ASIMILATING THE "PARADOX" OF TWINS

Many of the articles on this blog describe fundamental physics phenomena that may seem too abstract or complex for a general audience. In this article, I aim to "bring things down to earth" and show the reader, in a simple way, that it is not necessary to resort to "strange" or "abstract" phenomena based on complex laws of theoretical physics to demonstrate how incredible and fascinating our Universe is. A simple trip from Earth to a nearby location just a few light-years away is enough. Below, we will see that it is not necessary to resort to new physics or "exotic" phenomena to discover impressive phenomena. Our real Universe, with the currently known and established physics, is already a Universe full of fascinating and extraordinary phenomena.

Traveling to "X-Centauri"

Next we will begin our journey, the famous journey popularly known as "the twin paradox": we (twin brother Alex) begin our journey on January 1, 2022 to a hypothetical planet called "X-Centauri" located three light years from Earth while our twin brother Alfonso remains on Earth. We agreed with our brother that every year, exactly at 00:00 on January 1, we will send him an image of us with a New Year's greeting and we ask him to do the same. We will assume that we have a ship capable of traveling at 60% of the speed of light (3/5c). Alfonso, after doing the math, expected to receive his brother Alex's greetings every year on January 1 and meet him again on Earth after 10 years of travel (10 * 3/5c = 6 light years traveled). However, this is what happens when his brother returns:

- Alfonso: We agreed that you would send me an image once a year at New Year's and I have only received eight: one every two years the first eight years and the other four the last two years of travel!

- Alex: Impossible! I sent you the image every year right after the New Year's Eve grape harvest.

- Alfonso: In each image your watch was going slower than mine

- Alex: Impossible! It was your images that showed your watch was behind mine.

- Alfonso: Thank goodness you kept your promise and came back just in 10 years.

- Alex: 10 years? But I've only been gone for 8 years! Why are you making fun of me?

What the hell is going on here?

Of course, both brothers are perplexed. They then decide to call their physicist cousin Alberto, who agrees to explain what really happened. To understand what's happening without resorting to cumbersome formulas, we must draw, step by step, the space-time trajectory of the two reference frames: Alfonso's on Earth and Alex's inside the ship. But first, we must understand two fundamental laws of the Universe we inhabit.

The two pillars of relativity

The two fundamental laws are the following: every observer, regardless of their state of motion, will always measure the same speed of light in vacuum c, and every inertial reference frame (without acceleration) is equivalent to any other. These laws seem simple, but their implications will change our way of seeing the world forever. Consider two twins at rest. One fires a laser at instant t=0, and while one remains at rest, the other twin begins to chase the light ray at a speed of 0.6c. Both must measure the same speed for the light ray; that is, both will see that the light ray remains at the same distance , regardless of the speed at which the twin chasing the light ray travels! For something like this to be possible, and for the constant v=space/time=c to remain constant for both reference frames, there is only one possibility: the twin at rest must measure that, for the other twin, space and time decrease as v increases. This "reduction" in space and time "compensates" for the "extra speed" possessed by the twin moving towards the light beam.

Therefore, in each reference frame, the twin considered at rest will see that for the traveling twin, the path length decreases and time slows down by an amount given by the Lorentz factor. In our case, v = 3/5c, and knowing that the Lorentz factor is:

We obtain a Lorentz factor of 5/4 . Therefore, time for the relativistic system will be contracted by a factor of 1/(5/4) = 4/5. The following table shows the time contraction corresponding to the twin traveling on the ship:

Therefore, while for the system at rest the journey takes 10 years , for the system

In a relativistic way, the journey takes only 8 years . According to the second law of relativity, both systems have the "right" to be considered at rest and to consider the other system to be the one moving. This produces an apparent paradox: Which of the two twins should age faster? As we will see, the "paradox" is resolved by considering the fundamental fact that only one of the systems remains at rest throughout the journey: the Earth's reference system.

