Notes on Relativity and Cosmology - Physics Department, UCSB

130 CHAPTER 5. ACCELERATING REFERENCE FRAMES . . .
Z
A
In the new frame of reference, the rocket is at rest at event A. Therefore, the
rocket’s line of simultaneity through A is a horizontal line. Note that this line
passes through event Z.
This makes the line of simultaneity easy to draw on the original diagram. What
we have just seen is that:
Given a uniformly accelerating observer, there is an event Z from which
it maintains proper distance. The observer’s line of simultaneity through any
event A on her worldline is the line that connects event A to event Z.
Thus, the diagram below shows the rocket’s lines of simultaneity.
Z
1
0
-1
2
-2
Let me quickly make one comment here on the passage of time. Suppose that
events −2, −1 above are separated by the same sized boost as events −1, 0,
events 0, 1, and events 1, 2. From the relation θ = ατ/c 2 it follows that each
such pair of events is also separated by the same interval of proper time along
the worldline.
⋆⋆ But now on to the more interesting features of the diagram above! Note that
the acceleration horizons divide the spacetime into four regions. In the rightmost
region, the lines of simultaneity look more or less normal. However, in the
top and bottom regions, there are no lines of simultaneity at all! The rocket’s
lines of simultaneity simply do not penetrate into these regions. Finally, in the
left-most region things again look more or less normal except that the labels on
the lines of simultaneity seem to go the wrong way, ‘moving backward in time.’