EINSTEIN'S ANALYSIS OF THE TWIN PARADOX

EINSTEIN’S ANALYSIS OF THE TWIN PARADOX
UNNIKRISHNAN C. S.
Tata Institute, Mumbai (Bombay) &
LKB, Paris
GREX 2005
Paris

Outline:
1. Historical background to Einstein’s analysis in 1918
2. Einstein’s resolution in 1918
3. Discussion
4. Counter-examples

Background to my interest in the history of this problem
Results of of a theory of relativity with UNIVERSE
as a preferred massive frame, with gravitational
effects of moving through the massive universe is
properly accounted for:
1) Agrees with all known experimental data to date
2) A transported clock can run faster than a clock
stationary in the same frame (if the frame itself is
moving relative to the CMBR/average rest frame of
the matter in the universe, as supported well by
clock comparison experiments)

Background to my interest in the history of this problem
3) Closed path clock comparison is identical in physical content to
closed path interferometry –> Sagnac phase is a cosmic
gravitomagnetic effect, and it is the analogue of Aharonov-Bohm
phase for gravitational vector potential due to motion relative to the
galaxies and other matter
4) In all these physical effects, what matters is not the relative velocity
between observers, but the velocity relative to the matter/CMBR in
the Universe -> The correct theory of relativity seems to replace
Ether of Lorentz with the Universe containing matter and its gravity
Is it logically and physically consistent to work with a theory
meant for empty space in real universe filled with matter,
energy and fields (discovered much later than 1905) ?

"The gist of the principle of relativity is the following. It is in
no wise possible to detect the motion of a body relative to
empty space; in fact, there is absolutely no physical sense
in speaking about such motion. If, therefore, two observers
move with uniform but different velocities, then each of the
two with the same right may assert that with respect to
empty space he is at rest, and there are no physical
methods of measurement enabling us to decide in favour
of one or the other".
M. Planck, 1909

A
“If one of two synchronous clocks at A is moved in a
closed curve with constant velocity until it returns to A,
the journey lasting t seconds, then by the clock that has
remained at rest the travelled clock on its arrival at A
will be ½ tv 2 /c 2 second slow”
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A. Einstein, 1905
If two observers move with uniform but different velocities, then each
of the two with the same right may assert that with respect to empty
space he is at rest, and there are no physical methods of
measurement enabling us to decide in favour of one or the other".
M. Planck, 1909
So, in SR it is always the ‘other clock’ that runs slow

Experimentally,
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Time dilation depends on whether the frame in which the comparison
is being made moves relative to the matter in the universe or not
For example, if Einstein’s 1905 prediction of time dilation was tested
immediately by sending a clock in a closed path (as Einstein said)
around the earth in after 1905, the result would have gone against his
prediction! In fact, a clock send in the route Paris-NY-Mumbai-Paris
would have been seen running FASTER than a clock at rest in Paris
This would have agreed with Lorentz’s relativity though
Lack of such a test till the 1970s (atomic clocks) helped in generating a
firm belief in the theory of relativity based on relative velocities only
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u
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WHO ages less? (Both in linear motion for most part of their journey)
Each will infer that the other will age less according a theory of relativity
based on relative velocities, but this answer is paradoxical since they
would meet again…
A concludes that B will be ½ tv 2 /c 2 younger…
B concludes that A will be ½ tv 2 /c 2 younger…

Usual statement:
A
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A: B goes at velocity v for time T, very brief reversal,
and then comes back at v for time T. Total time dilation
is approximately -2Tv2 /2c2 . B ages less than me.
A
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B: A goes at velocity v for time T’, very brief reversal,
and then comes back at v for time T’. Total time dilation
is approximately -2T’v 2 /2c 2 . A ages less than me.
I did feel some force, while at rest, for a brief period.
Important: A’s estimate explicitly assumes that acceleration does nothing
to either clock. So, if both A and B use the same physical laws,
acceleration should be irrelevant (Einstein, 1911, 1914…)
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Some comments at this points:
Acceleration by itself does not generate time dilation,
and proper time is not affected by acceleration
Muons in storage ring at an acceleration of 10 14 cm/s 2
have the same time dilation as muons traveling in
straight line at the same velocity, for the same duration.
Note that there are SEVERAL “resolutions” of the twin clock problem
in text books and physics literature, using different physical and
logical reasoning:
A sure sign of confusion…
As confirmed by Einstein’s analysis much later.

There is no resolution of the twin paradox within special relativity
(Einstein 1918)
Einstein’s resolution of the twin paradox in 1918
A. Einstein, Naturewissenschaften 6, 697 (1918)
‘Dialogue about objections to the Theory of Relativity’
A reply, long overdue, to his vehement critics, like E. Gehrcke and P. Lenard
`The collected papers of Albert Einstein, Volume 7, The Berlin Years:
Writings, 1918-1921', Princeton University Press (2002)

Einstein on twin paradox before 1918:
No specific statement with a calculation available…
“The clock in uniform motion runs slower…but if it undergoes a
change in direction…,then the theory of relativity does not tell us
what happens. However, the longer the clock is moving
rectilinearly and uniformly with a given speed of forward
motion,… the smaller must be the effect of such a hypothetical
sudden change” (1911, Zurich, Also, certainly in 1914…)
So, in contrast to opinion of Langevin and some others, Einstein
in 1911 did not think that acceleration was important in the twin
paradox.

