3d visualizations of paradoxes in special...

3D Visualizations of Paradoxes in Special Relativity

Matthew R. Cook

Institute of Mathematics and Physics

Aberystwyth University

Astrophysics (MPhys.)

3rd Year Project Report

6th May 2014

The purpose of this report is to show the way in which the author investigated the use and design of visualizations to explain the resolutions of paradoxes in Special Relativity. The focus of the report is on relativistic buoyancy and the paradox known as Supplee‘s paradox; in which a neutrally buoyant bullet in a body of water traveling at relativistic speeds either floats or sinks depending on the reference frame. The paradox is resolved by the notion of the lake floor curving upwards in the bullet’s reference frame to meet the already rising bullet. With the outreach of Modern Physics to the wider public becoming evermore popular, the understandability of visualizations is of the utmost importance. In this report the resolution is first derived mathematically and then visualized in a number of manners. In the case of relativistic buoyancy, it was found that the best way to truly represent the resolution to Supplee’s paradox is by using a tri-‐axial Minkowski diagram, which is very much like the 2D Minkowski diagram but with a third axis of position in the y-‐direction.

Matthew R. Cook 110010603 mac41

3D Visualizations of Paradoxes in Special Relativity 2

Table of Contents INTRODUCTION & THEORY ................................................................................................. 3

VISUALIZATIONS ........................................................................................................................ 3 Scientific Visualizations ..................................................................................................... 3

SPECIAL RELATIVITY ................................................................................................................... 4 The History of Special Relativity ........................................................................................ 4

Maxwell’s Equations ................................................................................................................................... 4 The Aether & Early Relativistic Theories ..................................................................................................... 4

Einstein’s Postulates ......................................................................................................... 5 Consequential Effects of Special Relativity ....................................................................... 6

Time Dilation ............................................................................................................................................... 6 Length Contraction ..................................................................................................................................... 7

Paradoxes in Special Relativity ......................................................................................... 8 VISUALIZATIONS OF SPECIAL RELATIVITY ........................................................................................ 8

PROJECT OVERVIEW ........................................................................................................... 9 PROJECT OUTLINE ..................................................................................................................... 9

Supplee’s Paradox ............................................................................................................. 9 LITERATURE REVIEW .......................................................................................................... 9

THE BASICS OF SPECIAL RELATIVITY ............................................................................................ 10 PARADOXES IN SPECIAL RELATIVITY ............................................................................................. 10 SUPPLEE’S PARADOX ................................................................................................................ 11 VISUALIZATION OF PARADOXES .................................................................................................. 12 CONCLUSION TO THE LITERATURE REVIEW ................................................................................... 13

MATHEMATICAL DERIVATION OF THE SOLUTION ............................................................. 13 USE OF THE EQUIVALENCE PRINCIPLE .......................................................................................... 13 UNPRIMED REFERENCE FRAME .................................................................................................. 14 PRIMED REFERENCE FRAME ...................................................................................................... 14 ALTERNATIVE METHOD ............................................................................................................ 15 USING GRAVITATIONAL FORCE ................................................................................................... 16

DESIGN & PRODUCTION OF VISUALIZATIONS ................................................................... 16 GRAPHICAL VISUALIZATIONS ...................................................................................................... 16

Avizo ............................................................................................................................... 17 Grapher ........................................................................................................................... 18 Choosing Grapher over Avizo .......................................................................................... 19

ANIMATED VISUALIZATIONS ...................................................................................................... 19 Serif Draw ....................................................................................................................... 20

Export Issue Fix ......................................................................................................................................... 20 Animated Tri-‐axial Minkowski Diagram ......................................................................... 21

RESULTS ........................................................................................................................... 21 DISCUSSION ..................................................................................................................... 24

GRAPHICAL VISUALIZATIONS ...................................................................................................... 25 ANIMATED VISUALIZATIONS ...................................................................................................... 27

CONCLUSION ................................................................................................................... 27 ACKNOWLEDGEMENTS .................................................................................................... 28 REFERENCES ..................................................................................................................... 28 APPENDIX ........................................................................................................................ 30



Introduction & Theory

Visualizations Visualizing concepts is a fundamental part of the way in which a human brain

accepts and understands new information. It has a key role in society as the method of sharing an idea from the inventor’s mind to the rest of the world. History is strewn with early examples of ideas that have been conveyed through visualizations. From cave paintings and Greek geometry to Leonardo Da Vinci’s revolutionary approach to technical drawings for engineering and scientific purposes (see Figure 11).

In the modern world visualizations are used in all walks of life, even the simplest of maps are an attempt to help the viewer understand where they are in relation to everything else. Vision is one of the most important and evolutionarily advanced senses of the body and therefore it is logical to come to the conclusion that visual imagery is one of the most affective ways to help the viewer understand a complicated premise.

Scientific Visualizations Scientific visualization techniques are predominantly pertained to visualizations of three-‐dimensional phenomena in all branches of science. One of the most well known early attempts at a three-‐dimensional scientific visualization is that of Maxwell’s thermodynamic surface2. Maxwell’s thermodynamic surface is a sculpture made from clay that describes the various states of a fictitious substance with similar properties to that of water. The shape of the sculpture is governed by coordinates of volume (x), entropy (y) and energy (z), which are based on the graphical thermodynamics papers of Josiah Willard Gibbs from 18733.

With the age of modern technology in full swing, scientific visualizations no longer require vast amounts of time to construct using clayed sculptures. They can be created using numerous different types of software. Modern three-‐dimensional scientific visualizations have copious amounts of usefulness within the research, education and medical communities. They are used to show information about anything from the internal workings of the body in PET scans to the results of a simulation of a Rayleigh–Taylor instability caused by two mixing fluids (see Figure 24).

Figure 1 – An example of Da Vinci's visualizations of his warfare designs

Figure 2 -‐ A scientific visualization of a simulation of a Rayleigh–Taylor

instability caused by two mixing fluids



Special Relativity

The History of Special Relativity By the end of the 19th century, physicists around the world were rejoicing in the joint accomplishment of nearly solving everything there was to solve about the Universe. Only a few small problems were left to fix and add to the current theories governed by Newtonian mechanics and the science of the Universe would be complete. However, one seemingly small spanner that was thrown into the works turned out to be the catalytic inspiration for one of the greatest upheavals of our collective physical paradigm in the history of science.

Maxwell’s Equations This apparently insignificant spanner came in the form of Maxwell’s equations that were published in "A dynamical theory of the electromagnetic field."5. There was one constant in Maxwell’s equations that would very soon become a tremendously important factor in the birth of modern physics.

Maxwell’s equations were devised to describe the nature of the then newly discovered electromagnetic wave, and the constant c was the speed at which electromagnetic waves would propagate through free space. The scientists of the late 19th century were perplexed by the notion that the constant c was the same no matter what reference frame it was measured in. It was thought that light propagated as an electromagnetic wave due to the similarities in the propagating speed6, and from the equations it seemed as though c was given without reference to any inertial observer, this lead to some conflicting ideas with Newtonian mechanics and Galilean transformations between different reference frames.

For example, if there is a person on a train passing a station throwing a tennis ball down the carriage, and someone on the platform is throwing another tennis ball parallel to the train, assuming they both throw their respective tennis balls with the same amount of force, the tennis ball being thrown on the train will be travelling faster than the one on the platform, because its speed is the speed at which it is thrown plus the speed of the train. According to Maxwell’s equations, this is not the case when it comes to light. If, instead of the speed of the tennis balls, it were the speed of a beam from a torch that was being measured, the speed of the beam would be exactly the same for both the train passenger and the person on the platform. This even applies if the train was travelling at 99.9% the speed of light, the light on the train would be no faster than the light on the platform in each observer’s reference frame.

