The "Satellite Paradox": Twin paradox in orbiting satellites

I don't see how adding gravity to the Twin Paradox will make it simpler. If you take it out, and just have the twins move in circle, obviously their motion is symmetric.

To simplify it even more, you could remove the transverse motion, leaving 2 twins to execute one cycle of simple harmonic motion from the same starting point, but it on opposite directions.

If we call them the plus twin and the minus twin, then their speeds in the initial rest frame are:

$$ v_{pm} = vsin{omega t} $$

for $0 le t le frac{2pi}{omega}$.

At any time, their position is:

$$ x_{pm} = frac v {omega}(1 -cosomega )$$

Each sees the other moving at:

$$ v^{mp}_{pm} = frac{2vsinomega t}{1+frac{v^2}{c^2}sin^2omega t} $$

leading to a Lorentz factor:

$$ gamma = frac 1 {sqrt{1-frac{v^2}{c^2}}}= sqrt{frac{1+frac{v^2}{c^2}sin^2omega t}{1-frac{v^2}{c^2}sin^2omega t}}$$

by which each sees the others' clock ticking more slowly.

However, thought experiment twins don't travel in a vacuum. The have a lattice of comoving observers. When the twins are separated by a distance $x$, the right twin's comoving observer at the same position as the left twin (and vice versa) sees the clock's offset shift as the velocity changes. As he is moving away, the other twin's bias is moving forward. By the time they reach maximum separation, and both have $v=0$: the clock bias has exactly canceled the time dilation.

Now they begin coming home: as the move towards each other, $gamma$ causes the other's clock to tick more slowly. Meanwhile the large separation causes the other's clock bias to move forward in time. Hence, the twin's clock is ahead, but ticking more slowly.

When they rejoin, the clock rate and clock bias have exactly canceled, leaving the twins at the same age.

TLDR: the situation by design is completely symmetric, so both observers are the same age when they reunite. Furthermore, if they correctly apply relativity then they correctly predict that there is no time dilation between them.

The g-field of the planet has nothing to do with orbiting satellites since they are inertial, and thus GR corrections are not inserted.

This is not correct. This scenario involves curved spacetime so it requires general relativity. Specifically, if two objects move inertially (no acceleration) with different velocities in flat spacetime then they can meet at one single event at most. The only way to have them reunite after travelling without any acceleration is for the spacetime to be curved. GR is therefore unavoidable for this question.

The traditional gamma factor, attributed to one twin from the viewpoint of the other, is variable because the resultant velocity of one twin complying with the relativistic velocity addition is a variable measured from the viewpoint of the other one. However, whether or not the corresponding time dilation is varying, each twin anticipates the other one to be younger.

The traditional gamma factor is for special relativity only, and is not relevant here. To derive the relevant time dilation expression we can start with the Schwarzschild metric. Since we are working in a single orbital plane we can, without loss of generality, set $sin(theta)=1$ and $dtheta = 0$ and for convenience $R=2GM/c^2$. And assuming that the orbits are circular we can (with loss of generality) set $dr=0$ also. Thus we arrive at a simplified Schwarzschild metric $$ds^2 = -c^2 dtau^2= -left(1-frac{R}{r}right)c^2 dt^2+r^2 dphi^2 tag{1}$$

Now, for a spacecraft moving in a circular path in the plane we have $$frac{dphi}{dt}=pmfrac{v}{r} tag{2}$$ respectively for the two spacecraft and the spacecraft are inertial for $v^2=GM/r$.

Substituting $(2)$ into $(1)$ and solving for the time dilation we get $$frac{dt}{dtau}=frac{1}{sqrt{1-frac{R}{r}-frac{v^2}{c^2}}} tag{3}$$ Because one spacecraft moves at $v$ and the other moves at $-v$ we see from $(3)$ that the time dilation is equal for both spacecraft in the Schwarzschild coordinates. This means that at all times the spacecraft show the same time including at the moment of their reunion. This is as expected from the symmetry of the scenario.

Now, we would like to transform to the rest frame of one of the spacecraft and calculate the time dilation in the spacecraft's frame. We will use the coordinate transform $$t'=sqrt{1-frac{R}{r}-frac{v^2}{c^2}} t \ phi' = phi - frac{v}{r} t tag{4}$$ and differentiating $(4)$ gives $$ dt=frac{1}{sqrt{1-frac{R}{r}-frac{v^2}{c^2}}} dt' \ dphi = dphi' + frac{v}{rsqrt{1-frac{R}{r}-frac{v^2}{c^2}}} dt' tag{5}$$ Note that the expression for $phi'$ involves $t$. So the two terms will no longer be independent.

Substituting this $(5)$ into $(1)$ gives $$ds^2 =-c^2 dt'^2 + r^2 dphi'^2 + frac{2 r v}{sqrt{1-frac{R}{r}-frac{v^2}{c^2}}} dt' dphi' tag{6}$$

Because of the cross term in $(6)$ we cannot simply solve for $frac{dt'}{dtau}$. However, if we substitute $(5)$ into $(2)$ we get $$frac{dphi'}{dt'}=0 \ frac{dphi'}{dt'}=-frac{2v}{rsqrt{1-frac{R}{r}-frac{v^2}{c^2}}} tag{7}$$ respectively for the two spacecraft. Note that one twin is at rest, so this represents that twin’s rest frame. However, note that although this is the twin’s rest frame the rest twin is not at the center of the coordinate system. $phi$ is measured with respect to the planet not the twin.

Substituting $(7)$ into $(6)$ and simplifying we get $$ ds^2=-c^2 dt'^2 tag{8}$$ for both spacecraft. This indicates that in one spacecraft's reference frame the other spacecraft is never time dilated. Therefore the observers agree and expect that the clocks will read the same when they pass the other spacecraft. Because this is curved spacetime the gravitational time dilation and the velocity time dilation exactly cancel for these symmetric twins in the coordinates above representing one spacecraft's reference frame.

