Relativity Calculation: Two bodies traveling at relativistic speeds

[keeper blue]

Body 1 travels 48 light years from point A to point B at 82% of light speed.

Body 2 leaves point A 33 years after Body 1 and travels the 48 light years to point B at 99.99995% of light speed.

What I think I know:

• Body 1 takes 58 years to reach point B according to an earth observer.
• Body 1 takes 33 years to reach point B according to an on board observer.
• Body 2 takes 48 years to reach point B according to an earth observer.
• Body 2 takes 14 days to reach point B according to an on board observer.

What I do not know and I hope someone can tell me:

1. If Body 2 sends a message to Body 1 on their departure date
   1. When will Body 1 receive it (on board elapsed time)
2. According to Body 1, how much time will elapse between their arrival at point B and the arrival of Body 2?

This is research for a SF novel but one which adheres to the known laws of physics, so I need to get it right and really appreciate your help in doing so.

Mike

 

[PeterDonis]

according to an earth observer

Where is earth in your scenario? Is it point A? Point B? Somewhere in between?

• Body 1 takes 58 years to reach point B according to an earth observer.

$48 / .82 = 58.54$, so this looks ok (although I would round up to 59 years if we are working to the nearest year).

• Body 1 takes 33 years to reach point B according to an on board observer.

Gamma factor is $1 / \sqrt{1 - .82^2} = 1.747$, and $58.54 / 1.747 = 33.51$, so again this looks ok (although again I would round up to 34 years if we are working to the nearest year).

• Body 2 takes 48 years to reach point B according to an earth observer.

Yes, since the speed of Body 2 is close enough to $1$ for this purpose.

• Body 2 takes 14 days to reach point B according to an on board observer.

Not sure about this. I get a gamma factor of $1000$ for Body 2, which would mean travel time for the on board observer of $48 / 1000 = 0.048$ years, which works out to $17.53$ days.

What I do not know and I hope someone can tell me:

1. If Body 2 sends a message to Body 1 on their departure date
   1. When will Body 1 receive it (on board elapsed time)

Will the light arrive at Body 1 before or after Body 1 reaches Point B? (I'm assuming both bodies stop when they reach Point B.) Answering that should make it much easier to answer this question.

• According to Body 1, how much time will elapse between their arrival at point B and the arrival of Body 2?

What time in the Point A/B (earth) rest frame does Body 2 arrive at Point B? Answering that should make it much easier to answer this question.

 

[PeroK]

This is research for a SF novel but one which adheres to the known laws of physics, so I need to get it right and really appreciate your help in doing so.

Mike

These sort of calculations are all very well, but accelerating a large body to very nearly the speed of light is a bigger issue that getting the timings right to the nearest year.

 

[keeper blue]

These sort of calculations are all very well, but accelerating a large body to very nearly the speed of light is a bigger issue that getting the timings right to the nearest year.

Hi Pero, Absolutely correct so that is where we need some fiction in the science fiction. I am going with this unless you can suggest some other fiction? "Waldo Hasser perfects the Alcubierre warp drive with the integration of a quantum singularity to generate the necessary folded geometry of space-time."

 

[PeroK]

Hi Pero, Absolutely correct so that is where we need some fiction in the science fiction. I am going with this unless you can suggest some other fiction? "Waldo Hasser perfects the Alcubierre warp drive with the integration of a quantum singularity to generate the necessary folded geometry of space-time."

Then the basic calculations of SR in flat spacetime do not necessarily apply to onboard clocks. The point of Alcubierre is effective FTL.

 

[keeper blue]

Where is earth in your scenario? Is it point A? Point B? Somewhere in between?$48 / .82 = 58.54$, so this looks ok (although I would round up to 59 years if we are working to the nearest year).Gamma factor is $1 / \sqrt{1 - .82^2} = 1.747$, and $58.54 / 1.747 = 33.51$, so again this looks ok (although again I would round up to 34 years if we are working to the nearest year).Yes, since the speed of Body 2 is close enough to $1$ for this purpose.Not sure about this. I get a gamma factor of $1000$ for Body 2, which would mean travel time for the on board observer of $48 / 1000 = 0.048$ years, which works out to $17.53$ days.Will the light arrive at Body 1 before or after Body 1 reaches Point B? (I'm assuming both bodies stop when they reach Point B.) Answering that should make it much easier to answer this question.What time in the Point A/B (earth) rest frame does Body 2 arrive at Point B? Answering that should make it much easier to answer this question.

Hi Peter, Thanks so much for your help and pleased to see my calcs are not too far off the mark. Earth is Point A and a fictitious exoplanet called Arkaedis is point B. I am proposing that both Body 1 and Body 2 start at rest at Point A and finish at rest at point B. Hope that helps?

Mike

 

[keeper blue]

Then the basic calculations of SR in flat spacetime do not necessarily apply to onboard clocks. The point of Alcubierre is effective FTL.

To keep it psudo real, I have decided to impose a limit of 99.99995% for the Alcubierre drive. It keeps the calcs easier and somewhat understandable for readers. Going beyond light speed is to far into the unknown and is unnecessary. At light speed the onboard experience is arriving at the time you depart over any distance. Does FTL imply that you arrive before you leave... I have trouble with the logical implications of FTL travel. Do you have any ideas about it?

