A matter of time dilation: how much time has actually passed?

[bbbl67]

TL;DR Summary

How much time differential will two relativistic ships actually see between each other if they accelerate at different rates?

Two relativistic spaceships, A & B, are launched simultaneously from Earth towards Kepler 22b, 640 LY away. Ship A is a lighter ship, has no life support as all it carries is some powered-down androids, and it can accelerate at a constant 3g. Ship B is a more massive ship, it carries human crew, and it accelerates at only a constant 1g, to make it more comfortable for the crew. Both ships accelerate constantly for half the trip, and then flip around and decelerate at the exact same rate for the second half.

Ship A takes 640.6 years to reach Kepler 22b, as measured from Earth; on board, only 4.904 years have passed; it reached a maximum speed of 0.9999994919c at the half-way point before starting the deceleration phase. Ship B takes 641.9 years to reach it, as measured from Earth; on board, 12.586 years have passed; it reached a maximum speed of 0.9999954455c before deceleration.

So measured by each ship's onboard clock, there is a time differential of roughly 7.7 years between the arrival times of ships A vs. B. But measured from clocks on Earth, the time differential between each one's arrival at Kepler 22b is only 1.3 years. Assuming that the local time dilation between Earth and Kepler 22b are roughly equal, will the androids see the humans arriving on Kepler 22b: 7.7 years later, or 1.3 years later?

 

[DaveC426913]

Just extracting the data from the narrative for clarity:

Ship A:
a: 3g
v .9999994919c.
t(earth): 640.6 years
t(subjective), 4.904 years

Ship B:
a: 1g
v: .9999954455c
t(earth): 641.9 years
t(subjective): 12.586 years

time discrepancy (subjective): 7.7 years
time discrepancy (earth): 1.3 years


Did I miss anything?
I have not independently verified any of your numbers yet, just transcribed it.


One thing seems odd off the top of my head: Ship B is restricted to only 1g for the entire trip, yet manages to arrive a mere 1.3 years later than ship A?

 

[bbbl67]

Did I miss anything?
I have not independently verified any of your numbers yet, just transcribed it.

No, your summary seems right.

If you want to verify my numbers, I used to following online calculator:
https://www.omnicalculator.com/physics/space-travel

One thing seems odd off the top of my head: Ship B is restricted to only 1g for the entire trip, yet manages to arrive a mere 1.3 years later than ship A?

Yes, that's because the 1g and 3g start to converge once you get to relativistic speeds, since you can't exceed the speed of light.

 

[PeroK]

One thing seems odd off the top of my head: Ship B is restricted to only 1g for the entire trip, yet manages to arrive a mere 1.3 years later than ship A?

After about a year of 1g acceleration, ship A will be close to $c$ relative to Earth. So, ship A takes only about 2 years more than light to get to Kepler 22B (one extra year for the deceleration phase as well). This is largely independent of how long the journey is. Ship B takes only 0.6 years more than light, as it travels at close to $c$ for almost all the journey.

will the androids see the humans arriving on Kepler 22b: 7.7 years later, or 1.3 years later?

1.3 years.

 

[Orodruin]

TL;DR Summary: How much time differential will two relativistic ships actually see between each other if they accelerate at different rates?

Assuming that the local time dilation between Earth and Kepler 22b are roughly equal, will the androids see the humans arriving on Kepler 22b: 7.7 years later, or 1.3 years later?

You are confusing times here. 7.7 years is the difference between the proper travel times of the humans and androids. 1.3 years is the time passing at the destination between the androids arriving and the humans arriving. Therefore, the androids will have aged 7.7-1.3 = 6.4 years less than the humans when they meet again as the humans arrive.

 

[Ibix]

Without checking the numbers, the timeline using Earth clocks is:

• T=0 - departure
• T=640.6 - ship 1 arrives
  • Ship 1 clocks read 4.9
• T=641.9 - ship 2 arrives (1.3 Earth years after Ship 1)
  • Ship 1 clocks read 4.9 + 1.3 = 6.2 (because its clocks tick at the same rate as Earth clocks now it's at rest at its destination)
  • Ship 2 clocks read 12.6 (6.4 years more than Ship 1, consistent with @Orodruin's calculation)

So the first ship only has 1.3 years' head start in exchange for even more insane fuel costs than the 1g trip. Just send then earlier...

 

[Dale]

TL;DR Summary: How much time differential will two relativistic ships actually see between each other if they accelerate at different rates?

So measured by each ship's onboard clock, there is a time differential of roughly 7.7 years between the arrival times of ships A vs. B.

This isn't exactly correct. According to ship A there is a difference of 1.3 years between the arrival times of ships A and B. There is no similar measurement for ship B because ship B was not at both arrivals.

