###### By Don Lincoln, Ph.D., Fermi National Accelerator Laboratory

## This article attempts to explain the contradictions related to the famous paradox in special relativity, the twin paradox. The twin paradox assumes that one of the two twins travels to space at 99.9% the speed of light and comes back to Earth. It suggests that when the traveling twin returns she is younger than her twin sister who stayed on Earth.

### Relativity and The Twin Paradox

According to Einstein’s special theory of relativity, time does not pass in the same way for everybody and varies depending on the motion of the observer. To illustrate this, let us consider the case of two hypothetical twins, Abigail (Abby) and Gabrielle (Gabby).

Abby is someone who likes to stay at home while Gabby is an adventurous astronaut. When both of them are 30 years old, Gabby decides to travel to the star Tau Ceti, which is 12 light years away. She travels at a speed of 99.9% the speed of light in her ultra-fast spaceship.

The travelling twin returns significantly younger than her sibling who stayed at home. Using the concept of time dilation, we can explain how the twins experience a variation in the passage of time. Let us assume Abby to be the primed person and Gabby to be the unprimed one. As Gabby has been travelling at a high speed, she would have experienced the effects of relativity.

Thus, Gabby would have experienced just about 13 months of time and be 31 years old. However, Abby who lived on Earth would have waited for 24 years to see her twin and would be 54 years old when Gabby returned.

The equation of time dilation, 1 over the square root of the quantity (1 – v squared over c squared) can be used to calculate the times both women would experience.

The paradox arises when there is a deeper understanding of relativity. Relativity states that either person can consider themselves as stationary. So, what if Gabby considers herself as stationary with her spaceship and assumes that Abby on Earth traveled off at a high speed and returned. In such a scenario, it is the traveling twin that would be younger and it is illogical to state that both of them were younger than the other. Only one of them can be right and hence the contradiction, the twin paradox.

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### The Mistaken Theory on Acceleration

The common response to this paradox, even from physicists who don’t work with relativity would be that the two twins are different as one of them experiences acceleration. According to this theory, Gabby experiences acceleration to catch up the speed, decelerates to turn around, and then again decelerates to land on Earth.

So, if acceleration is the answer, it means that while the spaceship is freewheeling between the stars, both twins age equally and when the acceleration turns on, there is instant aging. However, there is only one problem, this theory is incorrect.

So, to resolve this, we shall presume that there are three observers: Abby, Gabby, and Tabby. As previously assumed, Abby is stationary on earth and Gabby is heading at a speed of 99.9% the speed of light to the star, Tau Ceti. Tabby, heading toward Earth at the same speed as Gabby, is 24 light years away and along the line of sight between the two stars. It is further assumed that the information on acceleration is not available and insignificant.

The time Abby will experience if she were to sit on Earth and watch the clock would be 24 light years or T_Abby. As for Gabby, if she zeros her stopwatch as she passes Abby, the amount of time the clock will display when she reaches Tau Ceti would be half of T_Abby divided by gamma. This is because the total travel time is 24 years, so half the time to reach the star would be 12 years.

Gamma is 22.4 and therefore, Gabby experiences a mere six and half months. Now the third person, Tabby heading toward Earth, arrives at Tau Ceti exactly the same time Gabby arrives. As Tabby heads toward Earth, she clears her clock and calculates the time duration to reach Earth from Tau Ceti.

Abby finds that both Gabby and Tabby experienced only about half a year in each of their legs, which meant the total time experienced by the travelers was just about a year when Abby waited for 24 years, proving that acceleration had no role to play at all.

Learn more about E=mc2 and Other Relativity Myths.

### Lorentz Transforms and the Twin Paradox

The easiest way to solve any problem on relativity is to use Lorentz transforms. Let us assume that ‘x’ denotes position and ‘t’ denotes time. The two equations to solve how two people in relative motion interpret the same event are x prime equals gamma times the quantity (x plus v times t) and t prime equals gamma times the quantity (t plus v over c squared times x). Hence, the co-ordinates can be represented as:

- When Gabby passes Abby the equation can be written as (x_ Abby, t_Abby) equals (0,0).
- When Gabby and Tabby cross each other at Tau Ceti, it can be represented as (x_Abby, t_Abby) equals (12 lightyears, 12 years).
- Finally, when Tabby passes Abby at Abby’s location, it can be written as (x_Abby, t_Abby) equals (0, 24 years).

Thus, while it could be established that the combined travel time was seen by Abby as 24 years and the travelers as one year, the core idea that was proved incorrect was that acceleration had little role to play. All the three observers moved at a constant velocity for the entire duration and yet Abby experienced more time than the travelers.

This proves the argument that acceleration is the key difference as incorrect. And therefore, it also ascertains the fact that the twin paradox is not a paradox at all.

Further, Abby sits in a single frame during the entire trip and her velocity doesn’t change. Gabby and Tabby, on the other hand have to add the times experienced by people who are moving at different speeds compared to one another. This is a crucial difference, which demonstrates that there is no paradox at all.

To summarize, anyone travelling to a distant star from Earth would definitely return younger than if they had stayed on Earth. Also, the twin paradox is not a paradox at all and the claim that it is acceleration that breaks the paradox is incorrect.

Learn more about can you go faster than light?

### Common Questions about is the Twin Paradox Really a Paradox?

**Q: What is a paradox?**

A proposition which has an apparently sound reasoning but leads to a contradictory conclusion is called a paradox.

**Q: Does the twin paradox exist?**

The twin paradox is real as the traveling twin will see the Earth clock moving as slowly as the twin on the Earth. Yes, it is real but shouldn’t really be called a paradox.

**Q: How is special relativity different from general relativity?**

Special relativity and general relativity are inter-related theories of Albert Einstein. Special relativity refers to physical phenomenon in the absence of gravity, and general relativity to the law of gravitation in relation to other forces of nature.