Response to E. Gehrcke, ~1914
“accelerations are irrelevant for the amount of time difference
between the two clocks, but their presence nevertheless
causes the slowing down of clock B and not clock A.”
Gehrcke continued his nasty criticisms of Einstein, to which Einstein
finally responded in the paper in 1918
He admitted that general theory of relativity and gravitational time
dilation are essential ingredients in the resolution of the paradox.
This part of the history of the problem is mixed with strong
German nationalist feelings and also explicit anti-Semitist
propaganda, supported by some German physicists and also
opposed strongly by some – helped in some balance that
allowed fair debates
There were even some anti-relativity public seminars, which Einstein
attended and responded to…
Also letters to his colleagues (Remarks about anti-Semitism)

Einstein’s resolution of the twin paradox in 1918
A. Einstein, Naturewissenschaften 6, 697 (1918)
‘Dialogue about objections to the Theory of Relativity’
Critic: People like me have often expressed their various doubts about
the theory of relativity in journals; but rarely has one of you relativists
responded…
Let K be an inertial coordinate system. Let U1 and U2 be exactly identical
clocks in K. If one of these clocks, U2 is in a state of uniform translatory
motion, then it shall, according to SR, go at a slower rate than U1… If U2 is
brought back it must be late relative to U1.
Relativist: I agree, absolutely…
Critic: Now comes the snag. According to the principle of relativity,
the entire process must occur in exactly the same way when
represented in reference the coordinate system K’ which partake in
the movement of the clock U2…Even the devoutest adherents of the
theory cannot claim that of two clocks, resting side by side, each one
is late relative to the other.

Relativist: …the entire line of reasoning is not legitimate because,
according to SR, the coordinate systems K and K’ are not at all
equivalent. The theory claims only equivalence of unaccelerated
systems…
Critic: …but if one accepts general theory of relativity, coordinate systems
of arbitrary states of motion are equivalent, and I can describe the
previous process as well with respect to K’ as I can with respect to K…
Relativist: It is certainly correct…But the coordinate systems K and K’ are
not equivalent…

FROM K
1. Clock U2 gets accelerated along
+x axis until it attains velocity V.
Clock U1 remains at rest.
U1
t
t’
U2
U1
2. U2 moves at constant velocity
along positive x for time T.
3. U2 is accelerated along
negative x and it reverses its
velocity to –V and then moves
uniformly.
U2
FROM K’
1. There arises a gravitational field
in the negative x axis in which clock
U1 falls accelerated until it attains
velocity V. Clock U2 is at rest in this
field. The g-field vanishes as soon
as U1 reaches velocity V.
2. U1 moves at constant velocity
along negative x for time T’.
3. There arises a homogeneous
gravitational field along positive x
in which U1 decelerates and falls
towards U2 and reaches velocity
V. U2 remains at rest in the field.
Then the g-field vanishes.
Rest is a repeat of these events in the opposite order.

Relativist: From the frame K’, during the step 2, the clock U1 moving
at velocity V has indeed a slower rate than the clock U2 which is at
rest. But this time lag gets overcompensated by the faster rate of U1
during step 3. Because, according to the general theory of relativity, a
clock has a more accelerated rate the higher the gravitational
potential is at the clock’s location; and during step 3, U1 is indeed at a
higher gravitational potential than U2.
Calculation shows that this running-ahead amounts to exactly twice as
much as the lag-behind during stages of inertial motion. This completely
clarifies the paradox you referred to.
Complete calculation:
C. S. Unnikrishnan
Current Science (Ind. Acad. Sci.), 2005
U1
U2
g
∆ T = Tgh/ c
2
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A
B

Some comments on Einstein’s analysis in 1918
1. He does not use concepts like ‘change in the line of simultaneity’,
asymmetric doppler shifts, preferential space-time diagrams etc…
2. He justifies a result predicted as part of SR in 1905 with physical
reasoning from a theory constructed in 1910 – 1915
3. He says explicitly that as much as SR is applicable in the situation
the time dilation is symmetrical, and the only asymmetry is
gravitational, to be treated by GR
4. He starts the article by saying that the paradox has not been
addressed adequately before 1918 by relativists
5. In the article he also makes a comment on the ether, that it is not
fully dead, and was in fact resurrected in a different form by GR
And, finally, the 1918 gravitational resolution does not succeed!

Counter example:
Freeze the clock reading whenever there are accelerations (easy
with real clocks)
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B
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Restart here
Hold here
No acceleration. The clock B is not running during its turn around, and
therefore the reading is identical to the one at “hold” when restarted.
Remember that B’s turning around cannot physically change A’s rate!
So, we have to conclude that, with real clocks, B’s turning around cannot
change either B’s rate or A’s rate…and therefore, neither rate is affected
due to turning around!

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1. Clocks are frozen in reading during any acceleration
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2. During relative uniform motion the time dilation is symmetric (SR)
3. Proper Time is an accumulated physical quantity on each clock
independently, and it is the sum of the time accumulated over
smaller parts of the journey. Therefore, no resolution that invokes
physical changes during turn-around works.
4. In particular, Einstein’s resolution in 1918 does not solve the
problem.
C. S. Unnikrishnan
Current Science (Ind. Acad. Sci.), 2005

CONCLUSIONS
1. Einstein’s analysis of the twin paradox rejects the idea that it can be
adequately addressed and resolved within special relativity itself
2. His resolution involving pseudo-gravitational time dilation, inspired by
the equivalence principle and general relativity crucially relies on
uninterrupted running of the clocks during accelerations, and this
requirement makes the Einstein resolution is 1918 ineffective,
because clocks can be stopped and restarted, and the result on time
dilation is not significantly changed
3. All standard resolutions of the twin-paradox suffer from a similar
problem and counter examples are easily generated for each
resolution
4. The resolution that the clock that moves more relative to the
matter in the universe ages less due to cosmic gravity works in all
cases without ambiguity, and is indeed the only resolution
consistent with GR and modern cosmology