According to the principles of special relativity, as an object approaches the speed of light, its apparent mass increases and its length contracts in the direction of motion. Additionally, time dilation occurs, causing time to pass more slowly for the moving object relative to an observer at rest.
In the scenario you described, where an object is traveling away from us at two-thirds the speed of light, we can still detect it, but there are several important factors to consider:
Detection of Light: If the object is emitting or reflecting light, we can detect it through conventional means, such as using telescopes or other optical instruments. However, the light emitted by the object will be subject to a phenomenon known as redshift. As an object moves away from an observer, the wavelength of the light it emits gets stretched, causing a shift towards longer wavelengths (towards the red end of the spectrum). This redshift can make the object's light more difficult to detect, particularly if it is moving at relativistic speeds.
Time Dilation: Due to time dilation, as observed from our frame of reference, time appears to pass more slowly for the object traveling at two-thirds the speed of light. This means that the object's movements and changes will appear to occur more slowly compared to an object at rest. However, this does not necessarily affect our ability to detect the object itself, although it may influence the observations we make of its behavior.
Relativistic Effects: At such high speeds, relativistic effects become significant. The mass of the object will increase, and its length will contract in the direction of motion, which can impact our measurements and observations. These effects must be taken into account when analyzing the data obtained from detecting and studying the object.
In summary, while there are challenges associated with detecting and studying an object traveling away from us at two-thirds the speed of light, it is theoretically possible to detect it using appropriate instrumentation and accounting for the relativistic effects involved.