Time can be a dimension and still be treated differently.
In fact, in special relativity, it is. When calculating the spacetime "distance" between two events, we use something that looks like the Pythagorean theorem, but we must always make sure to put a different sign in front of time than we do space. (If we sum the squares of space, we must subtract the square of time) (see: minkowski metric, really any metric)
Another point of interest would be the matrix representation of the Lorentz transformation (the way to go from one reference frame to another). The matrix "mixes" the values of the time coordinate with the space coordinates, and the space coordinates with each other, but time is mixed differently. (This is a nice one to bring up because its clear if you just look at the matrix.)
This is very significant for people wondering about time's status as a dimension. It is emphatically not the same kind of dimension as space, even if we choose to represent it as a component of the position vector.
Well, time does very concretely swap roles with one of the spatial dimensions, namely the radial one. They change signs in the metric, making the radial dimension timelike and the old temporal dimension spacelike.
While you're right that it's not a dimension in the sense that most people are used to, I think that it appropriately conveys the fact that time and space are intertwined. When you start traveling very quickly your sense of distance and time change in a way that is exactly proportional to each other. The fact that space and time need to be treated as a single entity was in fact one of the key conceptual advances that Einstein made.
One weird thing about time is that it’s sign is “flipped” in the spacetime distance equation. So, while in some sense it’s “just another dimension”, it behaves somewhat differently from the spatial dimensions when measuring distance.
(I’m not an expert, just remembering something I found interesting in Einstein’s book _Relativity_)
OTOH time is relative to the observer and certain universe phenomena, such as black holes, can distort how time passes in different points of the space dimension. The general idea that a point in time is how we define a particular snapshot of particles is probably right, but time as a dimension also appears to be a bit more complex than how we perceive it from Earth, which we should also not forget.
According to Einstein, time is relative. If I move really fast time is passing slower for me. So time and its effects vary, there should be more than 1 dimension for time in the existing model. Or time should be separated from spatial + other dimensions just because its an outcome of gravity (and other factors) that we cant override and change. This is what Einstein quote is claiming I suppose.
Time is very much a dimension: spacetime is well described as a 3+1 dimensional manifold.
Reading your link, I have no idea what those researchers are talking about (nor did it inspire me to look at their actual article). The "1 time + 3 space dimensions" structure of the universe is exceedingly well established experimentally by the essentially perfect verifications of the predictions of relativity that we use every day (in GPS, for instance). It's possible (though unlikely, from what I've seen here) that they have some deep new insight into the nature of time, sure. But any successful theory of time will have to be almost indistinguishable from standard 3+1 dimensional spacetime under "ordinary" conditions, or else it will contradict a huge body of experimental evidence. So even if what these folks are saying somehow makes sense, saying "time is not a dimension" would still be wrong in many important ways.
We know time and space are the same fabric, because different relativistic speeds let us see the same regions of space-time with different space-time coordinates.
The “angle” of time vs space is different based on perspective. Each perspective sees a different direction of time through the 4D space-time.
Yet from any perspective, the space time regions coordinates are still best represented with three real values and one imaginary value.
This is the form of coordinates in the space time interval equation, and reflects time dilation and space contraction.
So space and time are identical stuff, but from any vantage point you will see “your” time dimension acting differently than 3 spatial dimensions.
When evaluating the coordinate-invariant length of [x,y,z,t] (where time is just another number), we use the minkowski metric: xx+yy+zz - tt, where time is treated differently with the minus sign. (By xx, I mean x*x=x^2.)
So, that's essentially where, "time is a dimension" comes from, and that's where it goes.
Aside, if you're wondering what coordinate-invariant length really is: just think about the fact that looking at a house from different angles will not change the distance between the doors. This length is calculated with the Pythagorean theorem involving xx+yy+zz, and when you extend it to lengths which are also invariant under different choices of the time coordinate you arrive at xx+yy+zz-tt.
To complete what has turned into a brief introduction to SR, note how differences in the most natural choice of the direction of time might arise:
We want the path of a stationary object to involve no movement through space, only the required advance into the future (through time). So, we note that time is parallel to the path of a stationary object. However, people moving relative to each other will disagree about which objects are stationary. Therefore, they will end up thinking differently about which way time points.
Returning to my earlier analogy, this disagreement about which direction time points can be compared to the different viewers of the house, who may disagree about which direction forwards or left points.
It isn't too unnatural when you think about it, which is a good thing because it is, well, natural.
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