EINSTEIN'S ANALYSIS OF THE TWIN PARADOX
EINSTEIN'S ANALYSIS OF THE TWIN PARADOX
EINSTEIN'S ANALYSIS OF THE TWIN PARADOX
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EINSTEIN’S <strong>ANALYSIS</strong> <strong>OF</strong> <strong>THE</strong> <strong>TWIN</strong> <strong>PARADOX</strong><br />
UNNIKRISHNAN C. S.<br />
Tata Institute, Mumbai (Bombay) &<br />
LKB, Paris<br />
GREX 2005<br />
Paris
Outline:<br />
1. Historical background to Einstein’s analysis in 1918<br />
2. Einstein’s resolution in 1918<br />
3. Discussion<br />
4. Counter-examples
Background to my interest in the history of this problem<br />
Results of of a theory of relativity with UNIVERSE<br />
as a preferred massive frame, with gravitational<br />
effects of moving through the massive universe is<br />
properly accounted for:<br />
1) Agrees with all known experimental data to date<br />
2) A transported clock can run faster than a clock<br />
stationary in the same frame (if the frame itself is<br />
moving relative to the CMBR/average rest frame of<br />
the matter in the universe, as supported well by<br />
clock comparison experiments)
Background to my interest in the history of this problem<br />
3) Closed path clock comparison is identical in physical content to<br />
closed path interferometry –> Sagnac phase is a cosmic<br />
gravitomagnetic effect, and it is the analogue of Aharonov-Bohm<br />
phase for gravitational vector potential due to motion relative to the<br />
galaxies and other matter<br />
4) In all these physical effects, what matters is not the relative velocity<br />
between observers, but the velocity relative to the matter/CMBR in<br />
the Universe -> The correct theory of relativity seems to replace<br />
Ether of Lorentz with the Universe containing matter and its gravity<br />
Is it logically and physically consistent to work with a theory<br />
meant for empty space in real universe filled with matter,<br />
energy and fields (discovered much later than 1905) ?
"The gist of the principle of relativity is the following. It is in<br />
no wise possible to detect the motion of a body relative to<br />
empty space; in fact, there is absolutely no physical sense<br />
in speaking about such motion. If, therefore, two observers<br />
move with uniform but different velocities, then each of the<br />
two with the same right may assert that with respect to<br />
empty space he is at rest, and there are no physical<br />
methods of measurement enabling us to decide in favour<br />
of one or the other".<br />
M. Planck, 1909
A<br />
“If one of two synchronous clocks at A is moved in a<br />
closed curve with constant velocity until it returns to A,<br />
the journey lasting t seconds, then by the clock that has<br />
remained at rest the travelled clock on its arrival at A<br />
will be ½ tv 2 /c 2 second slow”<br />
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A. Einstein, 1905<br />
If two observers move with uniform but different velocities, then each<br />
of the two with the same right may assert that with respect to empty<br />
space he is at rest, and there are no physical methods of<br />
measurement enabling us to decide in favour of one or the other".<br />
M. Planck, 1909<br />
So, in SR it is always the ‘other clock’ that runs slow
Experimentally,<br />
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Time dilation depends on whether the frame in which the comparison<br />
is being made moves relative to the matter in the universe or not<br />
For example, if Einstein’s 1905 prediction of time dilation was tested<br />
immediately by sending a clock in a closed path (as Einstein said)<br />
around the earth in after 1905, the result would have gone against his<br />
prediction! In fact, a clock send in the route Paris-NY-Mumbai-Paris<br />
would have been seen running FASTER than a clock at rest in Paris<br />
This would have agreed with Lorentz’s relativity though<br />
Lack of such a test till the 1970s (atomic clocks) helped in generating a<br />
firm belief in the theory of relativity based on relative velocities only<br />
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WHO ages less? (Both in linear motion for most part of their journey)<br />
Each will infer that the other will age less according a theory of relativity<br />
based on relative velocities, but this answer is paradoxical since they<br />
would meet again…<br />
A concludes that B will be ½ tv 2 /c 2 younger…<br />
B concludes that A will be ½ tv 2 /c 2 younger…
Usual statement:<br />
A<br />
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A: B goes at velocity v for time T, very brief reversal,<br />
and then comes back at v for time T. Total time dilation<br />
is approximately -2Tv2 /2c2 . B ages less than me.<br />
A<br />
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2 B<br />
B: A goes at velocity v for time T’, very brief reversal,<br />
and then comes back at v for time T’. Total time dilation<br />
is approximately -2T’v 2 /2c 2 . A ages less than me.<br />
I did feel some force, while at rest, for a brief period.<br />
Important: A’s estimate explicitly assumes that acceleration does nothing<br />
to either clock. So, if both A and B use the same physical laws,<br />
acceleration should be irrelevant (Einstein, 1911, 1914…)<br />
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Some comments at this points:<br />
Acceleration by itself does not generate time dilation,<br />
and proper time is not affected by acceleration<br />
Muons in storage ring at an acceleration of 10 14 cm/s 2<br />
have the same time dilation as muons traveling in<br />
straight line at the same velocity, for the same duration.<br />
Note that there are SEVERAL “resolutions” of the twin clock problem<br />
in text books and physics literature, using different physical and<br />
logical reasoning:<br />
A sure sign of confusion…<br />
As confirmed by Einstein’s analysis much later.
