The Paradox is Resolved

The principles of special relativity hinge on two key postulates:

1. Do not distinguish between inertial frames of reference.
2. Maintain a constant speed of light across all frames.

An inertial frame is characterized by a net force of zero. For instance, a spacecraft moving in a straight line at a steady speed is indistinguishable from one at rest. Thus, we speak of velocity in relative terms rather than absolute ones.

The swift resolution to the paradox lies in recognizing that Alice does not remain in an inertial frame throughout her journey. She deviates from this state at three critical points:

1. When she accelerates to leave Earth.
2. When she turns around to come back.
3. When she decelerates upon her return.

We will examine Alice's journey from different reference frames, starting with what occurs, which will reveal that both Bob and Alice concur: Alice is the younger twin upon returning to Earth. Subsequently, we’ll derive some formulas to explain why this happens.

During our examination, we will delve into three effects of special relativity:

1. Time dilation — a moving clock ticks slower.
2. Length contraction — an object in motion appears contracted along its direction of travel.
3. Loss of simultaneity — a clock at the back of a moving object will be ahead of a clock at the front, assuming they were synchronized at rest.

The What

Bob’s Frame

Alice departs, moving at a velocity approaching light speed. Three clocks are synchronized:

• Bob’s clock: 0
• Alice’s clock: 0
• Alpha Centauri clock: 0

Alice’s 10-meter spacecraft contracts to 8 meters (length contraction). Due to time dilation, her clock ticks slower, measuring 4 years for every 5 years that pass for Bob.

Alpha Centauri lies 5 light years from Earth. When Alice arrives, approximately 8 years have elapsed, but only 6 years have passed on her ship due to her slower clock.

Upon her return, another 8 years pass. In total, she has been away for around 16 years but has only aged 13 years.

Alice’s Frame

From Alice’s perspective, she and her ship are at rest, while Bob and Earth recede from her at near-light speed. Alpha Centauri approaches her at a similar speed, causing both Bob and the star to contract in her frame. The distance to the star is perceived as 4 light years.

In addition to time dilation and length contraction, Alice notices that the Alpha Centauri clock is ahead of Earth’s clock by 3 years. However, both clocks run at the same rate, displaying 4 years for every 5 of Alice’s years.

After 6 years, Earth is now 4 light years behind Alice, and the star’s clock reads 8 years, having advanced by 5 years from its initial 3-year lead.

Upon returning to Earth, another 6 years pass. Alice’s total time away amounts to 13 years, while Bob's clock, which has been running slow, shows 16 years of elapsed time.

Common Frame

Despite differing perspectives on the sequence of events, certain occurrences remain frame-independent:

• Both clocks read 0 when Alice departs.
• Alice’s clock shows 6 years upon reaching Alpha Centauri.
• The Alpha Centauri clock reads 8 years when she arrives.
• Upon returning, Alice’s clock reads 13 years.
• Bob’s clock registers 16 years when she returns.

These events share a characteristic: they are determined when Alice is at either location, the star or Earth. If Alice were to crash her ship into Alpha Centauri, both clocks would remain stuck at their respective readings of 6 and 8 years.

The Why

The Gamma Factor

As Alice trains for the Periluminal Marathon, both she and Bob examine how motion affects time.

Bob and Alice each utilize identical running clocks. A light pulse travels upward, registering a tick at the top.

As Alice runs, Bob perceives that the light from the base of her clock must traverse a longer distance to reach the top. Nonetheless, the speed of light remains constant for both clocks. Bob’s clock records one tick before Alice’s clock, leading him to conclude that Alice’s clock is running slow.

Conversely, Alice perceives Bob's clock as the slower one.

Thus far, we have qualitatively described this time dilation effect. Now, we will quantitatively analyze both frames.

The following diagram illustrates distances covered in terms of velocity multiplied by time, with subscripts denoting the respective reference frame. The speed of light, c, remains frame-independent, as does the relative velocity, v.

Using Pythagorean principles, we can derive a conversion factor between the two frames.

The resulting conversion factor is vital in relativity calculations, earning it the name Gamma Factor, symbolized by the Greek letter ?. In the following expression, velocity is expressed as a fraction of light speed, with half the speed of light denoted as v = ½.

In Bob’s frame, Alice’s clock ticks slower as she departs from Earth. We can quantify just how much slower.

Loss of Simultaneity

Both Bob and Alice can observe the Alpha Centauri clock using their powerful telescopes. Each sees the clock reading -5 years and adjusts for the time taken for light to travel from the clock.

Bob’s Perspective

Bob is aware that Alpha Centauri is 5 light years away. Light from the clock took 5 years to reach him. Since that light left 5 years ago, the clock must now read 0.

Alice’s Perspective

Alice faces a more complex situation. She recognizes that Alpha Centauri is 4 light years away, but it’s also moving towards her at near-light speed. How long ago did it leave the clock?

Alpha Centauri has covered a distance equal to half the distance the light has traveled. The light has traversed an additional 4 light years to reach Alice’s telescope, indicating it has been on its journey for 10 years.

Alice also accounts for the slower clock and applies the Gamma Factor of 1/?2, concluding that the clock has advanced 8 years since it read -5 years, leaving her with a final reading of 3 years.

Here is Alice’s generalized calculation:

The derivation of this formula utilizes time dilation and length contraction principles, and can also be approached from first principles, as detailed in “The Andromeda Paradox Explained.”

Conclusion

In summary, we have thoroughly explored and clarified the twin paradox. During Alice's expedition, both she and Bob perceive the other's clock to be running slow. However, the circumstances are not symmetrical; Alice alters her velocity three times throughout her journey, resulting in her being younger than Bob upon her return.