Nature of project: **theory**, theory

Available to students on full-time physics degree schemes or joint students.

We are surrounded by objects that move much slower than light. For this, it is hard to grasp fully the laws of nature that properly describe the kinematics and dynamics of fast-moving objects, provided by the special theory of relativity. Since Albert Einstein published his works on special relativity, a radically new concept of space and time, and the beauty of the mathematical description of relativity have inspired countless physicists, mathematicians, philosophers, and others to explore a world that obeys relativistic laws. One of the most fascinating products of these studies is a long list of paradoxes. The pure comprehension of a paradox can already shed light into the theory. The resolution of it can further deepen our understanding of the significance and consequences of the relativistic equations. We can, however, exploit the most from relativistic paradoxes by working out a straightforward way of visualising the situation set up in the paradox. By that, the theory can be brought closer even to the wider public, who are fascinated by the peculiar world of relativity but lack the mathematical tools to solve the paradoxes by themselves. The student(s) will begin the project with an extensive literature review to read about paradoxes in special relativity. Particular attention should be given to the twin paradox. The project will be an extension of the twin paradox by adding complexity to the motion of the travelling twin to make the case more realistic. Once the mathematics of the problem has been established, the extended twin paradox will be resolved in a Minkowski space-time diagram and by using Lorentz transformations. Presenting relativistic events in two- and/or three-dimensional space-time diagrams is an effective way of studying and understanding a kinematic problem, and it eases our mental transition from traditional or classical thinking to the bizarre, nevertheless more realistic, relativistic comprehension.

*A successful project will develop beyond the above in one/some of the following directions:*

- Various cases of a triple can be investigated.

- The launch of the travelling twin can start at zero speed, and the acceleration can be constant, or the force on the rocket can be constant – they are different in relativity.

- A case study can be followed by a parameter study to find out which aspects of a journey contribute more to the different ageing of the twins.

- An effective way of visualisation could be the animation of the journey and the ageing of the twins as seen in different reference frames.

- The inverse problem can be solved: what dynamics of the rocket would reduce an age gap between two, or more, people to zero.

When considering where to take your project, please bear in mind the time available. It is preferable to do fewer things well than to try many and not get conclusive results on any of them. However, sometimes it is useful to have a couple of strands of investigation in parallel to work on in case delays occur.

*Additional scope or challenge if taken as a Year-4 project:* 2D and/or 3D motions add complexity to the problem, where even new paradoxes can be discovered and explored.

Please speak to **Balázs Pintér** (bap) if you consider doing this project.

*Initial literature for students:*

- Arunasalam, V., “Some remarks on the twin paradox in relativity”, 2008, Physics Essays, vol. 21, issue 1, pp. 6-8
- Grandou, T., Rubin, J. L., “On the Ingredients of the Twin Paradox”, 2009, International Journal of Theoretical Physics, 48, 1, 101-114
- Murphy, T. W. , “Confronting Twin Paradox Acceleration”, 2016, The Physics Teacher, 54, 5, 272-276
- de Wolf, D. A., “Aging and communication in the twin paradox”, 2016, European Journal of Physics, 37, 6, 065604

Although several versions of the twin paradox have been already considered, the relative mathematical complexity of the solutions makes it hard to pass the knowledge on to others, and the solutions are generally not available. The student(s) will build up a journey profile (speed versus time) and derive the answers to all the paradoxical questions for that case. Then, the results will have to be made publicly understandable by visualisation, which is a new objective. The Lorentz formulas are simple algebraic equations, but finding out the ways how to apply them to different paradoxical situations will make the project challenging. Difficulty also arises from the objective that the answers and explanations should be made understandable to others by visualisation.

Supervision and discussions can be done online.

milestone | to be completed by |
---|---|

Understanding the original twin paradox | end of October |

Designing a journey for one of the twins | end of November |

Representation of the journey in a suitable space-time diagram | end of February |

Creating clear, colourful, perhaps animated, illustrations of the journey and aging of the twins | mid-March |