Time Dilation: Object & Lightbulb

[Myslius]

Let's say there is an object at one place and the light bulb at another place. The light bulb switches on, when object sees that it starts to accelerate to the light bulb. After some time object reaches 0.866c relative to the light bulb, now the object and the light bulb is in inertial frame of reference for each other (there is no acceleration). The object starts the timer, the light bulb knows how much time is needed for an object to accelerate to the 0.866c so after that time light bulb starts the timer too. After some time object passes the light bulb and shares the information about the time passed.

Lorentz factor at 0.866c is 2 (time ticks two times slower)

How come the object sees that time for the light bulb ticks 2 times slower, and how come the light bulb sees that the time for an object ticks two times slower too?

 

[ghwellsjr]

Let's say there is an object at one place and the light bulb at another place. The light bulb switches on, when object sees that it starts to accelerate to the light bulb. After some time object reaches 0.866c relative to the light bulb, now the object and the light bulb is in inertial frame of reference for each other (there is no acceleration). The object starts the timer, the light bulb knows how much time is needed for an object to accelerate to the 0.866c so after that time light bulb starts the timer too. After some time object passes the light bulb and shares the information about the time passed.

Lorentz factor at 0.866c is 2 (time ticks two times slower)

How come the object sees that time for the light bulb ticks 2 times slower, and how come the light bulb sees that the time for an object ticks two times slower too?

I think you mean at the start, that the light bulb and the object are at rest with respect to each other but a great distance apart, then by prior arrangement and knowledge to each one, the object accelerates with a given profile toward the light bulb and after at a particular time that both can determine, they each start their respective timers and compare times when the pass. Correct?

First off, your implication that they are at rest in the same frame when they start their timers is not correct because they have a relative velocity between them (0.866c) even though they are not accelerating. Actually, you can analyze this scenario from any frame and the one where they both start at rest is the easiest. (Remember, all objects, timers, and light bulbs exist in all frames but you need to focus on one frame at a time.)

So if you define your scenario so that both timers start at the same time in the original rest frame, then because the object is traveling at 0.866c, its timer will be running at one-half the rate of the light bulb and when they get together it will have one-half the time on it that the light bulb timer has on it. It's a simple as that.

But, if you want you can look at this from another frame and see that you get the same answer OR you redefine your scenario so that the two timers start at the same time in a different frame and you get a different answer in any frame you analyze it in.