Twin paradox

The twin paradox, sometimes called the "clock paradox", stems from Paul Langevin's 1911 thought experiment in special relativity: one of two twin brothers undertakes a long space journey with a high-speed rocket at almost the speed of light, while the other twin remains on Earth. When the traveler returns to Earth, he is younger than the twin who stayed put. Or, as first stated by Albert Einstein (1911):

If we placed a living organism in a box ... one could arrange that the organism, after any arbitrary lengthy flight, could be returned to its original spot in a scarcely altered condition, while corresponding organisms which had remained in their original positions had already long since given way to new generations. For the moving organism the lengthy time of the journey was a mere instant, provided the motion took place with approximately the speed of light. (in Resnick and Halliday, 1992)

This appears to be a paradox if one expects that, according to relativity, either twin may validly claim to be "at rest", and thus each expects that the other twin will age slowly. Like most paradoxes, this occurs because of faulty assumptions. This one happens because one twin undergoes acceleration while the other does not, something that isn't taken into account when this twin-paradox story was invented.

Specific Example

Consider a space ship going from Earth to the nearest star system a distance $d=4.45$ light years away, at speed $v=0.866c$ (i.e. 86.6% of the speed of light). The round trip will take $t=2d/v=10.28$ years in Earth time (i.e. everybody on earth will be 10.28 years older when the ship returns). Ignoring the effects of the earth's rotation on its axis and around the sun (at speeds negligible compared to the speed of light), those on Earth predict the aging of the travellers during their trip as reduced by the factor $\epsilon ={\sqrt {1-v^{2}/c^{2}}}$ , the inverse of the Lorentz factor. In this case $\epsilon =0.5$ and they expect the travellers to be 0.5×10.28 = 5.14 years older when they return.

The ship's crew calculate how long the trip will take them. They know that the distant star system and the earth are moving relative to the ship at speed $v$ during the trip, and in their rest frame the distance between the earth and the star system is $\epsilon d=0.5d$ = 2.23 light years ("length contraction"), for both the outward and return journeys. Each half of the journey takes $2.23/v$ = 2.57 years, and the round trip takes 2×2.57 = 5.14 years. The crew arrives home having aged 5.14 years, just as those on Earth expected.

If a pair of twins were born on the day the ship left, and one went on the journey while the other stayed on earth, the twins will meet again when the traveller is 5.14 years old and the stay-at-home twin is 10.28 years old. This outcome is predicted by Einstein's special theory of relativity. It is a consequence of the experimentally verified phenomenon of time dilation, in which a moving clock is found to experience a reduced amount of proper time as determined by clocks synchronized with a stationary clock. Examples of the experimental evidence can be found on the time dilation page.

Origin of the Paradox

The strange result of twins of different physical age was not the central problem of the "twin paradox". As early as 1905, Einstein predicts that a clock which is moved away and brought back, will lag behind on stationary clocks. Einstein calls that result "peculiar", but the calculation is straightforward and the example is not presented as paradoxical, despite his suggestion in the introduction of the same paper that only relative motion between objects should matter.

In 1911, Langevin discusses the evolution of the concepts of space and time in physics and presents the principal aspects of special relativity. He suggests that special relativity implies the existence of a stationary ether and states: "Every change of speed, every acceleration, has an absolute sense". To stress this fact he provides several examples such as that accelerated electric charges emit electromagnetic radiation. He proceeds with a thought experiment similar to the specific example above. Langevin next explains the different aging of the twins: "Only the traveller has undergone an acceleration that changed the direction of his velocity." According to Langevin, acceleration is here "absolute", in the sense that it is the cause of the asymmetry (and not of the aging itself). Also in this discussion there seems nothing paradoxical about it.

However, it was Einstein's objective to show that all motion is "relative", and he thought to have reached that objective with General relativity. Thus he stated in 1916, in his introduction to that theory: "The laws of physics must be of such a nature that they apply to reference systems in any kind of motion" (including accelerated motion). And he concluded about two relatively accelerated systems K and K': "From the physical standpoint, the assumption readily suggests itself that the systems K and K' may both with equal right be looked upon as "stationary", that is to say, they have an equal title as systems of reference for the physical description of phenomena." In other words, all observers are equivalent, and no particular frame of reference is privileged.

