As \(v\rightarrow c\)
\(\left.\begin{array}{l}t=\cfrac{v}{c}x\\t=\cfrac{c}{v}x\end{array}\right\rbrace\rightarrow t=x\)
could be the reason why people think that if we travel at light speed we also travel through time. Or that time is at light speed, \(c\).
The last is true, only if light speed is the speed limit. Light speed is set as the speed limit when the scaling factor \(c\) was applied to Lorentz transform that scaled,
\(v\lt1\) to \(v\lt c\)
As for time travel (in the time dimension) at light speed in the space dimension...You cannot travel in the orthogonal direction \(y\) even if you are at light speed in the direction \(x\). Space and time are orthogonal dimensions. However, in a gravitational field, a change in gravity that changes your time speed that incidentally also propels you to light speed and back, may move you back and forth in time with respect to the rest of us. We are all traveling at the same time speed, already.
We are already traveling through time. To go forward by ten years. all you have to do is WAIT.
Note: Time, time speed and light speed are different things. \(Time\,speed\,=\,light\,speed\) is an assumption.
