A spaceship travelling at 0.8c

[smithg86]

Homework Statement

Suppose our sun is about to explode. In an effort to escape, we depart in a spaceship at v = 0.8c and head toward the start Tau Ceti, 12 light years away. When we reach the midpoint of our journey from the Earth, we see our Sun explode and, unfortunately, at the same instant we see Tau Ceti explode as well.

(a) In the spaceship's frame of reference, should we conclude that the two explosions occurred simultaneously? If not which occurred first?

(b) In a frame of reference in which the Sun and Tau Ceti are at rest, did they explode simultaneously? If not, which exploded first?

Homework Equations

Lorentz transforms, velocity transforms

1 lightyear = the distance light travels in one year = c*(1 year)

The Attempt at a Solution

I'm really confused about how to approach this problem. Any hints? Thanks.

 

[learningphysics]

Hint: For part a) consider the doppler effect... get the space-time coordinates of both explosions in the ship's frame...

b) just use lorentz transformations to go from the coordinates in the ship's frame to the coordinates in the rest frame...

 

[Doc Al]

When we reach the midpoint of our journey from the Earth, we see our Sun explode and, unfortunately, at the same instant we see Tau Ceti explode as well.

Another hint: Key is the statement "we see". What does that tell you? (Deduce everything you can.)

 

[smithg86]

correct me if I'm wrong, but this is what I'm thinking:

Let event A = Sun, event B = Tau Ceti, both from a stationary frame, S. Also let A' = Sun, B' = Tau Ceti, from the spaceship's frame, S'. Call the spatial distance between A and B = L, from a stationary frame. In S', A and B happen at the same time, t = 0.
A' = (x',t') = (-L/2, 0)
B' = (x',t') = (+L/2,0)

I want the coordinates for A and B = (x,t)

Applying the Lorentz transforms from S' --> S,

t = \gamma(t' + vx'/c^2)

with:
\gamma = 5/3
t' = 0
v = 0.8c
x' = +- L/2 = 6 light years

I got:
t_A = -8 yr
t_B = +8 yr

So A (The sun exploding) happened first, 16 yrs before B (Tau Ceti exploding)...(?)

-------
edit:
Doc Al, I assumed that 'we' meant the people on the ship traveling at 0.8c.

 

[Doc Al]

In S', A and B happen at the same time, t = 0.

How did you deduce this?

Doc Al, I assumed that 'we' meant the people on the ship traveling at 0.8c.

Yes, but I meant to emphasize the "see" part. What does the fact that they see the explosions tell you?

 

[smithg86]

Doc Al,
I thought the choice of a coordinate system was arbitrary, so I set the spaceship to (x,t) = (0,0)

 

[Doc Al]

In S', A and B happen at the same time, t = 0.

My question is: How did you deduce that those two events (the explosions) happen at the same time according to the spaceship frame?

 

[smithg86]

"When we reach the midpoint of our journey from the Earth, we see our Sun explode and, unfortunately, at the same instant we see Tau Ceti explode as well."

^^i thought that meant that the light from the two explosions reached the spaceship at the same time...but light always travels at c, no matter what observer measures its speed. so i thought that since they were at the midpoint of the trip, and the light from the 2 stars hit them at the same time, that the two events happened at the same time. I am confused.

 

[Doc Al]

"When we reach the midpoint of our journey from the Earth, we see our Sun explode and, unfortunately, at the same instant we see Tau Ceti explode as well."

^^i thought that meant that the light from the two explosions reached the spaceship at the same time...

Absolutely!

but light always travels at c, no matter what observer measures its speed.

Also true. But each observer measures the speed of light with respect to his own frame.

so i thought that since they were at the midpoint of the trip, and the light from the 2 stars hit them at the same time, that the two events happened at the same time.

Remember that "happen at the same time" depends on the frame doing the measurements. Simultaneity is relative. According to the frame of the stars, light travels the same distance to reach the midpoint. But what about the spaceship frame? Hint: where must the spaceship have been (approximately) when the explosions happened? Was it closer to the sun or the star?

 

[smithg86]

Doc Al,

the spaceship was closer to the sun when the explosions occurred, it took some time for the light to reach the ship. by the time the light hit the ship, the ship was at the midpoint (x = L/2).

if \Deltat is the time it took for the light to hit the ship at x = L/2, then c\Deltat = L/2, so \Deltat = L/(2c). Since the ship travels at speed v, was the ship at x = L/2 - \Deltatv = L/2 - (Lv)/(2c) = L(c-v)/(2c) = 1.2 light years from the sun when it exploded?

 

[Doc Al]

the spaceship was closer to the sun when the explosions occurred, it took some time for the light to reach the ship. by the time the light hit the ship, the ship was at the midpoint (x = L/2).

