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- I model the twin paradox with a straightedge and a compass.

To do this, I take the nonchalant trope that "the formula of the spacetime interval is like the formula of the Pythagorean theorem but with a minus sign" literally.

This is a response to a recent insight article regarding the twin paradox. The idea is to model the most basic scenario in a way, that the paradox disappears. This basic scenario and its kinematic effects are then modeled using only a straightedge and a compass.

The best way to read this post is to have a pen, paper, a compass, and a straightedge ready. Alternatively, the construction can be done in programs such as GeoGebra. The geometrically minded may also do the geometry in their head. For those with little geometric imagination, I provide a complete drawing at the end of the post that you can use for reference. However, I encourage you to really "do the geometry".

Before I model the twin paradox, I first strip the scenario of all elements that are usually part of the narrative. This includes the story of two twins, that are separated and reunited having aged differently. Secondly, I remove all references to observations (i.e. that one of the twins sees the other twin). Next, I remove all technical gimmicks such as rocket ships or telescopes that are used to flesh out the story. I will also remove something that might be unsuspicious of adding confusion: coordinate systems, coordinates, and frames of reference. Instead, I will model a spatial geometry on which observers who are embedded in this geometry must apply their relativistic coordinate systems. Lastly, I will remove the paradox from the title of the scenario: Instead of Twin Paradox, I name it the “three inertial particles scenario”.

In this scenario, I consider three particles ##P_1##, ##P_2##, and ##P_3## at three distinct events ##A##, ##B##, and ##C##. The events take place in a space without significant gravitational effects. For this model, an event is the coincidence of two particles in one place. To be in one place means, that two particles collide or pass each other very close. In this model, every event creates an electromagnetic signal. A signal propagates outward from the location of the event at constant speed in all directions. The three events are time-like related. This is what happens at and between the events:

At the first event ##A##, the particles ##P_1## and ##P_3## coincide.

After event ##A##, the particles ##P_1## and ##P_3## move in different directions.

Particle ##P_1## moves in the direction where the event ##B## will take place.

Particle ##P_3## moves in the direction where the event ##C## will take place.

At the second event ##B##, the particles ##P_1## and ##P_2## coincide.

After event ##B##, the particle ##P_2## moves on a trajectory that will eventually intercept particle ##P_3## at the event ##C##.

At the third event ##C##, the particles ##P_2## and ##P_3## coincide.

The model aims to determine the proper time the particles experience between the different events.

The particle ##P_1## experiences a proper time of ##t_1## between the events ##A## and ##B##.

The particle ##P_2## experiences a proper time of ##t_2## between the events ##B## and ##C##.

The particle ##P_3## experiences a proper time of ##t_3## between the events ##A## and ##C##.

To relate this scenario to the original "twin paradox" I compare the proper time ##t_3## with the combined proper time ##t_1 + t_2##.

In the intervals, ##AB##, ##BC##, and ##AC## between the events, the respective particles experience no acceleration. However, before or after the interval and at each event the particle may experience acceleration. This process of staging the particles for the three events is not part of the description.

If you do not yet have a compass and straightedge or GeoGebra ready, I urge you to do so. If you have time to spare, you should stop here and think for yourself how to draw the above scenario just with a compass and a straightedge. To prevent you from accidentally reading ahead, I have hidden further explanations behind the different spoiler buttons:

Hints

Modeling procedure,

related geometric concepts and graphical inspiration

Drawing and GeoGebra models

The first section "Hints" gives you just three very basic hints, which don't give away too much of the model.

The second section "Modelling procedure" is a complete step-by-step guide. I explain how to model the above scenario. If you just read this section you risk the feeling of disappointment that you did not even try to come up with the model yourself.

The third section "Related geometric concepts and graphical inspiration" can be read in two ways:

1) If you are not able to come up with a model by yourself you can read this section to get inspiration.

2) If you have either come up with the model by yourself or have read the previous section, you can check, if you can identify the related concepts before reading the section.

The final section "Drawing and GeoGebra models" contains just that: A drawing and links to two different GeoGebra models. Here you can check if your model is the same as mine without going through the complete modeling procedure.

If you decide to try and draw your own model, I give you three hints:

1) The idea is not to draw a Minkowski spacetime diagram.

2) The model requires only one drawing.

