Is momentum an invariant?

Question

Is the value of momentum an invariant?,
specificly for instance the momentum value $\mathbf p_{\text{lab}}[~\Lambda^0~]$ of a $\Lambda^0$ baryon (drifting from the (actual) interaction point of a collider experiment towards the beam pipe wall) with respect to suitable⁽¹⁾ constituents of the "lab" (of beam pipe wall, of detectors, of magnets)?

Or does the momentum value $\mathbf p_{\text{lab}}[~\Lambda^0~]$ depend on the assignment of coordinate values to the relevant (unique) events, such as the event of the $\Lambda^0$ baryon under consideration having been produced; or the event of the $\Lambda^0$ baryon under consideration passing the beam pipe wall, or the event of the $\Lambda^0$ baryon under consideration decaying?

Expressing the value of the momentum of the specific $\Lambda^0$ baryon under consideration wrt. the lab constituents as

$$\mathbf p_{\text{lab}}[~\Lambda^0~] := m[~\Lambda^0~] ~ c ~ \frac{\beta_{\text{lab}}[~\Lambda^0~]}{\sqrt{1 - (\beta_{\text{lab}}[~\Lambda^0~])^2}} ~ \mathbf e_{\text{lab}}[~\Lambda^0~], $$

where

• $m[~\Lambda^0~]$ denotes the invariant mass of the $\Lambda^0$ baryon under consideration,

• $c$ denotes the signal front speed, and

• $\mathbf e_{\text{lab}}[~\Lambda^0~]$ denotes the (normalized) direction of motion of the $\Lambda^0$ baryon under consideration wrt. the lab constituents,

is the corresponding real number value $\beta_{\text{lab}}[~\Lambda^0~]$ an invariant, too?
(Or else: How, explicitly, does the value $\beta_{\text{lab}}[~\Lambda^0~]$ depend on the assignment of coordinates?)

^{(1: Specifily, constituents of beam pipe wall, detectors, magnets which were remaining separate and at rest with respect to each other; i.e. constituting members of an inertial system in the sense of Rindler: "simply an infinite set of point particles sitting still in space relative to each other".)}

Answer 1

If I measure the speed of a particle in the lab and then write down in my notebook the value I measured, the number an observer in a different inertial frame reads from my notebook will be the same (although the numerals may be Doppler shifted, length contracted, etc.). In the same way, the name of my cat is "Mittenz" independent of any choice of coordinates. Perhaps you can give some more of the context that led you to consider this question so that we can give you a more satisfying answer.

Answer 2

Yes, any (measured) momentum value (of someone or something specific, with respect to a specific system of suitable participants) is invariant, i.e. unchanged by any and all coordinate transformations, such as Lorentz transformations; and so is any (measured) value of velocity, or of speed (of someone or something specific, with respect to a specific system of suitable participants).