According to the principles of relativity, the speed of light is constant for all observers, regardless of their state of motion. This means that the speed of light remains the same for both the traveler and the relatively static observer, even in the presence of time dilation.
Time dilation does not affect the speed of light itself. It is a phenomenon that affects the perception of time experienced by observers who are in relative motion or in different gravitational fields. The constancy of the speed of light is a fundamental postulate of special relativity.
To illustrate this, let's consider an example: Suppose there are two observers, one stationary on Earth and the other traveling in a spaceship at a high velocity relative to Earth. From the perspective of the stationary observer on Earth, they would measure the speed of light to be the same constant value, as is traditionally known, regardless of the motion of the spaceship. Simultaneously, the traveler on the spaceship would also measure the speed of light to be the same constant value, irrespective of their own motion.
Although the perception of time may be different for the traveler and the stationary observer due to time dilation effects, the constancy of the speed of light ensures that the observed speed of light remains the same for both. This is a fundamental consequence of the theory of relativity and has been confirmed by numerous experiments.