3) Since the three points are always coplanar, I can restrict the model to a plane without losing any relevant degree of freedom.
This is how I model:

I mark a point ##A## on the plane. From this point, I draw two rays ##b## and ##c## in arbitrary directions. Point ##A## marks the location where the particles ##P_1## and ##P_3## coincide (i.e. the event A). The two rays mark the directions of the particles ##P_1## and ##P_3## after event ##A## has taken place.

Next, I mark a second point ##B## that lies on one of the ray ##b##. This point marks where the particles ##P_1## and ##P_2## coincide (i.e. the event ##B##). The distance ##AB## is the spatial separation between events ##A## and ##B##.

In the next step, I model the temporal separation between the events. This temporal separation literally answers the question: How long after event ##A## does event ##B## take place? For this, I draw a circle ##c_1## around point ##A##. The radius ##r_1## of this circle must be greater than the distance between points ##A## and ##B## (i.e. ##r_1 > AB##). This circle signifies how far the electromagnetic signal emitted in the event ##A## propagates while the particle ##P_1## moves from ##A## to ##B##. If the radius of the circle is much larger than the distance ##AB##, the particle ##P_1## moves much slower than the signal propagates. The smaller the radius, the faster the particle moves. The speed of the particle can be expressed as ##v_1=AB \div r_1##. It should be noted, that this “speed” is a dimensionless ratio: The numerator (i.e. the distance ##AB##) and the denominator (i.e. ##r_1##) of the ratio are both distances.

After modeling the spatial and temporal separation, I can determine how much proper time ##t_1## the particle ##P_1## experiences as it moves from ##A## to ##B##: I apply the two distances ##AB## and ##r_1## to the formula of the spacetime interval:

$$\Delta s^2 =\Delta t^2 - \Delta x^2$$

Because the events ##A## and ##B## are time like related, I can plug in ##s = t_1##, ##\Delta t = r_1 ## and ##\Delta x = AB##. If I draw the square root, I get

$$t_1 =\sqrt{(r_1)^2 - (AB)^2}$$

The question is: Which distance or distances in the drawing represent (the square root of) the spacetime interval? If you don't see it immediately take your time. Draw some line segments. I will point it out in the next step. But once I tell it to you, the relation becomes obvious. Then the possibility for your own "aha moment" is gone.

To represent ##t_1## I draw a line segment that starts in point ##B## and that is perpendicular to the line segment ##AB##. The endpoint of this line segment lies on the circumference of the circle ##c_1##. It is not important to which side I draw this line segment. For reference, I will name the point where the line segment touches the circle ##c_1## point ##D##. I now have a right-angled triangle ##ABD##. The side ##AB## is the distance between the locations at which the two events take place. The side ##AD## of this triangle is the radius of the circle i.e. ##r_1## the distance the signal has propagated. This side ##AD## is obviously the hypothenuse of the triangle! The side ##BD## is the side for which the formula of the spacetime interval applies.

It is important to note, that the point ##D## and the line segments ##AD## and ##BD## have no spatial significance. Nothing moves along those line segments, nothing happens at point ##D##. The lengths of the line segments abstractly represent physical quantities: The line segment ##AD## represents the distance the signal has propagated (in all directions not only along the line segment). The length of the line segment ##BD## is proportional to the proper time that can be read of a clock moving with particle ##P_1##. I.e. ##BD = t_1##.

If the distance ##AB## is one light second, then ##t_1 = (BD \div AB) seconds##.

I can now repeat the same procedure for the particle ##P_2## and the interval ##BC##: First, I mark a point ##C##, that lies on the second ray ##c##. This point ##C## marks where the particle ##P_2## and ##P_3## coincide. I can connect points ##B## and ##C## with a line segment to visualize the trajectory of the particle ##P_2##. Next, I draw a circle ##c_2## with a radius ##r_2 > BC## around the point ##B##. I draw a line segment from point ##C##, that is perpendicular to ##BC## and that ends on the circumference of the circle ##c_2##. I name the point where the line segment and the circle meet ##E##. I have again a rightangled triangle ##BCE## The distance ##CE## represents the proper time ##t_2## that the particle ##P_2## experiences between the two events ##B## and ##C##.

The question is now: How much proper time ##t_3## does the particle ##P_3## experience between the two events ##A## and ##C##? The time that the particle experiences depends on the distance ##AC## and on the distance the electromagnetic signal from the event ##A## has propagated as the event ##C## takes place.

