Ben asked last time about definitions of force in special relativity. I did some hand waving about 4-vectors and \(d\tau\) as an invariant time ... but wasn't all that coherent.
The fundamentals are summarized at
Doing all this with the full calculus-with-four-vectors treatment is outside the scope of this class ... treat this as a preview of coming attractions or culture for now.
The stuff that you're responsible for in the context of this course is what's in the textbook, namely the stuff like
I also waved my hands a bit about the addition of veclocity formula, and claimed that combining two lorentz transformations is another lorentz transformation. I've attached a pencil scrawl with a bit more detail.
In the full four dimensional version the boosts may be in different directions things get more complicated. It turns out that boosts in two different directions are not the same as on boost - you need to also rotate the reference frames. If you include the three different sorts of spatial rotations (around the x, y, z axis) and the three different sorts of velocity boosts (along the x, y, z directions) then you get a 6 parameter set of transformations all of which are (sort of) "rotations" of space-time. That is the full Lorentz group.
If you also include translational motions, shifting the coordinate axis along (x, y, z, or t), you have another 4 sorts of transformations which should also leave the physics invariant. Those along with the Lorentz group is the Poincare group, a 10 parameter set of transformations of space-time.
Any description of fundamental particles like electrons and quarks and all that needs to built from math objects which behave in a consistent way under these transformations. The elements of the transformation (group) can be written as matrix - this is called a "representation" of the group, and is a well studied math property of groups. Different representations lead to different possibilities for consistent behaviors of fundamental particles ... and it turns out that this leads to some interesting constraints on how particles behave. In particular, the notions of "particles" and "anti-particles" and "spin" and "fermions" and "bosons" all arise from these constraints.
In other words, a sufficiently clever person, knowing just that the symmetries of physical laws are the Poincare group, could in principle predict some of the possible behaviors of quantum particles.
One last classic situation that is worth seeing - though a bit more math than I want to look at in full detail here - is a rocket which is undergoing constant acceleration as measured by someone in it.
The first point is that if you had such a rocket, you could go anywhere in the universe within your lifetime. (How much time would elapse on the earth is another story.) As you continued to accelerate, the remaining distance would shrink which would get you there faster.
The spacetime diagram for this situation is particularly cool and is intimately connected with the spacetime geometry of gravitational bodies (think black holes) which also produce uniform acceleration at a fixed distance from the center.
![[paper clip]](/cours/static/images/paper_clip_tilt.png)
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jim_addition_of_velocities.pdf | Sun Dec 22 2024 11:21 am | 321K |
