual perception of wavelengths lies in the
range between 380 and 780 nm. There are three types of
cones in the retina which are sensitive to red, green and blue
light. Hence, any visible color can be composed of three
primary colors.33
In 1931 the Commission Internationale de l’Éclairage
共CIE兲 defined three primary colors X, Y, and Z with the
corresponding color matching functions34 x̄共␭兲, ȳ共␭兲 and z̄共␭兲
共see Fig. 21兲.
Any color C = XX + YY + ZZ can be composed of these primary colors,35 where the components X, Y and Z follow from
the spectral intensity distribution I共␭兲 by convolution with
the color matching functions, for example,
X=k
冕
共C1兲
I共␭兲x̄共␭兲d␭.
Because we are interested only in the chromaticity values x,
y, and z with x = X / 共X + Y + Z兲 and y = Y / 共X + Y + Z兲 and z = 1
− x − y, the constant k cancels out. To find the RGB values for
a specific device, we need its primary chromaticity values
x共r,g,b,w兲 and y 共r,g,b,w兲 for red, green, blue and the white
Müller, King, and Adis
372
point.36 After the transformation from xyz to rgb, we normalize the rgb values according to their maximum,
共r,g,b兲 哫
共r,g,b兲
,
max共r,g,b兲
共C2兲
which has the effect that all colors have their largest possible
brightness.
a兲
Author to whom correspondence should be addressed. Electronic mail:
[email protected]
R. D’Inverno, Introducing Einstein’s Relativity 共Clarendon Press, Oxford,
1992兲.
2
C. E. Dolby and S. F. Gull, “On radar time and the twin ‘paradox,’” Am.
J. Phys. 69, 1257–1261 共2001兲.
3
S. P. Boughn, “The case of the identically accelerated twins,” Am. J.
Phys. 57, 791–793 共1988兲.
4
R. P. Gruber and Richard H. Price, “Zero time dilation in an accelerating
rocket,” Am. J. Phys. 65, 979–980 共1997兲.
5
W. Rindler, Relativity: Special, General, and Cosmology 共Oxford University Press, New York, 2001兲.
6
R. U. Sexl, and H. K. Urbantke, Relativity, Groups, Particles: Special
Relativity and Relativistic Symmetry in Field and Particle Physics
共Springer, Wien/New York, 2000兲.
7
R. Perrin, “Twin paradox: A complete treatment from the point of view of
each twin,” Am. J. Phys. 47, 317–319 共1979兲.
8
A. P. French, Special Relativity 共W. W. Nelson & Co, London, 1968兲.
9
R. H. Good, “Uniformly accelerated reference frame and twin paradox,”
Am. J. Phys. 50, 232–238 共1982兲.
10
H. und M. Ruder, Die Spezielle Relativitätstheorie 共Vieweg Studium,
Braunschweig/Wiesbaden, 1993兲.
11
See EPAPS Document No. E-AJPIAS-76-002803 for the Java applets,
RelSkyApplet and TwinApplet, and accompanying help manuals.
This document can be reached through a direct link in the online article’s
HTML reference section or via the EPAPS homepage 共http://
www.aip.org/pubservs/epaps.html兲.
12
One light year 共1 ly兲 equals the distance which is covered by light with
velocity c = 299, 792, 458 m / s in one year.
13
The negative acceleration of Tina away from the Sun equals the situation
where Tina has positive acceleration but looks contrary to her direction of
motion.
14
The limit can be calculated using l’Hospital’s rule.
15
G. D. Scott and H. J. van Driel, “Geometrical appearances at relativistic
speeds,” Am. J. Phys. 38, 971–977 共1970兲.
16
The Hipparcos star catalogue consists of about 118,000 stars most of
which are at a distance of roughly 100 pc. The data we are interested in
are right ascension 共H3兲, declination 共H4兲, magnitude in Johnson V 共H5兲,
trigonometric parallax 共H11兲, Johnson B-V color 共H37兲, and Henry
Draper 共HD兲 catalogue number 共H71兲. The digital catalogue I/239 can be
found at 具cdsarc.u-strasbg.fr/viz-bin/Cat?I/239典 1997HIP…C……0EEuropean Space Agency SP-1200.