The perception of paradox is rooted in a misunderstanding of the meaning of equivalent frames in relativity and therefore arises in both SR and GR. The Principle of Relativity states that the mathematical forms of the laws of physics – say, mechanics and electrodynamics — are identical in all frames of reference independent of their relative motion; i.e., all frames are equivalent in this restricted but important sense that the laws of physics are invariant from one frame to another, whether the frames are accelerating or not. In the twin problem, although the respective frames are equivalent in this sense, they are not dynamically symmetric since only the traveling twin experiences acceleration. However, if one assumes that the frames of the traveling and stay-at-home twin are dynamically symmetric and confuses dynamic symmetry with equivalence (and perhaps, additionally, falsely infers equivalence to mean identical calculations in all frames), one erroneously concludes the same result for each twin's time dilation as calculated by the other twin; i.e., one expects the traveler upon return, to have aged more and less than his twin – which is clearly impossible. In this connection, it is important to understand that relativity does not claim that all observers make identical measurements when observing the same phenomena; generally they do not. Nor does relativity claim that a time dilation calculation that each observer performs for his twin is a "law of physics" that must remain invariant from one observer or frame to another. However, these calculations would be expected to be identical for dynamically symmetric frames!

It is to be noted, however, that time dilation has no relationship whatsoever to the amount, direction, or duration of acceleration. Even the time dilation of particles in the Fermilab ring is determined only by their speed, in spite of the huge accelerations they undergo. In the usual resolution of the twin paradox, it is only the bare fact of acceleration which is invoked, and not any measured quantity.

The erroneous belief that the twins are in dynamically symmetric frames is likely rooted in Einstein's attempt to formulate a theory in which physical phenomena could be fully explained in terms of relative motion. In his landmark 1905 paper, “On the Electrodynamics of Moving Bodies”, when offering a motivating example for his Principle of Relativity, Einstein states that it is immaterial whether one considers the coil or the magnet as moving, and that only relative motion matters since the resultant induced current is the same. Thus, part of the misconception of what the Principle of Relativity means — which is essentially responsible for the perception of "paradox" — probably stems from these comments. That is, Einstein’s casually–stated claim that all motion is relative (as well as his original objective for relativity theory), can easily mislead in its suggestion that equivalent frames (with respect to invariance of the laws of physics) are necessarily symmetric with respect to motion.

Although the paradox is necessarily implied if one assumes dynamic frame symmetry, showing that it is non-existent due to the acceleration of one twin is just a necessary condition for resolution of the paradox, but not by itself sufficient. For sufficiency, one must perform the calculations to demonstrate that from the point of view of each twin, the traveler returns younger. Moreover, this must be done independently in both SR and GR, since the problem appears in both theories.

In a 1918 paper, Einstein reacts to objections to his relativity theory and issues arising in connection with the twin paradox. In that paper he reconfirms the differential aging prediction of special relativity, and points out that the equations of special relativity are only valid for inertial frames, that one twin is in an accelerating frame (if modeled in one frame throughout), and that the twins are not in symmetric frames for the reasons explained above. Next, after affirming the general principle of relativity, he sets out to show in a detailed example with clocks, that general relativity yields the same answer as special relativity and he claims that he thus "fully explained the paradox".

Resolution of the Paradox in Special Relativity

The usual resolution of the paradox as presented in physics text books ignores its origin (it only surfaced with general relativity, see above) and regards it as a problem due to misunderstanding of special relativity. Here the Earth and the ship are not in a symmetrical relationship: the ship has a "turnaround" in which it feels inertial forces, while the Earth has no such turnaround. Since there is no symmetry, it is not paradoxical if one twin is younger than the other. Nevertheless it is still useful to show that special relativity is self-consistent, and how the calculation is done from the standpoint of the traveling twin.

Of course the traveling twin comes home younger. Special relativity does not claim that all observers are equivalent, only that all observers in inertial reference frames are equivalent. But the space ship jumps frames (accelerates) when it does a U-turn. The twin on Earth rests in the same inertial frame for the whole duration of the flight (no accelerating or decelerating forces apply to him) and he is therefore able to distinguish himself as "privileged" compared with the space ship twin. The accepted resolution of the paradox is that the crew must make a different calculation from that above, a calculation which explicitly recognizes the change of reference frame, and the change in simultaneity which occurs at the turnaround.

There are indeed not two but three relevant inertial frames: the one in which the stay-at-home twin remains at rest, the one in which the traveling twin is at rest on his outward trip, and the one in which he is at rest on his way home. It is during the acceleration at the U-turn that the traveling twin switches frames. That's when he must adjust the calculated age of the twin at rest. Here's why.

