masudr said:
How do you synchronise clocks in an accelerated reference frame?
The stuff below, which I originally posted on sci.physics, uses the radar method to set up a coordinate system for an accelerated observer. Also, people interested accelerated observers might be interested in
https://www.physicsforums.com/showthread.php?t=110742".
Special relativity is robust enough to handle accelerated reference frames. These reference frames share some of the properties of coordinate systems in general relativity, i.e., they're not global, and they often possesses horizons. But they do this within the confines of special relativity.
First, a review of how an inertial observer establishes an inertial coordinate system using her wristwatch and light signals. Suppose P is any event in spacetime. The observer continually sends out light signals, and suppose the light signal that reaches event P left her worldline at time t_1 according to her watch. Upon reception of this signal, P immediately sends a light signal back to the observer, which she receives at time t_2. If x is the spatial distance of P from the observer's worldline, then, since the light goes out and back, the light travels a distance 2x in a time t_2 - t_1. Thus
2x = c (t_2 - t_1), or x = (t_2 - t_1)/2 with c = 1.
The light spends half the time going out, and half the time coming back. Therefore the time coordinate of event P is the same as the the event on the observer's worldline that is halfway (in time) between the observer's emission and reception events. Consequently, the time coordinate of P is
t = (t_2 + t_1)/2.
It is easy to convince oneself that this operational definition establishes a standard inertial coordinate system.
Note that t_1 and t_2 are proper times for the observer that sets up the coordinate system.
Assume that an accelerated observer uses the same procedure to establish a non-inertial coordinate system. Consider the case where an observer has constant acceleration. Let (t , x) be standard coordinates for a global inertial frame. Let the worldline for the accelerated observer be parameterized by her wristwatch (proper) time T so events on her worldline have coordinates
(t , x) = (a^{-1} \mathrm{sinh}(aT) , a^{-1} \mathrm{cosh}(aT))
This is one branch of the hyperbola t^2 - x^2 = - a^{-2}.
Now set up a coordinate system for the accelerated observer using the light signal procedure given above for an inertial observer. Let P be an event in spacetime that: receives a light signal that left the accelerated observer at proper time T_1; sends a light signal that the accelerated observer receives at time T_2. The accelerated frame coordinates are (t' , x') = ((T_2 + T_1)/2 , (T_2 - T_1)/2).
To get a handle on this coordinate system, find what curves of constant x' and curves of constant t' look like in the inertial coordinate system. Let Q and R be the emission and reception events of the accelerated observer, respectively. The inertial coordinates of P, Q, and R are
P: (t , x) , Q: ((a^{-1} \mathrm{sinh}(aT_1) , a^{-1} \mathrm{cosh}(aT_1)),
R: (a^{-1} \mathrm{sinh}(aT_2) , a^{-1} \mathrm{cosh}(aT_2))
QP lightlike gives
t - a^{-1} \mathrm{sinh}(aT_1) = x - a^{-1} \mathrm{cosh}(aT_1),
which gives
t - x = (a^{-1}) (\mathrm{sinh}(aT_1) - \mathrm{cosh}(aT1)) = -(a^{-1}) e^(-aT_1).
Similarly, PR lightlike gives
t + x = a^{-1} e^{aT_2}.
Multiplying these equations gives
t^2 - x^2 = -a^{-2} e^{a(T_2 - T_1)} = -a^{-2} e^{2ax'},
so curves of constant x' are hyperbolae in the inertial coordinates. The hyperbolae all have asymptotes t = x and t = -x. For x > 0, these hyperbolae successively become less sharply curved as x increases.
Dividing the equations gives
t = \frac{e^{2at'} - 1}{e^{2at'} + 1}x,
so curves of constant t' are straight lines that pass through the origin of the inertial coordinates. As t' \rightarrow \infty the lines approach t = x; as t' \rightarrow -\infty, the lines approach t = -x.
The accelerated coordinate system (t' , x') only covers the wedge x > |t| of the global inertial coordinate system, with the halflines t = x and t = -x playing the role of horizons.
These coordinates are related to Rindler coordinates, which when used for normal modes for quantum field theory give rise to the Unruh effect.
I wish I could have included diagrams. Misner, Thorne, and Wheeler, in 6.6 The Local Coordinate System of an Accelerated Observer, use a different method to derive this coordinate system. Their figure 6.4 illustrates the situation nicely.
PS I owe you a post in another thread.
