# Are spatial and temporal dimensions interchangeable?

According to general relativity, time is a dimension, one of four dimensions that form 4D spacetime - a structure which is mathematically symmetrical and homogeneous.

Should not all four dimensions, therefore, be mathematically interchangeable? Assuming that we are 3-dimensional bodies travelling through 4-dimensional space, perceiving 'slices' of the fourth dimension as instants or moments in time, surely there should be no reason to favour one particular dimension (the one we perceive as time).

Why, then, in the equations describing general relativity, can't duration (t) be interchanged with spatial distance? Why is there no mathematical framework that allows (as a simplistic example) the multiplication of duration and distance to produce area (dt ⇔ dd)?

I realise that the quantum mechanics describes time as fundamentally probabilistic, in which case this logic would not apply. But why does this logic not apply to the equations of GR?

If, indeed, time and space are separate structures (after all, the law of entropy would imply that time is asymmetrical, placing it at contrast with the homogeneity of space), is there any mathematical proof that this is the case?

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The equations of General Relativity do allow for space and time to reverse roles. For example, in the spherically symmetric Schwarzchild solution, time and space reverse roles sufficiently close to the center of mass (inside the event horizon of a black hole). Distance becomes the 'inevitable' dimension (since all objects fall towards the center), rather than the march forward of time which we regularly experience.

PeterDonis
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According to general relativity, time is a dimension, one of four dimensions that form 4D spacetime
That's the way it's often put in pop science treatments, but it's not quite correct. A better way to say it would be that, locally, at each event in spacetime, we can pick out one timelike and three spacelike vectors that form an orthonormal "tetrad". This tetrad can be interpreted physically as describing measuring devices carried by an observer whose worldline passes through that event; the timelike vector describes the rate of his clock, and the three spacelike vectors describe three mutually orthogonal rulers that pick out his spatial axes. Notice that I didn't say anything at all about "dimensions".

Why is there no mathematical framework that allows (as a simplistic example) the multiplication of duration and distance to produce area (dt ⇔ dd)?
Why do you think this can't be done? There is nothing stopping you from doing an integral, for example, over a surface that has one timelike and one spacelike dimension, so the area element for the surface looks like ##dt dx##, where ##dt## is the differential of a timelike coordinate and ##dx## is the differential of a spacelike coordinate.

What you can't do is treat timelike intervals (or vectors) the same as spacelike intervals (or vectors) or null intervals (or vectors). Those are three fundamentally different kinds of things, and the difference has to be recognized. (Physically, the difference is, heuristically, that we measure timelike intervals by the readings of clocks, spacelike intervals by the readings of rulers, and null intervals by the paths of light rays.)

I realise that the quantum mechanics describes time as fundamentally probabilistic
It does? Do you have a reference for this?

Orodruin
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What singles out timelike directions from spacelike in relativity is the metric tensor. Spacelike vectors will have a norm squared which is of the opposite sign from that of timelike vectors. (Which is positive and which is negative depends on your metric convention.)

Ultimately, this has to do with the symmetry group of Minkowski space being hyperbolic rotations and not the regular rotations you are used to from Euclidean geometry.

The equations of General Relativity do allow for space and time to reverse roles. For example, in the spherically symmetric Schwarzchild solution, time and space reverse roles sufficiently close to the center of mass (inside the event horizon of a black hole). Distance becomes the 'inevitable' dimension (since all objects fall towards the center), rather than the march forward of time which we regularly experience.
This is only an effect of your choice of coordinates and has little to do with "flipping" space and time. It just so happens that the r-coordinate inside the event horizon is timelike. In no way does this mean that you can locally make any sort of transformation to exchange what is time and what is space.

PeterDonis
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The equations of General Relativity do allow for space and time to reverse roles.
That's not really correct. What the equations allow is for coordinates to switch roles, heuristically speaking. But coordinates have no physical meaning in GR in and of themselves. Pop science treatments often talk as if "time and space are switching roles" when it's really just coordinates that are switching roles; but that's one of the reasons why we don't recommend trying to learn science from pop science treatments.

Dale
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Should not all four dimensions, therefore, be mathematically interchangeable?
The signature of the metric is (-+++), so the dimensions are not interchangeable. One dimension has the opposite signature of the other three.

I am not really sure about the reality of dimensionality, but it is perfectly mathematical and, in the case of extreme cosmological events (i.e. within BH Event Horizon) then Spatial dimensions can appear TIMELIKE at least. I am unsure of whether this is commutative in whether temporal dimensionality can appear SPACELIKE.

Then of course, there's the problems of whether what can be achieved mathematically has any actual bearing on reality and, of course, since some mathematics does describe reality, precisely where the boundary lies and, even why there is a boundary.

Orodruin
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