Relativistic rocket

Well, not exactly. I am not accounting for loss of mass for propulsion. Here I am just writing down the relativistic motion with constant proper acceleration. So Alice is riding a rocket that has some kind of propulsion such that she experiences a constant force. Her acceleration is: \(\alpha^{\prime\nu}=(0,\alpha)\), where \(\alpha\) that does not depends on proper time \( \tau \), which is a faithful parameter for the motion in all reference frame.

Let \( \beta \) be Alice's instantaneous velocity with respect to an observer Bob. The acceleration in Bob's frame is:

\[ \alpha^{\mu} = \Lambda_{\nu}^{\mu} \alpha^{\prime \nu} = \gamma \begin{pmatrix} 1 & \beta \\ \beta & 1 \end{pmatrix} \begin{pmatrix} 0 \\ \alpha \end{pmatrix} =\begin{pmatrix} \alpha \sinh w \\ \alpha \cosh w \end{pmatrix}\]

where \( \beta = \tanh w \) and \( w \) is the rapidity.

TBBT Physics Bowl

In one of the episodes of The Big Bang Theory, the gang enters into the department physics bowl. Near the end, Sheldon was stunned by the following Feynman diagram problem:

Considering that this is one of the basic diagram for QED, it would be impossible for the character to not even recognize the problem. Anyway, the janitor then answered that the correct answer is \( -8 \pi \alpha \). It is incorrect. Well, at least not without additional conditions not specified by the problem.

4-force

Classes in relativity usually avoid discussing force even though every physics student started in non-relativistic physics with . It is not immediately obvious if this is still true under relativity and if not what is the correct formula. Of course, discussion of regular 3-vector in relativity is not very convenience. It is more natural to construct relativistic (i.e. covariant) relations using 4-force , 4-momentum , 4-acceleration , 4-velocity , and with proper time . There is a problem: should we define 4-force by acceleration or momentum? .

Momentum is more fundamental than acceleration, so:

We still want Newton's law to hold in some sense. Consider the collision of two particles. Newton's third law would read like this:

But this is strange because it means both particles have the same proper time even though they might be travelling with very different velocities.

Relativity and photons

It's a relatively free weekend, so I ended up thinking about bunch of random stuff. Another problem I pondered: is number of photon a Lorentz invariance? That is, do observers in different frame count the same number of photons? From intuition, it seems photon number should be an invariance.