Nature of project: **theory**, software

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

One of the most fascinating ways of understanding the special and general theories of relativity is through studying their ever-increasing number of paradoxes.

The lock and key paradox particularly shows the importance of putting away our intuitions that originate in a world where relative speeds between objects are much less than the speed of light.

The main message of the lock and key paradox is simple: a rigid object (distances between parts of the object do not change) is only a model that can be used in classical dynamics but does not work when studying fast objects. The real challenge of the project is to give a correct description of how information propagates inside the different parts of the lock and the key and what is actually happening in detail when the two objects ‘collide’.

The lock-and-key system is fundamentally two dimensional, which makes it challenging to visualise the collision in a Minkowski space-time diagram but it is worth making the effort.

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

- The details of the collision, derived from the Lorentz equations and Minkowski diagrams, could be nicely visualised by converting the results into animation.

- An extensive parameter study can be added to the project to determine the physical and geometric properties of the lock and key that result in a collision without breaking the lock and/or the key.

- The speed of information propagation in the two objects relative to the object can be less than the speed of light. The mathematics of such a more realistic model can be expected to be more complicated.

- The details of the collision of the key and lock can be derived in a inertial frame in which none of the two objects is stationary.

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:* There are different levels of how realistically the propagation of information from the position of collision is modelled. For example, assuming finite speed of information propagation perpendicular to the line of motion would provide a more accurate understanding of the kinematics of the collision than the 'semi 1D' model, used in the Y3 project.

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

*Initial literature for students:*

- Rindler, W., “Length Contraction Paradox”, 1961, American Journal of Physics, Volume 29, Issue 6, pp. 365-366
- Soler, D., “Reference Frames and Rigid Motions in Relativity: Applications”, 2006, Foundations of Physics, Volume 36, Issue 11, pp.1718-1735
- Pierce, E., “The lock and key paradox and the limits of rigidity in special relativity”, 2007, American Journal of Physics, Volume 75, Issue 7, pp. 610-614
- Syla, N., Klinaku, S., “Rigid body and the ‘length contraction paradoxes’ in special relativity”, 2017, Physics Essays, 30, 1, 97-99

The lock and key paradox is far from fully exploited. Only a handful of papers are available on the basic paradox, and none of them provides a detailed description of the dynamics. The main difficulty of the project is to work out how to put together the different stages of the collision, which are in a complex cause-and-effect relationship with each other.

Supervision and discussions can be done online.

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

Understanding the paradox and its resolution | end of October |

Finding a working model to describe the details of the collisions | Christmas |

Setting up the model, the related Lorentz equations and the space-time diagram for the model | end of February |

Creating a space-time diagram of the dynamics of the collision | Easter |