# Time dilation formula vs Lorentz Transformation with light signal

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!

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.

• Thanks that does make sense. Shouldn't there be a way to use the Lorentz transformations/minkowski diagram to get the same answer, though? Did I define my events wrong? Apr 21, 2020 at 1:09
• This is Homework question and should be deleted? According to @David.Z
– Eli
Apr 21, 2020 at 6:09
• @Eli I feel they they've done all the work for both concepts. It's not that they're asking for the answer directly. They're struggling about the concept of which is the correct method. They've done the work for both already. I'd say it was a perfectly acceptable question. Apr 21, 2020 at 11:22