If a spaceship were to accelerate in proportion to its time dilation, it would experience several effects as predicted by special relativity. Let's consider a scenario where the spaceship is traveling relative to an observer at a constant velocity and then undergoes acceleration.
Time Dilation: According to special relativity, as the spaceship accelerates and its velocity increases, time dilation would occur. Time dilation means that time appears to pass more slowly for the spaceship as observed by the stationary observer. The faster the spaceship moves, the greater the time dilation effect. This means that clocks on the spaceship would appear to run slower relative to the observer's clocks.
Length Contraction: As the spaceship accelerates, it would also experience length contraction. Length contraction refers to the phenomenon where objects in motion appear to be shorter along their direction of motion as observed by a stationary observer. From the perspective of the stationary observer, the spaceship would appear to become shorter in the direction of its motion as its velocity increases.
Relativistic Doppler Effect: As the spaceship accelerates, there would also be a relativistic Doppler effect on the light emitted by the spaceship. The frequency of light emitted by a moving source is observed to be shifted based on the relative velocity between the source and the observer. As the spaceship moves faster, the light emitted from it would be shifted to longer wavelengths (redshifted) as observed by the stationary observer.
It's important to note that the effects of time dilation, length contraction, and the relativistic Doppler effect are interrelated and depend on the relative motion between the spaceship and the observer. As the spaceship accelerates, these effects would become more pronounced.