Will Gravity Affect the Speed of Rockets Traveling Parallel to Each Other?

[sirsmokealot]

hi, i got a question

Lets say we got 2 rockets X km appart, traveling parallel to each other in a straight line at the same constant speed. Now let's put a massive object like the Sun on one of sides on the plane the two spaceships make. Which one will get to the finish line first? Would the gravity of the Sun make one of the rockets get there first because gravity "distorts" time or will they pass the finish line simultaneously?

Sorry for my explanation, hard to explain myself in english when its not my native language. But this picture should clarify.

 

[russ_watters]

Welcome to PF...

You weren't very clear, but the way you worded the problem implies to me a very simple answer:

...traveling parallel to each other in a straight line at the same constant speed...

You didn't specify reference frames from which you would measure those, but if the speed is the same and the distance is the same, then the time is the same.

I suspect, though, that you were trying to ask something more complicated...

 

[sirsmokealot]

Ok, let me try to explain a little more. Let's say you were in the rocket farthest away from the sun, then your time would go faster then the other rockets time. You would use less time, thus finishing earlier. But you are both traveling the same length with the same speed so arent you getting there simultaneously?

 

[A.T.]

Lets say we got 2 rockets X km appart, traveling parallel to each other in a straight line at the same constant speed.

So they basically inertially moving projectiles, with no thrust?

Would the gravity of the Sun make one of the rockets get there first because gravity "distorts" time or will they pass the finish line simultaneously?

Hard to say for the setup in your picture, but let's make the situation simpler by having both rockets keep their straight flight path, and traveling initially at v << c:
Rocket A is far away from the mass, not affected by gravity.
Rocket B travels straight trough the center of the mass (imagine it has a tunnel), which is placed in the middle between start and finish.

In that case rocket B would be faster, due to greater average velocity. Even if its end velocity is the same as A's again, due to symmetry of acceleration & deceleration. This the same for Newtonian Gravity.

But if you replace the rockets with light signals, it's the opposite outcome in General Relativity. Signal B cannot be accelerated beyond c, but will be "slowed down" globally by space curvature. In fact it will simply have more space to cross between start and finish.

Maybe someone else knows at which speed between v << c and v = c they would arrive together. That would be the velocity at which the acceleration is exactly canceled out by the gravitational time dilation due to curved time and longer distance due to curved space.