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Another Relativity "Paradox"

Posted April 11, 2016 7:00 AM by Jorrie
Pathfinder Tags: acceleration relativity

A futuristic spaceship is stationed at its base, which is static at 10 light years (ly) from Earth in free space. The ship and base clocks are rate adjusted to run on Earth time.

The ship now departs from base and accelerates outward along the Earth radial at 1g on its accelerometer, for one year on its clock. Relativistic dynamics tells us that it achieves a speed of 0.76c relative to base and that the acceleration lasted for 1.175 base years. I also says that the ship was then 0.54 ly from base and 10.54 ly from Earth, all in base inertial coordinates.[1]

However, as observed in the ship's (new) inertial frame, that 10.54 ly Earth distance will be Lorentz contracted to 10.54(1-0.762)0.5 = 6.83 ly. This means that in ship coordinates, despite accelerating away from Earth for one year, Earth is 10-6.83 = 3.17 ly closer to the ship. What is more, this decrease in distance occurred at an average rate of 3.17 ly per year, which is effectively 3.17c.[2]

Paradoxical? Maybe. What do you think?

Regards, Jorrie

Notes (for the relativistically minded):

[1] Acceleration of 1g is very close to 1 ly per y2. Note that c=1 in these units. For 1g, the relative speed change is given by a simple hyperbolic tan function: Δv/c = tanh(Δt) = tanh(1) = 0.76, where Δt is the acceleration time as measured by the ship's clock.

In the base frame (X,T), time and distance traveled during the ship's acceleration is: ΔT = sinh(Δt) and ΔX = cosh(Δt)-1 respectively. It is obvious that we are here dealing with hyperbolic spacetime movement of the ship in the base frame. It is commonly known as Rindler coordinates. See https://en.wikipedia.org/wiki/Rindler_coordinates

[2] While we know that it isn't strange to find 'funny' speeds during acceleration, the initial and final distances were at least measured when the ship was inertial. It is true that after the acceleration, the ship could not have instantly measured the distance to Earth. But there could have been a handy inertial frame around, one that also moves at 0.76c away from Earth. The ship's captain could simply have asked a nearby stationary observer: "how far is Earth now?" The answer would undoubtedly have been 6.85 ly.

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#1

Re: Another Relativity "Paradox"

04/11/2016 8:10 AM

"Acceleration of 1g is very close to 1 ly per ly2." ? Think there's an "l" (or maybe "l2") too many in there.

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#2
In reply to #1

Re: Another Relativity "Paradox"

04/11/2016 8:36 AM

Yup, I'll fix that; thanks John.

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#3

Re: Another Relativity "Paradox"

04/11/2016 10:25 AM

You can't subtract the distance measured in the new inertial frame (6.83 lyr) from the distance measured with the base inertial frame (10 lyr).

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#5
In reply to #3

Re: Another Relativity "Paradox"

04/11/2016 12:14 PM

"You can't subtract the distance measured in the new inertial frame (6.83 lyr) from the distance measured with the base inertial frame (10 lyr)"

That's worth half a point.

If not, why not?

After all, we took the time on the ship's clock at the beginning and end and then subtracted them to find the proper time...

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#6
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Re: Another Relativity "Paradox"

04/11/2016 1:07 PM

The initial (base) frame and the final frame are moving with respect to each other. The base-earth distance appears to be 10 lyr from the base (and earth) frame. This same separation in space-time is foreshortened in the final reference frame which is moving with respect to the base/earth frame.

The two vectors of separation in space-time are not parallel. It's like trying to add two sides of a triangle and expecting to get the length of the third side.

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#7
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Re: Another Relativity "Paradox"

04/12/2016 1:42 AM

Rixter, you are right pf course, but at the risk of being overly pedantic, I'm still missing a clinching argument to resolve the 'paradox'. Apart from the invalid subtraction, there is red herring on the table...

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#10
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Re: Another Relativity "Paradox"

04/12/2016 8:50 PM

I fold!

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#13
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Re: Another Relativity "Paradox"

04/13/2016 2:04 AM

I wrote "Rixter, you are right of course, but at the risk of being overly pedantic, I'm still missing a clinching argument to resolve the 'paradox'. Apart from the invalid subtraction, there is [a] red herring on the table..."

Actually, you were only right on the "invalid subtraction" because of the arbitrarily chosen distance of 10 ly between Earth and base that I used. The 'red herring' is that it straddles the Rindler horizon associated with the acceleration.

