# Rigidly Constantly Accelerating Frame

We know that when a rigid frame, say a rocket undergoes constant proper acceleration, its worldline is hyperbolic. The equation is given by:

$$x^2 - c^2t^2 = \left( \frac{c^2}{a_0} \right)^2$$

Suppose P is such a worldline and worldine can also be written as:

I understand how these are derived up to this point. But it is the reasoning presented below that confuses me.

Things I don't understand:

• Why does ## X \cdot U = 0 ## imply that "4-vector from origin to particle is a line of simultaneity for the instantaneous rest frame" ?
• Why do these lines of simultaneity pass through (0,0)? Shouldn't they simply be ##x = \pm ct## based on the first picture?

For completeness, Fig. 9.16 showing the worldlines and lines of simultaneity are shown below:

Orodruin
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In the instantaneous rest frame, ##U## is pointing in the time direction, i.e., its components are (1,0,0,0). Anything orthogonal to this has time component zero and thus a line from the origin has zero change of dt in the instantaneous rest frame, i.e., it is a surface of simultaneity in that frame.

In the instantaneous rest frame, ##U## is pointing in the time direction, i.e., its components are (1,0,0,0). Anything orthogonal to this has time component zero and thus a line from the origin has zero change of dt in the instantaneous rest frame, i.e., it is a surface of simultaneity in that frame.
That makes sense. What about point 2? I thought based on the formula of the hyperbola these lines of simultaneity should simply be ##x = \pm ct##?

Orodruin
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##x = \pm ct## is not a surface of simultaneity in any frame. A surface of simultaneity is simply a set of events with the same time coordinate in a given inertial frame, which means they are by definition space-like. The light cone is by definition ... well ... light-like. :)

##x = \pm ct## is not a surface of simultaneity in any frame. A surface of simultaneity is simply a set of events with the same time coordinate in a given inertial frame, which means they are by definition space-like. The light cone is by definition ... well ... light-like. :)
So in fig 9.16, is there only one surface of simultaneity for different values of ##a_0##? How do I find an expression for that surface?

Orodruin
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