In the Michelson-Morley experiment, the expected arrival time difference between the two perpendicular light beams is not based on the concept of light traveling at different speeds in different directions. Instead, it is rooted in the assumption of an ether medium through which light propagates.
At the time of the Michelson-Morley experiment in the late 19th century, scientists believed that light waves required a medium called the "ether" to propagate, similar to how waves in water require a medium. According to the prevailing theory, Earth was thought to be moving through this stationary ether medium as it orbited the Sun.
Based on this assumption, it was expected that the speed of light in the direction of Earth's motion through the ether would be slightly faster than the speed of light in the opposite direction (perpendicular to Earth's motion). Therefore, the light beam traveling in the direction of Earth's motion would cover a greater distance in the same amount of time compared to the beam traveling perpendicular to Earth's motion. As a result, a time delay between the two beams would be observed.
However, the Michelson-Morley experiment famously demonstrated that there was no measurable difference in the speed of light along different directions, regardless of Earth's motion through the ether. This result challenged the prevailing theory and played a significant role in the development of Albert Einstein's theory of special relativity.
According to special relativity, the speed of light is constant and independent of the motion of the source or observer. This means that light always travels at the same speed, denoted by 'c' in a vacuum, regardless of the direction of motion. Therefore, in the context of special relativity, the two beams of light in the Michelson-Morley experiment would still arrive at the same time, irrespective of Earth's motion.