 

[DaveC426913]

After about a year of 1g acceleration, ship A will be close to $c$ relative to Earth. So, ship A takes only about 2 years more than light to get to Kepler 22B (one extra year for the deceleration phase as well). This is largely independent of how long the journey is. Ship B takes only 0.6 years more than light, as it travels at close to $c$ for almost all the journey.

Of course! :slaps forehead: view from Earth does not include any time dilation. Both journeys are effectively at c minus accel and decel legs, which are very brief, relatively.

 

[Halc]

I have not independently verified any of your numbers yet, just transcribed it.

I verified the numbers. 2 digit precision, but the numbers are correct.

One thing seems odd off the top of my head: Ship B is restricted to only 1g for the entire trip, yet manages to arrive a mere 1.3 years later than ship A?

Unintuitive, yea? The actual figure is closer to 1.31 years, Kepler time, but the g force figure was only given to 1 digit of precision, so technically all the results are over-precise.

This isn't exactly correct. According to ship A there is a difference of 1.3 years between the arrival times of ships A and B. There is no similar measurement for ship B because ship B was not at both arrivals.

Good point. Hence my reference to Kepler time since Kepler (and the android ship) are both present at and unaccelerated between the two referenced events.

So the first ship only has 1.3 years' head start in exchange for even more insane fuel costs than the 1g trip. Just send then earlier...

Send the androids earlier. I mean, a LOT earlier. Millennia of tereforming to do before the humans have any business arriving.
Your comment is of course an engineering problem, not a problem of mathematics or theory. Nobody ever actually asks the engineers (and biologists) how to go about colonizing some distant star system. Especially hollywood. Engineers are an annoyance to them.

 

[bbbl67]

Of course! :slaps forehead: view from Earth does not include any time dilation. Both journeys are effectively at c minus accel and decel legs, which are very brief, relatively.

No, acceleration is the entire first half of the trip, while deceleration is the entire second half. The very brief amount of time spent is the point at which they are switching from acceleration to deceleration.

 

[DaveC426913]

No, acceleration is the entire first half of the trip, while deceleration is the entire second half.

Of course, I was not suggesting otherwise. But it's not the point. It's not about the acceleration/deceleration (which, yes, each last half the trip), its about absolute velocity with respect to Earth.

At 1g or at 3g, they both reach ~c within leSs than a year. And then their velocities stay there for ~639 years before dropping back to zero.

So, for 99.8% of the entire journey, they are both moving wrt Earth at, effectively, c. There is no time dilation to factor-in when viewing the scenario from Earth's perspective. It would take sone pretty fancy telescope-foo to observe the difference in velocity between .9999994919c and .9999954455c.

It should be no surprise that they arrive almost simultaneously from Earth's perspective.

 

[bbbl67]

Send the androids earlier. I mean, a LOT earlier. Millennia of tereforming to do before the humans have any business arriving.

Well, Kepler 22b is an actual real exoplanet, and one of the first discovered in the habitable zone of its star. So hopefully, not a lot of terraforming required, since it's already in the habitable zone.

Also the ships are not necessarily cooperating with each other, they could be in competition with each other, sent from two hostile sides. Just launched roughly around the same time as each other.

Your comment is of course an engineering problem, not a problem of mathematics or theory. Nobody ever actually asks the engineers (and biologists) how to go about colonizing some distant star system. Especially hollywood. Engineers are an annoyance to them.

This is basically the premise of the Ridley Scott TV series called "Raised by Wolves".

 

[bbbl67]

You miss the point.

It's not about the acceleration/deceleration (which, yes, each last half the trip), its about absolute velocity with respect to Earth.

The velocity curve rises to ~c within about year and stays there for ~639 years and then drops back to zero in about a year.

View attachment 354740

So, for 99.8% of the entire journey, they are both moving wrt Earth at effectively, c.

The problem is that even though the graph looks pretty flat and the maximum speed looks pretty close, in the middle part of the graph from our very scaled out perspective; if you zoom in to specific parts of the middle of the journey, they are not actually very close. Ship A gets to a maximum of 0.9999994919c, while ship B gets to a maximum of 0.9999954455c, therefore there is a 0.0000040464c differential between their velocities relative to c. That produces a huge time dilation differential at those speeds. They are still accelerating and decelerating at 3g vs. 1g, respectively, even at those speeds.

 

[DaveC426913]

That produces a huge time dilation differential at those speeds.

Time dilation is irrelevant to Earth's perspective regarding arrival time. That's the specific scenario we are discussing in this sidebar (if you go back and check post 8).

Earth sees two ships - both reach c (to within many decimals) within a year, and then spend 639 years there. Their velocities - and therefore total trip duration - from Earth do not inolve time dilation at all.

So, the 0.9999994919/0.9999954455 difference results in a mere 1.3 years delay, as seen from Earth.

