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What I am looking for is a layman's explanation on the equations required to work out Time Dilation at high speeds including acceleration and deceleration of velocity. Or I would greatly appreciate it if you could point me in the direction of a website which explains each step required and all the variables involved in solving this problem if you feel that to outline it here would be too much work.

Many thanks.

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Suppose you know the coordinates of a particle as a function of time in some inertial reference frame. That is, you have $x(t)$, $y(t)$, and $z(t)$. This is called knowing the particle's worldline. Then from time $t_1$ to $t_2$, the particle ages by

$$\int_{t_1}^{t_2} \mathrm{d}t \sqrt{1 - \dot{x}^2 - \dot{y}^2 - \dot{z}^2} \equiv \Delta \tau$$

This quantity $\Delta \tau$ is called the proper time along the worldline.

For example, in a typical twin paradox, you might have $x(t) = \frac{4}{5}t$ for $0<t<5$ and $x(t) = 4 - \frac{4}{5}t$ for $5<t<10$. Also $y = z = 0$. Then $\dot{x} = \pm\frac{4}{5}$ and $\sqrt{1 - \dot{x}^2 - \dot{y}^2 - \dot{z}^2} = \frac{3}{5}$ Therefore

$$\Delta \tau = \int_0^{10} \mathrm{d}t \frac{3}{5} = 6$$

This says that if someone flies away at 80% the speed of light for five years, then turns around and comes back the same way, they have aged six years total, even though ten years have passed for you.

There's no fundamental modification needed to account for acceleration; you just need to change the value of $\dot{x}, \dot{y}$, and $\dot{z}$ to whatever they are for the accelerating object.

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