Find boost factor for general Lorentz Transformation

Here is my problem statement. I am looking at a moving particle with four momentum $$P_{\text{before}}^{\mu} = (E_p, 0, 0, p_z)$$ that is struck by a virtual photon with virtuality $$\Delta^{\mu} = (0, \Delta_x, \Delta_y, \Delta_z)$$. The resultant four momentum of the particle is therefore $$P_{\text{after}}^{\mu}=(E_{p+\Delta}, \Delta_x, \Delta_y, p_z+\Delta_z)$$. We have $$P^2_{\text{before}} = P^2_{\text{after}} = M^2$$. For convenience in analysis, I want to transform the frame of this interaction into the Breit Frame. Therefore, I want to find the correct boost factor $$\vec{\beta} = (\beta_x, \beta_y, \beta_z)$$ such that $$\underline{L}(\vec{\beta})^{\mu}_{\nu} P^{\nu}_{\text{before/after}} = P'^{\mu}_{\text{before/after}}$$ where

$$P'^{\mu}_{\text{before/after}} = (E_{p'\mp \Delta'/2}, \vec{p}' \mp \vec{\Delta'} / 2)$$

I can set up the problem simpler by considering the difference of before and after momenta

$$P_{\delta}^{\mu}=(E_{p+\Delta}-E_p, \Delta_x, \Delta_y, \Delta_z)\;;\quad P'^{\mu}_{\delta}=(0, \Delta'_x, \Delta'_y, \Delta'_z)$$

This isn't as ordinary as setting up some linear equation $$\underline{A}\cdot x = b$$ to solve for some vector b. So my question is, given $$P^{\mu}_{\delta}$$ and $$P'^{\mu}_{\delta}$$, find $$\vec{\beta}$$ that solves the linear equation

$$P'^{\mu}_{\delta} = L(\vec{\beta})^{\mu}_{\nu}P^{\nu}_{\delta}$$ when

$$L(\vec{\beta}) = \pmatrix{ \gamma & -\gamma\;\beta_x & -\gamma\;\beta_y & -\gamma\;\beta_z \\ -\gamma\;\beta_x & 1+(\gamma-1)\beta_x^2 & (\gamma-1)\beta_x\beta_y & (\gamma-1)\beta_x\beta_z \\ -\gamma\;\beta_y & (\gamma-1)\beta_x\beta_y & 1+(\gamma-1)\beta_y^2 & (\gamma-1)\beta_y\beta_z \\ -\gamma\;\beta_z & (\gamma-1)\beta_x\beta_z & (\gamma-1)\beta_y\beta_z & 1+(\gamma-1)\beta_z^2 } \\ \\ \gamma = \frac{1}{\sqrt{1 + \beta_x^2 + \beta_y^2 + \beta_z^2}}$$ ** I might be forgetting a factor of $$\beta^2 = \beta_x^2 + \beta_y^2 + \beta_z^2$$ in the expression above for $$L(\vec{\beta})$$ **

I ideally would like to write some kind of program that will solve for this. What kind of numerical method should I try to implement?

• you have 4 equations for the three unknowns $\vec v$ thus you can’t obtain unique solution
– Eli
Commented May 19, 2021 at 12:36

This could be useful:

A covariant four‐dimensional expression for Lorentz transformations
American Journal of Physics 50, 818 (1982); https://doi.org/10.1119/1.12748
Donald E. Fahnline

from the abstract (copied from above)

References on special relativity, despite denoting Lorentz transformations abstractly in covariant four‐dimensional notation, express the explicit formulas for these transformations only in the three‐vector form which treats space and time separately. Rewriting the explicit formulas covariantly yields an expression more natural for the four‐dimensional context where space and time are unified. A slight generalization valid in any inertial frame transforms any proper velocity four‐vector into any other.

Also possibly useful:
see the reference to Gourgoulhon's book in my answer for Geometric derivation of Lorentz boosts

(By the way, I think there is a sign error in your formula for $$\gamma$$.)