 

[PeroK]

Do you have any ideas about it?

Nothing more than is covered by Wikipedia:

https://en.wikipedia.org/wiki/Alcubierre_drive

 

[Orodruin]

Does FTL imply that you arrive before you leave...

In some reference frames, yes.

 

[keeper blue]

In some reference frames, yes.

I think my brain is melting.

 

[PeroK]

I think my brain is melting.

There's something here about it.

https://www.physicsforums.com/threads/near-light-speed-and-time.899454/page-3#post-5737549

The issue is that a FTL speed disrupts the mathematics of the Lorentz Transformation.

It's clear that if you send something FTL and it comes back FTL, it cannot come back before you sent it. The problem comes when you try to analyse that sequence in a reference frame moving relative to you. That's when the Lorentz Transformation breaks down, in the sense that causality is lost in that reference frame.

 

[PeterDonis]

Earth is Point A and a fictitious exoplanet called Arkaedis is point B. I am proposing that both Body 1 and Body 2 start at rest at Point A and finish at rest at point B.

Ok, good. Then, once each body arrives at point B, its clock ticks at the same rate as point B's clock, but its clock will have an offset because of the time dilation during the trip. So the obvious next things to figure out (and you should be able to do this) are:

What time is it on point B's clock when Body 1 arrives?

What time is it on Body 1's clock when Body 1 arrives at point B? (This, compared with the above, tells you the offset of Body 1's clock compared to point B's clock.)

What time is it on point B's clock when Body 2 arrives?

What time is it on Body 2's clock when Body 2 arrives at point B? (This, compared with the above, tells you the offset of Body 2's clock compared to point B's clock.)

 

[Ibix]

Incidentally, if you have an Alcubierre drive is there necessarily time dilation? I think it would depend on details of how the warp bubble worked, wouldn't it?

 

[PeterDonis]

if you have an Alcubierre drive is there necessarily time dilation?

IIRC in the original Alcubierre metric an observer at the center of the bubble has no time dilation relative to an observer at rest relative to the overall frame in which the metric is written, far outside the bubble.

 

[PeterDonis]

The issue is that a FTL speed disrupts the mathematics of the Lorentz Transformation.

No, that in itself is not the issue. Lorentz transformations transform spacelike vectors just fine.

It's clear that if you send something FTL and it comes back FTL, it cannot come back before you sent it.

No, it's not clear. That's the issue with FTL: when you "send something FTL", which spacelike trajectory does it follow? That will depend on the specific theory of "sending things FTL" that you adopt. But in any such theory in which the "FTL emission" depends on an invariant involving the emitter's worldline (for example, "FTL emission is always on a spacelike trajectory that is orthogonal to the emitter's worldline"), then it is possible for an FTL round trip between two emitters in relative motion to result in the original emitter receiving the return FTL item before it is sent.

The problem comes when you try to analyse that sequence in a reference frame moving relative to you. That's when the Lorentz Transformation breaks down, in the sense that causality is lost in that reference frame.

No. If the FTL item does not return to the original emitter before it was sent, that will be true in all reference frames, because the "emission" and "return" events are timelike separated (they must be, since they both lie on the same object's worldline), so their ordering is invariant.

 

[PeterDonis]

a FTL speed disrupts the mathematics of the Lorentz Transformation

It is true that you cannot transform to a frame of reference in which an FTL object is at rest; that would involve a Lorentz transformation with a relative speed greater than $c$, which doesn't work. But that is a different issue from the issue of under what conditions FTL motion raises causality issues. You don't need to define a "rest frame" for an FTL object to address the latter question.

 

[Ibix]

IIRC in the original Alcubierre metric an observer at the center of the bubble has no time dilation relative to an observer at rest relative to the overall frame in which the metric is written, far outside the bubble.

IIRC the original paper doesn't describe the formation or dissipation of a bubble, and that could induce all sorts of effects, I suppose. Although any effect on trip duration would presumably be a function of the formation and dissipation process rather than the distance travelled.

 

[PeterDonis]

IIRC the original paper doesn't describe the formation or dissipation of a bubble

The original paper assumes that the bubble exists forever. Of course this is not physically realistic, but it makes the math much easier. It also neatly sidesteps all of the issues with causality that we are discussing in this thread, since the bubble in the original paper never changes speed or direction so it cannot cause any causality issues. It's only when you have multiple bubbles that can be created or destroyed, or a bubble that can change direction, that you can get causality issues.

 

[keeper blue]

Peter, Pero and Ibix, FTL is a fascinating distraction and gives rise to all manner of paradoxical scenarios that would be interesting to incorporate into a SF novel, so I have saved this post and would like to come back to you for 'technical advice' on the subject after I finish this current book. In my current work, I propose using the Alcubierre principle to achieve near light speed (99.9999%) as this velocity is enough for the purpose of the story. Once I have finalised the detail, I will come back to you for a final check to ensure we are 'hard science' in respect of the calculations. Have a great day and thanks.