How do the Lorenz equations for time account for the movement of Alpha Centauri?

[arydberg]

They say that the light that reaches us from Alpha Centauri left that star 4 years ago but this assumes that both alpha centauri and the Earth are not moving relative to each other. In face Alpha Centauri s approaching us at about 10% of the speed of light. By the Lorenz equations for time T' = G * ( T - X*V/C^2) the X*V term become large so the assumption is not true. So how long ago did the light from Alpha Centauri leave.

 

[pervect]

It doesn't matter what the speed of Alpha Centaurus (henceforth AC) is. I'd be surprised if AC was moving towards Earth at such a high velocity, I would check your figures. But as long as one measure the distance to AC in the Earth frame, it doesn't matter what the velocity of AC is. The time (in the Earth frame) for light from AC to reach the Earth is the distance in the Earth frame, divided by c, the speed of light, irrespective of the speed of the source (in this case, AC).

 

[Ibix]

In face Alpha Centauri s approaching us at about 10% of the speed of light.

I doubt that very much. It'd be crashing into us in forty years. Wikipedia suggests about 18km/s, which is about 0.0062% of light speed.

In any case, if it were doing 10% of light speed you'd have to say which frame you were measuring the distance in and at what time in that frame in order to answer your question.

 

[Janus]

Alpha Centauri has a radial velocity of ~22 km/sec which is 0.0073% of the speed of light and no where near 10% of c.
But even if it were approaching at 10% of c, that would just mean that the image we now see for it originated when it was 4.3 light years away, and even if it is closer "now" that light still left it 4.3 years ago.

 

[Nugatory]

They say that the light that reaches us from Alpha Centauri left that star 4 years ago but this assumes that both alpha centauri and the Earth are not moving relative to each other.

It does not. The only assumption is that we're using a frame in which the Earth is at rest. In that frame the distance from Earth to alpha centauri at the moment that the light was emitted is four light-years (give or take some rounding errors) so it takes four years for the light to reach us. The motion of alpha centauri is irrelevant and the Lorentz transformations and time dilation formulas do not apply.

In face Alpha Centauri s approaching us at about 10% of the speed of light

Nowhere near that speed.

 

[arydberg]

It doesn't matter what the speed of Alpha Centaurus (henceforth AC) is. I'd be surprised if AC was moving towards Earth at such a high velocity, I would check your figures. But as long as one measure the distance to AC in the Earth frame, it doesn't matter what the velocity of AC is. The time (in the Earth frame) for light from AC to reach the Earth is the distance in the Earth frame, divided by c, the speed of light, irrespective of the speed of the source (in this case, AC).

My problem is how fast are the clocks on AC. No question that light living there takes 4 years to reach us. also seehttps://www.reddit.com/r/askscience/comments/30tvy8/how_fast_are_we_moving_relative_to_alpha_centauri/ for my 10% figure ( it may be wrong)

 

[arydberg]

My problem is how fast are the clocks on AC. No question that light living there takes 4 years to reach us. also seehttps://www.reddit.com/r/askscience/comments/30tvy8/how_fast_are_we_moving_relative_to_alpha_centauri/ for my 10% figure ( it may be wrong)

sorry my error I mixed 186,000 with 3 x 10^8

 

[russ_watters]

My problem is how fast are the clocks on AC.

I don't understand what this has to do with your original question. Could you please back up and try again?

sorry my error I mixed 186,000 with 3 x 10^8

I'm not seeing how you get to 10% even with that error...

 

[rootone]

If AC were moving towards the Solar system at a significant fraction of c, it's light would appear as hugely blue shifted.
It isn't hugely blue shifted.

 

[Drakkith]

My problem is how fast are the clocks on AC. No question that light living there takes 4 years to reach us.

Moving at roughly 0.0073% c, the clocks moving with AC are ticking at approximately 1.0000000026926 seconds for every 1 second here on Earth.
That's 99.99999973074 % the rate that clocks on Earth are ticking. Of course, an observer moving with AC sees our clocks ticking slower by the same amount.

Note that the above does not take into account gravitational time dilation.

 

[Janus]

My problem is how fast are the clocks on AC. No question that light living there takes 4 years to reach us. also seehttps://www.reddit.com/r/askscience/comments/30tvy8/how_fast_are_we_moving_relative_to_alpha_centauri/ for my 10% figure ( it may be wrong)

So, the question is how long would it take light from Alpha Centauri to reach Earth as measured by a clock at Alpha Centauri?
Okay, first we assume that the light leaves Alpha C when it is 4 light years from the Earth, As measured from the Earth.
This means that at that moment, as measured by Alpha C, it is 3.999999989 ly from Earth(length contraction). The light will be traveling away from Alpha C at c, and the distance between Earth and Alpha C will be decreasing at a rate of 0.000073 c, so it will take 3.999999989/(1-0.000073)=3.999708011 years or about 2.6 hours short of 4 years for the light to reach Earth by Alpha C's clock.

 

[Ibix]

By the Lorenz equations for time T' = G * ( T - X*V/C^2) the X*V term become large so the assumption is not true.

My problem is how fast are the clocks on AC.

Are you thinking that the time dilation formula is an approximation to the Lorentz transforms? It's not. Imagine a clock at rest in a frame S at position $x=X$. It ticks at time $t=T$ and time $t=T+\Delta T$. Obviously its proper tick rate is $\Delta t$. Now Lorentz transform the coordinates of the tick events and you get $\gamma (T-vX/c^2 )$ and $\gamma (T+\Delta T-vX/c^2)$. The difference is clearly $\gamma\Delta T$, independent of $X$.

The more general point is that the Lorentz transforms relate coordinates in different frames. They don't directly relate distances or durations. The time dilation and length contraction formulae do directly relate durations or distances in different frames, but only in the special case where the thing being measured was at rest in the initial frame.