Time Dilation: Travel from Solar System to Proxima Centauri

[Golf7]

If someone was to travel from the solar system to Proximi Centauri, I believe that from their point of view time would pass normally and regardless of how fast they were travelling, the speed of light would still be c. As I understand it, as the person traveling approaches c relative to Earth time relative to Earth will pass more slowly. How is it possible though that from the point of view of the person traveling they could reach Proxima Centauri in less than 4.24 years? Wouldn't that require the star to move towards them at speed that relativity doesn't allow.

 

[Nugatory]

Wouldn't that require the star to move towards them at speed that relativity doesn't allow.

No, not after you allow for length contraction as well.

A good exercise is to try writing down the position and time coordinates of the two events (ship leaves earth; and ship arrives at destination) in the Earth reference frame and then using the Lorentz transforms to convert the coordinates to the ship frame.

Departure event:
At t=0 and x=0 in the Earth frame, a ship zooms past as us at .5c heading towards Proxima Centauri which is rest relative to us. This is also t'=0 and x'=0 in the ship frame, in which the ship is at rest and Proxima Centauri is approaching the ship at .5c (and the Earth is receding behind the ship at the same rate). What is the x' coordinate of Proxima Centauri in the ship frame? That's the distance from earth, in the ship frame.

Arrival event:
At t=8.48 years, x=4.24 light-years in the Earth frame the ship zooms past Proxima Centauri (but note that an earth-bound telescope won't see this happen for another 4.24 years, when t=12.72 years). In the ship frame, the x' coordinate of this event is 0. What's the time coordinate in the ship frame? That's the amount of time that it took Proxima Centauri to cover the distance between it and the ship.

 

[ZikZak]

Time dilation is not the only relativistic effect. Lengths of bodies traveling at relativistic speeds also contract. So, the spaceship pilot, at rest in the spaceship, observes that the distance between the relativistically moving Earth and Proxima to be contracted. In other words, he does not have to travel the whole 4.24 light-years, but something significantly less.

 

[ghwellsjr]

Wouldn't that require the star to move towards them at speed that relativity doesn't allow.

No, it just requires that the coordinate system in which the traveler is at rest has a different assignment for how long distances are and that is why he measures the distance to be shorter. It's called length contraction.

Actually, from his rest frame, he's not reaching Proxima Centauri, but rather Proxima Centauri is reaching him.

Relative speeds, that is, the speed that object A measures object B to be traveling with respect to object A and vice versa are always the same in Special Relativity. So the traveler can measure how fast Proxima Centauri is approaching him and based on how long it takes he can determine how far away it was when it started moving toward him.

 

[sardar]

First of all, time dilation is a well-established concept in physics, and it is a consequence of Einstein's theory of relativity. It states that time passes differently for observers in different frames of reference, depending on their relative speeds and gravitational fields.

In the scenario described, the person traveling from the solar system to Proxima Centauri would experience time passing normally from their perspective, as you correctly stated. However, from the perspective of an observer on Earth, time would appear to pass more slowly for the traveler as they approach the speed of light.

This is because the closer an object gets to the speed of light, the more energy is required to accelerate it further. This means that as the traveler approaches the speed of light, their velocity relative to Earth would appear to decrease, causing time to slow down for them.

As for the possibility of reaching Proxima Centauri in less than 4.24 years, it is important to note that the speed of light is the maximum speed at which information can travel in the universe. This means that no object, including stars, can move towards the traveler at a speed faster than the speed of light.

However, due to the effects of time dilation, the traveler's perception of time would be different from that of an observer on Earth. For the traveler, the distance to Proxima Centauri would appear to shrink as they approach the speed of light, allowing them to reach their destination in less than 4.24 years.

It is also worth mentioning that this scenario assumes constant velocity, which is not possible in reality. In order to reach Proxima Centauri in less than 4.24 years, the traveler would have to constantly accelerate towards their destination, which would require a tremendous amount of energy.

In conclusion, the concept of time dilation explains how the perception of time can differ for observers in different frames of reference. While it may seem counterintuitive, it is a fundamental aspect of our understanding of the universe and has been confirmed through numerous experiments and observations.