This rather baffling and counter-‐intuitive thought experiment was to lay the foundations for Einstein to construct the two fundamental postulates of his theory of Special Relativity.

The Aether & Early Relativistic Theories The notion that c was the same for all observers was overlooked and neglected for 40 years or so, in the hope that an explanation would rise out of the new Quantum Theory being devised by Max Planck and various other attempts by well-‐known scientists to explain the black-‐body radiation problem 7 . However, no such



explanation was found within Quantum Theory. In the early 20th century, many physicists were working hard at trying to formulate a theory incorporating the fact that c was the same for all observers and they were getting very close to finding the answer. The first step that was to be taken was the understanding of the ‘Principle of Relativity’. This profound principle was conceived in order to explain the nonexistence of the mysterious ‘aether’, which at the time

was thought to be the medium in which light propagated as an electromagnetic wave.

Unfortunately, all experiments to find the aether had turned up negative, most famous of these being the Michelson-‐Morley experiment (see Figure 38) of 18879. Henri Poincaré was one of the first physicists to make the leap towards a theory

of relativity in his paper “The Measure of Time” (1898)10. The finite nature of c had to be taken

into account in the paper in order to develop a worldwide clock network that was synchronized using electric signals. Poincaré also gave suggestions to Hendrik Lorentz for

creating a formulation for electrodynamics, which explains the failure of all the aether drift measurement experiments. In Lorentz’s 1904 paper, “Electromagnetic phenomena in a system moving with any velocity smaller than that of light”11, he states the importance of the equations for transforming between two reference frames. This set of equations would come to be known as ‘Lorentz transformations’. Poincaré was the first to recognize that these equations belonged to a mathematical set, and came incredibly close to producing a full working theory of relativity. Unfortunately, Poincaré doesn’t get much recognition for this theory even though he anticipated a lot of Einstein’s approaches and terminology, because he still believed in the existence of the aether.

Einstein’s Postulates Einstein made a conscious decision to denounce the aether and abolished it from his theory. Instead, he postulated two fundamental principles12:

1. The Principle of Relativity – “The laws by which the states of physical systems undergo change are not affected, whether these changes of state are referred to the one or the other of two systems in uniform translatory motion relative to each other.”

2. The Principle of Invariant Light Speed – "... light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body."

These two postulates are the axiomatic basis for Einstein’s theory; other assumptions are to be made as well, such as the homogeneity, isotropy and memorylessness of space. With these two principles in mind, Einstein managed to derive the Lorentz transformation equations by using the axioms of relativity, instead of deriving them from a subset of the Poincaré symmetry group of isometrics

Figure 3 -‐ The setup of the Michelson-‐Morley Experiment



in Minkowski space-‐time, proving that his theory matched up with the mathematical frameworks of Lorentz and Poincaré.

Consequential Effects of Special Relativity The repercussions of Einstein’s theory are felt through all walks of life, one of the most important examples is the invention of the GPS or Global Positioning System. The reasons why Einstein’s theory is so essential for the accuracy of the GPS can be explained with a simple thought experiment to describe the first of many counter-‐intuitive effects that arise due to Special Relativity.

Time Dilation Time dilation is a genuine difference in elapsed time between two events as measured by observers moving relative to one another. Envisage a train travelling past a station platform at a speed close to the speed of light. In one of the carriages of the train, is a light clock. A light clock is a simple construction made of a mirror, a light source and a light detector; the premise of a light clock is simple, the light source produces a pulse of light in the direction of the mirror, which is then reflected back to the detector. The time taken for the light pulse to get from the source to the detector is then calculated using t=2L/c with L being the known distance between the source and the mirror (this can be seen in the top half of Figure 413). The effect of time dilation can be seen when observing the light clock from difference reference frames. From the point of view of a passenger on the train, the light clock is not moving and therefore the light pulse travels directly upwards and is reflected directly downwards. However, from the point of view of a person waiting on the platform, the light clock passes the station with a relative velocity v, and therefore the light pulse appears to travel a further distance than from the point of view of the passenger on the train (a diagram of this can be seen in Figure 4).

The obvious question arises that if the light has travelled further in the person on the platform’s reference frame and light has the same speed for all observers, how has the light travelled a greater distance when travelling at the same speed? The answer is counter-‐intuitive and rather beautiful in its intricacies. The consequential effect of the invariance of the speed of light is that time on the train is going slower than the platform in the platform’s reference frame. If time travels slower on the moving train then the light can travel a further distance even with the same speed. It can be said that time on the train has been expanded, stretched or dilated, hence the name of the phenomenon. The amount by which time is dilated is solely dependent on the relative velocity v, in the form of the Lorentz factor14:

Figure 4 -‐ (Above) View from inside the carriage (Bottom) View from the platform



The equation for time dilation is as follows and can be derived from the light clock thought experiment using simple algebra and Pythagoras’ theorem:

Where and are the time intervals for the light to reach the detector in the respective reference frames.

An easy way to remember the effects of time dilation is ‘the faster you travel through space, the slower you travel through time’.

Mindful of time dilation, the inventors of the GPS had to take into account that the satellites used would be travelling at high speeds relative to the user on the ground, and therefore would travel through time slower than the user. If this was not taken into account, the timings of the GPS would be incorrect and it would no longer give an accurate position for the user.

Length Contraction Another intriguing consequence of Special Relativity is that of length contraction. Length contraction can actually be derived from time dilation since the two effects are actually the same effect just seen from different reference frames15.

Picture a rod with proper length at rest in reference frame and a clock at rest in reference frame moving relative to one another along the length of the rod. The respective time intervals for the clock to travel along the length of the rod are

in and in , rearranging these equations gives:

By using the equation for time dilation from above the ratio between the two lengths can be found:

Therefore the length measured in is equal to:

This simple equation shows that the effect of time dilation on the clock in is interpreted as a length contraction of the rod in ( is always positive due to its definition, so a division by always equals a smaller number, hence a shorter or contracted length, which lends itself to the name of the phenomenon).



Paradoxes in Special Relativity Time dilation and length contraction lead to quite a few paradoxical thought experiments. This is usually down to the superficial application of the contraction formula. For example, the ‘Ladder’ paradox occurs due to the misinterpreted use of simultaneity16. The ladder paradox is the most commonly known paradox is Special Relativity; it involves a relativistically moving ladder and a garage with a front and a

back door. The ladder and the garage both have the same length when they are at rest, but the paradox arises when the ladder begins to move relativistically.

In the ladder’s reference frame, the garage is the one that is moving, and the garage is therefore length contracted, so the ladder no longer fits in the garage. Conversely, in the garage’s reference frame, it is the ladder that is moving, and the ladder is therefore length contracted, so the ladder easily fits in the garage. The resolution of the paradox is that the

events of the front and back doors seemingly trapping the ladder inside the garage to make it fit don’t actually occur at the same time in the ladder’s reference frame. The front door closes, with the back door still open (as the ladder’s rear is still outside due to the garage’s length contraction), then the front door opens and the back door closes (as the ladder’s front is now outside due to the garage’s length contraction, see Figure 517). Similarly, the Ehrenfest paradox is resolved by the concept of rigid bodies being incompatible with Special Relativity. The Ehrenfest paradox arises when a spinning disk of radius r spins relativistically; the circumference should length contract however the radius should not because it is perpendicular to the motion18.

