# Boost as a generator of constant acceleration

In Covariant Loop Quantum Gravity by Carlo Rovelli p204, Sec 10.4, there is a paragraph which states

..., a trajectory at constant acceleration is generated by the boost generator $$K_z$$ in the direction normal to the horizon surface. If we multiply $$K_z$$ by $$a$$ we obtain a transformation whose parameter along the trajectory is the proper time. Therefore $$H = a K_z$$ is the generator of proper time evolution, that is the hamiltonian, for the accelerated observer....

How does boost generate surfaces of constant acceleration ? How do we show that its Hamiltonian is given by the $$H = a K_z$$ ?

In a briefly at rest frame of the accelerating particle $$d\lambda = d\left(\frac{v}{c}\right)=\frac {a}{c} dt = \frac {a}{c} d \tau$$ where in the frame of the particle $$d\lambda$$ is a small change in the Lorentz boost parameter $$\lambda$$, $$dv$$ is a small change from zero velocity, and $$dt$$ is a small change in time. Since $$dt$$ is always in the frame of the particle, $$t$$ is the proper time $$\tau$$ of the particle along it's trajectory. It is a property of Lorentz boost parameters (also known as rapidity) in a single direction that they are additive as the particle's velocity changes (whereas you must add velocities using the Lorentz velocity addition formula). If we assume the acceleration $$a$$ in the frame of the particle is constant, then we can add up (integrate) all the little $$d\lambda$$ and $$d \tau$$ to get
$$\lambda = \frac {a}{c} \tau$$
$$e^{\lambda K}=e^{\frac {a}{c}\tau K}=e^{\tau H} \quad \text{where} \quad H=\frac{a}{c} K$$ Since $$H$$ appears to be the generator that moves the particle thru time, it is called the Hamiltonian.
It is incorrect to call boosts the generator of constant acceleration. $$K$$ is the generator of velocity boosts. $$H$$ is the generator of time translation. I don't know of any Lie group transformation $$e^{aA}$$ that would give an object an acceleration $$a$$ and where $$A$$ would be called the generator of acceleration. Maybe this is because when you do an acceleration you have to keep pushing on the object to maintain $$a$$. It doesn't stay done. This is unlike rotations and boosts where when you do them and take your fingers away they stay done.