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Let A and B be two points that are at rest relative to an inertial frame S. Let's say the distance between them is 1·109 m.
In the beginning you are at rest relative to S, but during exactly 1 second (counted by your wristwatch), you have managed to accelerate up to the velocity v=0.99c. Let's imagine you survived the acceleration.
Your velocity vector (relative to S) is parallel to the line segment AB.
Such that when your velocity has reached 0.99c relative to S, (From now on, we will only use your reference frame.) the distance between A and B must be (1·109/γ) m where γ=7.09 is the lorentz factor.
That means the distance between them is now 1.41·108 m.
So during a second, the distance between them decreased by 8.59·108 m.
Let's say A had the highest velocity during the time you accelerated. Then A could not have moved less than (8.59·108/2) m = 4.295·108 m which is about 4.3·108 m.
That means the highest velocity A had, could not have been less than about 4.3·108 m/s (since it lasted for 1 second).
And that velocity is the same as 1.4c, which is above the speed of light.
So my conclusion is that an object with uniform motion can have a velocity above c relative to an accelerating reference frame, but not relative to an inertial frame. Is this correct or have I done something wrong?
In the beginning you are at rest relative to S, but during exactly 1 second (counted by your wristwatch), you have managed to accelerate up to the velocity v=0.99c. Let's imagine you survived the acceleration.
Your velocity vector (relative to S) is parallel to the line segment AB.
Such that when your velocity has reached 0.99c relative to S, (From now on, we will only use your reference frame.) the distance between A and B must be (1·109/γ) m where γ=7.09 is the lorentz factor.
That means the distance between them is now 1.41·108 m.
So during a second, the distance between them decreased by 8.59·108 m.
Let's say A had the highest velocity during the time you accelerated. Then A could not have moved less than (8.59·108/2) m = 4.295·108 m which is about 4.3·108 m.
That means the highest velocity A had, could not have been less than about 4.3·108 m/s (since it lasted for 1 second).
And that velocity is the same as 1.4c, which is above the speed of light.
So my conclusion is that an object with uniform motion can have a velocity above c relative to an accelerating reference frame, but not relative to an inertial frame. Is this correct or have I done something wrong?