Formulating Position Vector with f(t)

[Philosophaie]

How do you formulate the Position Vector:

x^m = [t____x____y____z]

x^m = [f(t)____ax/2*t^2 + vx*t + x0____ay/2*t^2 + vy*t + y0____az/2*t^2 + vz*t + z0]

dx^m / dt = [df(t) / dt____ax*t + vx____ay*t + vy____az*t + vz]

What do you put in the"t" position "f(t)".

 

[tiny-tim]

Hi Philosophaie!

Perhaps I'm misunderstanding you, but shouldn't that be …

x^a = [t____ax/2*t^2 + vx*t + x0____ay/2*t^2 + vy*t + y0____az/2*t^2 + vz*t + z0]

dx^a / dt = [1____ax*t + vx____ay*t + vy____az*t + vz] ? ​

 

[Philosophaie]

I am basing my Metric Tensor on:


$$g_{ik}*\frac{dx^{i}}{dt}*\frac{dx^{k}}{dt} = -1$$


Do I have the right equation for being in a gravitational field say near a Black Hole with nothing else around?

I am trying to develop a general realistic Metric Tensor that has the acceleration, velocity and position all in a gravitational field (with GR rather than Newtonian).