17
B. C. Reed, “The composite observational-theoretical HR diagram,” J. R.
1
373
Am. J. Phys., Vol. 76, Nos. 4 & 5, April/May 2008
Astron. Soc. Can. 92, 36–37 共1998兲.
The bolometric magnitude of a star is defined by the flux density integrated over all wavelengths. However, a real telescope can only measure
the flux density within a small region of wavelengths. The bolometric
correction compensates the difference between the bolometric and the
visual magnitude 共Ref. 19兲.
19
Fundamental Astronomy, edited by Hannu Karttunen, Pekka Kröger,
Heikki Oja, Markku Poutanen, and Karl J. Donner 共Springer-Verlag,
Heidelberg, 2003兲.
20
We take the physical constants from the National Institute of Standards
and Technology, 具physics.nist.gov/cuu/Constants典.
21
R. W. Lindquist, “Relativistic transport theory,” Ann. Phys. 37, 487–518
共1966兲.
22
The absolute bolometric magnitude of the sun is M bol,䉺 = 4.83 and the
effective temperature is T䉺 ⬇ 5778 K, see also 具nssdc.gsfc.nasa.gov/
planetary/factsheet/sunfact.html典.
23
U. Kraus, “Brightness and color of rapidly moving objects: The visual
appearance of a large sphere revisited,” Am. J. Phys. 68, 56–60 共2000兲.
24
The constellations were obtained from the free planetarium software Stellarium, 具www.stellarium.org典.
25
J. C. Mather et al., “Calibrator design for the COBE far-infrared absolute
spectrophotometer 共FIRAS兲,” Astrophys. J. 512, 511–520 共1999兲.
26
For more information on x rays and ␥ rays we refer the reader to the
following missions. X-ray: ROSAT, 具wave.xray.mpe.mpg.de/rosat典, Chandra, 具chandra.harvard.edu典, XMM-Newton, 具sci.esa.int典; ␥-ray: INTEGRAL 具www.esa.int/esaMI/Integral/典.
27
The rotational velocity of a star in a galaxy does not follow Newton’s
law, but can be read from the rotation curve of the galaxy. See Ref. 28 for
the outer rotation curve of the Milky Way.
28
J. Brand and L. Blitz, “The velocity field of the outer galaxy,” Astron.
Astrophys. 275, 67–90 共1993兲.
29
The goal of the GAIA mission is to collect high-precision astrometric
data for the brightest one billion objects. A detailed description of the
ESA Science missions is available at 具www.esa.int/esaSC/典.
30
P. Anderson, Tau Zero 共Orion Publishing Group, London, 2006兲.
31
Data are taken from 具map.gsfc.nasa.gov典.
32
Single-precision floating-point numbers are stored in a 32 bit word,
whereas double-precision ones are stored in a 64 bit word. The word
itself is composed of a 23共52兲 bit mantissa, a 8共11兲 bit exponent, and one
bit for the sign. More information can be found in the IEEE standard for
binary floating-point arithmetic for microprocessor systems 共ANSI/IEEE
Std 754–1985兲.
33
This fact is known as the tristimulus theory. For more information see, for
example, Foley 共Ref. 35兲.
34
Color matching functions can be found at the Institute of Ophthalmology,
具cvrl.ioo.ucl.ac.uk/basicindex.htm典.
35
J. D. Foley, A. Van Dam, S. K. Feiner, and J. F. Hughes, Computer
Graphics: Principles and Practice 共Addison–Wesley, 1991兲.
36
We use the color rendering of spectra by John Walker, 具www.fourmilab.ch典. The white point is the set of chromaticity values 共xw , y w兲 that serve
to define the color white.
37
An excellent science fiction book on this topic is given in Ref. 30 which
might be a useful basis for discussion.
18
Müller, King, and Adis
373