The Distant Revolving Astronaut: When Time Runs
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At one point in the orbit, the ship will be traveling directly

At another point in the orbit, the ship will be traveling directly

If the Earth clocks read in the future at one moment, and in the past at a later moment, then at some point in between they must have run

If you want to skip all the math, you can just go right to the plot of the results.

Distances are measured in lightyears, time in years, and C is 1. Ship's MCRF coordinates will be (τ, ξ, ζ). The coordinates based on the planet the ship is circling will be (t,x,y). Letting the ship's current location be (t

We can also immediately write the formula for the ship's time as a function of the planet's time (see the revolving clock for the derivation).

The "primed" coordinates are rotated and translated relative to the unprimed frame, but they're still "planet coordinates".

In the primed coordinate system, the ship's coordinates are

and the coordinates of Earth are

'

The ship's MCRF is moving to the

All events which are simultaneous in the ship's MCRF, at time t

Applying (5) to (6) we obtain the coordinates of Earth in the primed Latin frame which correspond to a moment which is simultaneous with the ship in the ship's MCRF,

The x and y coordinates match the space coordinates of Earth, which is good -- it means we're probably not totally lost! The time in the primed frame differs from the time in the unprimed frame by t

Differentiating with respect to τ, we finally get,

This formula deserves some comment. Note what happens when θ = 0: Earth time proceeds γ times

The answer is in the geometry. Time dilation is a consequence of the motion of the "dilated" clock through the "stationary" frame of reference. If we work out how fast Earth is actually moving toward the ship when θ = 0, we find that it's

The data are plotted against θ, the angle of the ship's position with the Y axis (see figure 1). The thin blue line labeled ve(q) is the ship's velocity in the direction of Earth, which is v*cos(θ). Velocity units are lightyears/year. The thin magenta line labeled ts(q) is the current time, in the FoR of the planet, with units of years.

The red line, te(q), is the time on an Earth clock, in years, as seen by an observer flying past Earth in the MCRF of the ship. Earth time starts out at +2 years, when ship's time is at zero. However, the time line for Earth starts falling immediately, and we see that it actually follows a rising sine curve. It's trending upward, but with a peak each time the ship's velocity

And that brings us to the Big One: The green line, dte_dtau(q), is the

This effect is important, because it is at the very heart of the "twin paradox". The traveling twin

When he turns around to come home, the clocks on Earth, which had been loafing along due to

The issue is really "relativity of simultaneity". What's simultaneous with what depends on what (inertial!) frame of reference you happen to be in. When the traveler accelerates, he

But it's possible to use lots of words in an effort to explain this, without ever really improving on what the equations already say on their own.

At the same time, keep in mind that this effect, dramatic as it seems, is

At a particular moment, and

Now, consider what is meant by the

So the difference in Earth time we compute is taken between measurements done by

There is absolutely no way for a

One thing nearly everyone agrees on: This effect is confusing.

As they accelerate toward Earth, the Doppler shift they see will be, as always, a combination of the relative motion of Earth toward them, and the apparent slowing of Earth's clocks due to time dilation. The distance to Earth will seem to decrease due to Fitzgerald contraction, and the time they

The point to carry away from this is that the MCRF view, though it contains the key to the twin paradox, can only be