Is it Possible to Substitute Time for a Dimension of Space and See no Change?

Question

Put into more words, to my knowledge we see 3 dimensions of space moving through a 4th dimension of time. My thought then is that if there was a hypothetical creature/device that could see time via replacing one of the spatial axes with time, let´s say X. So essentially instead of seeing the X, Y, Z coordinates through all the unique time T coordinates, they´d see all the T, Y, Z coordinates through a string of X coordinates. The kind of foundation of this question is then would the observed physics differ between the two observers due to time having some distinction from space I don´t fully understand or would they observe the same physics implying that time is only significant due to our biology assigning it significance. For reference I understand that they won´t exactly see the same thing as particles would then be placed in their different spatial and time coordinates, I´m just curious if the physics would be the same, kind of like a principle of relativity but for observing unique dimensions, if I swap the X and Y axes on a graph the laws of physics still apply, is the same true for swapping the T and X axes?

Answer 1

Time is geometrically distinct from space. The metric (in units where c=1) in a local inertial frame is $$ds^2=-dt^2+dx^2+dy^2+dz^2$$ Notice that time has the opposite sign as space. Time is also distinguished by the fact that there is only one time dimension while there are three space dimensions. Those two facts together accounts for all of the differences between time and space that are not related to entropy/thermodynamics.

EDIT: So what physical features does this $ds^2$ encode?

First, if $ds^2<0$ then the interval is timelike, meaning that it will be physically measured with a clock. Conversely, if $0<ds^2$ then the interval is spacelike, meaning that it will be physically measured with a ruler.

Second, because there are multiple positive-signature coordinates it is possible to make a closed loop in space. You can draw a circle at an instant of time that loops back on itself and is everywhere spacelike (measured with a ruler not a clock).

Third, because there is a single negative-signature coordinate it is not possible to make a closed loop in time. Any closed loop will necessarily be spacelike somewhere.

Fourth, related to the third, there is a geometric distinction between future and past. You can rotate a "left" vector into a "right" vector through a series of small rotations, but you cannot rotate a "future" vector into a "past" vector through a series of small rotations. Surfaces defined by a constant $ds^2$ form hyperboloids of one sheet for $0<ds^2$ and hyperboloids of two sheets (future and past) for $ds^2<0$.

Finally, the invariance of $ds^2$ directly encodes Einstein's 2nd postulate. For $ds^2=0$ we get $$dt^2=dx^2+dy^2+dz^2$$ which is the equation of a sphere whose radius is expanding at a speed of $1=c$.

So, what you can do is to swap your coordinate labels so that you have $$ds^2= dt^2 - dx^2 + dy^2 + dz^2$$ Then $x$ is time and $t$ is one of the three spatial dimensions. Intervals in $x$ are measured with clocks and there is a future $x$ and a past $x$ direction. Physics doesn't care about our coordinate labels.

What you cannot do is to change the sign so that all are positive $$ds^2 \ne dt^2+dx^2+dy^2 + dz^2$$ This would mean that you have only space and no time. There is no future or past and no quantity is measured with a clock.