This document discusses the Lorentz transformation of intrinsic orbital momentum. It defines intrinsic orbital momentum as the angular momentum of a system measured from its center of mass in the center of momentum frame where its total momentum is zero. The intrinsic orbital momentum transforms like a spacelike vector under Lorentz transformations, experiencing contraction in the direction perpendicular to the boost velocity rather than parallel to it. As an example, it calculates the intrinsic orbital momentum of a two-particle system with one particle orbiting the other.

1. Lorentz Transforma on on Intrinsic Orbital Momentum
Sergio Prats, April 2023
The angular momentum 𝑳 = 𝒓 × 𝒑 is a conserved quan ty for an isolated system, however it
has a “gauge” depending on where is the origin of coordinates in the posi on and momentum
spaces. The intrinsic orbital momentum of a system is its angular momentum measured at the
center of mass and choosing as source inner al reference frame the one in which the system has
zero momentum (aka “center of momentum”). Therefore, for a system with N par cles
{(𝑚 , 𝒙𝟏, 𝒑𝟏), … , (𝑚 , 𝒙𝑵, 𝒑𝑵)} the center of mass and center of momentum 𝒙𝑶 and 𝒑𝑶 are
defined as:
𝒙𝑶 = 𝑚 𝒙𝒊 ∗
1
∑ 𝑚
≡
1
𝑀
𝑚 𝒙𝒊
𝒑𝑶 =
1
𝑀
𝒑𝒏
Once the center of mass and momentum is known, the intrinsic orbital angular momentum can
be obtained with the usual angular momentum formula:
𝑳𝒊𝒏𝒕𝒓 = 𝒓𝒏 × 𝒑𝒏
Where the 𝒓𝒏 and 𝒑𝒏 are now rela ve to the center of mass and SRI in which the system is at
rest.
The intrisic orbital momentum can help to describe a complex system in which we do not have
clear knowledge on what is inside. The orbital movement brings no momentum net momentum
to the system, but it increases the energy of the system, therefore it can be a way to dis nguish
two systems that externally seem iden cal.
We may want to calculate the intrinsic angular momentum from an iner al reference frame
different than the center of momentum, for that we should apply the velocity addi on law and
measure in rela on to the movin center of mass.
The simplest system to calculate the intrinsic orbital momentum is one which two point-
masses, one infinitelly big in the center and another mass m orbi ng with posi on
𝒙 = 𝑟(𝑐𝑜𝑠(𝜑)𝑥 + 𝑠𝑖𝑛(𝜑)𝑦) and velocity 𝒗 = 𝑟(−𝑠𝑖𝑛(𝜑)𝑥 + 𝑐𝑜𝑠(𝜑)𝑦). Since the big mass will
not be affected by the effect of the smaller mass, all the intrinsic angular momentum will come
from the small mass and will be:
𝑳𝒊𝒏𝒕𝒓 = −𝑚𝛾(𝑣)𝑣 𝑧̂
Now, in order to see how this intrinsic orbital momentum transforms, we can transform both
the velocity and the posi on to the new system. If the lab sees the orbi ng system moving with
velocity u, the instant velocity for the orbi ng par cle can be calculated by using the velocity
addi on law and the rela ve posi on can simply be obtained by applying a contrac on of
length to the rela ve posi on respect the center, therefore we will have:

2. 𝒗 = 𝒗⨁(−𝒖)
𝒙 = 𝒙 − (𝒙 · 𝑢) ∗ (1 −
1
𝛾(𝑢)
)𝑢
𝑳𝒊𝒏𝒕𝒓′ = 𝑚𝛾(𝑣′)𝒙′ × 𝒗′
Where 𝛾(𝑢) = 1/ .
Some mathema cal calcula ons show a rather curious rela on between 𝑳𝒊𝒏𝒕𝒓 and 𝑳𝒊𝒏𝒕𝒓′.
When the intrinsic angular momentum is boosted to a different iner al reference frame, it gets
a contrac on, however it is not the direc on of u which gets contracted, but in the direc on of
𝑳𝒊𝒏𝒕𝒓 that is perpendicular to u:
𝑳𝒊𝒏𝒕𝒓 = 𝑳𝒊𝒏𝒕𝒓 − 1 −
1
𝛾(𝑢)
(𝑳𝒊𝒏𝒕𝒓 − (𝑢 · 𝑳𝒊𝒏𝒕𝒓)𝑢)
The intrinsic orbital momentum behaves like a space like vector, except for that it contracts
the perpiducular direc on instead of the parallel direc on when it is transformed to a
different frame.
Calcula ons can be found here:
h ps://github.com/SergioPratsL/PhysicsStuff/blob/main/HacerTuTeoria/Probaturas_Momento
_Angular.m