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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 $H = a K_z$ ?

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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

$$ \lambda = \frac {a}{c} \tau $$

The particle is boosted by the operator

$$ e^{\lambda K}=e^{\frac {a}{c}\tau K}=e^{\tau H} \quad \text{where} \quad H=\frac{a}{c} K $$ 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.

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