Visualizations of Special Relativity The best way to visually express the solution to a paradox or any other relativistic situation is by using a Minkowski space-‐time diagram. Illustrating the paradox using Minkowski diagrams often allows for easier, more effective studying and understanding of the situation. Figure 619 depicts a typical Minkowski diagram showing an event E in two reference frames, S and S’, S’ is moving relative to S with velocity v. S is represented by the x and t axes and S’ is represented by the x’ and t’ axes. The blue line is the v=c line showing the theoretical limit for the speed of an object

Figure 5 -‐ The Ladder Paradox

Figure 6 -‐ A typical Minkowski diagram



in Special Relativity. As the speed of S’ increases, as does the angle , reaching its maximum at 45˚ matching the v=c line and its minimum when v=0, making S’ and S both at rest relative to one another and therefore have the same axes. Reading values from the S’ axes are just like reading values from the S axes, for example, by drawing a line parallel to the t’ axis from point E a value for x’ can be obtained. Likewise, drawing a line parallel to the x’ axis from point E gives a value for t’. This method of showing different reference frames for the same event can easily portray the differences in measured time and position due to the effects of time dilation and length contraction.

Project Overview

Project Outline The main aim of the project was to find a way of utilizing 3D imagery so as to describe paradoxes in Special Relativity in simple terms, so that the public could look at the visualization and understand what was going on. In order to allow for more focus on the 3D visualization side of the project, it was decided that only one paradox would be chosen to study. The key goals of the project were to derive a solution to the chosen paradox and design a 3D visualization of that derived solution. The 3D visualization would be constructed using relevant computer software and a 3D monitor.

Supplee’s Paradox The paradox that was chosen to be studied in detail in this report is known as Supplee’s paradox (or the ‘Submarine’ paradox). Supplee’s paradox is different in many ways to a lot of the better-‐known paradoxes of Special Relativity because it takes relativistic buoyancy into account. This area of Special Relativity is scarcely covered in many textbooks and has some rather intriguing outcomes. The paradox consists of a bullet (or a submarine) in a body of water, with the assumption that the bullet and the water are both neutrally buoyant (the bullet neither sinks nor floats upwards). The paradox arises when the bullet starts to move relativistically; in the water’s reference frame the bullet is length contracted, and therefore the density of the bullet increases, making it heavier than the water so it sinks and hits the bottom of the body of water. However, in the bullet’s reference frame it is the water that is moving and the water is therefore length contracted, increasing the water’s density, making it heavier than the bullet, so the bullet floats upwards. The bullet cannot sink and float at the same time; this is the essence of Supplee’s paradox. A more in depth description of the paradox can be found at the start of the Appendix.

Literature Review This section has already been marked previous to this report and is only included for completeness.

The only logical place to start the literature search was at Einstein’s paper on Special Relativity itself from 190520. It proved to be a very insightful paper showing the shear brilliance of Einstein at his very best in the middle of his Annus Mirabilis (Miracle



Year). The very premise of Special Relativity is laid out in this paper along with various equations describing the transformations in detail. However, as it is the first paper on the subject of Special Relativity, there is only a small number of references in it that lead to more useful papers. For further reading on the basics of Special Relativity, books were the way to go. Thus, a number of books were borrowed from the Physical Sciences library on the University of Aberystwyth’s main campus.

The Basics of Special Relativity A.P. French’s “Special Relativity” (CRC Press 1968) was a very good benchmark to start reading. It was filled with explanations and diagrams helping the reader to understand exactly what French was trying to convey. It also had concise and useful definitions of the consequences of Special Relativity, such as Time Dilation and Length Contraction. J.H. Smith’s “Introduction to Special Relativity” (Dover Publications Inc. 1996) was also crucial to understanding the basics of Special Relativity and helping the reader to comprehend the notation of Lorentz transformations. Further to the explanations from Smith, W. Rindler’s “Special Relativity” (Interscience Publishers Inc. 1960) had a very thorough mathematical approach to the workings of Special Relativity. It also had exceptionally in-‐depth explanations for all the phenomena of Special Relativity, albeit without many diagrams and illustrations. The literature search also lead to L.D. Landau’s “The Classical Theory of Fields” (Pergamon Press 1975); however, this only covered the same material if not less than the aforementioned books as it covered a wider range of physical problems than just Special Relativity; therefore, the authors did not spend much time with this text.

Hinckfuss’s “The Existence of Space and Time” (Oxford University Press 1975) has a more philosophical approach to explaining Special Relativity, starting with questions like ‘What is Space?’ and explaining the different essences of the word’s meaning. This is a very good read for the general public if they are interested in this field, ideas could be taken from this book as to how to approach the explanation of resolved paradoxes in the future. There was very little mathematics in the book itself, but rather good illustrations along with thorough explanations.

Paradoxes in Special Relativity The emphasis of the literature search was then changed to paradoxes in Special Relativity. The results of this search were many; there were dozens of different paradoxes to be studied, so the right place to start was with YQ Gu’s paper “Some Paradoxes in Special Relativity and the Resolutions” (2011)21. In Gu’s paper relativity is treated as nothing but geometry and a number of different paradoxes are resolved, namely the Ladder paradox, Ehrenfest’s rotational disc paradox, the Twins paradox and Bell’s spaceship paradox. The general conclusion of the paper is that all the paradoxes are caused by a misinterpretation of the relativistic concepts themselves; in other words, the paradoxes only arise because of our unwillingness to let go of our wrongly founded ideal of global simultaneity.

While on a trip to the library, one book stood out from the others in terms of its title, and that was Terletskii’s “Paradoxes in Special Relativity” (Springer 1968). This book proved to be very useful for learning about the paradoxes themselves but didn’t



really shed any light on anything that wasn’t already known from reading the other books in this literature review, and therefore was a little disappointing. Had this book been found first then it might have seemed to be a little more useful. However, it did give an equation for the hyperbolae used in Rindler charts, by calling them ‘Calibration Hyperbolae’.

At first it was decided that looking into a paradox that we had already come across would prove beneficial, so that we could understand the processes in resolving it mathematically. The paradox that was chosen for this purpose was the Ladder paradox; having come across the Ladder paradox in the form of the train in the tunnel, it was very easy to comprehend this paradox and visualize it mentally. Focusing on this paradox lead to W Rindler’s “Length Contraction Paradox” (1961). Rindler had resolved the Length Contraction paradox in question by creating what is now

known as a Rindler chart (see Figure 722). Rindler charts are based on Minkowski diagrams; moreover they are Minkowski diagrams for objects moving with a hyperbolic motion, using a co-‐ordinate system called Rindler co-‐ordinates, which represents a part of flat space-‐time. This different approach to visualizing the paradox was very intriguing, and leads the literature search to a number of different books by Rindler.

The first book of Rindler’s that was found was “Essential Relativity” (Van Nostrand Reinhold company 1969). This book is a very detailed account of everything to do with Relativity (Special and General). However, there was very little mention of the Rindler charts that had been seen in Rindler’s 1961 paper. It became very apparent that Rindler was a very well known scientist in the field of Relativity and that most of his books would be incredibly helpful. Rindler’s “Relativity” (Oxford University Press 2006) is technically the 2nd edition of the aforementioned “Essential Relativity”; however it proved to be much more useful. Providing numerous pages on Rindler charts, with equations for the hyperbolae and various diagrams showing Rindler charts and their characteristics.

