# Problem 4 – Hint 5

As a final hint, we’ll provide step-by-step instructions on how to solve the problem. So, we’ll consider the following reference frames and 4-velocities:
Frame (A), the laboratory frame, and its 4-velocity $u$;
Frame (B), the spaceship after the first acceleration stage, and its 4-velocity $v$;
Frame (C), the spaceship after the second acceleration stage, and its 4-velocity $w$;
Frame (D), the spaceship after the second acceleration stage, and its 4-velocity $s$.
To begin with, we’ll derive a formula for the velocity components of the spaceship after two stages of acceleration, i.e. the components $P$ and $Q$ of $w$ in Frame (A) (c.f. Hint 4). To get familiarized with the method, and to see that three-dimensional boosts are not commutative, it is also advisable to derive the components of $u$ in the frame (C), although this formula can be directly deduced from the previous one by reversing the order and direction of the acceleration stages. To that end, we equate the Lorentzian inner product of $u$ and $w$ (i.e. the Lorentz invariant constructed from these two vectors) in Frame (B) (for which we know all the necessary components) with the inner product of these two vectors in Frame (A). Since we have two unknowns, $P$ and $Q$, we need one more equation which is provided by the fact that the inner square (i.e. the Lorentzian inner product with itself) of any 4-velocity is equal to $c^2$.

Now we know the components of $u$ in Frame (C), and based on similarity, we can also write down the components of $s$ in Frame (C) (be careful here!). So, we can equate the inner product of $u$ and $s$ in Frame (C) to its expression in Frame (A). This equation yields us immediately the gamma factor of $s$ in Frame (A), which is all what we need: it must be equal to the gamma factor of $v$ in Frame (A). This equality is what we use to find $v$.

Please submit the solution to this problem via e-mail to physcs.cup@gmail.com. There will be no more hints for the Problem 4, but it will remain open for submissions until at least 23rd April 2023.