[1/5] Here’s a fun relativistic thought experiment.
In your coordinates, a particle of rest mass 1 has velocity v_x=0.5, v_y=0.5, in units where c=1.
Its mass-energy is 1/√(1–0.5^2–0.5^2) =√2.
You bounce a laser pulse off it, aimed along the y-axis, increasing v_y to 0.6.
In your coordinates, a particle of rest mass 1 has velocity v_x=0.5, v_y=0.5, in units where c=1.
Its mass-energy is 1/√(1–0.5^2–0.5^2) =√2.
You bounce a laser pulse off it, aimed along the y-axis, increasing v_y to 0.6.
[2/5] You’ve contributed a change in momentum solely in the y direction, and you’ve caused the particle to speed up along the y-axis.
But by increasing its total speed, haven’t you increased its mass-energy—and so increased its momentum in the x direction?
How is it conserved?
But by increasing its total speed, haven’t you increased its mass-energy—and so increased its momentum in the x direction?
How is it conserved?
[3/5] As always, the best thing to do in relativity is:
Look at the 4-vectors!
The initial 4-momentum of the particle is, in (t,x,y) coords:
p_1 = (√2,√0.5,√0.5)
This has Lorentzian squared length –m^2=–1. If we change the y component so that v_y = p_y/p_t = 0.6, we get:
Look at the 4-vectors!
The initial 4-momentum of the particle is, in (t,x,y) coords:
p_1 = (√2,√0.5,√0.5)
This has Lorentzian squared length –m^2=–1. If we change the y component so that v_y = p_y/p_t = 0.6, we get:
[4/5]
p_2 = (1.531, 0.707, 0.919)
where we’ve Lorentz-rotated the t and y components, preserving the total length of the vector, and leaving p_x completely alone.
We now have a mass-energy of 1.531 rather than 1.414. And the velocities are:
v_x = 0.707/1.531 = 0.462
v_y = 0.6
p_2 = (1.531, 0.707, 0.919)
where we’ve Lorentz-rotated the t and y components, preserving the total length of the vector, and leaving p_x completely alone.
We now have a mass-energy of 1.531 rather than 1.414. And the velocities are:
v_x = 0.707/1.531 = 0.462
v_y = 0.6
[5/5] So, in our coordinate system, the particle has actually *slowed down* in the x-direction, which is how we see momentum conserved in that direction, even as the total mass-energy we would measure for the particle has increased.