For the 1g ship, the horizon permanently stays at 1/g=1 ly behind the accelerating ship. If I do the sums for an initial Earth distance of say 0.8 ly base-Earth distance, I find that after the acceleration, Earth's distance has increased from 0.8 to 0.87 ly in the ship's frame. The apparent paradox then disappears.

Just like the Schwarzschild metric that breaks down at the event horizon, the Rindler metric breaks down at the Rindler horizon. In fact if the acceleration did not stop, the 0.8 ly distance would have grown to asymptotically approach the horizon, but never reach it, as observed by the ship.

-J

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#4

Re: Another Relativity "Paradox"

04/11/2016 11:55 AM

Have you discovered an intradimensional wormhole...?

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#8
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Re: Another Relativity "Paradox"

04/12/2016 3:27 AM

"Have you discovered an intradimensional wormhole...?"

"A wormhole is not really a means of going back in time, it's a short cut, so that something that was far away is much closer," NASA's Eric Christian wrote.

So yea, I made Earth disappear at 10 ly distant and reappear one year later at 6.85 ly.

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#9

Re: Another Relativity "Paradox"

04/12/2016 5:07 AM

Questions like this are enough to make the brain hurt. <:-(

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#14
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Re: Another Relativity "Paradox"

04/13/2016 4:25 AM

PWSlack, do not feel alone. This one has stumped a few CERN physicists for a while. (Learned that from Don Lincoln, Fermilab/Cern)

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#11

Re: Another Relativity "Paradox"

04/12/2016 9:59 PM

Before commenting on the "paradox", let's clear up a possible error. You wrote:

Earth is 10-6.83 = 3.17 ly closer to the ship.

Shouldn't it say:

Earth is 10-6.83 = 3.17 ly away from the ship.

?

Also, is the first paradox the twin paradox?

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#12
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Re: Another Relativity "Paradox"

04/13/2016 12:57 AM

Should have rather said:

"Earth is 10-6.83 = 3.17 ly closer to the ship than it was before the acceleration".

Yes, AFAIK Einstein originated the twin paradox indirectly. I think his was a 'one-way-and-stop" thought experiment.

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#15

Re: Another Relativity "Paradox"

04/13/2016 10:02 AM

Lorentz contractions aren't real, they are an optical illusion to an observer. Since the ship probably could not be see through a telescope at that distance, the problem seem mute. Still it is a result I wouldn't have expected.

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#17
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Re: Another Relativity "Paradox"

04/13/2016 11:44 AM

I suppose "aren't real" is more philosophical than scientific, but the science says that the ship would always have seen Earth through a good telescope. The reason is that the acceleration stopped after some time, causing the Rindler horizon to disappear.

If the ship did carry on accelerating, maintaining the Rindler horizon, Earth's light would have been quickly redshifted out of existence and effectively have 'disappeared'.

Anyway, like discussed with Rixter, the paradoxical contraction result happened because Earth was deliberately placed on the other side of the Rindler horizon and hence the subtraction of distances were false.

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#16

Re: Another Relativity "Paradox"

04/13/2016 11:06 AM

Something's been bugging me about this for a while, but I couldn't put my finger on it. Then I realized the problem:

"A futuristic spaceship is stationed at its base, which is static at 10 light years (ly) from Earth in free space. The ship and base clocks are rate adjusted to run on Earth time."

The premise that two objects can be locked in the same inertial frame of reference when separated by cosmic differences is a logical impossibility. Once two objects are distant enough to prevent 'real-time' communication between them, then their position, velocity, and time rates are mere approximations from the other's point of view. Even if the Earth and the station were sending a constant data stream with clock information, you cannot be certain if the variations are due to differing inertial frames, or from other bodies warping the space-time the signals are passing through.

But perhaps I'm reading too much into the problem due to the setting. Maybe I should look at it using the simplified model of the 'cosmic pool table,' with the Earth, The station, and the ship being the only objects on the table, and assuming a 'cosmic now' that allows the clocks to be confirmed to be running at the same rate.

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#18
In reply to #16

Re: Another Relativity "Paradox"

04/13/2016 12:56 PM

"The premise that two objects can be locked in the same inertial frame of reference when separated by cosmic differences is a logical impossibility."

