Time (or rather, $dt$) as a Vector?

Question

I want to preface this by saying that I'm by no means experienced with the field of physics, but while on my commute this morning, I started thinking:

Basically, I thought about time being a fourth dimension, time dilation, and whatnot (it was very early), and I thought about how if we're moving at one second per second through that fourth dimension, that means we must have some kind of constant velocity through it. Then I realized that if time really was a fourth dimension, then movement through the other three dimensions would alter where that "time velocity" vector lands, theoretically explaining why time dilation is experienced at higher velocities in our three dimensions.

My question is: is this idea supported by physics? I can't imagine that no one's thought of this before, so I'm mainly just wondering if I'm wrong.

Answer 1

I have encountered this idea in my reading elsewhere. (I don't recall where.) You need to relate your units of distance to units of time. (1 meter/(3e8 m/sec) = 1 sec.) Then any motion, represented by a 4D vector in space-time exhibits a time dilation as predicted by special relativity.

Answer 2

First, some basic physics. Conventionally, your black arrow along the Time axis is not a velocity through time but a displacement in time. It is the reference by which velocities through space are measured. Your orange arrow actually has a different length in Minkowski spacetime: $\sqrt{ (x^2 + y^2 + z^2 - t^2) }$. It is known as a spacetime interval.

If you really wish to give meaning to velocity through time in $s/s^{-1}$, that presupposes a second time dimension, as say $dt_1/dt_2$. Your original logic soon leads you to ask what is your velocity in that second time dimension, which leads on to an infinite regress of time dimensions. The consequences of this, including a similar regress for the conscious observer, were explored long ago by J.W. Dunne in his bestselling An Experiment with Time and its even more dubious sequel The Serial Universe. They lead nowhere useful, unless you are a theologian or a parapsychologist.

Alternatively, one might come up with a way to break the regress after a chosen number of iterations; "That's it! There's no velocity though this dimension!" Systems with more than one time (or time-like) dimension have been studied, but none to date has been able to derive the observed nature of physical reality. The ability to reverse cause-and-effect is just one of several problems they raise.