 

[Dale]

The problem is that even though the graph looks pretty flat and the maximum speed looks pretty close, in the middle part of the graph from our very scaled out perspective; if you zoom in to specific parts of the middle of the journey, they are not actually very close. Ship A gets to a maximum of 0.9999994919c, while ship B gets to a maximum of 0.9999954455c, therefore there is a 0.0000040464c differential between their velocities relative to c. That produces a huge time dilation differential at those speeds. They are still accelerating and decelerating at 3g vs. 1g, respectively, even at those speeds.

Why is that a problem?

 

[DaveC426913]

@Dale: bbbl67 and I are sort of talking past each other, I think. And it's my fault.

All this arose because in my first post - post 2 - I said "One thing seems odd off the top of my head: Ship B is restricted to only 1g for the entire trip, yet manages to arrive a mere 1.3 years later than ship A?"

In post 8, I realized why:

Of course! :slaps forehead: view from Earth does not include any time dilation. Both journeys are effectively at c...

So I see why the two trips arrives almost simultaneously.

But - I phrased it poorly:

... minus accel and decel legs, which are very brief, relatively.

bbbl67 picked up on my misspeak. And now that I read it a third time, I see what he means:

No, acceleration is the entire first half of the trip, while deceleration is the entire second half. The very brief amount of time spent is the point at which they are switching from acceleration to deceleration.

He is correct. My comment was poorly-phrased and seems to suggest the accel/decel legs of the trip only lasted a short time.

What I should have said is that the effective velocity from Earth's perspective is ~c for 99.6% of the journey - and that virtually all the velocity incease/decrease happens in the first and last year only (as per my diagram).

So, in answering my own question: You'd have tough time measuring the difference between the two ships, one going at 0.9999994919c and the other going at 0.9999954455c. Which is why they arrive almost simultaneously from Earth's perspective.

From Earth's persective and measurements, acceleration (i.e. delta v over delta t) is, in fact, observed to drop off dramatically very rapidly in the trip, as it approaches c (even though the engines are still firing the whole time) and likewise, decel only really starts slowing from c in the last year.

Thanks to @bbbl67 for calling out that poor choice of words. It was unintentionally misleading.

We now return you to your regularly scheduled programming.

 

[Nugatory]

@bbbl67 There is a serious misunderstanding buried in the title of your thread, where you are asking about the “actual time”. There’s no such thing, as least the way you seem to be thinking.

The only physically meaningful time is that measured by a single clock, and it is meaningful only for someone colocated with and at rest relative to that clock. So in your setup there are five “actual times”:
1) the time between ship A departure and arrival as measured by A’s shipboard clock
2) the time between ship B departure and arrival as measured by B’s shipboard clock
3) the time between ship A arrival and ship B arrival as measured by a clock at the destination. This could be a clock that we somehow prepositioned there, or it could be A’s clock if ship A is stopped at the destination, and if so we might as well set it to zero on arrival because we only care about the time from then to B’s arrival.
4) the time between what the earth clock reads when both ships take off and the time that the earth clock reads at the same time that A arrives at the destination.
5) the time between what the earth clock reads when both ships take off and the time that the earth clock reads at the same time that B arrives at the destination.


The ratio between #1 and #4 is A’s time dilation relative to earth

The ratio between #2 and #5 is B’s time dilation relative to earth

#3 is unrelated to the difference between #1 and #2 even though both ships took off at the same time.

There is in general no reason why #3 should be equal to the difference between #4 and #5. It turns out that in this setup it we can make it come out that way, but only because earth and the destination are at rest relative to one another and - crucial but easily overlooked! - we have chosen to define “at the same time” in a way that makes it come out that way.

 

[Mister T]

The title of this thread indicates a possible misconception. How much time "really passes" seems to indicate a belief in some kind of real time that actually elapses compared to the times that elapse aboard the ships. There is no distinction. Any amount of proper time that elapses is just as real as any other. There is no universal actual time.

 

[DaveC426913]

True. Although his ultimate question is still valid:

"...will the androids see the humans arriving on Kepler 22b: 7.7 years later, or 1.3 years later?"

The answer is 7.7 years later.

(It is Earth that sees them arrive 1.3 years apart.)

 

[Dale]

"...will the androids see the humans arriving on Kepler 22b: 7.7 years later, or 1.3 years later?"

The answer is 7.7 years later.

The answer is 1.3. When the B ships arrive their clocks differ from A’s by 6.4 years, but the reading on the B clocks does not impact the duration on A clocks between the two landings.

 

[Nugatory]

The answer is 7.7 years later.

There’s an ambiguity here. Are we talking about the time measured by ship A clock from takeoff plus the time measured at destination between A arrival and B arrival, or just the time between A arrival and B arrival? You’re answering the former, @Dale is answering the latter which I think was what OP meant:

But measured from clocks on Earth, the time differential between each one's arrival at Kepler 22b is only 1.3 years. Assuming that the local time dilation between Earth and Kepler 22b are roughly equal, will the androids see the humans arriving on Kepler 22b: 7.7 years later, or 1.3 years later?