In special relativity there is no concept of absolute present. A present is defined as a set of events that are simultaneous from the point of view of a given observer. The notion of simultaneity depends on the frame of reference (see relativity of simultaneity), so switching between frames requires an adjustment in the definition of the present. If one imagines a present as a (three-dimensional) simultaneity plane in Minkowski space, then switching frames results in changing the inclination of the plane.

File:Twins paradox diagram.png

Twins paradox Minkowski diagram

In the spacetime diagram on the right, the first twin's lifeline coincides with the vertical axis (his position is constant in space, moving only in time). On the first leg of the trip, the second twin moves to the right (black sloped line); and on the second leg, back to the left. Blue lines show the planes of simultaneity for the traveling twin during the first leg of the journey; red lines, during the second leg. Just before turnover, the traveling twin calculates the age of the resting twin by measuring the interval along the vertical axis from the origin to the upper blue line. Just after turnover, if he recalculates, he'll measure the interval from the origin to the lower red line. In a sense, during the U-turn the plane of simultaneity jumps from blue to red and very quickly sweeps over a large segment of the lifeline of the resting twin. The resting twin has suddenly "aged" very fast, in the reckoning of the traveling twin.

What it looks like: the relativistic Doppler shift

Now, how would each twin observe the other during the trip? Or if each twin always carried a clock indicating his age, what time would each see in the image of the distant twin and his clock? The solution to this observational problem can be found in the relativistic Doppler effect. The frequency of clock-ticks which one sees from a source with rest frequency $f_{\mathrm {rest} }$ is

f_{\mathrm {obs} }=f_{\mathrm {rest} }{\sqrt {\left({1-v/c}\right)/\left({1+v/c}\right)}}

when the source is moving directly away (a reduction in frequency; "red-shifted"). When the source is coming directly back, the observed frequency is higher ("blue-shifted") and given by

f_{\mathrm {obs} }=f_{\mathrm {rest} }{\sqrt {\left({1+v/c}\right)/\left({1-v/c}\right)}}

This combines the effects of time dilation (reduction in source frequency due to motion by factor ε) and the Doppler shift in received frequency by factor (1 $\pm$ v/c)^-1, which would apply even for velocity-independent clock rates. For the example case above where $v/c=0.866$ , the high and low frequencies received are 3.732 and 0.268 times the rest frequency. That is each twin would see the image of his sibling aging at a rate 3.732 or 0.268 times his own rate of aging. Note that 3.732 = 1/0.268.

LIGHT PATHS FOR IMAGES EXCHANGED DURING TRIP
Left: Earth to ship. Right: Ship to Earth.
RED LINES INDICATE LOW FREQUENCY IMAGES ARE RECEIVED
BLUE LINES INDICATE HIGH FREQUENCY IMAGES ARE RECEIVED

The

x-t

(space-time) diagrams at left shows the path of light signals (carrying the image of the twin and his age-clock) sent between Earth and the Ship during the journey. The vertical black line is the Earth's path through space time and the other two sides of the triangle show the Ship's path through space time (as in the Minkowski diagram above). One diagram shows the signals (carrying images) sent from Earth to Ship, while the other shows the signals sent from Ship to Earth. The images (or signals) are sent at equal intervals in the time of the sender; they are received at unequal intervals in the time of the receiver.

At first each twin would see 1 second pass in the received image of the other twin for every 3.73 seconds of his own time. That is, each would see the image of the other's clock going slow, not just slow by the ε factor, but extra slow because of the Doppler observational effect. This is shown in the figures by red light paths. At some point the images received by each twin change so that each would see 3.73 seconds pass in the image for every second of his own time. That is, the received signal has been increased in frequency by the Doppler shift. These high frequency images are shown in the figures by blue light paths.

The asymmetry in the Doppler shifted images

The asymmetry between the earth and the space ship is manifested in this diagram by the fact that more blue-shifted (fast aging) images are received by the Ship and more red-shifted (slow aging) images are received by Earth. Put another way, the space ship sees the image change from a red-shift (slower aging of the image) to a blue-shift (faster aging of the image) at the mid-point of its trip (at the turnaround, 2.57 years after departure); the Earth sees the image of the ship change from red-shift to blue shift after 9.59 years (almost at the end of the period that the ship is absent). In the next section, you will see another asymmetry in the images: the Earth twin sees the Ship twin age by the same amount in the red and blue shifted images; the Ship twin sees the Earth twin age by different amounts in the red and blue shifted images.

