Is the speed of light constant even for an object which is accelerating in free space?
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Yes, but only instantaneously. If at any instant the acceleration were to stop, the speed of light would be $c$ relative to the observer who was accelerating. Acceleration means to continually change inertial frames. In each of those frames the speed of light is $c$ relative to the frame. Mathematically this may be stated $\frac{dc}{d\tau}=0$.
Steven Thomas Hatton
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