There is no resolution of the twin paradox within special relativity<br />
(Einstein 1918)<br />
Einstein’s resolution of the twin paradox in 1918<br />
A. Einstein, Naturewissenschaften 6, 697 (1918)<br />
‘Dialogue about objections to the Theory of Relativity’<br />
A reply, long overdue, to his vehement critics, like E. Gehrcke and P. Lenard<br />
`The collected papers of Albert Einstein, Volume 7, The Berlin Years:<br />
Writings, 1918-1921', Princeton University Press (2002)
Einstein on twin paradox before 1918:<br />
No specific statement with a calculation available…<br />
“The clock in uniform motion runs slower…but if it undergoes a<br />
change in direction…,then the theory of relativity does not tell us<br />
what happens. However, the longer the clock is moving<br />
rectilinearly and uniformly with a given speed of forward<br />
motion,… the smaller must be the effect of such a hypothetical<br />
sudden change” (1911, Zurich, Also, certainly in 1914…)<br />
So, in contrast to opinion of Langevin and some others, Einstein<br />
in 1911 did not think that acceleration was important in the twin<br />
paradox.
Response to E. Gehrcke, ~1914<br />
“accelerations are irrelevant for the amount of time difference<br />
between the two clocks, but their presence nevertheless<br />
causes the slowing down of clock B and not clock A.”<br />
Gehrcke continued his nasty criticisms of Einstein, to which Einstein<br />
finally responded in the paper in 1918<br />
He admitted that general theory of relativity and gravitational time<br />
dilation are essential ingredients in the resolution of the paradox.<br />
This part of the history of the problem is mixed with strong<br />
German nationalist feelings and also explicit anti-Semitist<br />
propaganda, supported by some German physicists and also<br />
opposed strongly by some – helped in some balance that<br />
allowed fair debates<br />
There were even some anti-relativity public seminars, which Einstein<br />
attended and responded to…<br />
Also letters to his colleagues (Remarks about anti-Semitism)
Einstein’s resolution of the twin paradox in 1918<br />
A. Einstein, Naturewissenschaften 6, 697 (1918)<br />
‘Dialogue about objections to the Theory of Relativity’<br />
Critic: People like me have often expressed their various doubts about<br />
the theory of relativity in journals; but rarely has one of you relativists<br />
responded…<br />
Let K be an inertial coordinate system. Let U1 and U2 be exactly identical<br />
clocks in K. If one of these clocks, U2 is in a state of uniform translatory<br />
motion, then it shall, according to SR, go at a slower rate than U1… If U2 is<br />
brought back it must be late relative to U1.<br />
Relativist: I agree, absolutely…<br />
Critic: Now comes the snag. According to the principle of relativity,<br />
the entire process must occur in exactly the same way when<br />
represented in reference the coordinate system K’ which partake in<br />
the movement of the clock U2…Even the devoutest adherents of the<br />
theory cannot claim that of two clocks, resting side by side, each one<br />
is late relative to the other.
