In the scenario you described, where both you and I synchronize our clocks and then move towards each other with the same values for acceleration and velocity, the principle of special relativity can help determine the readings on our clocks when we meet at the midpoint.
According to special relativity, when two observers are in relative motion, their perception of time can be different. Specifically, time dilation occurs due to the relative velocity between the observers. As you and I move towards each other, the relative velocity increases, resulting in time dilation effects.
In this case, since we both have the same values for acceleration and velocity, we would experience symmetrical time dilation. This means that the time dilation effects on our clocks would be equal and opposite.
As a result, when we meet at the midpoint, our clocks would still be synchronized according to our respective frames of reference. However, from an external observer's perspective who is stationary relative to the midpoint, they would observe that our clocks are not synchronized due to the time dilation effects.
It's important to note that the exact readings on our clocks when we meet would depend on the specific values of acceleration, velocity, and the distances involved. The calculations involving time dilation in special relativity can be quite complex, requiring the use of the Lorentz transformation equations and accounting for factors such as proper time and proper length.
Nevertheless, in the scenario you described, both of our clocks would still be synchronized according to our individual perceptions when we meet at the midpoint, while an external observer might perceive a difference due to the effects of time dilation.