[A-wal (OP)]

I want to get the time that passes on an accelerating clock from within an inertial frame for an extended period of constant acceleration using only a simple algebraic expression without any exponentials or integrals.

I've come up with a formula for simultaneity shifts that works for any extended acceleration profile (unless there's a change in direction in which case it's multiple shifts) just using the endpoints but I'm struggling with this even for constant acceleration.

If v' is the starting velocity and v'' is the end velocity then some velocity between v' and v'' will give the velocity of a single constant transformation that can be applied to over the duration of the acceleration to give the correct time difference that in addition to the simultaneity shift will give the clock difference between the accelerating frame and an inertial frame, I just need an algebraic way to find that velocity.

Without the scaling factor is would obviously be the velocity half way between v' and v'' and with the scaling factor it gets pushed closer to v''. I tried
v = ((v'(√(c^2/(c^2−v'^2))))+(v''(√(c^2/(c^2−v''^2)))))
/((√(c^2/(c^2−v'^2)))+(√(c^2/(c^2−v''^2))))
but apparently that doesn't work.

[Halc]

I want to get the time that passes on an accelerating clock from within an inertial frame for an extended period of constant acceleration using only a simple algebraic expression without any exponentials or integrals.

In any natural units, assuming the accelerating clock starts from a stop in said inertial frame,
τ = arcsinh(at) / a

Where
τ is time on the accelerating clock
t is time passed in inertial frame since onset of acceleration
a is proper acceleration

It's that easy.

[A-wal (OP)]

I don't want to use hyperbolics, their values are defined by which one is needed to make the formula spit out the answer that you want it to. I want to do it properly.

If it can be done for simultaneity shifts with any arbitrarily wobbly worldline I don't see why this can't even be done for constant acceleration.

[paul cotter]

What's wrong with using hyperbolic functions?, they are commonly used in engineering and general science.

[Bored chemist]

"How to get the right slowing factor of a clock that's constantly accelerating?"
Given that the clock on my wall is effectively constantly accelerating at about 9.8 m/s/s I'd have thought this was a well studied question with a well known answer.

[A-wal (OP)]

What's wrong with using hyperbolic functions?, they are commonly used in engineering and general science.

Formulas that use them don't actually show what's going on, it want it terms of x, t and c only.

I was able to do it with simultaneity shifts because it's only the end points that matter but with this it's the average of a constant acceleration that's scaled at √((c^2−v^2)/c^2).

I suppose this isn't even physics. As a a pure maths question how do get the average over a range that increases with a non‐linear scale?

The problem is a constant acceleration isn't a constant acceleration when it's multiplied by √((c^2−v^2)/c^2) so I need to scale the acceleration by √(c^2/(c^2−v^2)) first but I tried that and it didn't work.

"How to get the right slowing factor of a clock that's constantly accelerating?"
Given that the clock on my wall is effectively constantly accelerating at about 9.8 m/s/s I'd have thought this was a well studied question with a well known answer.

It's acceleration as in an increasing scale of relative velocities that I need to find the weighted average of.