Suppose a tall (100m high) rocket sits on the launch pad. It is equipped with launch boosters and a sustainer motor that can give the rocket a prolonged 1g length-wise acceleration in free space. Amongst others, it is also fitted with the following sensors: two identical, synchronized atomic clocks, one in the nose and one near the tail, plus three accurate identical accelerometers, one in the nose, one in the tail and one midway between the first two.
While on the launch pad (waiting for a long delayed launch) you monitor all the sensors and determine that the nose clock is marginally gaining time on the tail clock. You satisfy yourself that this is normal due to gravitational time dilation and amounts to 1 part in about 10^14 (coming from dt/t = gL/c^2, where g = 9.81 m/s^2 and L is the difference in height between the clocks). You also verify that the higher accelerometers read marginally lower accelerations than the lower ones, in agreement with the inverse square law of gravitational acceleration (a = -GM/r^2, where G is Newton’s gravitational constant M the mass of the Earth and r the distance from Earth’s center).
Eventually the system is launched into free space and all the boosters fall away. After verifying that everything operates as designed and synchronizing the nose- and tail clocks, you ignite the 1g-propulsion system at the back. After a fair time of monitoring exactly 1g of acceleration at the tail of the rocket, you read all your sensors again. Will the clocks and accelerometers be able to tell you that you are now being linearly accelerated at 1g in free space and no longer sitting stationary on Earth’s surface?