I know that the signal from the event ##A## propagates the distance ##AD## while the particle ##P_1## moves from ##A## to ##B##. After event ##B## the signal propagates the distance ##BE##, while the particle ##P_2## moves from ##B## to ##C##. So, I can conclude, that between the events ##A## and ##C##, the signal from event ##A## propagates a total distance of ##AD + BE##. Consequently, I draw a circle around the point ##A## with a radius ##r_3 = AD + BE##. Next, I draw a line segment that starts in point ##C##, that is perpendicular to ##AC## and that ends on the circumference of the circle ##c_3##. I name this point ##F##. I have again a right-angled triangle ACF. The distance ##CF## expresses the proper time ##t_3##, the particle ##P_3## experiences between the events ##A## and ##C##.

You can create different variations of this model and see, how the proper times ##t_1 + t_2## (i.e. ##BD + CE##) and ##t_3## (i.e. ##CF##) relate to each other.

If I assign a numerical value to ##t_1##, I can calculate ##t_2##, ##t_3## and the distances.

By replacing all circles with spheres, I can imagine the above scenario in a voluminous space. However, for visualization, it is easier to stick to a plane. If I look at the model on a plane, it is as if I look into a Minkowski spacetime diagram from the top.

In a way this model is lazy. It tacitly assumes, that there are clocks inside the particles that somehow register “the passage of time” without considering what that means. However, this laziness is due to the need to keep the model clear and simple. I could model the particles as moving light clocks in which electromagnetic signals propagate up and down or forward and backward. The distances ##BD##, ##CE##, and ##CF## express the proper distance that electromagnetic signals “move around” within the particles as the particles traverse the spatial separation between two events. If the particles are modeled to this level of detail, it is necessary to model particles contracted along their direction of movement.
In this model, the spacetime interval between two events is related to the concept of the power of a point as introduced by Swiss mathematician Jacob Steiner in 1826.

On a more fundamental level, the spatial representation of the spacetime interval is related to the theorems of intersecting chords and intersecting secants

The visualization is inspired by the works of French scientist Étienne-Jules Marey. Especially his studies of motion.
Drawing of the three particle scenario as described above:

Basic GeoGebra model: https://www.geogebra.org/classic/u99bxf5p

Animated https://www.geogebra.org/classic/pzgjv6qv
This model is simple. It is literally as simple as ##a^2 + b^2 = c^2## or rahter ##a^2 = c^2 - b^2##. It is so simple, that everybody who knows the math of SR kinematics could have come up with it. However, simplicity doesn't imply that it is a good or even a true model.

I am aware of the objections that can be brought up against this model. However, I leave it to others to bring forward these objections.

Regardless of whether you think this model is good or bad, true or false, I hope you can at least enjoy the model for its simplicity.

The best way to read this post is to have a pen, paper, a compass, and a straightedge ready. Alternatively, the construction can be done in programs such as GeoGebra. The geometrically minded may also do the geometry in their head. For those with little geometric imagination, I provide a complete drawing at the end of the post that you can use for reference. However, I encourage you to really "do the geometry".

Before I model the twin paradox, I first strip the scenario of all elements that are usually part of the narrative. This includes the story of two twins, that are separated and reunited having aged differently. Secondly, I remove all references to observations (i.e. that one of the twins sees the other twin). Next, I remove all technical gimmicks such as rocket ships or telescopes that are used to flesh out the story. I will also remove something that might be unsuspicious of adding confusion: coordinate systems, coordinates, and frames of reference. Instead, I will model a spatial geometry on which observers who are embedded in this geometry must apply their relativistic coordinate systems. Lastly, I will remove the paradox from the title of the scenario: Instead of Twin Paradox, I name it the “three inertial particles scenario”.

In this scenario, I consider three particles ##P_1##, ##P_2##, and ##P_3## at three distinct events ##A##, ##B##, and ##C##. The events take place in a space without significant gravitational effects. For this model, an event is the coincidence of two particles in one place. To be in one place means, that two particles collide or pass each other very close. In this model, every event creates an electromagnetic signal. A signal propagates outward from the location of the event at constant speed in all directions. The three events are time-like related. This is what happens at and between the events:

At the first event ##A##, the particles ##P_1## and ##P_3## coincide.

After event ##A##, the particles ##P_1## and ##P_3## move in different directions.

Particle ##P_1## moves in the direction where the event ##B## will take place.

Particle ##P_3## moves in the direction where the event ##C## will take place.

At the second event ##B##, the particles ##P_1## and ##P_2## coincide.

