Twin Paradox: Understanding Relativity and Time Dilation

The twin paradox theory is a special relativity thought experiment involving identical twins, one of whom travels into space in a high-speed rocket and returns home to discover that the twin who stayed on Earth has aged more.

This finding is perplexing since each twin perceives the other as passing, and so each should paradoxically find the other to have aged less as a result of an erroneous and naive implementation of twin paradox time dilation and the theory of relativity.

Twin Paradox Explained With an Example

Jane and Joe are identical twins. Jane flies to a distant location in a straight line at relativistic speed v. He then slows down and returns. Joe, his twin brother, remains on Earth. On both the outbound and return legs, Joe notices that Jane's onboard clocks, including his biological one, which runs at Jane's proper time, run slowly. As a result, Jane concludes that when he returns, he will be younger than he is now. Jane notices Joe's clock ticking slowly on the outer leg, and he notices it ticking slowly on the return leg.

Can Jane come to the conclusion that Joe has aged less? And who is right if he does? 

According to the paradox's proponents, there is symmetry between the two observers, so each can predict the other's age simply by plugging in the equations of relativity. This cannot be valid for both at the same time, so relativity is incorrect if the logic is right.

However, within the normal context of special relativity paradoxes, this situation can be resolved: the trajectory of the travelling twin requires two separate inertial frames, one for the outbound journey and one for the inbound journey. Another way to think about it is to consider that the travelling twin is accelerating, making him a non-inertial observer. There is no symmetry between the twin’s spacetime paths in either view. As a result, the twin paradox definition is not a logical contradiction in the traditional sense.

History of Twin Paradox Theory

• Albert Einstein deduced in his famous paper on special relativity in 1905 that if two clocks were brought together and synchronised, and then one was pushed away and returned, the clock that had travelled would be found to be lagging behind the clock that had stayed put. This Einstein twin paradox, was a logical consequence of special relativity, not a paradox, as some had claimed, and he restated and expanded on this finding in 1911.

• Paul Langevin described the storey of a traveller making a trip at a Lorentz factor of 100 in 1911, calling it a "striking illustration". The traveller spends one year as a projectile and then reverses course. When the traveller returns, he will discover that he has aged two years and that 200 years have passed on Earth. Both the traveller and the Earth continue to transmit signals to each other at a constant pace during the journey, placing Langevin's tale among the Doppler shift variants of the twin paradox. The different ageing rates are explained using relativistic effects on signal rates. Since any change in velocity or acceleration has an absolute value, the asymmetry that existed because only the traveller experienced acceleration is used to justify why there is any difference at all.

• Langevin's explanation was expanded upon by Max von Laue. Laue went on to show that the world lines of inertially moving bodies increase the proper time elapsed between two cases, using Hermann Minkowski's space-time formalism. He also said that the asymmetric ageing is fully explained by the fact that the astronaut twin flies in two different frames while the Earth twin stays in one, and that the period of acceleration can be rendered arbitrarily small in comparison to the time of inertial motion.

• By introducing the three-brother technique, Lord Halsbury and others were able to eliminate any acceleration. The travelling twin moves his clock reading to a third, which is moving in the opposite direction. The use of the relativistic Doppler effect is another way to avoid acceleration effects.

• Such findings were not found troublesome by Einstein or Langevin. Langevin viewed it as a product of absolute acceleration, while Einstein just called it peculiar. Both men said that no self-contradiction could be derived from the time difference depicted in the tale of the twins. In other words, neither Einstein nor Langevin saw the tale of the twins as posing a threat to relativistic physic self-consistency.

Analysis of Twin Paradox Theory

Here to analyse the twin paradox solution consider the example of identical twins Jane and Joe explained earlier.

To better understand this problem, consider making clear the observations of each other's clocks. For example, the ticks of a clock may be transmitted from one twin to the other through electromagnetic radiation (EMR). In theory, both of them could look at the other's clock through a telescope. To make the diagram easier to understand, the clocks only tick once a year, and each twin sends the other a greeting message on their separation anniversary.

Let's make graphs of position vs. time for the twins' voyages, as well as their anniversary posts. Space-time diagrams are what they're called. There are three space-time diagrams in this problem, one for each of the three separate inertial reference frames involved.

Joe's diagram has one plot whereas Jane's diagrams have two plots, with the return journey diagram drawn above the outward journey diagram. For v = 0.66 c, these diagrams are drawn to scale. In Joe's frame, the gap is 2.67 light-years in both directions. As a result, Joe estimates that the journey will take 2.67 light-years /0.66 c = 4 years in either direction.

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The travel length is shorter in both of Jane's frames by the factor \[\frac{1}{🇾}\] = \[\sqrt{(1-\frac{v^{2}}{c^{2}})}\] = 0.75, where v is the speed of the ship and c is the speed of light. So the gap is just 0.75x2.67 = two light-years for him, and it takes him only 3 years in each direction at 0.66c. When Joe and Jane are reunited, he will be 8 years old and Jane will be 6 years old, according to Joe. Jane's version of events can be determined by how cautiously he applies Special Relativity.