Well, 10 ly distance in free space is hardly a "cosmic difference". The expansion of the universe is utterly negligible in this case. I do not know what you mean by "real time communication", but no communication is instantaneous, that's the norm.

What I do agree with is that when there is a horizon (Rindler, gravitational or cosmic) involved, one must be careful when you define inertial frames that include them, but it can still be done with due caution.

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#19
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Re: Another Relativity "Paradox"

04/13/2016 3:07 PM

"Real Time communication" means communication where the delays are short enough that the participants feel like the conversation is a dialog rather than correspondence. It is communication where the phrase 'synchronize watches' can make sense.

A "Cosmic distance" as I was defining it, is one where the time-of-flight delays in light (and radio waves) make "Real Time communication" impossible.

Perhaps a simplified example might help. Alice and Bob each have a clock, a video camera aimed at the clock, and a monitor receiving the other person's video feed. They have "Real Time communication" when Alice can look at her monitor, set her clock to be in sync with the image she sees, and Bob, looking at his monitor, can agree that the clocks are now in sync. When they are distant enough that Alice has to set her clock ahead of the one she sees in order for Bob to confirm that he sees the two clocks in sync, then they are at a "Cosmic Distance," and each only knows the other's past, as Alice's Now won't bee seen by Bob until Bob's Future, and vice versa.

Once there is no way to have a 'shared Now,' then all the observations get tangled up into a wibbly-wobbly, timey-wimey ... ball.

With 10 ly of distance, the station cannot even be sure Earth still exists, all they can cay with certainty is "it was there ten years ago."

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#20
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Re: Another Relativity "Paradox"

04/13/2016 3:21 PM

To 'solve' the paradox, once the ship is far enough from the station that "Real Time communication" is no longer feasible, Then instead of two Nows, we have three Nows, each only knowing the past of the other two, but not its present, until the ship gets close enough to Earth for their Nows to interact and become the same Now.

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#21
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Re: Another Relativity "Paradox"

04/13/2016 3:55 PM

"Real Time communication" means communication where the delays are short enough that the participants feel like the conversation is a dialog rather than correspondence."

Now that's a subjective definition if I've ever heard one.

Are the Moon and the Sun then also at 'cosmic distances'?

"Once there is no way to have a 'shared Now,' then all the observations get tangled up into a wibbly-wobbly, timey-wimey ... ball."

Relativity is one of very few theories about spacetime where things are not 'wibbly-wobbly'. The spacetime interval is one of the true invariants...

Do I detect a measure of anti-relativity sentiment?

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#22
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Re: Another Relativity "Paradox"

04/14/2016 10:12 AM

"Now that's a subjective definition if I've ever heard one."

Well, the line between 'using Newtonian Physics because it's easier math' and 'using Einsteinian Physics because it's more accurate.' is a bit subjective. Just how fast must an object be moving before Newtonian is no longer accurate enough? It's like the 'sand heap paradox,' how many grains of sand must you place together to make it a heap? How many grains of sand can you remove from a heap until it becomes 'not a heap?'

"Are the Moon and the Sun then also at 'cosmic distances'?"

Technically, yes, if the Sun suddenly vanished at the same time as I hit 'Preview Comment,' nobody on Earth would know about it for about eight minutes.

Even the GPS satellites are technically at a 'cosmic distance,' that's why we need a signal from FOUR to locate ourselves on the planet. We use three to trilateralate(1) our position, and the fourth we use as the reference clock to compare the others against to get the 'time of flight' calibration for the signals.

"Do I detect a measure of anti-relativity sentiment?"

No, you just failed to detect a measure of Doctor Who fan sentiment. I was exaggerating the level of uncertainty between isolated inertial frames to shoehorn in a quote.

After all, one of the key premises of astronomy is, oversimplified, Distance = Time. The farther away something is, the further back in time our observations of it are. We see the Sun as it was eight minutes ago, we see Alpha Centauri as it was four years ago, looking out to the edges of the Observable Universe is looking back in time almost to the Universe's birth.

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#23
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Re: Another Relativity "Paradox"

04/14/2016 12:53 PM

I have a problem with your definition of "cosmic distance". It surely has little to do with whether things are 'Newtonian' or 'Einsteinian'. Relativistic effects are about very high speeds and/or strong gravitational fields, irrespective of distance scale.

