Time dilation formula vs Lorentz Transformation with light signal

Question

I'm confused about when to properly use the time dilation formula $\Delta \tau=\frac{\Delta t}{\gamma}$ or the Lorentz transformation $c\Delta t = \gamma (c\Delta t-\beta \Delta x)$, especially when it comes to defining events for this problem.

A spacecraft leaves the Earth going at 0.99c. If it is 3*10^11 m away when Earth sends a light signal to it, how long will it take the signal to reach the ship a) for an observer on Earth and b) for a passenger on the ship?

For a) no special relativity is needed, and you get 100,000 seconds for the Earth's perspective.

For b) I tried doing it two ways:

1. According to the time dilation formula, 100,000 seconds for Earth is 14,106 seconds for the craft (using $\gamma = 7.088$)

2. Defining coordinates for each event and using the transformation. I let S be the Earth frame and S' be the ship frame, where time starts when the ship leaves earth. Event 1 is when Earth sends the signal after the ship is far from the earth ($t_1, x_1$) = (1010, 0). Event 2 is when the signal reaches the ship ($t_2, x_2$) = ($101010, 3*10^{13}$). If you use the transformation, however, you get a different $\Delta t'=7090$.

I tried using a Minkowski diagram, and the difference seems to boil down to whether you put event 1 on the Earth's axis (x=0), or the ship's axis (x=3*10^11). However both lines of reasoning sound valid to me.

What am I missing here? Which one is really correct? Thank you so much to anyone who can help!

Answer 1

It depends how you have defined $\gamma$. It is usually

$$ \gamma = \frac{1}{\sqrt{1-v^2/c^2}} \tag{1} $$ So that $\gamma \geq 1$.

In which case $$\Delta t_A = \gamma \Delta t_B. \tag{2}$$ B is the one travelling close to the speed of light so his clock slows down according to outside observers. We know how many seconds it takes for A, so

$$ \Delta t_b = \frac{\Delta t_A}{\gamma} \tag{3} $$ Which should be less than A. It's much easier to use $\gamma$ than the Lorentz transformation as the question is specifically about the time between events, and the Lorentz transformation involves lengths scales not time.