User:Garfield Garfield/sandbox

Einstein's 1907 derivation of a redshift
In 1907 Einstein gave an ingenious argument, using three different coordinate systems, for the gravitational redshift. It is restricted to small velocities, small uniform accelerations, and small time intervals. Consider first two coordinate systems S(x,y,z,t) and Σ(ξ,η,ζ,$$\tau$$) which at one time are coincident and which both have velocity $$v$$ = 0. At that one time, synchronize a network of clocks in S with each other and with a similar network in Σ. The time of coincidence of S and Σ is set at $$t = \tau = 0$$. System S remains at rest, while Σ starts moving in the x direction with a constant acceleration $$\gamma$$. Introduce next a third system S'(x'.y',z',t') which relative to S moves with uniform velocity $$v$$ in the x direction in such a way that, for a certain fixed time t, we have x' = ξ, y' = η, and z' = ζ. Hence $$v = \gamma t$$. Imagine further that at the time t of coincidence of S' and Σ all clocks of S' are synchronized with those of Σ.

I. Consider a time interval $$\delta$$ after the coincidence of S' and Σ. This interval is so small that all effects $$O(\delta^2)$$ are neglected. What is the rate of the clocks in S' relative to those in Σ if $$\gamma$$ is so small that all effects $$O(\gamma^2)$$ can also be neglected? One easily sees that, given all the assumptions, the influence of relative displacement, relative velocity, and acceleration on the relative rates of the clocks in S' and Σ are all of second order or higher. Hence in the infinitesimal time interval $$\delta$$ we can still use the times in the local Lorentz frame S' to describe the rate of the Σ clocks. Therefore, "the principle of the constancy of the light velocity can be ... used for the definition of simultaneity if one restricts oneself to small light paths" Why three coordinate systems? On the one hand, S and S' are inertial frames and so one can use special relativity. On the other hand, during a small time interval $$\delta$$ the measurements in S' can be identified with those in Σ up to higher order effects.

II. How do clocks in two distinct spatial points of Σ run relative to each other? At $$t = \tau = 0$$, the two Σ clocks were synchronous with each other and with the clocks in S. The two points in Σ move in the same way relative to S. Therefore the two Σ clocks remain synchronous relative to S. But then (by special relativity) they are not synchronous relative to S' and thus by (I), not synchronous relative to each other. We can now define the time $$\tau$$ of Σ by singling out one clock in Σ---say, the one at the origin---and for that clock setting $$\tau = t$$. Next, with the help of (I) we can define simultaneity in Σ by using S': the simultaneity condition for events 1 and 2 is

(9.2)   $$t_1 - c^{-2}vx_1 = t_2 - c^{-2}vx_2$$

where again, $$v = \gamma t = \gamma \tau$$. Let 1 correspond to the origin of Σ and 2 to the spatial point (ξ,0,0) where the clock reading is called $$\sigma$$. Introduce one last approximation: The time $$\tau$$ of the coincidence of S' and Σ is also taken to be small so that $$O(\tau^2)$$ effects are negligible. Then $$ x_1 - x_2 = x'_1 - x'_2 = \xi, t_1 = \tau, t_2 = \sigma $$, so that Equation (9.2) becomes

(9.3)   $$ \sigma = \tau(1 + c^{-2} \gamma \xi) $$

Applying the principle of equivalence to this equation gives us

(9.4)   $$ \sigma = \tau(1 + c^{-2}\Phi)$$

where $$\Phi$$ is the gravitational potential difference between (ξ,0,0) and (0,0,0).

Argument from Bohr's book
The weighing of the box may be accomplished to accuracy $$\Delta m$$ by attaching suitable loads that restore the pointer to the zero position. The essential point is now that any determination of this position with a given accuracy $$\Delta q$$ will involve a minimum latitude $$\Delta p$$ in the momentum of the box, where $$\Delta p \Delta q$$ ≈ h.