At high school we learn that there is a correspondence:

- linear momentum - angular momentum
- mass - moment of inertia
- velocity - angular velocity
etc.

For relativistic motion we have E^2 - p^2c^2 = m^2 c^4.

and p = v E/c^2

What are the corresponding expressions using
angular momentum, moment of inertia,
and rotational energy? Do they exist at all?


Thanks for any help!

Tom Helmond

Subject: Rotational energy and angular momentum in special relativity?
From: (Tom Helmond)
Date: 8/16/03 12:54 AM US Mountain Standard Time
Message-id:

At high school we learn that there is a correspondence:

- linear momentum - angular momentum
- mass - moment of inertia
- velocity - angular velocity
etc.

For relativistic motion we have E^2 - p^2c^2 = m^2 c^4.

and p = v E/c^2

What are the corresponding expressions using
angular momentum, moment of inertia,
and rotational energy? Do they exist at all?


Consider cylindrical coordinates ( ct, z, r, theta ) and flat spacetime. The
conserved angular momentum is
L = (E/c^2)(r^2)(dtheta/dt) = r x p
which you can say is
L = I*omega
if you take I to be
I = (E/c^2)(r^2)
You can't break up the energy itself into translational and rotational parts
like you can in Newtonian mechanics where you can do so because the kinetic
energy is proportional to v^2, but equations of motion are often written in
terms of the conserved "kinetic energy parameter" T instead of the energy E.
T = (p^2)/2m
T = gamma*[(1/2)(E/c^2)(dz/dt)^2 + (1/2)(E/c^2)(dr/dt)^2 +
(1/2)(E/c^2)(r^2)(dtheta/dt)^2]
If you say I is (E/c^2)(r^2) as above then the rotational kinetic energy
parameter is
T = (1/2)gamma*I*omega^2

L = (E/c^2)(r^2)(dtheta/dt) = r x p

should be

L = (E/c^2)(r^2)(dtheta/dt) = |r x p|
with mag bars as I'm only discussing a magnitude.

(Tom Helmond) wrote in message m...
At high school we learn that there is a correspondence:

- linear momentum - angular momentum
- mass - moment of inertia
- velocity - angular velocity
etc.

For relativistic motion we have E^2 - p^2c^2 = m^2 c^4.

and p = v E/c^2

What are the corresponding expressions using
angular momentum, moment of inertia,
and rotational energy? Do they exist at all?


Thanks for any help!

Tom Helmond

Speaking of which... how is the kinetic energy of rotation comparable
to linear kinetic energy?

We have the extra d theta squared term that we didn't have earlier...
so how do we transfer energy from one system to the other? This
ultimately means that the units of rotation which give useful results
are predetermined. IE, it might end up being radians.

(...Starblade Riven Darksquall...)