 

[pervect]

True. Although his ultimate question is still valid:

"...will the androids see the humans arriving on Kepler 22b: 7.7 years later, or 1.3 years later?"

The answer is 7.7 years later.

(It is Earth that sees them arrive 1.3 years apart.)

Surely, if the androids look at their wristwatches when they land, and again when the humans land, their wristwatch will keep Kepler 22b time, which is assumed to be the same as Earth time as it is assumed they are essentially at rest and there is no significant time dilation.

Since the Earth time difference is only 1.3 years, (assuming the calculation is right, it seems reasonable), the Kepler 22b time difference should be the same.

I assume that's the question the original poster was asking? I.e. the proper time measured by the androids on kepler between the two e event (androids land on kepler) to the event (humans land on kepler)?

 

[Halc]

Well, Kepler 22b is an actual real exoplanet, and one of the first discovered in the habitable zone of its star. So hopefully, not a lot of terraforming required, since it's already in the habitable zone.

Being in the habitable zone just means that the planet (say our moon) is the right distance from the sun for water to exist at the surface. It doesn't mean there is actually any water there, or especially, that the planet is actually habitable. You seem to be thinking a class-M planet, one where you can step out of the ship unsuited without immediate death. Class M planets only exist in fiction where the TV show lacks the budget to depict an actual exoplanet.

Ship A gets to a maximum of 0.9999994919c, while ship B gets to a maximum of 0.9999954455c, therefore there is a 0.0000040464c differential between their velocities relative to c. That produces a huge time dilation differential at those speeds.

Depends on your definition of 'huge'. The trips take a subjective duration of 5 and 12.5 years respectively. That's a factor of 2.5, which is a lot, but not 'huge'. The arrive with their watches only about 6.3 years out of sync. There's plenty of stories about slower trips to stars far closer that involve decades of differential aging.

 

[DaveC426913]

Depends on your definition of 'huge'. The trips take a subjective duration of 5 and 12.5 years respectively. That's a factor of 2.5, which is a lot, but not 'huge'. The arrive with their watches only about 6.3 years out of sync. There's plenty of stories about slower trips to stars far closer that involve decades of differential aging.

Perhaps an appropriate time to post this:

 

[Orodruin]

True. Although his ultimate question is still valid:

"...will the androids see the humans arriving on Kepler 22b: 7.7 years later, or 1.3 years later?"

The answer is 7.7 years later.

(It is Earth that sees them arrive 1.3 years apart.)

This is wrong. On Kepler they also see the humans arriving 1.3 years later. It is the proper time of the journey itself that is 7.7 years longer for the humans. Since the androids also spend 1.3 years on Kepler before the humans arrive, they will be 6.4 years younger.

 

[Orodruin]

Depends on your definition of 'huge'. The trips take a subjective duration of 5 and 12.5 years respectively. That's a factor of 2.5, which is a lot, but not 'huge'. The arrive with their watches only about 6.3 years out of sync. There's plenty of stories about slower trips to stars far closer that involve decades of differential aging.

The maximal gamma factors will differ significantly, but the thing is that this is irrelevant. The part of the journey that corresponds to a large gamma factor contributes comparatively little to the elapsed proper time since it is suppressed by the large gamma factor. The larger contribution to elapsed time along each ship’s world line is going to come from the start of the acceleration phase and the end of the deceleration phase.

 

[pervect]

A few more thoughts on a slightly different, more formal analysis of the problem. When we ask "how much time has actually passed", my thoughts are that we want is a physically significant proper time. It's possible the OP wants something different, some sort of coordinate time perhaps, but in that case, they'd need to specify the coordinate system they want to use. If one want something independent of a particular coordinate choice (and one should!), proper time is the correct approach.

To measure proper time, we need a worldline with two events on it. Then the proper time is simply the Lorentz interval between the two events integrated along the worldline. It's the "length of the curve" connecting the two points.

I believe the OP specified that we want the time for the androids, so that's whose worldline we need to use. And the two events on the worldline of significance are the time that the androids land, and the time that the slower Earth spaceship lands, we've defined the problem.

Because the worldline is straight, finding it's "length" is easy.

Starting from the general equation, where I've used units such that c=1 for convenience, we get:

$$d\tau = \sqrt{dt^2 - dx^2}$$

when we integrate we get

$$\tau = \sqrt{(t2 - t1)^2 - (x2-x1)^2}$$

And the OP has already computed the coordinates of these events in the Earth frame (the choice of coordinates doesn't matter, as we've mentioned the proper time is coordinate independent), we have everything we need to work the problem. Because x2=x1=640 for this problem (using units of years for time and light years for distance the answer is just the time difference between the Earth coordinates.