Calculation of elapsed time from the Doppler diagram

The twin on the ship sees low frequency (red) images for 2.57 years. During that time he would see the Earth twin in the image grow older by 2.57/3.73 = 0.69 years. He then sees high frequency (blue) images for the remaining 2.57 years of his trip. During that time, he would see the Earth twin in the image grow older by 2.57×3.73 = 9.59 years. When the journey is finished, the image of the Earth twin has aged by 0.69 + 9.59 = 10.28 years (the Earth twin is 10.28 years old).

The Earth twin sees 9.59 years of slow (red) images of the Ship twin, during which the Ship twin ages (in the image) by 9.58/3.73 = 2.57 years. He then sees fast (blue) images for the remaing 0.69 years until the Ship returns. In the fast images, the Ship twin ages by 0.69×3.73 = 2.57 years. The total aging of the Ship twin in the images received by Earth is 2.57+2.57 = 5.14 years, so the Ship twin returns younger (5.14 years as opposed to 10.28 years on Earth).

The distinction between what they see and what they calculate

To avoid confusion, note the distinction between what each twin sees, and what each would calculate. Each sees an image of his twin which he knows originated at a previous time and which he knows is Doppler shifted. He does not take the elapsed time in the image, as the age of his twin now. And he does not confuse the rate at which the image is aging with the rate at which his twin was aging when the image was transmitted.

If he wants to calculate when his twin was the age shown in the image (i.e. how old he himself was then), he has to determine how far away his twin was, when the signal was emitted - in other words, he has to consider simultaneity for a distant event.
If he wants to calculate how fast his twin was aging when the image was transmitted he adjusts for the Doppler shift. For example, when he receives high frequncy images (showing his twin aging rapidly), with frequency $f_{\mathrm {rest} }{\sqrt {\left({1+v/c}\right)/\left({1-v/c}\right)}}$ , he does not conclude that the twin was aging that rapidly when the image was generated, anymore than he concludes that the siren of an ambulance is emitting the frequency he hears. He knows that the Doppler effect has increased the image frequency by the factor $1/\left(1-v/c\right)$ . He calculates therefore that his twin was aging at the rate of

f_{\mathrm {rest} }{\sqrt {\left({1+v/c}\right)/\left({1-v/c}\right)}}\times \left(1-v/c\right)=f_{\mathrm {rest} }{\sqrt {1-v^{2}/c^{2}}}\equiv \epsilon f_{\mathrm {rest} }

when the image was emitted. A similar calculation reveals that his twin was aging at the same reduced rate of $\epsilon f_{\mathrm {rest} }$ in all low frequency images.

Simultaneity in the Doppler Shift calculation

It may be difficult to see where simultaneity came into the Doppler shift calculation, and indeed the calculation is often preferred because one does not have to worry about simultaneity. As seen above, the Ship twin can convert his received Doppler-shifted rate to a slower rate of the clock of the distant clock for both red and blue images. If he ignores simultaneity he might say his twin was aging at the reduced rate throughout the journey and therefore should be younger than him. He is now back to square one, and has to take into account the change in his notion of simultaneity at the turn around. The rate he can calculate for the image (corrected for Doppler effect) is the rate of the Earth twin's clock at the moment it was sent, not at the moment it was received. Since he receives an unequal number of red and blue shifted images he should realise that the red and blue shifted emissions were not emitted over equal time periods for the Earth twin, and therefore he must account for simultaneity at a distance.

Resolution of the Paradox in General Relativity

In general relativity we have the effect of gravitational time dilation. This means that if we have a clock above us in a fictitious field we will see that it runs faster. (In GR the word gravitational is used frecuently to speak about fictitious forces)

With this effect in mind we can explain the point of view of the travelling twing. When he starts accelerating towards his brother he will see a fictious field, and the whole universe will be free-falling in this field, including the brother at home.

In a fictious field of forces we can speak about "up and down", as if it were the gravitational field at earth. The travelling twing accelerates towards his brother at home, and therefore, he will be always see the brother at home "up above".

Therefore he will conclude that the clock at home will have to run faster that his accelerated clock, and this will be confirmed when both brothers meet.

