mertcan said:
Let me a little bit clear, I just want to know under which circumstances ##x^2 - t^2 = \text{constant}## means a uniform acceleration ?
Here is a hint. ##dx^2 / dt^2## is not constant, what is constant is ##dx^2 / d\tau^2## where ##\tau## is proper time.
http://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html may be somewhat helpful, though it just gives the formulas, and not the derivation.
So a more precise statement would be that the hyperbola is a curve of constant proper acceleration.
To work out the problem for yourself, you need the following definition of proper time ##\tau## from special relativity.
$$c^2 \,(\Delta \tau)^2 = c^2 (\Delta t)^2 - (\Delta x)^2$$
[add] While this would work, I can be clearer. Let
$$\beta = \frac{v}{c} \quad \gamma = \frac{1}{\sqrt{1-\beta^2}}$$
Then ##\Delta t = \gamma \Delta \tau##, proper time is different from coordinate time because of "time dilation".The rest is a lot of algebra. Probably the easiest thing to do is to verify the pre-existing results from "The Relativistic Rocket" link. If you're not familiar with proper time, you may have to research that a bit first, though I tried to give you the equation that you really needed.
A detour into "rapidity" could be helpful, see for instance
https://en.wikipedia.org/wiki/Rapidity
Rapidities, denoted by w in the Wiki article, have the property that rapidities add, unlike velocities. So a curve of constant proper acceleration is a curve where, rather than v= at, we have ##w = a \tau##.
[add]
Note that from the wiki
$$ \beta = \tanh w \quad \gamma = \cosh w$$
When v <<c, ##w \approx \beta## because ##\tanh w \approx w - w^3/3 + ...## via a Taylor series expansion. Similiarly when v<<c, ##d\tau = dt##.
To calculate proper acceleration, we can calculate dv/dt in a frame in which the object is at rest. But this is the same as ##dw/d\tau## in a frame where the object is at rest.
Because velocities do not add ##\left( v(t+\Delta t) \,(minus)\, v(t) \right) / \Delta t ## requires a relativistic velocity subtratction operation when the object is not at rest, because velocities do not add in special relativity. But rapidities do add, so ##\left( w(\tau + \Delta \tau) - w(\tau) \right) / \Delta \tau## uses a simple arithmatical subtraction. Furthermore ##\tau## is observer independent, so the calculation is much easier.
