Saturday, February 24, 2007

Time Dilation to AC

Images of Sol and AC components from: http://upload.wikimedia.org/wikipedia/commons/thumb/1/17/Alpha_Centauri_relative_sizes.png/350px-Alpha_Centauri_relative_sizes.png


Obviously not to scale. Above diagram shows our notional starship in three phases of flight from Solar System to Alpha Centaurian System.

I. Accelerate for 300 days at g-force. This achieves a high cruise speed to shorten flight duration and also simulates gravity for contents and occupants. Since g-force comes from particle flow leaving aft of vessel, occupants feel a gravity like force from decks (axial planes intersecting the longitudinal axis). Thus, "up" is toward forward end of vessel which is pointing toward AC, our destination.

II. Cruise for 4.2 years at .866c. Since there's not enough fuel to accelerate for entire flight, flight planners must determine another way to simulate Earth gravity. This is done by rotating craft in such a manner to produce centrifugal g-force against inside of outer hull. Thus, from Phase I to Phase II, ship must transform from living area on decks (floors perpendicular to exhaust flow of particles from aft end of vessel) to living area on hull. Thus, occupants must quickly adjust from decks being floors and hull being wall to decks becoming walls and hull becoming floor. Thus, "up" is toward the central, longitudinal axis of the cylindrical vessel. (While Phase I's up would be parallel for all occupants, Phase II's up would differ for every occupant.)

III. Decelerate for 300 days at g-force; ship must slowdown from enormous velocity (86.6% of light speed) to operational velocity required to manuever around AC system. Thus, hull becomes a wall again, and decks once more becomes a floor. Of course, exhaust flow must point toward destination because the exhaust force must now counteract motion of vessel. (Of course, "up" once again points forward, but the forward end of vessel is now pointing toward Sol, our place of departure.)

Finally, onboard observed times will differ from Solar observed times because Special Relativity has been shown to "slow time" at high speeds. Using Lorentz Transform, we determine that onboard observed times are:

Phases I and III: 300 days becomes 270 days. (Note: For both phases, we use the linear "average velocity" between 0 and .866c. Of course, this is the midvalue, .433c.)

Phase II: Inflight cruise time is observed by ship occupants to slow down considerably. From the Lorentz Transform, we calculate that ship crew and passengers observe time to slow down to one half the rate of time as observed from a relative speed of zero; mainly, flight controllers back on Earth observe inflight cruise time taking 4.2 years to travel the cruise distance of 3.64 LYs; however, onboard observers observe inflight cruise time taking 2.1 years.

Paradox: Does the ship arrive at AC earlier then a photon that left at same time? (Recall light particles would by definition take 3.64 years to travel 3.64 LYs.)

I think not. Since I need enhanced understanding to better answer this question, I sent an email to a reknown astrophysicist who has this understanding as well as the eloquence to communicate the answer in a very understandable way. Email to Dr. Tyson follows; hopefully, I'll be able to post his REPLY soon. (Note: Since I'm sure Dr. Tyson has considerable demands on his time, I changed parameters to simplify above thot problem. Concept remains the same.)

Sunday, Feb 25, 2007, sent following email to Neil D. Tyson:
I just saw your Cspan piece which I greatly enjoyed. I also bought and read your previous book, "Adventures of Urban Astrophysicist" (Forgive me if I got title wrong.)
I have a question for which I'm sure you can provide the answer, or even better, a pointer to where you've already answered it. (i.e., I'm more then willing to buy another of your excellent books, if that's where the answer is.)
Subjection: Time Dilation due to Special Relativity.
From casual reading, I understand that time slows down when one travels at significant fraction of light speed, c. Furthermore, I've heard of various thought experiments such as the "Twins Paradox" where one twin departs at a young age on a journey at near light speeds for a year. The twin then returns to find his/her twin a senior citizen.
For me to pose my question about time dilation, consider the following thought experiment. Assume that we have a capable interstellar vehicle that can achieve very high percentages of c. This vessel departs from Sol and quickly accelerates to 86.6% of light speed (.866c). It then maintains .866c for 4.0 light years (4.0 LY). Finally, it decelerates to a near zero speed to accomplish its mission at Alpha Centauri stellar system.
We know the following items about the 4.0 light years that we've just cruised at .866c.
1. A photon of light will take 4 years to travel this distance.
2. An observer at the departure point will observe the spaceship taking 4.615 (=4.0/.866) years to travel that distance.
3. Onboard observers observe cruise portion to take 2.3075 years. (By using the Lorentz Transform, we determine that observers on the spaceship will observe a much shorter timespan (due to "time slowing"). t' = t SQRT(1-v**2/c**2). Since we've chosen v = .866c, I calculate onboard time to be half of time observed by observers back at Sol.)
Thus, it appears that our notional spaceship has beaten the photon to our destination.
I doubt that our notional spaceship actually exceeded lightspeed, but I'm also sure that my understanding of the time dilation phenomena needs to improve.
Can you provide pointers to help?
Many thanks,
Jim O

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