Cosmological effects are about distances large enough so that the expansion of the universe has to be taken into account, typically from 300 million light years outwards. Then distance is not linear with time, e.g. the light from near the 'edge' of the observable universe took some 13.8 billion years to each us. When that light was emitted, the 'edge' was only some 42 million years away and today it is some 45 billion light years from us.

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#24
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Re: Another Relativity "Paradox"

04/14/2016 2:20 PM

Since the original problem was looking at clocks starting out 'in sync' while distant enough that their being in the same inertial frame of reference would be in doubt, I have been structuring my responses along the lines of inertial frames and being able to confirm that the clocks are running at the same rate. After all, if one cannot confirm the initial conditions of an experiment, one cannot be sure if the result means what it is appearing to mean.

I'm really getting convinced that the 'setting' of the problem was what was throwing me off; that things would be easier if the extra variables were removed from the description: "On the Cosmic Pool Table are three balls: Cue Ball A, Cue Ball B, and Marble C, all three have Cesium-accurate chronometers on them, and all three are starting 'at rest' on the pool table, with Cue Balls A and B 10 light years apart, and Marble C adjacent to Cue Ball B." Now we've removed the problem of 'distant clocks in sync,' while defining all chronometers to run at identical rates. We're not supposed to care about Earth-Station sychronization, just elapsed time on each observation point. And when Marble C goes through its acceleration and deceleration to travel from B to A, it becomes clear which formulas need to be used.

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#25
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Re: Another Relativity "Paradox"

04/14/2016 2:55 PM

Yes, it is so that clocks in free space cannot really be synchronized with Earth clocks, unless they are rate adjusted. This is due to the inevitable differences in gravitational time dilation between Earth and free space.

However, with a rate adjusted clock and a known distance, it is possible to reasonably verify the accuracy of the synchrony. And with the times involved in the test, the required accuracy is quite low.

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#26

Re: Another Relativity "Paradox"

04/16/2016 10:29 AM

As a simple novice engineer I cannot argue with the logic - I take what the experts say at face value. But for me there is no paradox. It is the experts that seem to disagree amongst themselves.

The discrepancies in results to my simplistic mind can easily be explained by my 'view' on the world of relativity is that time (the speed of time) itself is changing. This gives rise to a force of acceleration that we 'feel' as gravity and is indistinguishable from gravity.

Gravity itself causes light to curve. Thus everything we 'see' that we think is in a perfectly straight line (in this universe) but is actually curved - coupled to which we have no reference or target point in which to travel to take a short cut.

But we can draw pictures on paper to demonstrate the logic. Your spaceship and Earth separated by 10 ly when measured by the clocks (synchronised?) in a line that is indistinguishable from a perfectly straight line, are in fact two points on a curve.And 10 ly is the length of arc.

It is when you juggle the numbers in your formula that you get a number of 6.83 that equates to the shortest distance between the ship and Earth. ie, a short cut along the chord of the circle.(of a diameter we know not of).

In our reality we have no way of knowing how to travel along this short cut, because whatever route we took would follow the arc, and if we pointed our ship in a different direction (a random guess at the short cut) we would simply travel along the arc of a smaller diameter size circle - and thus further.

We 'live' in a world where light travels on a composite curve of infinite gravity circles, collectively on average of almost infinite diameter and thus the difference only shows up in calculations rather than in practice.

Which incidentally, a triangle drawn to connect with an intermediate point on the infinite arc between the spaceship and Earth would have angles of 0, 180, 0 where the long side would in fact be equal to the length of the other two sides added together.

I imagine.

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#27

Re: Another Relativity "Paradox"

06/07/2016 11:06 AM

"However, as observed in the ship's (new) inertial frame, that 10.54 ly Earth distance will be Lorentz contracted to 10.54(1-0.762)0.5= 6.83 ly."

There really isn't any paradox going on here. The problem is the manufacturer of the spaceship forgot to etch following warning into the glass:

"Objects in the mirror are farther than they appear!"

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#28

Re: Another Relativity "Paradox"

01/28/2017 6:26 AM

In the reference frame of the base (x=0), the ship always occupies some space-time event (x,t). And the base knows that the ship thinks the distance is only x/γ.

The time-rate-of-change of that quantity is:

d/dt (x/γ) = x'/γ - x γ'/γ2

Is it possible to judiciously choose an acceleration profile such that the above always equals zero ?

x'/x = γ'/γ

Then, in such a "Sisyphian" case, the ship would perpetually accelerate, but never perceive itself to be distancing itself from the base !

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