Relativist: …the entire line of reasoning is not legitimate because,<br />
according to SR, the coordinate systems K and K’ are not at all<br />
equivalent. The theory claims only equivalence of unaccelerated<br />
systems…<br />
Critic: …but if one accepts general theory of relativity, coordinate systems<br />
of arbitrary states of motion are equivalent, and I can describe the<br />
previous process as well with respect to K’ as I can with respect to K…<br />
Relativist: It is certainly correct…But the coordinate systems K and K’ are<br />
not equivalent…
FROM K<br />
1. Clock U2 gets accelerated along<br />
+x axis until it attains velocity V.<br />
Clock U1 remains at rest.<br />
U1<br />
t<br />
t’<br />
U2<br />
U1<br />
2. U2 moves at constant velocity<br />
along positive x for time T.<br />
3. U2 is accelerated along<br />
negative x and it reverses its<br />
velocity to –V and then moves<br />
uniformly.<br />
U2<br />
FROM K’<br />
1. There arises a gravitational field<br />
in the negative x axis in which clock<br />
U1 falls accelerated until it attains<br />
velocity V. Clock U2 is at rest in this<br />
field. The g-field vanishes as soon<br />
as U1 reaches velocity V.<br />
2. U1 moves at constant velocity<br />
along negative x for time T’.<br />
3. There arises a homogeneous<br />
gravitational field along positive x<br />
in which U1 decelerates and falls<br />
towards U2 and reaches velocity<br />
V. U2 remains at rest in the field.<br />
Then the g-field vanishes.<br />
Rest is a repeat of these events in the opposite order.
Relativist: From the frame K’, during the step 2, the clock U1 moving<br />
at velocity V has indeed a slower rate than the clock U2 which is at<br />
rest. But this time lag gets overcompensated by the faster rate of U1<br />
during step 3. Because, according to the general theory of relativity, a<br />
clock has a more accelerated rate the higher the gravitational<br />
potential is at the clock’s location; and during step 3, U1 is indeed at a<br />
higher gravitational potential than U2.<br />
Calculation shows that this running-ahead amounts to exactly twice as<br />
much as the lag-behind during stages of inertial motion. This completely<br />
clarifies the paradox you referred to.<br />
Complete calculation:<br />
C. S. Unnikrishnan<br />
Current Science (Ind. Acad. Sci.), 2005<br />
U1<br />
U2<br />
g<br />
∆ T = Tgh/ c<br />
2<br />
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Some comments on Einstein’s analysis in 1918<br />
1. He does not use concepts like ‘change in the line of simultaneity’,<br />
asymmetric doppler shifts, preferential space-time diagrams etc…<br />
2. He justifies a result predicted as part of SR in 1905 with physical<br />
reasoning from a theory constructed in 1910 – 1915<br />
3. He says explicitly that as much as SR is applicable in the situation<br />
the time dilation is symmetrical, and the only asymmetry is<br />
gravitational, to be treated by GR<br />
4. He starts the article by saying that the paradox has not been<br />
addressed adequately before 1918 by relativists<br />
5. In the article he also makes a comment on the ether, that it is not<br />
fully dead, and was in fact resurrected in a different form by GR<br />
And, finally, the 1918 gravitational resolution does not succeed!
Counter example:<br />
Freeze the clock reading whenever there are accelerations (easy<br />
with real clocks)<br />
A<br />
B<br />
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Restart here<br />
Hold here<br />
No acceleration. The clock B is not running during its turn around, and<br />
therefore the reading is identical to the one at “hold” when restarted.<br />
Remember that B’s turning around cannot physically change A’s rate!<br />
So, we have to conclude that, with real clocks, B’s turning around cannot<br />
change either B’s rate or A’s rate…and therefore, neither rate is affected<br />
due to turning around!
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1. Clocks are frozen in reading during any acceleration<br />
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2. During relative uniform motion the time dilation is symmetric (SR)<br />
3. Proper Time is an accumulated physical quantity on each clock<br />
independently, and it is the sum of the time accumulated over<br />
smaller parts of the journey. Therefore, no resolution that invokes<br />
physical changes during turn-around works.<br />
4. In particular, Einstein’s resolution in 1918 does not solve the<br />
problem.<br />
C. S. Unnikrishnan<br />
Current Science (Ind. Acad. Sci.), 2005
CONCLUSIONS<br />
1. Einstein’s analysis of the twin paradox rejects the idea that it can be<br />
adequately addressed and resolved within special relativity itself<br />
2. His resolution involving pseudo-gravitational time dilation, inspired by<br />
the equivalence principle and general relativity crucially relies on<br />
uninterrupted running of the clocks during accelerations, and this<br />
requirement makes the Einstein resolution is 1918 ineffective,<br />
because clocks can be stopped and restarted, and the result on time<br />
dilation is not significantly changed<br />
3. All standard resolutions of the twin-paradox suffer from a similar<br />
problem and counter examples are easily generated for each<br />
resolution<br />
4. The resolution that the clock that moves more relative to the<br />
matter in the universe ages less due to cosmic gravity works in all<br />
cases without ambiguity, and is indeed the only resolution<br />
consistent with GR and modern cosmology