The inaccurate interpretation of the term "paradox" refers to the assumption that the situation is completely symmetrical. If that's the case, Jane's diagram is simply a mirror image of Joe's. However, the results obtained in Special Relativity were limited to the relationships between inertial frames of reference. In this regard, the twins' situations are unquestionably asymmetrical. Joe remains in a single inertial frame throughout, but Jane does not.

Jane is well aware that he has not been in the same inertial frame for the duration of the journey. He had to turn the ship around and fire the engines hard and long to stop the outward journey and begin the return. During that time, he realised he wasn't in an inertial frame because all of the objects, including himself, were suddenly squashed towards the end of the spaceship with the engines. Then, as the ship approached Earth at speed 'v,' he cut the engines, and he was back in another, different, inertial frame.

Jane receives a slew of anniversary messages from Joe in the second of his inertial frames leading up to the return journey. If he pretends to have been in this same frame of reference the entire time, that is, if he believes he has been travelling towards Earth at constant v for six of his years, he may infer that Joe has been sending them for eight of his years. This is an odd assumption since he can also say that he and Joe had never been together and that Jane and Joe had never worked together before.

Jane would calculate that when he began the journey, Joe was four light-years away, had been travelling towards him for two years, and had already sent a couple of anniversary messages, based on this odd assumption. Jane observes a sudden shift in the pace at which the messages arrive shortly after he turns, which is another asymmetrical observation.

Let's pretend Jane isn't naive, that he understands relativity, that he remembers the acceleration, that he remembers being with Joe at the start of the journey, and that he applies this understanding to his interpretation of the space-time diagrams. First, he will use Special Relativity until he has left the Earth and is travelling without acceleration towards his destination. Jane notices that the distance between his destination and the Earth has shrunk.

It has shrunk by the factor \[\frac{1}{🇾}\] = \[\sqrt{(1-\frac{v^{2}}{c^{2}})}\] = 0.75, so he will only have to fly for three years of his life. Similarly, the distance is shorter on his return trip in another inertial frame so he can use Special Relativity again, so he only has to fly for three years.

Role of Acceleration in Twin Paradox Relativity

• While some solutions relate the effect to the travelling twin's acceleration at the time of the turnaround, others point out that the effect can also be seen if two different travellers, one outbound and one inbound, pass each other and synchronise their clocks at the point corresponding to a single traveller's turnaround. 

• The physical acceleration of the travelling clock has no bearing in this model. The question is not how twisted the world-lines are, but how long they are. 

• The Lorentz-invariant length or proper time interval of a trajectory, which corresponds to the elapsed time measured by a clock following that trajectory, is the length referred to here.

• Even if these only mean accelerations of the same size separated by different amounts of time, the travelling twin must feel a different history of accelerations from the earthbound twin in Minkowski spacetime. 

• Nevertheless, in formulations of the twin paradox in curved spacetime, where the twins will fall freely along space-time geodesics between meetings.

Non-Space-Time Approach to Twin Paradox Theory

• An out-and-back twin paradox adventure could include the transition of a clock reading from one astronaut to the next, effectively removing the impact of acceleration. 

• Physical acceleration of clocks, according to the so-called clock postulate, does not add to the kinematical effects of special relativity. 

• Rather, as discussed in Einstein's original 1905 relativity paper as well as all subsequent kinematical derivations of the Lorentz transformations, the time differential between two reunited clocks is provided solely by uniform inertial motion. 

• Since spacetime diagrams incorporate Einstein's clock synchronisation with their lattice of clocks methodology, there will be a necessary jump in the reading of the Earth clock time by a suddenly returning astronaut who inherits a new meaning of simultaneity in keeping with a new clock synchronisation dictated by the transfer to a different inertial frame, as explained in Spacetime Physics by Einstein.

• If instead of incorporating Einstein's clock synchronisation, the outgoing and incoming astronauts, as well as the Earth-based team, send radio signals that fly at light speed, both parties will notice an incremental accumulation of asymmetry in time-keeping, starting at the turn around the stage. 

• Prior to the turnaround, each party believes the other's clock is recording time differently than his own, but the disparity is symmetrical between the two.

• The noted variations are not symmetrical after the turnaround, and the asymmetry increases incrementally before the two parties are reunited. This asymmetry can be seen in the actual disparity between the two reunited clocks after they have been reassembled.

Conclusion

The twin paradox is only evident because special relativity does not adequately address the case. To return to Earth, the spacecraft must change course, which contradicts special relativity's central condition of steady straight-line motion. General relativity, which shows that there will be an asymmetrical change in the time between the two twins, is needed for a complete treatment. As a result, the paradox has no bearing on how special relativity describes time, as various studies have shown.