How Does Time Dilation Affect Light Speed from a Moving Spaceship?

[Giulio B.]

A spaceship is moving, then it launches a ray of light behind itself.
if time in the spaceship is slower than in other frames that are moving less fast, shouldn't the ray appear much faster than c?

this thing works with a ray launched in front of the ship, but behind, shouldn't the time go faster to let us see the ray at c speed?
thanks.

 

[selfAdjoint]

A spaceship is moving, then it launches a ray of light behind itself.
if time in the spaceship is slower than in other frames that are moving less fast, shouldn't the ray appear much faster than c?

this thing works with a ray launched in front of the ship, but behind, shouldn't the time go faster to let us see the ray at c speed?
thanks.

No. First of all "The time on the ship" relative to the ship isn't slowed at all. As far as the crew of the ship are concerned, they might as well be at rest. That's why their state is caled the "Rest Frame" of the ship. Ship time is only slowed as experienced by an observer for whom the ship is moving fast.

So the ship, at rest with respect to itself, emits a light beam in any direction, and the crew measures the beam's light speed, and whaddayaknow, they get c.

 

[Mickey]

Someone traveling behind the ship, catching up to it with greater speed, would meet the beam of light head on and also measure the light's speed at c.

A third person watching the two ships both whiz by from a nearby planet would also measure the light's speed at c.

 

[Riposte]

if time in the spaceship is slower than in other frames that are moving less fast

This isn't quite right. In SR, everything's relative. One frame can not be said to be moving faster or slower than another. From the spaceship's frame, time is moving slower both ahead and behind it. From a planet's frame (as the spaceship goes whizzing by) time is moving slower on the spaceship. Both frames see a time contraction and a length contraction that force c to be the same speed in any reference frame.