Observing from Earth the clock in a spaceship

In summary: You can't just look up the time on the clock at (Te, Xe) and use that as the answer because the light takes time to travel from the spaceship to the observer. So you need to calculate the time that the light was emitted from the spaceship and account for the time it takes to reach the observer. That's what the Lorentz transformation is for. It transforms the time at which the light was emitted from the spaceship into the time at which it reaches the observer.
  • #1
giuliopascal
5
0

Homework Statement


A spaceship leaves Earth at time ##t=0## with constant speed ##u##. Its clock is synchronized with the terrestrial one. At time T an earthling reads with an optical telescope the clock inside the spaceship. What value does he read?

Homework Equations


Lorentz equations.

The Attempt at a Solution


The solution says:
##c(T-T_e)=u T_e##
##T_e=\frac{T}{1+u/c}## and ##X_e=\frac{uT}{1+u/c}##
Apply a Lorentz transformation:
##t=\sqrt{\frac{1-u/c}{1+u/c}}T## at ##x=0##
And ##t## is the answer.

However when I tried to solve it I considered also the time needed for light to reach the Earth from the spaceship. As a consequence my solution would be ##t'## such that:
##t'=t+T-T_e##
Is it wrong?

Thank you very much.
 
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  • #2
Can you explain in words what's being done in the solution? It appears you don't fully understand what those equations mean.
 
  • #3
Ok, I think I have taken a few steps forward.

I misunderstood which points was the Lorentz equation applied to. I thought there was a time dilatation in ##(T_e,X_e)##, but it is wrong, isn't it? In the Earth reference frame, when the spaceship emits the beam of light the clock time is ##T_e##, without any correction. If that is true, I applied a useless Lorentz transformation, and by chance I found the correct answer.

So:
solving ##x=ut## (spaceship) and ##x=-c(t-T)## (beam of light), I find ##(T_e,X_e)##: that is when the beam of light is emitted. Then I need to go back to ##(T,0)##, where the observer is. But why do I need a Lorentz transformation with boost ##u##? (I have to use that, haven't I?)

Thank you.
 
  • #4
giuliopascal said:
Ok, I think I have taken a few steps forward.

I misunderstood which points was the Lorentz equation applied to. I thought there was a time dilatation in ##(T_e,X_e)##, but it is wrong, isn't it? In the Earth reference frame, when the spaceship emits the beam of light the clock time is ##T_e##, without any correction. If that is true, I applied a useless Lorentz transformation, and by chance I found the correct answer.

So:
solving ##x=ut## (spaceship) and ##x=-c(t-T)## (beam of light), I find ##(T_e,X_e)##: that is when the beam of light is emitted.
That's when and where the light is emitted as measured by the Earthbound observer.

Then I need to go back to ##(T,0)##, where the observer is. But why do I need a Lorentz transformation with boost ##u##? (I have to use that, haven't I?)
As long as light is emitted at ##(T_e, X_e)##, it will reach the observer at (T, 0). The question now is what is the image that was emitted?
 

FAQ: Observing from Earth the clock in a spaceship

1. How does time dilation affect the clock in a spaceship?

Time dilation is a phenomenon in which time runs slower for objects that are moving at high speeds. This means that the clock on a spaceship moving at high speeds will appear to run slower compared to a clock on Earth. This effect is predicted by Einstein's theory of relativity and has been confirmed through experiments.

2. Can we accurately observe the clock on a spaceship from Earth?

Yes, with advanced technology and precise measurements, we can accurately observe the clock on a spaceship from Earth. However, the effects of time dilation and distance can make it challenging to accurately measure the time on a clock in a spaceship.

3. How does the distance between Earth and the spaceship affect the observation of the clock?

The distance between Earth and the spaceship can affect the observation of the clock due to the time it takes for light to travel between the two objects. This is known as the time delay effect and can cause a slight difference in the time observed on the clock in the spaceship compared to the time on Earth.

4. Would the clock on a spaceship be affected by gravitational time dilation?

Yes, the clock on a spaceship would also be affected by gravitational time dilation. This is because gravity can also affect the flow of time, causing it to run slower for objects in areas with stronger gravitational forces. This effect is also predicted by Einstein's theory of relativity.

5. Can observing the clock on a spaceship from Earth provide us with valuable scientific insights?

Yes, observing the clock on a spaceship from Earth can provide us with valuable scientific insights. By studying the effects of time dilation and the time delay effect, we can gain a better understanding of the relationship between time and space and how it is affected by factors such as speed and gravity. This can also help us make more accurate predictions and calculations in the field of astrophysics.

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