Your confusion is because you are missing something here…time aboard the spaceship.
For simplicity sake I’ll assume instant acceleration. If you do that to near light speed then the trip to Alpha Centauri, as perceived by people on earth, will take about 4.5 years. The trip as perceived by those aboard the spaceship will take less than a second. Say our explorers immediately turn around and return to earth at the same speed and the whole trip will have taken them a matter of a few seconds but 9 years will have passed on earth.
The reason time gets funky like this isn’t too hard to explain. When you measure the speed of light you will always get the same answer. Imagine an experiment on earth (which we’ll say is not moving even though that is not correct) where you shoot a photon at a target some distance away and time it. Do the math and you get 186,000 miles per second. Now do the same experiment aboard a spaceship moving at near light speed. Your photon leaves the emitter and races to its target but the target is moving away from you (the photon does not gain energy due to the speed of the ship like a baseball would if thrown in the ship). The photon is racing down the length of your spaceship but the target at the other end is moving in the same direction just a little slower than your photon. As a result it takes longer for the photon to reach the target. The thing is, when I take my reading I get 186,000 miles per second for the speed of my photon. Look at the math here: Speed=Distance/Time. I know my speed and I know my distance between the emitter and the target. What’s left is time and it needs to be adjusted to allow the 186,000 miles/second answer to be true. In this case you need to slow time down to get your answer.
It gets more complicated than that (length also shortens at relativistic speeds for instance…go fast enough and the entire Universe would seem to be inches across) but that should be enough to give you the sense of it.
Right- I’m confused now, and I thought I understood this stuff.
Your spaceship has got a heck of a lot of fuel, and has to continue to accelerate for 68 years at 1g-
that is the equivalent of a rocket supporting itself and its gigatonnes of fuel motionless in earth’s gravity, say 100metres off the ground for 68 years…
but to the crew, it has only been 4.8 years of acceleration at 1 gee…
and the fuel has been time dilated too, so do they only use 4.8 years of fuel or 68years?
I am pretty sure they use 68 years worth of fuel, but that means that when the crew monitor the rocket motor they see it burning more than ten times as fast in order to provide a subjective 1 gee at 0.999c …
everything isn’t relative, after all.
The rocket has to burn a lot faster.
Or am I wrong?
Whack-a-Mole
It takes several minutes for the light from the sun to reach the earth… How can someone get to AC and back in a few seconds. Even using ship time as the benchmark?
Nevermind, I get it now, I re-read your post. The time on the ship would slow WAY down as you approach relativistic speeds - that coupled with the length arguement- helps explain.
Lets see if I understand this now; Our observed time here on Earth is directly effected (though we dont notice) by our planets speed through the universe.
People on a different planet, which may be traveling much faster then ours, will still measure the speed of light on their planet as 186,000 miles/sec. But, because they are traveling faster than us their second is longer than our second. Substitute planets for any mode of transportation such as rockets or airplanes.
Actually, the way to think about it is that as you approach c, time stops. It has no meaning. So, if you travel at ver close to c, essentially no time elapses.
As for accelerating to c, the problem is pretty simple to understand if you look at the equation for relativistic mass:
m® = m(0)/sqrt(1-(v/c)^2)
m(0) is the rest mass. As v approaches c, m® approaches infinity. You quickly run out of force to accelerate an infinite mass.
Everything moving at near-c speeds relative to earth would experience time dilation, including the spaceship itself and all of its fuel. So in the example, they would use 4.8 years worth of fuel, not 68 years.
I don’t know the answer to this but I’ll speculate and see if it sparks an answer in others…
I wonder if this quandry relates to my question in the OP about the fuel itself gaining mass. You need 68 years worth of mass but only start with 4.8 years of mass. As you accelerate your fuel gains mass that in the end becomes equivalent to the 68 years you need as defined on earth.
Time aboard the ship is the only thing that matters at any moment as far as the ship is concerned. Your mass and fuel, as well as the air, food, supplies, that you need is 4.8 years worth to reach 99.99% C.
People on earth see you doing the same thing, but in their perspective it takes longer for everything to happen. It’s all a matter of reference frames.
Exactly the same things happen at exactly the same rates. It’s just that people interpret those rates differently depending on their reference frame.
And for whoever asked this question, it’s totally a matter of speed, not acceleration. Whether you get to 99.99% C at 1 g or 3 g or 1,000,000 g it’s all the same when you get there.
The equation relating time and velocity in a stationary frame is:
v = at / [1 + (at/c)[sup]2[/sup]]
You can solve this equation for t or just check out these examples for a = 1g acceleration.
v = .7700000c…ship time = 1 year……stationary time=1.19 year
v = .9700000c…ship time = 2 year……stationary time=3.75 year
v = .9999300c…ship time = 5 year……stationary time=83.7 year
v = .9999998c…ship time = 8 year……stationary time=1840 year
You can find the gory details somewhere in the phsics faq.
I didn’t want to solve the above equation because it looked like I’d have to complete the square and I didn’t want to do that. But I forgot that my calculator has a solve function. Here’s what it says:
t = (c*v/a)[-1/(v[sup]2[/sup]- c[sup]2[/sup]][sup]1/2[/sup]
For my favorite fictional treatment of this subject, see Poul Anderson’s novel Tau Zero. (It currently looks as if the universe is not going to collapse into another Big Bang, but it’s still a good story.)
That Albert Einstein… man, he was some kind of genius or something.
I’ll second Tau Zero as a good story, but IIRC Anderson says it takes about a year at 1G to get close to light-speed, and that the ship will cover about half a lightyear during that time. He never specifies how close to light-speed he means (or whether that year is local time), but it sounds like he used the linear equation for velocity.
No matter really, since the main action takes place quite a bit later anyway.
According to the formulas I gave before (which I plugged into a spreadsheet to make it easy to answer these variables when I was struggling with this) at 1 g you get up to 77% C after 1 year of ship’s time (1.19 year’s of earth time). My numbers, I see, agree with Ring’s using a different equation. Always good to see independent verification. And yes, you travel .56 light year in that time.
I think the equation I used can be derived from your much more complete and elegant set. The only reason I used the one I did is because a lot of peoples eyes tend to glaze over when they see hyperbolic functions.
i just skip the maths, i dont even know what half the symbols mean hehe.
how much fuel you’d need in a hyphothetical situation is rather irrelevant.
the proof about the atomic clock will possibly be flawed by the fact that the atoms in the clock could too be affected by the speed(which proves or disproves very little), or maybe that IS the proof?
but anyways, say “matter” is ‘affected’(for a lack of better wording) depending on your relative speed, so that it’s not as much the fact that time passes faster or slower, but rather that your perception of it. i.e. you would actually move slower aboard the ship?
is this a halfway explanation? or is it plain babble? hehe