The Earth's reference system

From the Earth's reference frame, the Earth is at rest, and the traveler is moving in space-time. We'll draw the spatial dimension on the x-axis and the temporal dimension on the y-axis. In the Earth's reference frame, the trip will last 10 years, so we divide the vertical axis into 10 parts:

Next we place our planet X-Centauri 3 light years away on the X axis. For the traveler the trip takes only 8 years (4 there and 4 back) so in the previous diagram we must draw two straight lines with 4 divisions each. The speed of 3/5c corresponds, in units of light years, to a straight line with a slope of x/y=3/5 therefore:

Now all that remains is to add the paths of the light rays and the lines of simultaneity between both reference systems. The path of a light ray corresponds to a slope of x/y=1, i.e., a 45° inclination, while the lines of simultaneity between both reference systems will be straight lines with a slope of 1/(3/5)=5/3. Drawing them gives us our complete space-time diagram:

The brown and blue 45° arrowed lines are the light ray paths sent between Earth and the spacecraft. The gray lines are straight lines with a 5/3 inclination and represent the lines of simultaneity from the traveler's point of view. The lines of simultaneity with respect to Earth are simply horizontal lines (not drawn).

Just by glancing at the space-time diagram above, we suddenly stumble upon that "strange" and fascinating Universe we mentioned at the beginning of this article. The first "strange" phenomenon is the following: if we look at the lines of simultaneity between both reference systems, we find that both the traveler and the Earth-bound observer find that time moves more slowly for the other. For example, the first year for the traveler occurs when only 0.8 years have passed on Earth (gray line), and vice versa (draw a horizontal line from the first division on the vertical axis).

How is this possible? Who's right? What the hell does this mean?

Strange phenomenon number two is no less striking: observing the vertical axis that represents the Earth's reference system, we can verify that the terrestrial twin receives a signal (annual from the point of view of the traveling twin) every two years for the first eight Earth years and two signals per year for the last two years.

This is a total of only 8 signs! How so? Can you imagine the Earthling's perplexity? The first thing he'll think is that his twin brother is cheating to outwit him.

Strange phenomenon number three is also shocking: right at the traveler's return point in the fourth year of the journey, the lines of simultaneity suddenly jump from 3.2 to 6.8 Earth years, that is, right at the instant when Alex changes his direction, Alfonso suddenly ages several years!

Strange phenomenon number four is already well known: when the twin brother returns to Earth and both brothers compare their chronometers, one finds that 10 years have passed and the other only eight years!

Below we will try to provide answers to these strange phenomena without resorting to complex formulas.

The spacecraft's reference system

Next, we'll draw the space-time diagram from the traveling twin's reference frame. Alex's journey consists of two very different parts:

Outward journey

On the outward journey, the "symmetry" with respect to the previous case is maintained: one system can be considered at rest and the other moving at constant speed. Now, Alex is considered at rest, and the Earth is moving away from him at -3/5c. Therefore, we draw the spaceship on the vertical axis and the Earth traveling through space on a straight line with a slope of -3/5c:

As can be seen, the graph is consistent with the previous case: the 4 years of the outward journey for Alex are only 3.2 years for Alfonso and in that time the Earth has moved 2.4 light years away (draw a horizontal line from the value 3.2 on the y-axis in the previous graph).

Return journey

But once we reach our destination, the "symmetry" with respect to the previous cases is broken: we must brake, turn, and accelerate again until we reach the same speed as on the outward journey but in the opposite direction.

We will assume that the braking-turning-acceleration intervals are very short, comparable to the rest of the trip (even though it would take 9 months to reach that speed accelerating at 1g), and we can ignore them (although, as we will see, these phenomena are essential to understanding the whole process). Exactly at the instant in which the return trip begins, the reference frame has changed; now the two systems are moving towards each other (since both systems end up meeting at the same "point" in space-time) . Because of this, the relative velocity between both systems is no longer -3/5c but -(3/5c)-(3/5c), which produces, using the formula for adding relativistic velocities, a velocity of -15/17c.

This explains why Alex receives messages from Alfonso more frequently when he returns!

This can be seen more intuitively by completing the previous graph: after 10 years from the Earth's point of view, both reference systems must meet. Therefore, we must draw a straight line from the point y=4 to the tenth division of the Earth's straight line. The final space-time diagram will be:

We can see that the slope of the line we've drawn is exactly -15/17. If we compare both diagrams, we see that they are the same: Alfonso receives the first four messages at a rate of one every two years and the other four at a rate of two each year, while Alex receives the first two messages every two years and the next eight at a rate of two per year. One of the reasons for this asymmetry is easy to understand: on the outward journey, Alex moves further away from Alfonso, and therefore the light beams take longer to arrive, while on the return journey, both systems move closer together and suddenly receive the missing light beams.