Supplee’s Paradox As stated in the Project Outline, one paradox had to be chosen in order to allow for more focus on the 3D Visualization of the paradox’s solution. J.M. Supplee’s “Relativistic Buoyancy” (1989) paper really stood out from the rest of the paradoxes because it was more than just a Length Contraction or Time Dilation problem; there was gravity and a buoyant force to take into account. The number of relevant papers

Figure 7 -‐ A Rindler chart



on this subject is incredibly low in comparison to the previous success that had been seen with the other paradoxes. Supplee had stated the paradox and resolved in one short paper, and therefore it must have been assumed that no further work was needed on the matter. Supplee’s 1989 paper was very self-‐contained; everything that was needed to understand the paradox was in the paper, along with all the mathematics used to resolve the paradox. There was however a section for discussion in which Supplee states that because gravity is involved, General Relativity should be used instead of just Special Relativity in order to fully show that the paradox is resolved.

It wasn’t for another 14 years until G.E.A. Matsas published his paper “Relativistic Archimedes law for fast moving bodies and the General-‐relativistic resolution of the Submarine paradox” (2003)23. The mathematics in this paper was very hard to follow given that the authors of this review had never studied General Relativity before. However, there is one part of the paper that is incredibly useful and that was a Rindler chart of the paradox that had been used to model the solution. Matsas came to the same conclusion as Supplee did in his paper; the submarine would sink and hit the bottom of the body of water in question and also that the shape of the body of water would change so that the bottom of the body of water comes up to meet the submarine. Therefore, allowing for the differences in density due to different reference frames while keeping the actual outcome the same.

Visualization of Paradoxes Having found all the papers needed in order to solve the Submarine paradox, the literature search turned towards researching how the solution was going to be portrayed. This was primarily going to be done by using Minkowski diagrams and Rindler charts. Finding literature on Minkowski diagrams is exceedingly easy, being an essential part of Special Relativity naturally it is in almost every piece of literature on Special Relativity and therefore the books and papers mentioned previously in this review were suffice in explaining the concepts. Literature on Rindler charts on the other hand, is remarkably scarce. Even in Rindler’s books themselves, the idea of Rindler charts wasn’t given the pride of place that perhaps would have been expected. Furthermore, the majority of references to Rindler charts were in context to 2D Rindler charts. 3D Rindler charts are incredibly rare. The only reference to 3D Rindler charts that was found throughout the entire Literature search was in GEA Matsas’s 2003 paper.

Additionally, there will be an illustration of the paradox created in order to help the

general public understand what is actually happening in the paradox itself. The programming language in which this is to be done is yet to be determined, but there are numerous tutorials online that explain everything there is to know about 3D rendering.

Once the programming language has been determined, then a few coding ‘bibles’

Figure 8 -‐ Length Contraction in "The New World of Mr. Tompkins"



shall be selected such as K. Sierra’s “Head First Java” (O'Reilly Media 2005) or J. Zelle’s “Python Programming: An Introduction to Computer Science” (Franklin, Beedle & Associates Inc. 2010). G. Gamow’s “The New World of Mr. Tompkins” (Cambridge University Press 1999) provided a great deal of ideas on how to illustrate the strange world of Special Relativity (see Figure 824), the protagonist of the book travels to a world where the ‘speed limit of nature’ is in the region of 20 mph and therefore there are many different visual effects that the protagonist can see; for instance when he observes someone on a bike, or when he himself rides a bike.

Conclusion to the Literature Review In conclusion to this literature review, many valuable sources of information were found in the literature search. The basics of Special Relativity are very widely and readily available in countless books, papers, and articles and are now fully understood by the authors. Papers on paradoxes in Special Relativity are numerous but only a few get a good deal of coverage, namely the Ladder paradox, Ehrenfest’s paradox and the Twins paradox. The paradox chosen for this project is Supplee’s paradox (also known as the Submarine paradox or Bullet in Water paradox). There are only a couple of papers on this paradox, but they are incredibly concise and extremely useful. Literature on Rindler charts is scarce but Minkowski diagrams (that Rindler charts are based on) are well documented.

Mathematical Derivation of the Solution

Use of the Equivalence Principle Although the Equivalence Principle is founded within General Relativity, it is not implausible to use it in Special Relativity; the reason why it does not often occur in Special Relativity is due to the fact that most of the situations and paradoxes do not need to take into account the effects of gravity. However, in Supplee’s paradox

relativistic buoyancy is one of the key factors, and therefore gravity plays a vital role. The Equivalence Principle was stated as thus by Einstein in 1907:

“Assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system.”25

A clear example of this can be seen in Figure 926. There is no actual physical difference between the ball appearing to fall to the Earth due to gravity and the ball appearing to hit the bottom of the accelerating rocket ‘elevator’.

Figure 9 -‐ An example of the Equivalence Principle



Unprimed Reference Frame In order to work out what really happens in Supplee’s paradox, the mathematics of the situation was to be worked out in both reference frames. Following Supplee’s notation, in this report the unprimed reference frame is denoted by and the origin is defined at the base of the lake under the bullet when the bullet is fired. The bullet is said to be moving with relative velocity , and using the Equivalence Principle to represent gravity, the lake floor can be said to be accelerating upwards with acceleration . In order to calculate the position and time at which the bullet strikes the lake floor the vertical component of the buoyancy force was considered along with Newton’s 2nd Law of Motion27.

The vertical acceleration of the bullet in the unprimed reference frame was found to be:

Where is the Lorentz factor.

This acceleration is clearly less than that of the lake floor, thus making the bullet sink with relative acceleration:

Where .

Using the relative acceleration that the bullet sinks at, it was possible to determine the time at which the bullet strikes the lake:

Where is the starting height of the bullet.

A simple rearrangement and use of t yields the position of the impact:

The derivation for the above is in Appendix A.

Primed Reference Frame The primed reference frame was then to be considered. In accordance with Supplee’s notation, the primed reference frame is denoted by . The origin is defined as being at the base of the lake floor again, however, is moving with relative horizontal speed so the bullet is always at = 0 (viz. the origin

is coincident with ). The upward acceleration of any fixed point, at a constant , on the lake floor is . Just as before by using the Equivalence Principle, it can be said that gravity is just the equivalent to the acceleration.



The same method as the unprimed frame was used to find the acceleration of the bullet, the time of impact and the horizontal distance travelled before the bullet strikes the lake floor.

The vertical acceleration of the bullet in the primed reference frame was found to be:

The upward acceleration of the bullet is more than the upward acceleration of the lake, causing the bullet to appear to float.

In this reference frame the bullet does actually end up striking the lake floor even though it appears to float, because the shape of the lake floor has changed. In this reference frame the shape of the lake floor was found to be:

The reasoning for this can be seen in Appendix B.

Due to this reference frame moving with the bullet, the impact event happens at:

Therefore making the impact time:

If the equations for the impact’s time and position are Lorentz transformed to the unprimed reference, they equal the equations given for the impact’s time and position in the unprimed equations worked out above and in Appendix A.

This shows that the paradox is resolved; the bullet strikes the bottom of the lake at the same relative time and distance in both reference frames.

The derivation for the above can be seen in Appendix B.

