Special Relativity: Astronaut & Earth Light Speed Paradox

[Gary Boothe]

I would like to make sure my interpretation of special relativity is correct. Is the following valid?

An astronaut is traveling away from Earth at 95% of light speed. She turns on a headlight in the nose of the space ship. Question 1: How fast does she measure the light traveling from the headlight? Question 2: How fast is the light from the headlight traveling away from the Earth?

Answer to question 1: c.

Answer to question 2: c.

This apparent paradox is due to the fact that distance and time are distorted in a reference frame that is moving with respect to another reference frame. Earth would see the clocks on the spaceship running slow, and the distances contracted, so that light seems to creep slowly from the headlight at 5% of c, but moving at c with respect to the Earth. Similarly, the astronaut would see the clocks on Earth running slower than her clock, and the speed of light would appear sluggish, but with respect to the spaceship it would have a velocity of c.

 

[Orodruin]

This apparent paradox is due to the fact that distance and time are distorted in a reference frame that is moving with respect to another reference frame.

I do not think "distorted" is a good word to use here as it seems to imply that it is not "distorted" in the Earth frame. Space and time are different in the two frames, but no frame is special.

 

[Joe Ciancimino]

I might suggest the study of the Lorentz Transformation as a primer before getting deep into relativity.

 

[Janus]

I would like to make sure my interpretation of special relativity is correct. Is the following valid?

An astronaut is traveling away from Earth at 95% of light speed. She turns on a headlight in the nose of the space ship. Question 1: How fast does she measure the light traveling from the headlight? Question 2: How fast is the light from the headlight traveling away from the Earth?

Answer to question 1: c.

Answer to question 2: c.

This apparent paradox is due to the fact that distance and time are distorted in a reference frame that is moving with respect to another reference frame. Earth would see the clocks on the spaceship running slow, and the distances contracted, so that light seems to creep slowly from the headlight at 5% of c, but moving at c with respect to the Earth. Similarly, the astronaut would see the clocks on Earth running slower than her clock, and the speed of light would appear sluggish, but with respect to the spaceship it would have a velocity of c.

There is a third "effect" that has to be taken into account, the Relativity of Simultaneity. Put your light in the middle of the ship and have the light going towards both the nose and tail of the ship. For the astronaut, the light arrives at the nose and tail simultaneously. However, for the Earth, they do not. The light travels at c in both directions, and the nose is "running away" from it, and the tail is "running" towards" it. Thus the light hits the tail before it hits the nose.
To illustrate, consider the two clocks below:

Light is emitted from a point halfway between them and starts each clock when it hits each clock. Each clock starts simultaneously and then runs in sync.

Now consider things from a frame where the clocks are moving at a good fraction of the speed of light to the right.

The light is still emitted from the halfway point and travels outward at c. But now the light hits and starts the left clock before hitting and starting the right clock. Once both clocks are running, they run at the same speed, but are out of sync.

The above example does not take into account time dilation or length contraction, but even with them included, the basic argument remains the same. In the frame of the clocks, they are in sync, and in the frame in which they are moving, they are not.

With our spaceship, If we put clocks in the nose and tail that are synchronized according to the astronaut, the light arrives simultaneously and when the clocks read the same. According to the Earth, the light does not arrive simultaneously, but neither are the clocks in sync, with the end result being that the Earth agrees that the readings on each clock when the light reaches it are the same for both clocks.

 

[harrylin]

I would like to make sure my interpretation of special relativity is correct. Is the following valid?

An astronaut is traveling away from Earth at 95% of light speed. She turns on a headlight in the nose of the space ship. Question 1: How fast does she measure the light traveling from the headlight? Question 2: How fast is the light from the headlight traveling away from the Earth?

Answer to question 1: c.

Answer to question 2: c.

This apparent paradox is due to the fact that distance and time are distorted in a reference frame that is moving with respect to another reference frame. Earth would see the clocks on the spaceship running slow, and the distances contracted, so that light seems to creep slowly from the headlight at 5% of c, but moving at c with respect to the Earth. Similarly, the astronaut would see the clocks on Earth running slower than her clock, and the speed of light would appear sluggish, but with respect to the spaceship it would have a velocity of c.

Yes indeed. Note that it is necessary to set up or refer to an inertial reference system (a supposed "rest" system) that is co-moving with the space ship, as elaborated by Janus. And the astronauts may instead choose to keep using the ECI frame (as they probably would do for short trips); in that case they simply agree with the Earth measurements.

 

[jartsa]

Earth would see the clocks on the spaceship running slow, and the distances contracted, so that light seems to creep slowly from the headlight at 5% of c, but moving at c with respect to the Earth.

Actually it's like this:

Earth would see the clocks on the spaceship running slow, and the distances contracted, in such way that light would be measured to distance itself from the headlight at c, if those distorted clocks and measuring sticks were used to measure the rate of the increase of the distance.

Using non-distorted and Earth located measuring devices the light would be measured to have speed c, and the spaceship would be measured to have speed 0.95 c. Then simple arithmetics says that light would increase its distance from the headlight at rate 1.0 c - 0.95 c.

 

[Gary Boothe]

Thanks all, for your illuminating comments.