[varsigma]

In case of symmetrical motions, average relative velocity is zero.

?? I don't think you can just claim that separate motions are symmetrical. What do you mean symmetrical? What symmetry?

[hamdani yusuf (OP)]

In case of symmetrical motions, average relative velocity is zero.

?? I don't think you can just claim that separate motions are symmetrical. What do you mean symmetrical? What symmetry?

Let two identical clocks stationary to each other. Accelerate both clocks at the same rate but opposite direction. They maintain the same distance to the original position at all time. After predetermined distance, they both return to the original position at the same speed.
Both clocks experience symmetrical acceleration and speed at any time. They only differ in direction. Any difference in clock reading at the reunion can be attributed to difference of the original position to a preferred frame of reference, assuming that neither of clocks are defective.

[hamdani yusuf (OP)]

The asymmetry can then be used to determine absolute frame of reference.

Or, to put it another way, relative velocity. There is no absolute frame of reference.

If that's the case, then there will be no asymmetry. And both clocks will still be synchronized when they reunite.

[hamdani yusuf (OP)]

(Personally, I think the OP is diving off into Twins' Paradox etc, with ever increasing complexity, when they don't really grasp the significance of the basics of relativity. I think the explanations need to get simpler, not more complex.)

Perhaps you can point out which basics of relativity I've missed out in my calculations with symmetrical twin travellings?

I expected someone to come out using space time diagram to explain away the seemingly paradoxical symmetrically travelling twins thought experiment.

[alancalverd]

If two clocks have identical histories, you'd expect them to show the same time when they re-unite.

[hamdani yusuf (OP)]

If two clocks have identical histories, you'd expect them to show the same time when they re-unite.

Then relativity of simultaneity alone isn't adequate to explain the asymmetrical time dilation experienced by the twins in the original twin paradox.

[alancalverd]

The twins in the "original paradox" do not have identical histories.

As I have pointed out several times previously, time dilation is always asymmetric except in the case where v_rel = 0, and now we might add "or the clock histories are symmetric and result in v_rel = 0", but you'd have to qualify that statement too: if they accelerate equally in opposite directions and then stop, they would appear to an observer at the starting point to be synchronised to one another, but neither could tell the time on the other's clock (or the observer's) simply by looking at his own.

[hamdani yusuf (OP)]

if they accelerate equally in opposite directions and then stop, they would appear to an observer at the starting point to be synchronised to one another, but neither could tell the time on the other's clock (or the observer's) simply by looking at his own.

Is there a formula to calculate the other's clock, when their relative position and motion are known?
If not, is it a random value? Is there a high or low limit for that random value?

[hamdani yusuf (OP)]

What if there is another twin travel to the opposite direction with the same speed? And another one in perpendicular direction?

The formula for perpendicular velocity addition is:

https://en.wikipedia.org/wiki/Velocity-addition_formula
In our case, vx=vy=0.866c.
For convenience, I used spreadsheet to calculate.
putting the number to the left formula gives 0.247c
putting the number to the right formula gives 1.732c
Either result doesn't seem to be correct. It shouldn't be slower than the individual velocity. It shouldn't be higher than c either. Can someone show where the error is?

I searched and found this formula in quora.

...
This messy matrix is not a Lorentz boost. It is, in fact, a combination of a Lorentz boost (corresponding to some velocity in the  x−y plane) and a spatial rotation (again in the  x−y plane.)

However, at least we can get the magnitude of the resulting velocity from this matrix, as the upper left component of the matrix is not affected by the spatial rotation. It is determined purely by the Lorentz boost. A little bit of trivial algebra tells you the magnitude:
v=√(vx^2 + vy^2 - vx^2.vy^2/c^2)

Plugging in the value, where vx=vy=0.866c, v is 0.968c.
The gamma factor is then around 3.98. For simplicity, I'll just round it up to 4.

[hamdani yusuf (OP)]

Just like before, we can calculate the time difference between the twins.
In chronological order,  here's what a travelling twin's timeline.

At the beginning, all clocks are 0.

