Questioning Constant Acceleration & Sanity - Is Something Wrong?

[somethingswrong]

I'm starting to question my sanity.

This wikipedia article claims a ship with a constant acceleration of 1g will cross 100000 light-years in 24 years by traveller's clock: https://en.wikipedia.org/wiki/Space_travel_using_constant_acceleration

Numerous calculators over the web give you the same value (example: http://nathangeffen.webfactional.com/spacetravel/spacetravel.php).

I get a sense something is wrong here. Are they using the acceleration measured in observer's frame? But that's nonsensical, it's not what "constant acceleration" means by any sane definition, right?

Aren't proper acceleration, traveller's time and observer's distance connected with a simple Newtonian s=½at²? The result is about 622 years, not 24.

Am I missing something?

 

[PeroK]

I'm starting to question my sanity.

This wikipedia article claims a ship with a constant acceleration of 1g will cross 100000 light-years in 24 years by traveller's clock: https://en.wikipedia.org/wiki/Space_travel_using_constant_acceleration

Numerous calculators over the web give you the same value (example: http://nathangeffen.webfactional.com/spacetravel/spacetravel.php).

I get a sense something is wrong here. Are they using the acceleration measured in observer's frame? But that's nonsensical, it's not what "constant acceleration" means by any sane definition, right?

Aren't proper acceleration, traveller's time and observer's distance connected with a simple Newtonian s=½at²? The result is about 622 years, not 24.

Am I missing something?

Yes, you're missing quite a lot! The most fundamental thing is that, due to time dilation, large distances can be traveled by a spacecraft (or high energy particle) in relatively short time in the traveller's frame. If we leave out acceleration for a moment and simply look at a spacecraft traveling at high speed away from the Earth.

In the Earth's frame of reference, the spacecraft will cover a distance $s = vt$. But, the traveller's time will be slower due to time dilation and the traveller will reach a certain destination in less time (in their frame). The faster the craft is traveling the less time will elapse for the travellers between leaving the Earth and reaching their destination. In principle, you could travel as far as you want in any given time. Although, there are overwhelming engineering problems to build a spacecraft that could get anywhere near the highly relativistic speeds required.

To take acceleration. For something traveling at a high speed, there are two measures of acceleration:

In the Earth's frame the acceleration is the usual $dv/dt$.

In the traveller's frame, the acceleration is called "proper" acceleration. What they are talking about in the webpage is constant proper acceleration. But, the accleration as measured on Earth would get less and less as the craft got faster. On Earth, the speed would asymptotically tend towards the speed of light as an unreachable terminal velocity. In the craft, however, they could (in principle) keep on accelerating constantly indefinitely. Strange, but true!

 

[Nugatory]

They mean proper acceleration, which is the acceleration measured by a shipboard accelerometer.

To get the distance covered, you have to allow for time dilation, length contraction, and relativistic velocity addition. When these are included, the Newtonian $s=at^2/2$ doesn't work, which is not surprising when you consider that it's derived by integrating $v=at$ - and that formula cannot be correct because it predicts speeds far greater than the speed of light.

 

[somethingswrong]

In the traveller's frame, the acceleration is called "proper" acceleration. What they are talking about in the webpage is constant proper acceleration. But, the accleration as measured on Earth would get less and less as the craft got faster. On Earth, the speed would asymptotically tend towards the speed of light as an unreachable terminal velocity. In the craft, however, they could (in principle) keep on accelerating constantly indefinitely. Strange, but true!

I understand that. What I have trouble understanding is where does the result 24 years come from? It's too fast! How can we arrive 25 times faster than we would on Newton's terms?

This is what Wikipedia has to say on proper velocity:

change in proper velocity is the integral of proper acceleration over map time i.e. Δw = αΔt for constant α

I'm confused.

 

[Ibix]

In short, time dilation. To an observer on Earth, the minimum time needed for a ship to reach a star 20 light years away is 20 years, assuming the capability to acceletate instantly to arbitrarily close to lightspeed. In practice, obviously, it'll take longer, but that's the limit.

Aboard ship, however, the time taken can be made as small as you like by sufficiently brutal acceleration. The observer on Earth notices that the clocks on the ship tick slower the faster it goes, so they tick fewer times during the crossing. At 99% c, for example, clocks on the Earth tick seven time for every one tick on the ship. So what the Earth observer sees as taking about 20.2 years takes less than 3 (20.2/7) aboard the ship.

The maths is more complex for a sane acceleration, but the principle is the same.

 

[PeterDonis]

a ship with a constant acceleration of 1g will cross 100000 light-years in 24 years by traveller's clock

I've bolded the key phrase. It will take more than 100,000 light-years by clocks at rest at the origin and destination--in other words, more time by those clocks than it would take a light ray to travel between the same two points.

Aren't proper acceleration, traveller's time and observer's distance connected with a simple Newtonian s=½at²? The result is about 622 years

You do understand that Newtonian physics is not valid for objects traveling close to the speed of light, right? You need to use special relativity. That's what the online calculators you used are doing. SR says it takes more than 100,000 years according to clocks at rest at the origin and destination, as above--and it also says that, because of time dilation, only 24 years elapses according to clocks on the ship.

The SR math is described in this article:

http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html

 

[black hole 123]

If we leave out acceleration for a moment and simply look at a spacecraft traveling at high speed away from the Earth.

i thought the whole effect of time dilation happens because of acceleration (e.g. in the twin paradox). because velocity is always continuous, the faster u accelerate to speed v the greater the acceleration the greater the time dilation.

 

[Ibix]

i thought the whole effect of time dilation happens because of acceleration (e.g. in the twin paradox). because velocity is always continuous, the faster u accelerate to speed v the greater the acceleration the greater the time dilation.

No - differential aging (as in the twin paradox) is not a result of acceleration. It's due to the differing intervals along the paths followed by the two twins. Basically, the elapsed time for an observer is a measure of the distance they have traveled through space-time between two events. Since the two twins do not follow the same path the length of the paths, and the elapsed time along the paths, may be different.

That said, you are correct that the final age difference is in general dependent on the acceleration(s) used. This is, however, because different accelerations produce different paths with different lengths (i.e., different elapsed times for observers following the path), rather than because of the acceleration per se.

The scenario being discussed here is only one leg of a twin paradox. Since the ship isn't returning, this isn't quite the same as a twin paradox. One can say unambiguously how long shipboard clocks will say it takes to make the trip. However, how long the trip takes according to Earth clocks depends on the frame of reference used.