Alternative Method For the primed reference frame, the same conclusion can be reach by another derivation. In the primed frame the lake floor is no longer flat and therefore the second derivative of the equation for the lake floor with respect to time is the upward acceleration of the lake floor and it is greater than the upward acceleration of the bullet. Therefore, the bullet sinks with a relative acceleration. So it follows from the calculations being done at that the time of the impact in the primed frame is:



This equation matches the equation derived in the ‘Primed Reference Frame’ section above. Thus proving this method is also correct and the paradox is resolved.

The derivation for the above is in Appendix C.

Using Gravitational Force The effects of gravity dominate Supplee’s paradox and therefore it is also necessary to provide a result that has used gravitational forces to reach the same conclusions. The net force acting on the bullet is the force due to buoyancy minus the gravitational force; if the net force is set as equal to the rate of change of the vertical component of the momentum then it is possible to determine the vertical acceleration of the bullet and was found to be:

From this equation the same method is used as in the unprimed reference frame to discover the time and position of the bullet’s impact with the lake floor.

The mathematical derivation for the above can be found in Appendix D.

Design & Production of Visualizations

Graphical Visualizations Once the derivation of the solution had been completed, the design and production of the graphs that would be used to explain the resolution could commence. The strategy for creating the graphical visualization was to produce a tri-‐axial Minkowski diagram using the equations derived in the previous section. Much like the standard 2D Minkowski diagram, a tri-‐axial Minkowski diagram has two pairs of time axes and x-‐axes for position in the x-‐direction, but with the addition of a third axis for position in the y-‐direction. The reason why there isn’t a pair of y-‐axes is because the bullet/water (depending on the reference frame) isn’t moving at a relativistic speed in the y-‐direction (its just accelerating due to gravity and the net buoyancy force). Therefore, and there is no need for a axis. Throughout the design process, many different people (scientists and members of the public alike) were asked on their preferences for the colours used in the Minkowski diagram with 90% of them saying the diagram was easier to read with a black background and brightly coloured lines.

For simplistic reasons, a few definitions were defined in order to create the graphs. The height at which the bullet would be fired from was defined as , the speed of the bullet was defined as , giving a Lorentz factor of . It is also common in Minkowski diagrams and Special Relativity calculations to take the speed of light as , this makes the graphs a sensible size to work with and simplifies



many of the equations for ease of use and was therefore also applied here. With all these definitions, the impact times for both reference frames were found to be:

The term ‘tri-‐axial’ is being used to describe this type of Minkowski diagram so as to not confuse the use of ‘3D’ (which is used to signify that the image appears to have visual depth). The reason for this necessary separation in expressions is to allow for the definition of a 3D projection onto a 2D plane. For example, drawing a cube on a piece of paper. The cube is a 3D object, but the drawing or projection of the cube is in 2D, therefore it would be incorrect to label the drawing of the cube as a 3D image because it shows no visual depth. Similarly, calling the tri-‐axial Minkowski diagram a 3D Minkowski diagram could cause confusion because it could be viewed on a 2D screen and would not actually be in 3D (in the same way the cube can be viewed on the paper, but is not actually in 3D).

Given the 3D aspect of the project it was required that a 3D monitor and some specialized 3D software would be needed to construct such a graph. The IMAPS department at the University of Aberystwyth already had a 3D monitor capability and thus, that set up was to be used in order to construct the graphs.

Avizo The software that was to be used was called ‘Avizo’ and was developed by FEI’s Visualization Science Group. Avizo is a very powerful piece of software with many functionalities, its main showpiece being that it can take a number of flat images (for example, a number of PET Scan images) and ‘volume render’ them together to create a 3D image. Mostly used for designing engine parts and other complicated 3D objects for Computer-‐Aided Design, Avizo also had a graphical section to its functionality that would plot data points and construct wonderfully coloured, smooth & seamless 3D graphs.



Figure 10 -‐ A screenshot of a typical Avizo 'Network'

In order for Avizo to manage to plot the lines on a graph a dataset of each line would first have to be made. This was simply done by creating an excel spreadsheet with a few hundred values and then exporting it as a comma separated variable (.csv) file. However, the method of getting Avizo to accept this data file proved to be a little bit more trouble. It appeared that Avizo only had protocols to deal with and accept two-‐dimensional .csv files and therefore did not interpret the three-‐dimensional .csv file it was being given correctly. The only remedy was to create a text (.txt) file with the data in it separated by spaces. This way the data could be given to Avizo as raw data. Once raw data is inside the ‘network’ of a project file it is much easier to manipulate the data format in order for Avizo to recognize it as a ‘line set’. Once Avizo recognizes the line set in can then be plotted in a ‘bounding box’, which has the dimensions of the maximum and minimum values of the dataset (see Figure 10). From the project’s ‘network’, local axis and labels can be added afterwards along with illuminated effects for when the project is in ‘3D Mode’.

However, there was a technical setback in the form of the 3D monitor’s drivers not working and the distribution of Scientific Linux the computer had as an operating system was unstable. This pushed the project back 3 weeks passed its planned date to start working with Avizo in mid-‐February 2014 while the problem was fixed.

Grapher During the 3 weeks without Avizo, another program was used to create a quick look at the equations derived in the resolution to the paradox. This program was called ‘Grapher’. Grapher is a free in-‐built piece of software that comes with every version of Mac OS X. It is a rather unknown piece of software due to its tucked-‐away location in the ‘Utilities’ folder and seemingly nonexistent advertising. However, it is an incredibly useful program.



Figure 11 -‐ A screenshot of a typical Grapher file

It was chosen due to its function plotter capabilities; instead of using a dataset formed of numbers from the equations to plot the graphs, this piece of software could simply be given the equations themselves and it would plot a continuous line from specified start and end values saving vast amounts of time (see Figure 11). However, there is no capability that allows for the creation of 3D images of the graphs and there is no functionality for labeling the axes, as strange as it may sound because it is such a simple functionality but the phenomenon is well documented on the internet forums with no sign of an implementation coming any time soon.

Choosing Grapher over Avizo 3 weeks of in-‐depth practice with Grapher and the slow, complicated processes needed to create the graphs in Avizo lead to the unfortunate decision of having to phase out the use of Avizo to create the final tri-‐axial Minkowski diagram. Had more time been available, all efforts would have been in creating a working 3D tri-‐axial Minkowski diagram animation on Avizo; however, Grapher was chosen to construct the final tri-‐axial Minkowski diagram as a 3D projection on a 2D plane (normal computer screen).

Animated Visualizations Although tri-‐axial Minkowski diagrams are the best way to express the resolution mathematically, not everyone is completely versed in reading Minkowski diagrams. Therefore, another form of visualization is needed to explain the paradox to the wider public. A common form of visualization used by scientists to explain concepts to the public is a simple animation of the situation. It was decided that two separate animations showing the paradox from the lake floor’s reference frame and bullet’s reference frame would be the best way in which the resolution was conveyed to the public. Bubbles would be used to depict if the water was moving or not, with bubbles floating upwards in the lake floor’s frame and bubbles moving sideways in the bullet’s frame. With the possibility of a few more animations showing the length



contractions and changes in density for each reference frame if there was time at the end of the project.

Serif Draw The program that was going to be used to create these animations was called ‘Serif DrawPlus X6’ and was developed by Serif Europe. The animations are created in a similar way to how a PowerPoint presentation works. There are a number of ‘frames’ (a lot like slides) that contain the objects that are to be animated (see Figure 12). The objects can be placed into different position on different frames and then all the frames are run together (like a really quick slideshow or a flipbook) making them look as though they are moving.

