For those who are interested, here are some even more details concerning the uniformly accelerating spacecraft problem.
Suppose we have a uniformly accelerating intergalactic spaceship whose acceleration w.r.t. a sequence of instantaneously coincident inertial frames is g along the +z direction. Suppose we make our lab frame K to be at rest w.r.t. the spaceship (because the lab is on board the ship). This is a *non*inertial frame. Let's also consider a fully inertial frame K' whose velocity relative to frame K is momentarily at rest after which the K frame accelerates away from the the inertial Minkowski K' frame along the + z' direction. In this case the invariant time-like square space-time (square proper time) interval between a pair of two infinitesimally time-like separated space-time events as seen in both the inertial Minkowski K' frame and the rocket's non-inertial frame K is given by:
Both frames are each static in the sense that in each frame the metric coefficients do not depend on the time-like coordinate for either of them. This makes all motions and behaviors invariant w.r.t. the chosen moment when either/and t = 0, & t' = 0 happens to be chosen.
Frame K is rigidly accelerating in the sense that all fixed locations at rest w.r.t. the spaceship maintain the same spatial distances with respect to each other in K over time t as the ship accelerates w.r.t. the inertial K' frame. But when viewed from the inertial K' frame those (supposedly fixed) relative spatial separations narrow down along the z' direction over time t' as the ever faster spaceship becomes ever more length-contracted along its direction of motion. This continually increasing amount of length contraction causes the 'bottom' of the spaceship to accelerate slightly faster than the 'top' of it as the amount of the length contraction increases causing the bottom of the ship to gradually catch up with the top of it making its overall length ever shorter when the ship is moving at a significant fraction of speed c.
The time-like parameter t' measures the proper time experienced for any observer at rest in K', and the time-like parameter, t measures the proper time experienced by any observer at rest in K *and* whose z coordinate is 0. But for other observers at rest in K with *other* fixed z coordinates, they have proper times that are different but proportional to the time kept by the parameter t. In particular, for an observer at rest in K the proper time at a fixed height z in the spaceship that observer's own proper time is measured by (1 + g*z/c^2)*t. Thus clocks at rest on the spaceship tick faster if they are higher up in z in the spaceship than clocks that are situated at lower values of z. (This is very similar to the case of a real gravitational field and can be to be so from the Equivalence Principle.)
To make some further observations for this pair of reference frames we choose specific coordinate systems for each frame and figure out the transformation equations between them so as to find the space-time coordinates for any event in one frame from the space-time coordinates of the same event in the other frame. We pick some fixed point, O on/in the spaceship to represent the spatial origin of the *non*inertial Cartesian space-time coordinate system for the K frame in which the ship always remains at rest. Let O' be some point in space at rest in some locally instantaneously coincident *inertial* Minkowski-Cartesian coordinate system for K' such that the path of O passes directly over O' for the spaceship's trajectory in the inertial K' frame, and at the moment when it does so O & O' are not moving w.r.t. each other. We let O' be the spatial origin of the inertial K' frame's space-time coordinate system. For convenience we can choose both coordinate systems to have parallel corresponding spatial basis vectors so that the relative motion between the frames is a continuously variable pure boost along the z-direction with no spatial rotation between the frames to muck things up. In this case the coordinate transformation between the 2 frames is given by:
x' = x
y' = y
z' = (z + (c^2)/g)*cosh(g*t/c) - (c^2)/g
t' = (z/c + c/g)*sinh(g*t/c) .
The event of O and O' being momentarily coincident marks the zero of both time-like coordinates, t & t' for each corresponding frame. At the moment t = t' = 0 all the spatial coordinates of both frames coincide, and both frames are momentarily at rest w.r.t. each other.
The inverse of the transformation above is given by:
x = x'
y = y'
z = (z' + (c^2)/g)*sqrt(1 - (t'/(z'/c + c/g))^2) - (c^2)/g
t = (c/g)*arctanh(t'/(z'/c+ c/g)) .