The other factor responsible for the asymmetry is purely relativistic: the jump in the lines of simultaneity is due to the sudden change of reference at the start of the return . At that instant, the twin changes from a "world line" with a certain sequence of space and time values to a different "world line" with a different space-time sequence. In fact, the Lorentz contraction factor on the return is 17/8, much larger than the one on the outward journey, 5/4. Therefore, at the very instant Alex begins the return, the values of space and time are instantaneously updated, which produces the jump in the lines of simultaneity.

Understanding the twin "paradox"

At this point, the reader (like the author) will surely have a strange feeling: How the hell is this possible? This feeling is due to the head-on collision between our everyday experience and the reality revealed by the laws of physics.

relativistic. How is it possible that both clocks don't coincide at the meeting point?

A more "intuitive" way to see it is to assimilate that our Universe is not the three-dimensional world that our senses show us, the Universe has 4 dimensions.

Let's look at the outward journey in the terrestrial reference system, for example:

The blue vector represents the Earth's space-time interval, and the red vector represents the spaceship's. The Earth twin "sees" 5 "time units" and 0 "space units," while the traveling twin "sees" only 4 "time units" and 3 "space units." Therefore, the traveling twin travels "slower" in the time dimension and therefore ages more slowly. Both twins see different "projections" of the same entity: an invariant four-dimensional vector.

A system at rest like the Earth travels practically 100% in time and 0% in space, while a traveler in a spaceship at 3/5c travels 3 parts of space for every 5 parts of time. This tells us that when they meet, the traveling twin has traveled less time and is therefore younger. However, the four-dimensional space-time interval does not vary ; it is the same for all observers. This can be easily seen by considering the four-dimensional interval (the magnitude of the vector in four dimensions), which is:

Considering only the x-axis we have that the four-dimensional space-time interval for the Earth is:

(S1)^2= (time^2,x-axis^2,y-axis^2,z-axis^2) (S1)^2=(5^2,0,0,0)= 25

While for the ship's reference system we have:

(S2)^2=(time'^2,x-axis'^2,y-axis'^2,z-axis'^2) (S2)^2=(4^2,3^2,0,0)=16+9= 25

For a four-dimensional being the space-time interval is always the same!

The "projections" of the same 4-dimensional vector (red line) in a reference frame at rest (green line) and in a relativistic reference frame with rotated axes (blue line)

Finally, we will try to answer the main questions that may arise regarding this apparent twin paradox:

- If both twins "see" that the other is aging more slowly, how is it decided which of the two is older at the meeting point?

The answer is that the situation is not as symmetrical as it appears: one of the twins is always in an inertial frame, while the other is subject to acceleration forces that can be measured locally. It is the latter twin who has aged the least and who will be younger at the meeting point. For this reason, the "twin paradox" is not really a paradox but is perfectly explainable by the theory of special relativity, since the two frames are not equivalent.

- What role does acceleration play?

Some works claim that it is the acceleration required to change reference frames that produces space-time dilation, since, due to the equivalence principle, it is as if the traveler's clock were immersed in a gravitational potential. However, it can be shown (for example, using two inertial reference frames traveling in opposite directions and meeting and exchanging clocks at the point of return) that this phenomenon can be explained using special relativity alone.

- Has time "really" slowed down for the traveling twin? That is, does the "biological" time that the traveling twin experiences also slow down?

All matter existing in the Universe, including of course our biological bodies, is made up of atoms. All atomic physical processes (except for a few nuclear phenomena) are electromagnetic processes based, at a fundamental level, on the exchange of photons traveling at the speed of light. Therefore, any magnitude of atomic distance, such as the Bohr radius, or any magnitude of atomic time, such as the transition time between different energy levels in an atom, are subject to the relativistic processes of space contraction and time dilation.

- Does the twin who remains on Earth have more time to "do things"? That is, if we recorded both journeys second by second, would we actually see that the traveling twin has lived less time?

Indeed, in the meeting, Alfonso has lived two years longer than Alex and has had more useful time than him, as demonstrated by the fact that he was able to send 10 messages, unlike Alex, who was only able to send 8.

If the reader is still not convinced at this point, they can verify that all this is correct by using the relativistic Doppler formula. The frequency shift will be:

For example, if we consider an average speed of v=12/13c we have that:

Therefore the average frequency of messages received by Alex is:

Which correctly tells us that during that period, at that average speed, Alfonso would have aged 13/5 more than Alex.

Sources: The twin paradox space time diagrams