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I'm trying to learn how to draw a momentum diagram but I didn't find any specific method to do so. I following the The Geometry of space-time by Tevian Dray.

What I know is that The geometry of 2-Momentum looks like :

enter image description here

But I'm not able to use this in different problems for different observers. If there are any specific steps to draw, please specify them.

Suppose for the problem the momentum diagram for two head-on colliding lumps of clay. Assuming mass(rest) to be $m_1$ and $m_2$. Other parameters you can take as you required.

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Update: It’s unfortunate that $\beta$ (Instead of $\theta$ or $\phi$ ) was chosen for the rapidity in the diagram above, since $\beta$ is traditionally the dimensionless velocity factor $v/c$ (as I use below). [I think Einstein used $\beta$ for the time-dilation factor.]

For simplicity, assume all motions occur in the $x$-direction.

Draw the [planar] energy-momentum diagram for a collision.

  • For two particles before the collision, I usually draw the sum of their 4-momenta $\tilde m_1 + \tilde m_2$ tip-to-tail with first tail at the origin.
  • ( $\tilde m_1$ is a timelike vector with [in natural units]
    magnitude $m_1$ (the invariant [rest-]mass) and
    rapidity $\theta_1=\mbox{arctanh}(\beta_1)$ where $\beta_1=\tanh\theta_1$ is the dimensionless-velocity,
    timelike leg $E_1=m_1\cosh\theta_1 =\gamma_1 m_1$, and
    spacelike leg $p_1=m_1\sinh\theta_1 = v_1\gamma_1 m_1$....
    and similarly for the other 4-momenta.)
  • For two particles after the collision, then draw the sum of their 4-momenta $\tilde m_3 + \tilde m_4$ tip-to-tail with first tail at the origin.
  • For conservation, the tip of $(\tilde m_3 + \tilde m_4)$ must coincide with the tip of $(\tilde m_1 + \tilde m_2)$.

Now, treat this as if it were "an equilibrium problem [using a "free body diagram"] involving 4 coplanar forces" in Newtonian mechanics.... but using hyperbolic trig-functions instead of the ordinary Euclidean trig-functions.

Given some knowns [among various known components, known magnitudes, or known rapidities], find the unknowns.

You may use components, or you can use vector-methods involving dot-products.

From a post of mine ( Mass Addition in Special Relativity)
here is the diagram for a particular totally inelastic collision.
$$M=(m_1\cosh\theta_1+m_2\cosh\theta_2)=(\gamma_1 m_1 + \gamma_2 m_2).$$ Go there to see that specific problem worked out.

robphy-collision

Here's are links to more complicated problems I worked out.
enter image description here
( Momentum in center of mass-frame out of knowledge kinetic energy in lab-frame )
( COM frame in Relativistic particle physics )
( How to determine particle energies in center of momentum frame? )


robphy-RRGP-doppler

( On conservation of momentum and doppler shift in special relativity )
(For an elastic collision using a new method of mine, visit the end of https://www.physicsforums.com/insights/relativity-rotated-graph-paper/ )

[update:
I think the reference to Tevian Dray is either his book "The Geometry of Special Relativity" [on Amazon] or his article "The geometry of relativity" in the American Journal of Physics 85, 683 (2017); https://doi.org/10.1119/1.4997027 . You might be able to find drafts on his website.]

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