Note (from the equation for t') a clock at rest w.r.t. the spaceship at z = -(c^2)/g is infinitely dilated and stopped relative to frame K'. Such a clock is moving at speed c in the K' frame. Thus the K frame has a coordinate singularity/horizon at z = -(c^2)/g, and any events in the K frame region where z < -(c^2)/g are un-physical and have the direction if time t reversed from the direction of time t' in the inertial frame K'. Also the space-time labeled by frame K coordinates is incomplete in the sense that a physical part of space-time does not correspond to sensible coordinate values in frame K. In particular, any events in the perfectly normal space-time region of K' for which their coordinates have
|t'| > |z'/c + c/g| do not have sensible real values for their corresponding non-inertial coordinates z & t in frame K, but instead correspond to complex coordinate values in K.
If we pick a point at rest in frame K whose fixed height in the spaceship is z_0 and watch its trajectory in the inertial frame K' (starting at t' = 0) its trajectory is given by:
The first form here for v' explicitly shows that the fixed point on the rocket has an initial acceleration of g (plus terms of order 1/c^2) as seen in K'. The second form explicitly shows that the upper bound limit, c on v' is approached from below at infinite time.
OTOH, if we switch points of view and conversely imagine a particle dropped/released (at time t = 0) by an observer at rest in the spaceship, and suppose this freely falling particle is released from a height z_0 on the ship at that moment. Since at t = t' = 0 both the K & K' coordinate systems are momentarily coincident, and are at rest w.r.t. each other, this means the released/dropped particle simply maintains its fixed z' value over time in K' as it always remains at rest in that inertial frame, i.e. z'(t') = z_0 for all t'. But if the dropped particle is instead observed in the non-inertial K frame, it is observed to be accelerated downward, and its trajectory in K is given by:
z(t) =(z_0 +(c^2)/g)*sech(g*t/c) - (c^2)/g .
The velocity v(t) in K of this dropped particle is given by:
v(t) = -(c + z_0*g/c)*tanh(g*t/c)*sech(g*t/c) .
Note this dropped particle is initially accelerated downward at an acceleration -g (plus terms of order 1/c^2 when z_0 ≠ 0). But it achieves a maximum speed and then slows down and approaching a halt at the horizon at z = - (c^2)/g. Note that since the horizon is so far from the origin O on the ship in frame K the ship needs to be equipped with a hole in the bottom of it to let the dropped particle fall through it into intergalactic space. The maximum downward speed achieved by the dropped particle is:
v_max = (c + z_0*g/c)/2
and it reaches this maximum speed at time t_max =ln(1 + sqrt(2))*c/g after being dropped. After reaching the maximum speed the dropped particle begins to slow back down, coming to a halt approaching the horizon after an infinite amount of time. BTW, this speed is measured using a clock which keeps *coordinate* time, i.e. time as seen by a clock at rest w.r.t. the ship at height z = 0. If a local clock is used which keeps time as seen at the height of the point of drop, then the maximum speed is just c/2. Because of some possible local quirks of various coordinate systems speeds measured with a time-like coordinate that doesn't actually measure local proper time may actually violate the v <= c constraint. That is not a violation of the principles of relativity; it is merely an artifact of using a locally un-physical time to measure speeds. BTW, the same caveat applies to speeds that are measured with spatial coordinates that don't actually measure local proper distances, either.
The existence of the horizon at z = - (c^2)/g is sort of analogous to the event horizon in the Schwarzschild space-time geometry. In both cases a dropped object accelerates downward and then slows back down to a halt when approaching the horizon when viewed from an external coordinate system experiencing a constant-in-time gravitational field. But two main differences are that the Schwarzschild geometry has spherical symmetry, whereas this problem has rectangular symmetry. And, more importantly, the Schwarzschild geometry is a *real* gravity field with real space-time curvature, whereas this accelerating spaceship problem has an intrinsically flat space-time geometry isomorphic (actually diffeomorphic) to the ordinary Minskowski space-time of SR, which it is when seen in the inertial K' frame and the gravitational field experienced in the non-inertial frame is caused solely by the non-inertial character of the frame, and *not* by any nearby mass, energy, momentum, or other stress.
Also the coordinate horizon at z = - c^2/g in K for the accelerating spaceship problem is analogous to the horizon that occurs in a non-inertial uniformly rotating reference frame at a distance of r = c/ω from the rotation axis.