Figure 12 -‐ A screenshot of a typical Serif DrawPlus X6 project

Export Issue Fix Unfortunately, the exporter was not functioning correctly within the program when it was asked to export the animation as a video. Instead of converting the running animation into a .mov file it would generate a black and white pixelated version of the animation with no background (see Figure 13).

This was unacceptable as the animation that would be shown to the public. The fix for this problem was to take a screen capture video of the running animation while it was open in Serif DrawPlus itself. The program that was used to take the screen

Figure 14 -‐ The result of the Export issue Figure 13 -‐ A screenshot of the GifCam fix



recording was called ‘GifCam’; GifCam records an area of the screen and then converts it into a .gif file (see Figure 14). Given the popularity of using .gif files to view videos quickly and easily over the internet, the decision was made that the final animations would stay as .gif files so that if needs be they could be posted on all forms of webpage instead of needing a flash player like the .mov file did.

Animated Tri-‐axial Minkowski Diagram Another fantastically useful feature about Grapher was its ability to allow for a variable to be defined as a ‘parameter’ and then allowing that parameter’s value to be changed with a slider (see Figure 15).

Figure 15 -‐ A screenshot showing the parameter slider in Grapher

This feature was used to allow for easy manipulation of the value for in the tri-‐axial Minkowski diagram. The value for could be set at 0 and then the slider would then be able to change the value of to anything up to the time of impact; this would then change the plots of the graphs to the appropriate time of , allowing for the creating of an animation of the graph itself showing how it changes as time progresses. However, this animation could not be exported from the Grapher window and was recorded using a piece of software called ‘QuickTime’ to record the screen.

Results The mathematical derivation of the solution to Supplee’s paradox shows that the paradox is indeed resolved by the notion of the bullet always hitting the lake floor with the lake floor curving upwards to meet the already rising bullet in the bullet’s reference frame and the bullet sinking to hit the lake floor in the lake floor’s reference frame. Mathematical proof of this can be seen in Appendix E.



The tri-‐axial Minkowski diagram (see Figure 16) proved to be a fantastic way to visualize the paradox and was edited in many ways to produce different views of the resolution to the paradox regarding different values of and and an excellent explanation of why the lake floor curves upwards in the bullet’s reference frame.

Figure 16 is an angled view of the tri-‐axial Minkowski diagram created using Grapher and the equations derived in the above section of this report. The orange surface represented the shape of the lake floor in ANY reference frame and the red surface going across it is constructed of all the different values of that intersect the lake floor (orange surface). The blue line that starts at is the path of the bullet itself with a blue sphere to denoted the position of impact with the lake floor. The purple line that runs across both the orange and the red surfaces is the shadow of the bullet on the lake floor (if a light source was directly above the bullet so the shadow is directly below it). The other variations of the tri-‐axial Minkowski diagram can be seen in the Discussion section.

A very primitive version of a 3D tri-‐axial Minkowski diagram was also created on Avizo, but was not so clear in showing the resolution of the paradox (see Figure 17).

Figure 16 -‐ The tri-‐axial Minkowski diagram

x

t

y

x’ t’ v=c

Figure 17 -‐ A 3D image of the primitive tri-‐axial Minkowski diagram



The animations of the paradox clearly show the two reference frames that are in question and depict the resolution of the curved lake floor very distinctly. The links to where the full animations can be found are:

http://tiny.cc/lakefloorRF

http://tiny.cc/bulletRF

In Figures 18 and 19, screenshots of the animations can be seen. Figure 18 shows the start and finish points of the lake floor’s reference frame animation and Figure 19 shows the start and finish points of the Bullet’s reference frame animation.

The animation of the tri-‐axial Minkowski diagram also clearly shows the movement of the lake floor and the bullet, and portrays the paradox quite eloquently. The link to where the animated tri-‐axial Minkowski diagram can be found is:

http://tiny.cc/MinkDiag

A screenshot of the animated tri-‐axial Minkowski diagram can be seen in Figure 20.

Figure 19 -‐ Screenshots from the Lake floor's reference frame animation

Figure 18 -‐ Screenshots from the Bullet's reference frame animation



The lines in Figure 20 are just single values of and unlike the surfaces of every value of and from Figure 16. However, they still represent the same things:

• The path of the bullet in blue (with the blue sphere denoting the bullet’s location)

• The path of the bullet’s shadow in purple (with the purple sphere denoting the shadow’s location)

• The shape of the lake floor in the unprimed reference frame in orange (intersection of the orange surface from Figure 16 with a horizontal plane for a given value of )

• The shape of the lake floor in the primed reference frame in red (intersection of the orange surface from Figure 16 with an angled plane parallel to the axis for a given value of )

The animation of the tri-‐axial Minkowski diagram shows the upwards of acceleration due to gravity (using the Equivalence Principle) of the lake floor and the point of impact very evidently.

Discussion The mathematical derivation of the resolution to the paradox successfully proved that the paradox could be resolved by showing that the bullet does hit the lake floor in both reference frames, there are not many improvements that could be made to this process although it would be interesting to see the paradox resolved using

Figure 20 -‐ A screenshot from the animation of the tri-‐axial Minkowski diagram

x

t

y

v=c



General Relativity instead of Special Relativity in order to take gravity into account using the best theory of gravity at present and then construct a visualization using that resolution.

Graphical Visualizations The tri-‐axial Minkowski diagram was a complete success in clearly showing the resolution to the paradox. Many different variations were produced from the diagram allowing for multiple ways to explain the paradox.

The most useful accomplishment that was achieved by the tri-‐axial Minkowski diagram was the clear explanation as to why the lake floor is curved in the primed reference frame. This incredibly useful image can be seen in Figure 21.

The reason why the lake floor is curved in the primed reference frame is evident in Figure 21, the white plane shows all the points on the graph where (time of the impact in ). This means that anywhere were the shape of the lake floor (orange) intersects this plane is at . So at this specific time in the primed reference frame the shape of the lake floor is curved. This method also works in explaining the rather obvious notion of why the lake floor is flat in the unprimed reference frame. If a plane of (time of impact in ) were to be plotted, the points at which that intersects the orange surface would be the shape of the floor at

. Seeing as though is the z-‐axis, the plane would be horizontally across the orange surface and therefore the intersection (and the shape of the floor) would be a straight line (orange lines in other figures of the diagram).

Figure 21 -‐ Two different views of the tri-‐axial Minkowski diagram showing the curved nature of the lake floor in the primed reference frame

(Left) – A bird’s eye view of the diagram showing the XY plane and the time axis pointing out of the page (Right) – An angled view of the same diagram showing the intersection of the t’ = 0.218 plane more clearly

x

y

t

v=c y

x

t v=c



The Minkowski diagram also revealed why the paradox seemingly exists in the first place, ‘does it sink or does it float?’ being the question on every person’s lips when they first hear the paradox. Figure 22 uses information from the tri-‐axial Minkowski diagram in order to demonstrate the reason why the bullet appears to sink in the lake floor’s reference frame.

The key to understanding the paradox is getting the reference frames right. Reference frame is the frame in which everything else moves relative to, NOT the lake floor’s reference frame. It is important to stress this because the lake floor is accelerating upwards with acceleration to simulate gravity.