Good. Since the ship was closer to the sun before reaching the midpoint, there's no way that the explosions could have happened at the same time according to the spaceship frame. If they did happen at the same time, then the ship would have seen the explosion of the sun before seeing the explosion of the star. (You should be able to deduce which explosion occurred first according to the spaceship--no need to do any calculations.)

if \Deltat is the time it took for the light to hit the ship at x = L/2, then c\Deltat = L/2, so \Deltat = L/(2c). Since the ship travels at speed v, was the ship at x = L/2 - \Deltatv = L/2 - (Lv)/(2c) = L(c-v)/(2c) = 1.2 light years from the sun when it exploded?

Good--as long as you realize that all of this calculation is from the view of the sun-star frame. The spaceship is 1.2 light years from the sun when both explode according to the sun-star observers. Since the spaceship concludes that the explosions happened at different times, he must have been at a different location for each explosion--according to spaceship observations.

 

[learningphysics]

I was making the problem way more complicated than it needed to be... very nice explanation Doc Al.

 

[smithg86]

Since the spaceship concludes that the explosions happened at different times, he must have been at a different location for each explosion--according to spaceship observations.

I thought the spaceship saw both stars explode at the same time? Thus the spaceship would think it was in the same spot for both explosions?

"When we reach the midpoint of our journey from the Earth, we see our Sun explode and, unfortunately, at the same instant we see Tau Ceti explode as well."

Also, the ship is 1.2 ly away from the sun (according to the sun-star observers) when the sun explodes, but how far away is the ship from the sun according to the spaceship? Can I say this: (?)

let:
location of ship according to stars at time of explosion = x_sun
location of ship according to spaceship at time of explosion = x_ship

x_sun = (x_ship) \gamma

so x_ship = 1.2/(5/3) = 0.72 light years?

 

[learningphysics]

The thing is that two events that are seen at the same time in a frame of reference... don't necessarily happen simulatenously in that frame...

The spaceship is aware that he is in motion relative to the stars and uses that knowledge to figure out the actual times of the explosions in his frame of reference...

 

[Doc Al]

I thought the spaceship saw both stars explode at the same time? Thus the spaceship would think it was in the same spot for both explosions?

"When we reach the midpoint of our journey from the Earth, we see our Sun explode and, unfortunately, at the same instant we see Tau Ceti explode as well."

That's why I wanted you to focus on the word "see" and what it means. The fact that the spaceship saw both explosions at the same time (and just as it reached the midpoint between the sun and star) means that the light from both the sun and the star reached the midpoint at the same time. Just because the light from the explosions reaches the ship at the same time does not mean that the explosions happened at the same time.

Also, the ship is 1.2 ly away from the sun (according to the sun-star observers) when the sun explodes, but how far away is the ship from the sun according to the spaceship? Can I say this: (?)

let:
location of ship according to stars at time of explosion = x_sun
location of ship according to spaceship at time of explosion = x_ship

x_sun = (x_ship) \gamma

so x_ship = 1.2/(5/3) = 0.72 light years?

No, it doesn't work that way. Realize that sun-star observers and spaceship observers disagree on where the the spaceship was at "the moment" the sun exploded.

What is true is that when the ship is 1.2 light years away from the sun (as measured by the sun-star observers), the spaceship measures his distance from the sun to be only 0.72 light years from the sun.

 

[smithg86]

thanks for your help, Doc Al and learningphysics! one more thing:
do the two observers (one on the spaceship and one in the frame of the stationary stars) agree when the ship has reached the midpoint? do they agree that the same spot is the midpoint?

 

[learningphysics]

thanks for your help, Doc Al and learningphysics! one more thing:
do the two observers (one on the spaceship and one in the frame of the stationary stars) agree when the ship has reached the midpoint? do they agree that the same spot is the midpoint?

good question! the answer is yes! The observer on the ship sees the length between the two stars shrink... by length contraction... imagine that there was a huge ruler/scale between the two stars... say the length is R with divisions marked down... imagine the 0 of the scale is at earth...

from the ships point of view, the scale shrinks by a factor of gamma to R/gamma... he sees the distance between the two ends of the scale as R/gamma... he also sees the distance between 0 and R/2 shrink by gamma... hence he sees the R/2 division at R/(2gamma)

in other words... he sees the R/2 division... at R/(2gamma) from the 0 end... which is half the scale length he sees (R/gamma).

They both agree that where the scale says R/2... is the midpoint...

 

[learningphysics]

Also wanted to just mention... you can find the time the explosions took place in the stars rest frame... then use lorentz transformations to get the times for the explosions in the ship's frame...

It is unnecessary for solving the problem, but it will confirm everything discussed.

 

[Doc Al]

one more thing:
do the two observers (one on the spaceship and one in the frame of the stationary stars) agree when the ship has reached the midpoint?

Not sure what you mean by agreeing on "when", since ship and stars are in relative motion and measure time differently.

do they agree that the same spot is the midpoint?

Absolutely!

Furthermore, all observers agree that the spaceship reached the midpoint at the exact moment that the light from each exploding star reached the midpoint. (As long as two events happen at the same time and place, everyone will agree that they happened simultaneously. Of course, they will measure that time and place in different coordinate systems.)