Acceleration is not the Issue
Special relativity is not only a mysterious interesting topic that stimulates nearly everyone’s curiosity but also offers a set of mathematical tools that sometimes are used for data projects. Mathematical objects such as Minkowski’s space are used in speech projects. Thus being familiar with this domain is not only intellectually intriguing but can provide data science’s virtues.
Why do I tell you this?
Several days ago, I heard in our corridor a discussion between two employees about the twin paradox (Yes, I am working with extremely educated and curious dudettes and dudes). During the conversation, one of the attendees said “it is not a paradox since the traveler accelerates“. Although most of my physics knowledge has been evaporated, I heard an alert sound in my mind. However, when I began to google “twin paradox” I saw a massive number of web sights that aim to clarify the paradox and focus on acceleration too. Indeed acceleration can help to create a time gap and it sounds like a coherent clarification… but it is erroneous (For the readers that are not familiar with the twin paradox, wait for the next section). As I read more websites, I found out that there is an absence of visualization for these explanations. This post aims to improve it and provide a more tangible explanation.
What is the Twin Paradox?
in 1905, shortly after he published the special relativity, Einstein wrote that If we take two twins and send one of them on a journey at a velocity that is near the speed of light while his brother remains on rest when the traveler will return from his journey he will be younger. This statement was given close to the prominent “announce” of relativity: ” everything is relative”. If so, then the traveler sees himself rest while earth and the resting twin went on a journey, thus the latter has to be younger as well: A paradox.
Well, you can guess that physics seldom allows paradoxes and Einstein is seldom wrong. However, when we encounter a topic that healthy brain people believe is a paradox, it is reasonable that it has an intellectually exciting and cool solution.
Before we move forward, I wish to point out that I will avoid using formulas in this post, but I recommend that the readers will be familiar with some notions:
- Gamma Factor
- Lorentz Transformation -The equation that conducts the transitions between two coordinate systems in the spacetime
- Spacetime interval. – The length of an interval in spacetime. Out of trivial linear algebra, the length of this interval is invariant under coordinate transformation.
- Simultaneity –A concept in relativity that shows that two simultaneous events by one observer are not neseecerily by another
- Time dilation
We are now to introduce spacetime. For simplicity we will assume a 1D space, nevertheless, it is easy to deduce that the entire rules hold for 3D space as well.
In the graph above we see the vanilla shape of spacetime. The yellow sloping lines represent the light speed. The x-axis represents the space (commonly in light years) and the Y-axis is time or more accurate time multiplied by the speed of light C
Since nothing can move faster than the speed of light, physical events can take place only in the upper and lower sections as seen below:
In order to move forward with the discussion on the twin paradox, we need to present the concept of coordinate systems. In special relativity, we always consider the invariant property of problems. For the twin paradox we consider the unprime and the prime system
Unprime refers to the twin that remains on earth and doesn’t move. It is also denoted as the rest frame
In this graph the magenta vertical line represents the rest frame. Since he assumes that he is fixed, only its clock varies forward.
In this graph we added the red sloping line that represents a traveller (the prime system) from the rest frame perspective. The dotted blue line represents a time level curve in which the resting twin observes the traveller.
Since we discuss special relativity I will represent the prime system perspective as well
Here we see the red line as our reference: The traveler is fixed in space and varies only on time. He observes the rest frame (magenta) going away at a fixed velocity. The dotted cyan line is a time-level curve according to the traveler’s coordinate system.
What is correct about the paradox?
As we wrote above the time in the watch of a high-velocity traveler moves slower than the time in his journey as it is measured by the rest frame. Since the traveler can consider himself a rest frame we obtain the vice versa result.
in the following section, we will analyze these phenomena and provide an objective illustration. For this study we take Wikipedia’s numbers: Assume a spaceship that moves in space at a velocity of 0.8*C. As it reaches the earth it preserves its velocity and synchronizes its watch with a person on earth. Now it flies to a star whose distance is 4 light years according to earth. Note that in the scenario we study there is no acceleration at all!!.
Let’s analyze from both perspectives:
We will use the notations that we already met: The magenta vertical line represents the observer on earth and the red line the spaceship traveler. According to the guy on earth, it will take the traveler 5 years to reach the star. The traveler will need according to his watch only 3 years. We can see this using two methods:
- Simply use Lorentz transformation which is presented by the cyan dot line.
- Using Spacetime interval: According to the guy on earth the path length is 3C (origin to (4,5) ), thus it must be 3C according to the traveler as well. But the traveler assumes that he is fixed thus its space coordinates are 0. Hence he experiences a path of length 3C which depends only on time. namely, it takes 3 years.
We can use the previous graph to explain his perspective
The traveler is again represented by the red line. He assumes that he doesn’t move and sees earth with the rest frame (the guy on earth) moving away at 0.8*C. The earth’s guy stands after five years in his position:(0,5). How much time according to the traveler passed on earth? We can use both spacetime interval or Lorentz transformation: The path’s length is 5C according to earth so it will be 5C according to the traveler. How much exactly? it appears that 8 years and 4 months and it is in the point (-20/3.25/3) according to the traveler system, which its length is exactly 5C.
We can summarize:
In every system, the observer’s watch moves slower than the watch that measures him in the other system
The graph beyond shows the general scenario: We have two travelers one with big speed the another with slow. We can see that every traveler measures a shorter time on his watch than the time in which he is measured by the other guy. We see however, that the measurements are not equal: the time measured by the prime guy on the unprimed guy is not equal to the time measured by the unprime on the prime
Relativity is relative but it is not a mirror!
Well then what is the solution?
In order to describe how things actually work we will observe the next graph
In the graph above we can see the timelines according to both systems: The rest frame (cyan horizontal lines) and traveler (red sloping line) The distance between every two adjacent lines is 1 with respect to the relevant system. We can see that the distance between two red neighbors is bigger than the distance between two cyans, This reflects that the clock of the traveler moves slower. Moreover, we can deduce that as the velocity increases the time goes slower as we see in the next graph
Here we added a slower travel object that is represented by magenta lines.
The reader who wishes to check this mathematically simply needs to play with the Lorentz equation to achieve these lines.
Summary & Conclusions
In this post, I have shown that in contrast to the common opinion on the web, time gaps between twins are not an outcome of acceleration. Furthermore one doesn’t need to be familiar with the notion of” inertial” in order to grasp this idea. It is merely about understanding Spacetime manners.
I wish to thank Irina Shalem for raising the challenge of this question and forcing me to think coherently.
The code of the plot can be found here.