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Moving in time is a literal interpretation of the space and time merger proposed by Minkowski.
It’s also another perspective of the moving light clock. Light is emitted from a source in a direction p, ...
...perpendicular to x, the direction of motion, and reflects from a mirror a distance d=1, to a detector/counter. For the clock to function, the photon path must have an x and p component. The x component compensates for the motion of the clock at speed v. The p component becomes the active part of the clock. Since the photon speed is constant, its path in any direction generates a circular arc for the 90º between the p axis and x axis.
This means the relative photon speed along p = c*sqrt(1-(v/c)^2) = c/γ, i.e. the clock ticks slower, the faster it moves past an observer.
Line 3 being so similar to line 2, allows a metaphorical interpretation as popularized by Briane Green and others. The clock moves in a 1-dimensional space, while/(simultaneously) the photon moves in a 2-dimensional space.
Thus, saying that something is "moving in time" is quite meaningless.
This is also incorrect and it's another reason why you can't discuss this in this forum and you'll have to move it to the New Theories sub forum. You have it that the speed of light is a function of v. The problem with that is that the speed if light has the same value in all inertial frames of motion.