Final answer:
In special relativity, an object is considered to be in a geodesic when the covariant derivative of its 4-velocity is zero. However, when measuring the 4-velocity from a non-inertial reference frame, such as an accelerating car, the object may appear to be in a geodesic even though it is clearly in a non-inertial frame.
Step-by-step explanation:
In special relativity, an inertial reference frame is one that is not accelerated or rotating. When the covariant derivative of a 4-velocity is zero, it indicates that the object is moving in a geodesic, which is essentially a straight line in spacetime. However, if you measure the 4-velocity of an object from a non-inertial reference frame, such as an accelerating car, you may see that the 4-velocity is constant relative to you. This can create a paradox because the object appears to be in a geodesic as measured by you, even though it is clearly in a non-inertial reference frame.
This paradox can be resolved by understanding that the concept of a geodesic depends on the reference frame being used. From the perspective of the non-inertial reference frame, the object may appear to be in a geodesic, but from an inertial reference frame, the object is actually accelerating. In the example of passengers in an accelerating car, they may feel as though they are moving in a straight line, but in reality, their motion is the result of the car's acceleration.
Therefore, the apparent paradox arises from a difference in reference frames and the understanding that what may appear as a geodesic in one frame can be understood as acceleration in another frame.