# Time dilation and length contraction

VertexOperator
I am confused about these two concupiscences of the consistency of light.
One of my books/notes says that time dilation and length contraction do not happen at the same time because they are 'the same thing' which kind of makes sense but the other says "when we reach relativistic velocities both time dilation and length contraction allow us to reach our destination faster because less time passes for the astronauts and less distance needs to be covered. However, mass dilation is one of the reasons that achieving relativistic speed is very difficult, as more and more energy is converted into mass as the speed increases..."
So can someone please explain this to me :)
Thank you!

clamtrox
... but the other says "when we reach relativistic velocities both time dilation and length contraction allow us to reach our destination faster because less time passes for the astronauts and less distance needs to be covered. However, mass dilation is one of the reasons that achieving relativistic speed is very difficult, as more and more energy is converted into mass as the speed increases..."

I wouldn't read this source if I were you. It's very confusing, and using an archaic formalism.

Consider a spaceship traveling from Earth to Alpha Centauri. From Earth's inertial frame, the distance the rocket covers is ~4 ly, but the clocks on board the spaceship run slower. This is time dilation (moving clocks run more slowly)

From the spaceship frame, the distance Earth and Alpha Centauri is shorter than 4 ly. This is length contraction (distances are shorter in a moving reference frame).

VertexOperator
Wow, I had a huge conceptual misunderstanding. Thank you so much :)
This is what I wrote in my notes:
"The nearest star to us, Proxima Centauri, is about 4.3 light years away. Travelling at the fastest speed any space probe has ever gone, it would take us over 42,000 years to reach it. At these speeds, it would take about 265 million years to reach the centre of our galaxy, and traveling beyond our galaxy would take inconceivable amounts of time.
However, space scientists have not given up. Designers of a new kind of spacecraft called a light sail make the claim that these craft could achieve speeds of up to 0.1c. In the future, engineers may be able to design spacecraft that can reach speeds much closer to the speed of light, and this would mean that the effects of time dilation and length contraction would become significant, and work in our favour. From the Earth's inertial frame, the distance the rocket covers is 4.3 light years, but time passes slower on the spacecraft than on Earth, so for the astronaut’s, the journey would take less time.. This is time dilation (moving clocks run more slowly). From the spaceship’s frame, the distance Earth and Alpha Centauri would have contracted to a shorter distance than 4.3 light years. This is length contraction (distances are shorter in a moving reference frame). Therefore the distance is shorter, which they cover in less time.
However, assuming that such speeds are technically possible, the energy costs of achieving them would be prohibitive, since acceleration is always the most costly phase of a space mission This issue is made even worse by the effect of mass dilation, because as speeds approach the speed of light, even greater force and energy input is needed for only marginal increases."

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clamtrox
That's correct. Note that observers in both inertial frames agree on how much the astronauts have aged; just the cause of the lack of aging is different in different frames.

VertexOperator
That's correct. Note that observers in both inertial frames agree on how much the astronauts have aged; just the cause of the lack of aging is different in different frames.

So ultimately both time dilation and length contraction lead to the same result. That is what they meant by 'they are the same'. This is a much better explanation :) Thank you!

clamtrox
So ultimately both time dilation and length contraction lead to the same result. That is what they meant by 'they are the same'. This is a much better explanation :) Thank you!

Yes. A "deeper" explanation would be that there are things which do not depend on the coordinate system (like for example how wrinkly are the astronauts when they reach their destination). In different coordinate systems, space and time coordinates are different, and therefore the thing one coordinate system considers as "time" can be a mixture of "space" and "time" in another. This is why one observer calls the phenomenon time dilation, whereas another calls it length contraction.

Suraj Nehra
I wouldn't read this source if I were you. It's very confusing, and using an archaic formalism.

Consider a spaceship traveling from Earth to Alpha Centauri. From Earth's inertial frame, the distance the rocket covers is ~4 ly, but the clocks on board the spaceship run slower. This is time dilation (moving clocks run more slowly)

From the spaceship frame, the distance Earth and Alpha Centauri is shorter than 4 ly. This is length contraction (distances are shorter in a moving reference frame).

As far as I know it is the length of moving object that contract and not that of the reference point at rest

Staff Emeritus