- #1

Freixas

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Summary:

- Someone claimed that an accelerated observer's clock (observer at 1g from Earth to halfway to Alpha Centauri, then -1g the rest of the way) would run slower compared to a clock on Earth (or Alpha Centauri, which appears to be treated as being motionless relative to the Earth).
- @PeroK said that "Time dilation is symmetrical. You could equally say that time is going slower on Earth than in the rocket."
- Later, @PeroK introduced a row of clocks, presumably also motionless with respect to the Earth, placed along the travel path. He said "The crew on the rocket would measure each of these clocks to be running slow as they passed it..."

**The gamma solution**

One approach to the problem of comparing clock rates would be to just look at the traveler's instantaneous gamma, which is derived from the instantaneous velocity and call it a day. Since the velocity is a relative velocity (Earth vs. rocket), the time dilation is the same from either viewpoint.

The problem is that, at the end of the journey, the traveler and Earth are back in the same inertial frame and the final time tallies are not symmetrical. From each observer's view, the other's clock always ran slower (except at the start and end), yet the traveler's clock is the one with less time.

**The "line of simultaneity" solution**

When dealing with observers moving with different

*constant*velocity, we can compare clock rates by using lines of simultaneity with respect to each observer's inertial frame. We draw the line of simultaneity for one observer at one point and then one time unit later. These two lines intersect the other's time axis and we read the time interval there. The ratio establishes the rate difference. It is always symmetrical.

I can draw the instantaneous line of simultaneity for an accelerated observer and find the corresponding point on Earth's t axis. I could try to draw a second line of simultaneity one traveler's time unit later to find the rate difference. The problem is that I would be trying to compare two quantities measured relative to two different inertial frames. And the clock rates computed by the traveler with this method would show Earth time sometimes slower and sometimes faster than the traveler's own clock. The Earth observer would come up with the traveler always having a slower rate.

**The row of clocks**

@PeroK introduced a row of clocks along the travel path, presumably to make it clearer that the time dilation was symmetrical. The idea was for the traveler to compare his clock tick to the tick of the one of the clock he passes.

I am unsure how this brings any clarity to the problem, nor how any method the traveler tried to use to measure the passing clock's tick wouldn't involve measurements that start and end in different inertial frames. Perhaps @PeroK will be willing to walk me through his thinking in this forum.

I may be handicapped by trying to understand the answer using S.R. I suspect this may be more of a G.R. topic. Nothing ventured, nothing gained...