After event ##B##, the particle ##P_2## moves on a trajectory that will eventually intercept particle ##P_3## at the event ##C##.

At the third event ##C##, the particles ##P_2## and ##P_3## coincide.

The model aims to determine the proper time the particles experience between the different events.

The particle ##P_1## experiences a proper time of ##t_1## between the events ##A## and ##B##.

The particle ##P_2## experiences a proper time of ##t_2## between the events ##B## and ##C##.

The particle ##P_3## experiences a proper time of ##t_3## between the events ##A## and ##C##.

To relate this scenario to the original "twin paradox" I compare the proper time ##t_3## with the combined proper time ##t_1 + t_2##.

In the intervals, ##AB##, ##BC##, and ##AC## between the events, the respective particles experience no acceleration. However, before or after the interval and at each event the particle may experience acceleration. This process of staging the particles for the three events is not part of the description.

If you do not yet have a compass and straightedge or GeoGebra ready, I urge you to do so. If you have time to spare, you should stop here and think for yourself how to draw the above scenario just with a compass and a straightedge. To prevent you from accidentally reading ahead, I have hidden further explanations behind the different spoiler buttons:

Hints

Modeling procedure,

related geometric concepts and graphical inspiration

Drawing and GeoGebra models

The first section "Hints" gives you just three very basic hints, which don't give away too much of the model.

The second section "Modelling procedure" is a complete step-by-step guide. I explain how to model the above scenario. If you just read this section you risk the feeling of disappointment that you did not even try to come up with the model yourself.

The third section "Related geometric concepts and graphical inspiration" can be read in two ways:

1) If you are not able to come up with a model by yourself you can read this section to get inspiration.

2) If you have either come up with the model by yourself or have read the previous section, you can check, if you can identify the related concepts before reading the section.

The final section "Drawing and GeoGebra models" contains just that: A drawing and links to two different GeoGebra models. Here you can check if your model is the same as mine without going through the complete modeling procedure.

1) The idea is not to draw a Minkowski spacetime diagram.

2) The model requires only one drawing.

3) Since the three points are always coplanar, I can restrict the model to a plane without losing any relevant degree of freedom.

I mark a point ##A## on the plane. From this point, I draw two rays ##b## and ##c## in arbitrary directions. Point ##A## marks the location where the particles ##P_1## and ##P_3## coincide (i.e. the event A). The two rays mark the directions of the particles ##P_1## and ##P_3## after event ##A## has taken place.

Next, I mark a second point ##B## that lies on one of the ray ##b##. This point marks where the particles ##P_1## and ##P_2## coincide (i.e. the event ##B##). The distance ##AB## is the spatial separation between events ##A## and ##B##.

In the next step, I model the temporal separation between the events. This temporal separation literally answers the question: How long after event ##A## does event ##B## take place? For this, I draw a circle ##c_1## around point ##A##. The radius ##r_1## of this circle must be greater than the distance between points ##A## and ##B## (i.e. ##r_1 > AB##). This circle signifies how far the electromagnetic signal emitted in the event ##A## propagates while the particle ##P_1## moves from ##A## to ##B##. If the radius of the circle is much larger than the distance ##AB##, the particle ##P_1## moves much slower than the signal propagates. The smaller the radius, the faster the particle moves. The speed of the particle can be expressed as ##v_1=AB \div r_1##. It should be noted, that this “speed” is a dimensionless ratio: The numerator (i.e. the distance ##AB##) and the denominator (i.e. ##r_1##) of the ratio are both distances.

After modeling the spatial and temporal separation, I can determine how much proper time ##t_1## the particle ##P_1## experiences as it moves from ##A## to ##B##: I apply the two distances ##AB## and ##r_1## to the formula of the spacetime interval:

$$\Delta s^2 =\Delta t^2 - \Delta x^2$$

Because the events ##A## and ##B## are time like related, I can plug in ##s = t_1##, ##\Delta t = r_1 ## and ##\Delta x = AB##. If I draw the square root, I get

$$t_1 =\sqrt{(r_1)^2 - (AB)^2}$$

The question is: Which distance or distances in the drawing represent (the square root of) the spacetime interval? If you don't see it immediately take your time. Draw some line segments. I will point it out in the next step. But once I tell it to you, the relation becomes obvious. Then the possibility for your own "aha moment" is gone.