The tri-‐axial Minkowski diagram shows this by using the path of the bullet (blue) and the shadow of the bullet on the lake floor (purple). If the y component of the purple

line were to be taken away from the y component of the blue line, the result would be the white line in Figure 22. This is taking away the simulation of gravity, and therefore the lake floor is shown as being flat at in Figure 22. The white line is the path the bullet would take in the lake floor’s reference frame NOT in , thus showing that the bullet does sink in the lake floor’s reference frame, but not in the bullet’s reference frame.

Figure 23 shows a different angle of the diagram looking down the axis ( ). This figure shows that the bullet (blue) doesn’t move horizontally in the primed reference frame, further verifying the validity of the tri-‐axial Minkowski diagram. This same motion can be seen in the primed reference frame animations produced using Serif DrawPlus X6.

The only obvious improvements that could be made to this diagram would be to create a 3D image of it using Avizo, had more time been available this would have been a good way to get the public involved in understanding Special Relativity and its paradoxes by enticing them with 3D images, which have always seemed to fascinate the human mind.

Figure 22 -‐ 'Does it sink or does it float?'

y

t

x

Figure 23 -‐ Looking down the t' axis, showing the bullet is always at x' = 0

y

x t

t’ v=c



Animated Visualizations When the animations of the paradox were shown to the public they were very well received. A short talk on the essence of the paradox and the animations themselves proved to be a great combination in explaining the paradox to a wider audience. The simple yet aesthetically pleasing look of the animations is very easy on the eye and doesn’t scare the average person away from a complicated concept, which they otherwise might not have shown any interest in.

The animations themselves couldn’t be improved much more, perhaps only in the addition of other graphics such as shadows and reeds floating in the water at the bottom of the lake. However, more animations of the situation could be created to improve the ability of explaining the paradox to the public; most probably animations of length contraction and what is meant by ‘density’ and ‘net buoyancy force’ would be useful additions. For example, an animation of the length contraction of the bullet showing the molecules getting closer together and therefore making the bullet more dense, which decreases the net buoyancy force making the bullet sink.

Conclusion In conclusion to this report, a mathematical derivation of the resolution to Supplee’s paradox has been found using numerous methods including the Equivalence Principle and gravitational forces, a 3D projection of a tri-‐axial Minkowski diagram has been created using the mathematically derived equations of the resolution of Supplee’s paradox (a feat never achieved before), and a number of animations of the situation have been created in order to aid in explaining the resolution of the paradox to a wider audience.

The mathematical derivation of the resolution using Lorentz equations reaches the same conclusion found by Supplee himself and states that the lake floor curves up to meet the already rising bullet in the primed reference frame in order for the impact to take place at the same relative time and location in both reference frames.

The tri-‐axial Minkowski diagram proved to be an invaluable way of visually expressing the mathematical reasoning behind the lake floor being curved and aiding the viewer in grasping the concept of the paradox. Multiple variations of the Minkowski diagram were created in order to show different values of and allowing for an animation of the Minkowski diagram and various other forms of features that verified the validity of the diagram.

The animations of the paradox proved to be a priceless asset in facilitating the explanation of the resolution of the paradox to the public and the only improvements to the animations that could be made are aesthetic. The use of these simplistic animations in explanations of the paradox was well received by a number of members of the public with no prior knowledge of Special Relativity. Thus confirming the idea that simple visualizations are one of the best ways to convey a complicated concept to another person.



Acknowledgements The Author would like to thank:

• Project partner Mr. Timothy E. A. Powell for his collaboration throughout the project, his input in the visualization of the animations and his enthusiasm towards the outreach of science to the public.

• Dr. Balázs Pintér for priceless advice throughout the project, the use of three of his books during the literature review and his passion for the visualizations of paradoxes in Special Relativity that made this project conceivable.

• Mr. Dave Price for his incredible work ethic in fixing and setting up the Avizo software needed for this project.

• Dr. James M. Supplee for general guidance and feedback during the visualization stage of the project.

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15 Halliday, David, et al. “Fundamentals of Physics, Chapters 33-‐37”. John Wiley & Sons, (2010). 16 Rindler, Wolfgang. “Length Contraction Paradox”. American Journal of Physics 29 (1961): 365f 17 Wikimedia – Ladder Paradox diagram http://upload.wikimedia.org/wikipedia/commons/thumb/2/25/Ladder_Paradox_LadderScenario.svg/500px-‐Ladder_Paradox_LadderScenario.svg.png 27/4/2014 18 Grøn, Øyvind. “Einstein's General Theory of Relativity”. Springer (2007): 91 19 Relativity – David Eckstein http://www.relativity.li/uploads/images/K/K11_1.jpg 27/4/2014 20 Einstein, Albert. Op. cit. 21 Gu, Ying-‐Qiu. "Some Paradoxes in Special Relativity and the Resolutions." Advances in Applied Clifford Algebras 21.1 (2011): 103-‐119. 22 Rindler, Wolfgang. “Essential Relativity”. Van Nostrand Reinhold Company (1969) 23 Matsas, George EA. "Relativistic Archimedes law for fast moving bodies and the general-‐relativistic resolution of the “submarine paradox”." Physical Review D 68.2 (2003): 027701. 24 Gamow, G. “The New World of Mr. Tompkins”. Cambridge University Press (1999) 25 Einstein, Albert. "How I Constructed the Theory of Relativity". Translated by Masahiro Morikawa from the text recorded in Japanese by Jun Ishiwara, Association of Asia Pacific Physical Societies (AAPPS) Bulletin 15.2 (2005): 17-‐19. 26 Wordpress http://thinkingscifi.files.wordpress.com/2012/07/loadbinary.gif 27/4/2014 27 Newton, Sir Isaac. “Philosophiae naturalis principia mathematica” (1687)

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Smith, J. H. “Introduction to Special Relativity” (Dover Publications Inc. 1996)

Rindler, W. “Special Relativity” (Interscience Publishers Inc. 1960)

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Appendix Consider a bullet and a body of water with equal density . The bullet would be neutrally buoyant when it is fully submersed and at rest in the water. Now consider the same bullet being fired horizontally through the water at a relativistic speed. Due to the speed at which the bullet is travelling its relativistic mass ( ) is increased by the Lorentz factor , and it is length contracted in the direction of motion by . Therefore, the density of the bullet traveling at relativistic speed is

, whilst the density of the water stays the same, as it is at rest. Owing to the relativistic density of the bullet is greater than that of the water it is traveling through and thus the bullet sinks.

If the inertial frame at which this situation described above is changed to that of the bullet having an initial speed of zero, the resulting density of the bullet is the rest density, , and of the moving water is . It follows that the bullet now has a density less than that of the moving water, causing it to float. This is the Relativistic Buoyancy paradox. The solution to this paradox is that the calculations in the two inertial frames are in agreement.

This paradox addresses a theoretical exercise in relativity and simplifies the problem by ignoring viscosity and wake.

A. As stated above the unprimed reference frame considers the paradox when the bullet is moving with relativistic speed horizontally and the body of water is at rest. The upward acceleration of the lake is . The bullet is fired at .

The buoyant force on the bullet is:

(A.1)

Where is the buoyant force and is the rest volume. The rest density is:

(A.2)

Substituting (A.2) into (A.1) gives:

(A.3)

This buoyant force is the force acting upward on the bullet in the water. The force acting downward on the bullet is due to Newton’s second law of motion:



(A.4)

(A.5) When the bullet is at rest , therefore (A.3) becomes:

Therefore, when the bullet fully submersed and at rest in the body of water it is neutrally buoyant.

Now the component of Newton’s second law needs to be found. It can be written as:

(A.6)

Where is the speed in the direction. As both and depend on the product rule has to be applied.

