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.