Here’s a little physics riddle. It’s really meant as a moment of self-reflection for physics teachers (I invite you to compare what answers you’d give within Relativity Theory).
We’re in the context of Newtonian mechanics.
There are three small bodies. In the inertial coordinate system (t, x, y, z), we know the following about the three bodies (at a given instant of time):
- The first has mass 3 kg
- The second has velocity (1, 0, 0) m/s
- The third has momentum (2, 0, 0) kg⋅m/s
Now consider a new coordinate system (t’, x’, y’, z’) related to the first by the following transformation (a Galileian boost):
t’ = t, x’ = x - u⋅t, y’ = y, z’ = z with u = 1 m/s
Questions:
- What is the mass of the first body in the new coordinate system?
- What is the velocity of the second body in the new coordinate system?
- What is the momentum of the third body in the new coordinate system?
Can you give definite answers to these three questions, and motivate your answers with simple physical principles? Note that by “definite answer” I don’t necessarily mean an answer with a definite numerical value.
The new coordinate system is moving one meter per second in the x-direction. This is small enough compared to the speed of light that we can ignore relativity. So the mass is the same and the velocity and momentum just get shifted by one meter per second, no?
Yes we’re considering Newtonian mechanics in any case. What I’m especially curious about is what physical principles people use to motivate their answers.
Regarding the third body, consider the case where its mass is, say, 2 kg, and the case where it’s 1 kg instead (the momentum being the same).
You dont have sufficient information to calculate the momentum of the third body in the new system. If you are treating the system as Newtonian you need the mass of it to calculate its momentum in the new frame, if you are treating it as relativistic you need its total energy.
Completely agree, which I think is very interesting. In Newtonian mechanics, some scalar and vector quantities such as mass, internal energy, contact forces (stress), heat flux are frame-indifferent. Others, such as velocity and acceleration, are frame-dependent but we do have transformation rules for them. Some quantities – and quite important ones – such as momentum, are in a sort of limbo: they are frame-dependent, but there’s no clear transformation rule for them.
From the point of view of relativity theory, it’s interesting to note that for this particular case of coordinate transformation – note that it is not a Lorentz boost – we can actually calculate the spatial components of momentum in the new coordinate system, if the reported momentum is expressed as a covariant vector (p_µ). This is because its unknown temporal component (energy) gets multiplied by zero in this transformation. But the text is ambiguous on whether the reported components are covariant or contravariant.