To represent ##t_1## I draw a line segment that starts in point ##B## and that is perpendicular to the line segment ##AB##. The endpoint of this line segment lies on the circumference of the circle ##c_1##. It is not important to which side I draw this line segment. For reference, I will name the point where the line segment touches the circle ##c_1## point ##D##. I now have a right-angled triangle ##ABD##. The side ##AB## is the distance between the locations at which the two events take place. The side ##AD## of this triangle is the radius of the circle i.e. ##r_1## the distance the signal has propagated. This side ##AD## is obviously the hypothenuse of the triangle! The side ##BD## is the side for which the formula of the spacetime interval applies.

It is important to note, that the point ##D## and the line segments ##AD## and ##BD## have no spatial significance. Nothing moves along those line segments, nothing happens at point ##D##. The lengths of the line segments abstractly represent physical quantities: The line segment ##AD## represents the distance the signal has propagated (in all directions not only along the line segment). The length of the line segment ##BD## is proportional to the proper time that can be read of a clock moving with particle ##P_1##. I.e. ##BD = t_1##.

If the distance ##AB## is one light second, then ##t_1 = (BD \div AB) seconds##.

I can now repeat the same procedure for the particle ##P_2## and the interval ##BC##: First, I mark a point ##C##, that lies on the second ray ##c##. This point ##C## marks where the particle ##P_2## and ##P_3## coincide. I can connect points ##B## and ##C## with a line segment to visualize the trajectory of the particle ##P_2##. Next, I draw a circle ##c_2## with a radius ##r_2 > BC## around the point ##B##. I draw a line segment from point ##C##, that is perpendicular to ##BC## and that ends on the circumference of the circle ##c_2##. I name the point where the line segment and the circle meet ##E##. I have again a rightangled triangle ##BCE## The distance ##CE## represents the proper time ##t_2## that the particle ##P_2## experiences between the two events ##B## and ##C##.

The question is now: How much proper time ##t_3## does the particle ##P_3## experience between the two events ##A## and ##C##? The time that the particle experiences depends on the distance ##AC## and on the distance the electromagnetic signal from the event ##A## has propagated as the event ##C## takes place.

I know that the signal from the event ##A## propagates the distance ##AD## while the particle ##P_1## moves from ##A## to ##B##. After event ##B## the signal propagates the distance ##BE##, while the particle ##P_2## moves from ##B## to ##C##. So, I can conclude, that between the events ##A## and ##C##, the signal from event ##A## propagates a total distance of ##AD + BE##. Consequently, I draw a circle around the point ##A## with a radius ##r_3 = AD + BE##. Next, I draw a line segment that starts in point ##C##, that is perpendicular to ##AC## and that ends on the circumference of the circle ##c_3##. I name this point ##F##. I have again a right-angled triangle ACF. The distance ##CF## expresses the proper time ##t_3##, the particle ##P_3## experiences between the events ##A## and ##C##.

You can create different variations of this model and see, how the proper times ##t_1 + t_2## (i.e. ##BD + CE##) and ##t_3## (i.e. ##CF##) relate to each other.

If I assign a numerical value to ##t_1##, I can calculate ##t_2##, ##t_3## and the distances.

By replacing all circles with spheres, I can imagine the above scenario in a voluminous space. However, for visualization, it is easier to stick to a plane. If I look at the model on a plane, it is as if I look into a Minkowski spacetime diagram from the top.

In a way this model is lazy. It tacitly assumes, that there are clocks inside the particles that somehow register “the passage of time” without considering what that means. However, this laziness is due to the need to keep the model clear and simple. I could model the particles as moving light clocks in which electromagnetic signals propagate up and down or forward and backward. The distances ##BD##, ##CE##, and ##CF## express the proper distance that electromagnetic signals “move around” within the particles as the particles traverse the spatial separation between two events. If the particles are modeled to this level of detail, it is necessary to model particles contracted along their direction of movement.

On a more fundamental level, the spatial representation of the spacetime interval is related to the theorems of intersecting chords and intersecting secants

The visualization is inspired by the works of French scientist Étienne-Jules Marey. Especially his studies of motion.

Basic GeoGebra model: https://www.geogebra.org/classic/u99bxf5p

Animated https://www.geogebra.org/classic/pzgjv6qv

I am aware of the objections that can be brought up against this model. However, I leave it to others to bring forward these objections.

Regardless of whether you think this model is good or bad, true or false, I hope you can at least enjoy the model for its simplicity.