Consecuences of this explanation

It was Einstein's dissatisfaction with special relativity as incomplete which prompted the development of general relativity. His dissatisfaction stemmed, at least in part, from the fact that asymmetrical aging did not depend only on the kinematics (the relative motion of the bodies). See also the essentially equivalent example of two separated but synchronized clocks being unsynchronized when brought together (Einstein 1905). The kinematics did indeed appear to be symmetrical, and special relativity had to treat inertial frames as "privileged". This point of view can be clearly seen in Einstein's preamble to his 1916 paper on general relativity (section 2).

Einstein's general relativity solution was as follows.

From the viewpoint of the traveller, the calculation for each separate leg equals that of special relativity, in which the Earth clocks age less than the traveller. For example, if the Earth clocks age 1 day less on each leg, the amount that the Earth clocks will lag behind due to speed alone amounts to 2 days. Now the accelerated frame is regarded as truly stationary, and the physical description of what happens at turn-around has to produce a contrary effect of double that amount: 4 days' advancing of the Earth clocks. Then the traveller's clock will end up with a 2-day delay on the Earth clocks, just as special relativity stipulates.

According to Einstein's equivalence principle, acceleration is equivalent in its effect to a gravitational field. What might be called inertial forces felt by the turnaround twin can be treated as gravitational forces, and according to Einstein at that time, they are real, induced gravitational forces for the "traveller". Thus the gravitational field replaces the acceleration as cause for the asymmetry. While this uniform gravitational field exists for the turnaround twin, he is at a lower gravitational potential than people on Earth. He accepts from the equivalence principle that if his clock is at a lower gravitational potential in a gravitational field, it will be running slow compared to clocks at a higher potential - his clock runs slow compared to the Earth clocks while he feels the gravitational force. He can calculate that after the field disappears, the Earth clocks have gained time compared to his momentarily very slow clock. The Earth clocks have advanced exactly twice as much relative to his clock as the Earth clocks will in total retard due to speed.

It is sometimes claimed that the twin paradox cannot be resolved without the use of general relativity, by which it might be meant that age difference cannot be calculated by the travelling twin without general relativity, something we have tried to show can be done. On the other hand, the claim that general relativity is necessary, may be a claim that someone who doesn't believe the argument which rejects the first (erroneous) calculation by the space-ship crew is strong enough to convince the crew that they should perform the second correct calculation instead. The general relativity explanation says: if you are going to claim your reference frame is good, and deny the implications of changing reference frames, you will need to consider the inertial forces as equivalent to gravity forces and then account for the physical effect of gravity.

This explanation was popular among a number of physicists (Møller 1952) and continues to find adherents today. However, that calculation and its related interpretation have met with serious criticism. For example, after remarking that the general relativity calculation only corresponds to perceived reality for the traveler, Builder (1957) writes that Einstein's solution of the twin paradox is erroneous because the induced field must appear everywhere at the same time:

The concept of such a field is completely incompatible with the limited value c for all velocities [...], so that the specified field would have to be created simultaneously at all points in S' and be destroyed simultaneously at all points in S₀. Thus the principle of equivalence can contribute nothing of physical significance to the analysis.

Thus in later years, physicists have increasingly treated general relativity as a theory of gravitation, rejecting its aspirations to be a theory about relative motion including acceleration; consequently, acceleration is commonly regarded as "absolute" after all.

References

Builder, G. (1957) "The Resolution of the Clock Paradox", Australian J. Phys. 10, pp246-262.
Einstein, A. (1905) "On the Electrodynamics of Moving Bodies", Annalen der Physik, 17, p891, June 30 1905 (English translation)
Einstein, A. (1916) "The Foundation of the General Theory of Relativity ." Annalen der Physik, 49 (English translation)
Einstein, A. (1918) "Dialog über Einwände gegen die Relativitätstheorie", Die Naturwissenschaften 48, pp697-702, 29 November 1918
French, A. P. (1968). Special Relativity. W. W. Norton: New York.
Ives, H (1937) "The aberration of Clocks and the Clock Paradox" JOSA 27 p.305
Langevin, P. (1911) "L’évolution de l’espace et du temps", Scientia, X, p31
Møller, C. (1952). The theory of relativity. Clarendon press: Oxford.
Resnick, Robert and Halliday, David (1992). Basic Concepts in Relativity. New York: Macmillan.{{cite book}}: CS1 maint: multiple names: authors list (link)
Tipler, Paul and Llewellyn, Ralph (2002). Modern Physics (4th ed.). W. H. Freeman. ISBN 0716743450.{{cite book}}: CS1 maint: multiple names: authors list (link)

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