I just want to point out, that the time passed for any object traveling from event $A$ to event $B$ on a curve $gamma(t)$ (with tangent vector $dot{gamma}(t)$) in spacetime with metric $g$ is: $$tau=int_A^B sqrt{-g(dot{gamma},dot{gamma})}dt,$$ or when you put it in a more intuitive language, it is given by time-like "length" of the curve.

Due to symmetry in the case of negligible rotation, the length of the curve is same wheter it goes clockwise or counterclockwise and the twin will thus age the same.

In general relativity you must be aware that there is no global lorentz frame and thus in general you do not have natural procedure for comparing two (random) clocks. So the twin paradox cannot be even formulated the way you do it, because the sentence "twin attributes a variable time dilation to the other one" makes no sense.

All the issue about time dilation in the twin paradox comes from the expression (using $c = 1$) for $$frac{dt}{dtau} = gamma = (1-v^2)^{-1} $$ valid for calculations in inertial frames.

In the case of a circular orbit, using the Schwarzschild solution: $$dtau^2 = left(1 - frac{2GM}{r}right)dt^2 - left(1 - frac{2GM}{r}right)^{-1}dr^2 - r^2dtheta^2 - r^2sin^2(theta)dphi^2$$

If we assume a circular orbit on the equator to simplify the calculations, ($dr = dtheta = 0$, and $sin(theta) = 1$), and dividing by $dt^2$ we get:

$$frac{dt}{dtau} = left(1-frac{2GM}{r} - omega^2r^2right)^{-1/2}$$

The time dilation is the same for both twins, because they have the same values for $omega$ and $r$.

Of course, it is also necessary to calculate the metric in coordinates where each twin are at rest, to confirm that the values are the same. But I can not see how it would be different due to the symmetry of the situation.

Initially I did not understand your question, but the interpretation of your question by WillO makes sense, so I will go with the same interpretation.

while you do not state so, it appears what you have in mind is the case of orbital motion at constant altitude. I will discuss that case.

In your description you added a starting point in time, an instanteous launch, horizontally, at orbital level. However, such elaborate history obscures the point you are trying to make. It is not necessary to describe how the orbital motion started. It is sufficient to state that it is a case of orbital motion.

So, we have a celestial body, with two satellites in orbit, counter-orbiting. That is, the two satellites are in the same orbital plane, very close in orbital altitude. The two satellites pass each other twice every orbit. We can designate one such pass as $t = 0$

This case does not change when each orbit is filled with many satellites. I increase the number of satellites to help with a comparison that will come later.

The two fleets of satellites can set up a procedure using only the exact moment of passing each other to iterate to a synchronized fleet time that is common to all the satellites.

Let me use an arbitrary number of satellites in each orbit, let's say 12 in each orbit. Let the clocks initially be unsynchronized. Each time two satellites pass each other by they transmit the time readout of their own clock to the other satellite. On reception each satellite adjusts its own time towards the middle of the two readouts. Iterated over multiple orbits that procedure will bring all the clocks of that fleet to a synchronized fleet time. That is, at every passing the time readout received will be the same as the time on their own clock.

The above case is one where the motion is arranged in a loop. In special relativity, when you close a loop things get very interesting.

It's time now to contrast the above case - motion along a loop - with the case where the motion is linear.

Let there be two fleets of spaceships, I will arbitrarily call them the 'red fleet' and the 'green fleet'. Each fleet is arranged in a single line. The two fleets pass by each other, like trains moving in opposite direction over parallel tracks.

In this case, the linear motion case, there are far less means available to arrive at a synchonized fleet time. The ships of the red fleet can use pulses of light to establish a synchronized fleet time for the ships of the red fleet (using Einstein synchronization procedure ), and similarly the ships of the green fleet can established a synchronized fleet time.

However, the synchronized red fleet time will not have the same simultaneity as the synchronized green fleet time. That is: in the case of linear motion there is no escape from relativity of simultaneity.

That raises the question: how does the escape from relativity of simultaneity come about in the case of closing-a-loop?

To see that we look at what we need to remove to prevent the escape from relativity of simultaneity.

Consider a fleet of satellites, all in the same orbital plane, at the same orbital altitude. Let these satellites arrive at a synchonized fleet time (using pulses of light), but disallow them from closing the loop. Let's say there are 12 satellites, numbered 1 to 12. Each satellite exchanges pulses with its orbital neighbour only. Satellite 12 and satellite 1 are neighbours, but now we disallow sending or receiving between 12 and 1. Without closing the loop the procedure is once again the same as Einstein synchronization procedure.

This is a profound difference between Newtonian space and time on one hand and Minkowski spacetime on the other hand. In Newtonian space and time: being able to close a loop makes no difference. But in Minkowski spacetime it does make a difference.

The answer to the question you've asked is this: When the twins come back together (half way around the world from where they started), it's clear from symmetry that their clocks must agree. But that's so obvious that I have to suspect you meant to ask something else.

Perhaps this is a question about what happens at some other point along the journey, before the reunification? If so, the answer is that the twins are in motion with respect to each other and therefore each one's clock runs slow in the other's (instantaneous) frame. So maybe you meant to ask this: If each twin sees the other's clock as running slow all along the way, how can the clocks agree when they meet?

The answer to that is: Each twin's instantaneous frame keeps changing, so he keeps changing his description of where the other twin is "right now". Those changing descriptions end up canceling the effects of the slow-running clock. Again, it's obvious from symmetry that this cancellation must occur, though it can be instructive to confirm this with a calculation. If you have any remaining confusion, that calculation should resolve it.