(A.7) The dot in (A.7) denotes differentiation with respect to time. Equation (A.6) is in immediate agreement with equation (2) from Supplee’s Relativistic Buoyancy paper.

In (A.7) has both and components, but when is constant (which it is in this scenario) the component drops out as the time derivative of in the direction is 0 when is constant. The derivation for in the direction is as follows:

(A.8)

(A.9)

The method used to differentiate (A.8) was the chain rule and can be seen in (A.10):

(A.10)

Where:



(A.11)

(A.12)

To differentiate with respect to you need to use the product rule:

(A.13)

The differentiation of with respect to :

(A.14)

Combining (A.13) and (A.14):

(A.15)

Equation (A.15) is in immediate agreement with equation (3) from Supplee’s Relativistic Buoyancy paper. Substituting (A.15) into (A.7):



(A.16)

Following that then:

Which means:

(A.17)

Equation (A.17) is in immediate agreement with equation (4) from Supplee’s Relativistic Buoyancy paper.

Combine (A.3) and (A.17) to find the upward acceleration of the bullet:

(A.18)

Equation (A.18) is in immediate agreement with equation (5) from Supplee’s Relativistic Buoyancy paper. As the lake is accelerating upwards at and the bullet at the bullet sinks and impacts the bottom of the lake. The bullet sinks at a rate of:

(A.19) . Where is rate at which the bullet sinks and . From the rate at which the



bullet sinks it is possible to work out the time and position the bullet strikes the lake floor. This is done by using the constant acceleration equations:

(A.20)

Where is the height at which the bullet is fired from, is the vertical acceleration of the bullet and is the time the bullet strikes to floor of the lake. Equation (A.20) can only be used when starting from rest at position zero. (A.20) can be rearranged to equal :

(A.21)

In this situation and . Substitute (A.19) into (A.21):

(A.22)

Equation (A.22) is in immediate agreement with equation (7) from Supplee’s Relativistic Buoyancy paper. The total horizontal distance, , (the velocity of bullet multiplied by time) can be calculated using (A.23) and (A.22):

(A.23)

(A.24)

Equation (A.24) is in immediate agreement with equation (8) from Supplee’s Relativistic Buoyancy paper.

This is the end of the derivations for the unprimed reference frame.



B. As stated above the primed reference frame considers the paradox when the bullet is at rest and the body of water is moving with relativistic speed horizontally. The upward acceleration of any fixed point on the lake floor is:

(B.1)

In the primed frame the water’s density is

(B.2) and the buoyant force on the bullet is

(B.3)

The bullet has it’s rest volume, , in the primed frame (B.3) can be combined with (B.2), (B.1) and (A.2):

(B.4)

Equation (B.4) is in immediate agreement with equation (12) from Supplee’s Relativistic Buoyancy paper. From (B.4) it can be determined that the upward acceleration of the bullet is . This means that the upward acceleration of the bullet is larger than the upward acceleration of the lake floor resulting in the bullet to float in the primed frame.

This paradox can be resolved by considering the shape of the lake floor. In order to discover the distance the lake floor travels upwards, the double integral of the vertical acceleration with respect to time is calculated:

(B.5)

Where are all constants.

When , :



Leaving (B.5) being:

(B.6)

As the speed in the vertical direction is then , making the equation of the floor of the lake, (B.6):

(B.7)

(B.7) can be Lorentz transformed. The Lorentz transformation equations for and are:

(B.8)

(B.9)

Using (B.8) and (B.9) with (B.7) yields:

(B.10)

Standard kinematics can now be used (as to determine if, and when and where the bullet strikes the floor of the lake. The constant acceleration equations can be used to find the bullet’s location:

(B.11)

Where is the location of the bullet, is the height, , at which the bullet is from the lake bottom, is the vertical acceleration of the bullet and is the time the bullet strikes the lake floor. Taking all of this into account, (B.11) can be rewritten:

(B.12)

The impact happens at

(B.13)

(As the coordinates of the primed frame ‘stay with the bullet’) and when the coordinates of the bullet (B.12) and the lake floor (B.10) are equal to each other.

(B.14)

As , (B.14) becomes:



(B.15)

Rearrange (B.15) to find the time the bullet impacts the lake floor.



(B.16)

Equation (B.16) is in immediate agreement with equation (19) from Supplee’s Relativistic Buoyancy paper and is the time the bullet strikes the lake floor in the primed floor.

C. There is a second method to reach the same result when working in the primed frame. This method considers the shape of the lake floor, as it is no longer flat. The second derivative, at constant , of the equation of the lake floor (B.10) is the upward acceleration of the lake floor. Lorentz transformation needs to be used at is constant but the calculations are in the primed frame. The Lorentz transformation equation for into is:

(C.1)

Substitute (C.1) into (B.10):

(C.2)

The easiest way to differentiate (C.2) is to use the chain rule:

(C.3)

Where:

(C.4)



(C.5)

Combine (C.2) and (C.4):

(C.6)

Substitute (C.6) and (C.5) into (C.3):

(C.7)

Differentiate (C.7) with respect to time:

(C.8)

Equation (C.8) is the same as equation (B.1) and equation (20) from Supplee’s Relativistic Buoyancy paper.

If you ‘stay with the bullet’ and find the second derivative, at constant , of the equation of the lake floor (B.10), the upward acceleration of the lake floor in the primed frame is found. As equation (B.10) can be rewritten:

(C.9)

Differentiate (C.9) with respect to time:

(C.10)

The upward acceleration of the lake floor (C.10) is greater than the upward acceleration of the bullet ( ). Therefore, the bullet sinks with relative acceleration:

(C.11)



Using (C.11), the height the bullet is fired from, , and simple kinematics in the form of the constant acceleration equation (A.20):

,



(C.12)

Equation (C.12) is in immediate agreement with equations (B.16), and (23) from Supplee’s Relativistic Buoyancy paper.

D. The solution can be derived another way, not using the equivalence principle, but by using gravitational forces. Here the force acting on a particle in a constant gravitational field is:

(D.1)

The gravitational field is a weak uniform field; this leads to the following conclusions:

(D.2)

(D.3)

(D.4)

Where is the gravitational potential. The gradient operator, , is the usual Cartesian operator.

Substitute (D.2), (D.3) and (D.4) into (D.1):

(D.5)

Since (D.5) becomes:

(D.6)



The net force on the bullet is the buoyant force (upward) (A.3) minus the gravitational force (downward) (D.6):

(D.7)

Setting the net force equal to the time rate of change of the vertical component of the momentum is:

(D.8)

Equation (D.8) agrees with (A.19). The minus sign is due to the direction the positive direction is taken. Therefore, the bullet strikes the bottom with coordinates given above ((A.22) and (A.24)).

E. The Lorentz transforming equations for and are:

(E.1)

(E.2)

Using (E.1) to Lorentz transform (B.13) into the unprimed frame gives:



therefore:

(E.3)

(E.3) is equal to (A.23) therefore, the bullet impacts the floor of the lake at the same position in the Unprimed reference frame and the Primed reference frame.

Using (E.2) to Lorentz transform (B.16) into the unprimed frame gives:

(E.4)

Substitute in (A.23) into (E.4)

(E.5)

(E.5) is equal to (A.22) therefore, the bullet impacts the floor of the lake at the same time in the Unprimed reference frame and the Primed reference frame.

The calculations between the two inertial frames agree. Therefore, the paradox is solved.

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