How an accelerated object sees another accelerated body in special relativity?

Question

Assume two objects are moving with a constant acceleration $a_1$ and $a_2$, which are the measured accelerations by respective object (or constant force being applied to each of the objects). My problem is how one object sees the motion of the other one. I know that the accelerations are transformed according to the relation (Why proper acceleration is $du/dt$ and not $du/d\tau$?), $$ a = \frac{a'}{\gamma^3} $$ where $a'$ is the measured acceleration and $a$ is the acceleration in a inertial frame.

Answer 1

Depending on the magnitudes of the separate constant accelerations of the two objects, depending on the angle between their trajectories and depending on their "initial configuration" (initial separation and initial speeds, as determined by members of one suitable inertial system) there are indeed qualitatively distinct outcomes how such two object would have "seen another":

(1) Either they both "kept sight" of each other throughout the experiment: each signal indication stated by one of them was observed by the other; and consequently, to each signal indication stated by one object it eventually also observed the corresponding reflection from the other.

(2) Or: one of them "lost sight" of the other (but not vice versa): i.e. the one object so described observed only some "early" subset of all signal indications of the other object; although the other object in turn observed all signal indication stated by the one object.

(3) Or: they both "lost sight" of each other; i.e. each observed only some some "early" subset of all signal indications of the other.

Each of these cases may be further differentiated quantitatively. For example, within case (1) ("signal round trips mutually observed throughout") the two objects may

• either find that the round trip durations they determined between each other remained constant (and the two objects remained in the chrono-geometric sense rigid to each other);

• or they found their round trip durations varying (which applies for instance if the two objects remained rigid to each other in the sense of Born).

Answer 2

Einstein says:

"If, relative to K, K' is a uniformly moving co-ordinate system devoid of rotation, then natural phenomena run their course with respect to K' according to exactly the same general laws as with respect to K."

Apparently, "uniform movement", i.e. one without accelerations, is the assumption behind the whole SR Theory that allows us to compare two different frames of reference. Therefore, and despite Alfred Centauri's comment and link above, SR precludes acceleration.

Incidentally, when Einstein was deriving the field equations for GR he went to infinitesimals, because, as he claimed, only then he could apply SR equations, as gravity does not apply at this level (appropriate citations can also be found here). And yet, the GR was based on the equivalence principle stating that:

"The local effects of motion in a curved space (gravitation) are indistinguishable from those of an accelerated observer in flat space, without exception."

EDIT: MCGayan - take a look at the explanation given in the accepted answer to the question you cited:

"... we can think of an accelerating object as an object that 'jumps' between inertial frames. So at any instant, we can say that the object is located in a particular inertial frame, where an inertial observer will see the object accelerating with a⃗ ′(t′)... In an infinitesimal time interval dt′ he will see the object moving with a velocity dv⃗ ′=a⃗ ′dt′."

Now, how can you simultaneously assume that: 1) There is no time change (instant), so that the accelerating object can be treated as in an inertial frame, and also that 2) the inertial observer will see acceleration that requires time change (time interval)?

One needs to decide, whether we are talking about an inertial frame (which in our case requires stopping the time, and therefore there is no acceleration observed) or a non-intertial frame (which requires time change, and therefore there can be acceleration observed). You can't eat the cake and have the cake at the same time - you must decide, either one or the other.