A simple SR question, but confusing to me

[jnorman]

three spaceships, two moving at 0.9C and the 3rd stationary relative to the first two. the two moving ships are 1 LY apart but moving in the same direction toward the 3rd ship. as the first of the two moving ships passes the 3rd (stationary) ship, the ship which is now 1 LY from both the other ships emits a photon.

since the distance from the emitting ship and the stationary ship is 1 LY, it will take one year for the photon to reach the stationary ship, acccording to the clock on the stationary ship. however, because time measurement is different for the stationary ship and the ship which passed it as the photon was emitted, it will also take only 1 year for the photon to reach the other ship (by its on-board clock).

1. according to the clock on the stationary ship, it will take nearly two years for the photon to reach the moving ship - correct?.

2. according to the moving ship's clock, it only took 0.1 year (? - something well less than a year, at any rate) for the photon to reach the stationary ship - correct?

from the stationary ship's perspective, the distance between the two moving ships is a constant 1 LY - why wouldn't it measure exactly one year for a photon to move from one ship to the other?

from the moving ship's perspective, it was clear that the stationary ship was exactly 1 LY from the other ship when the photon was emitted - why should it measure less than a year for the photon to reach the stationary ship? how can it reconcile the idea that the photon traveled 1 LY in less than a year?

thanks.

 

[bcrowell]

however, because time measurement is different for the stationary ship and the ship which passed it as the photon was emitted, it will also take only 1 year for the photon to reach the other ship (by its on-board clock).

I think this is incorrect. If the 1 LY separation between the two moving ships is 1 LY in the stationary ship's frame, then it's more than 1 LY in the moving ships' frame. Therefore in the moving ships' frame, it will take more than 1 year to get from the first ship to the second ship.

 

[Dale]

three spaceships, ...

The straightforward way to do this is to write down the worldline of the photon and each ship, and then do Lorentz transform into the other frame. Then the times between different events in each frame can be determined simply by finding the intersections of the various worldlines. For convenience we can use units of years for t and light-years for x so that c=1.

For the first ship:
$$x_1 = 0.9t+1.0$$
$$x'_1 = 2.29$$

For the second ship:
$$x_2 = 0.9t$$
$$x'_2 = 0.0$$

For the stationary ship:
$$x_3 = 1$$
$$x'_3 = -0.9t'+0.44$$

For the light pulse:
$$x_0 = t$$
$$x'_0 = t'$$

The rest is just finding the intersections of the various worldlines.

 

[ZikZak]

three spaceships, two moving at 0.9C and the 3rd stationary relative to the first two. the two moving ships are 1 LY apart but moving in the same direction toward the 3rd ship. as the first of the two moving ships passes the 3rd (stationary) ship, the ship which is now 1 LY from both the other ships emits a photon.

since the distance from the emitting ship and the stationary ship is 1 LY, it will take one year for the photon to reach the stationary ship, acccording to the clock on the stationary ship.

Correct so far.

however, because time measurement is different for the stationary ship and the ship which passed it as the photon was emitted, it will also take only 1 year for the photon to reach the other ship (by its on-board clock).

Incorrect. The distance between the two ships in their own rest frame is greater than 1 light-year. In fact, it is 1*sqrt(1-0.9^2) = 2.29 light-years. The 1 light-year measurement made by the "stationary" ship is a contracted version of the distance between the ships in their rest frame. The light ray will take 2.29 light years to reach the front ship in their frame.

1. according to the clock on the stationary ship, it will take nearly two years for the photon to reach the moving ship - correct?.

No, it takes 10 years, because the light ray travels at c, and the forward ship travels away from it at 0.9c, leaving a closing rate of 0.1c to travel a 1 light-year separation.

2. according to the moving ship's clock, it only took 0.1 year (? - something well less than a year, at any rate) for the photon to reach the stationary ship - correct?

Now it's more complicated, because the two events of (ship #1 fires light ray) and (ship #2 alongside ship #3), which are simultaneous in the frame of ship #3, are not simultaneous in the frame of ships 1 and 2. Suppose that ships 2 and 3 set their clocks to zero when they pass each other and then ship #2 synchronizes clocks with ship #1 in their frame. In that case, the light ray was sent when ship #1's clock read a time v*l/c = 0.9*2.29 = 2.061 years. In other words, in its own reference frame, the rear ship sends the light ray 2.061 years AFTER ships #2 and 3 were coincident.

So the distance between ship #1 that emits the flash and ship #3 that receives it when it was sent was 2.29 light-years less the distance traveled at speed 0.9c by ship #3 in 2.061 years, for a total of 0.4351 light-years.

In that frame, the light ray travels at c, and ship #3 travels towards it at 0.9c, for a total closing rate of 1.9c, so the light ray reaches ship #3 in 0.4351/1.9 = 0.229 years. (Note, that in 0.4351 years, the light ray will reach the location ship #3 was when the light ray was sent.)

from the stationary ship's perspective, the distance between the two moving ships is a constant 1 LY - why wouldn't it measure exactly one year for a photon to move from one ship to the other?

Because the light ray travels at c, but the front ship moves ahead of it at 0.9c. It takes exactly one year (in that frame) for the light ray to get from ship #1 to where ship #2 was when the light ray was sent.

from the moving ship's perspective, it was clear that the stationary ship was exactly 1 LY from the other ship when the photon was emitted

No, you must be extremely careful when talking about "whens" between two reference frames. Actually, as I have shown you above, the distance was significantly less than 1 light year, AND on top of that, ship #3 was moving into the light ray.

why should it measure less than a year for the photon to reach the stationary ship? how can it reconcile the idea that the photon traveled 1 LY in less than a year?

thanks.

In the rear ship's reference frame, it traveled 0.229 light-years in 0.229 years. No problem.