During outbound journey, his own clock increases by 10 years. In his calculation, earth clock increases by 5 years. While the other travelling twin's clock increases by 10/4 years.

During short period of turn around, earth clock jump from 5 to 35 years. While the other travelling twin's clock jump from 10/4 to 70/4 years.

During return journey, his clock increases by 10 years. In his calculation, earth clock increases by 5 years. While the other travelling twin's clock increases by 10/4 years.

At the end of the journey, both travelling clocks show 20 years, while earth clock shows 40 years.

[hamdani yusuf (OP)]

Just like in the case of opposite directions twins, Earth clock jumps by 30 years during the turn around. But this time the other travelling clock jumps by 120/4 =15 years.
How are those numbers calculated?
They should depend on the distance and change of velocity. When the distance is zero, there's no time jump, just like at the start of the journey.
The other travelling clock is further than the earth clock during the turn around. But somehow it undergoes less time jump. Where did I miss?

[Halc]

Your numbers in both posts are good except for that bit about 120/4 being 15.Not sure where 120/4 comes from. The 4 was from the perpendicular case, but the case being discussed was not the perpendicular one.

How are those numbers calculated?

The numbers come from integrating the interval of the remote worldline between the event where it is simultaneous with the acceleration event in the first frame, and the the event where it is simultaneous with the acceleration event in the second frame. This is pretty easy for an inertial object, but your traveling clock is not inertial, so it is best done in two pieces.

For an inertial object (like Earth) and an instantaneous frame change,  t' = Lv
where t' is the inertial clock's time jump, v is the velocity component change towards the inertial thing as measured in the inertial object's frame, and L is the distance component between the two as measured by the inertial thing. Velocity component is negative if the accelerating thing accelerates away.

Things get more complicated (and require calculus) if the acceleration is smooth and not abrupt.

They should depend on the distance and change of velocity.

Yes, Einstein called it 'moment of acceleration', kind of like moment of inertia which is zero for mass at the axis, and increases proportionally with the distance of the mass from the axis.

The other travelling clock is further than the earth clock during the turn around. But somehow it undergoes less time jump. Where did I miss?

You missed the fact that the Earth clock is inertial between the two events of the 'jump' and the traveling clock is not, so its worldline is half the temporal length that it would have had had it been inertial between its two events.

[hamdani yusuf (OP)]

Your numbers in both posts are good except for that bit about 120/4 being 15.Not sure where 120/4 comes from. The 4 was from the perpendicular case, but the case being discussed was not the perpendicular one.

15 years is the time jump required to compensate the time dilation on the other twin's clock to match with first twin's clock when they reunite. Where does the number come from? That's what I was asking.


As I mentioned when quoting answer from quora, 4 is the gamma factor for perpendicular case. For twins going on opposite direction, the gamma factor is 7,as I posted earlier.

[hamdani yusuf (OP)]

You missed the fact that the Earth clock is inertial between the two events of the 'jump' and the traveling clock is not, so its worldline is half the temporal length that it would have had had it been inertial between its two events.

Where does the number "half" come from?

[alancalverd]

Is there a formula to calculate the other's clock, when their relative position and motion are known?

Yes. Time dilatation formula. It's sufficiently accurate to allow GPS navigation. 

[hamdani yusuf (OP)]

Is there a formula to calculate the other's clock, when their relative position and motion are known?

Yes. Time dilatation formula. It's sufficiently accurate to allow GPS navigation. 

Then why you said this?

if they accelerate equally in opposite directions and then stop, they would appear to an observer at the starting point to be synchronised to one another, but neither could tell the time on the other's clock (or the observer's) simply by looking at his own.

[alancalverd]

Because it's true. The difference between the two scenarios is "simply by looking at his own." 

[hamdani yusuf (OP)]

Because it's true. The difference between the two scenarios is "simply by looking at his own." 

Even when both twins accelerate equally in opposite directions and then stop?

[alancalverd]

How does A know what has happened to B, simply by looking at his own clock?

[hamdani yusuf (OP)]

How does A know what has happened to B, simply by looking at his own clock?

By calculating according to the Time dilatation formula., and assuming that B moves according to the planned sequence.