Relativity in 2+1 or 4+1 dimensions

Whereas we have experience of relativity working with 3 spacial dimensions and one of time would there be similar rules affecting a two dimensional space and one of time or even with a 4 space and 1 of time?

• Can you clarify what you are asking? Are you asking if special (or general) relativity can be formulated in spacetimes with a number of space dimensions different from three? – John Rennie Feb 13 at 9:51
• You seem to have understood my question, but would these alternate sets necessarily have relativity as part of their properties? – John Webb Feb 13 at 10:00

Special relativity is a geometrical property of a spacetime. You're no doubt familiar with Pythagoras' theorem. This tells us that in regular three dimensional space if we move a distance $$dx$$ in the $$x$$ direction, $$dy$$ in the $$y$$ direction and $$dz$$ in the $$z$$ direction then the total distance moved is given by:

$$ds^2 = dx^2 + dy^2 + dz^2 \tag{1}$$

We tend to take Pythagoras' theorem for granted because we learn it at an early age, but in fact this is an extremely important equation. It is called the metric and it defines the geometry of the space. In this case it defines the geometry of our 3D space to be Euclidean.

The Euclidean metric (1) is not the only metric a 3D space can have. Starting with the work of Gauss, Lobachevsky and Bolyai we found that 3D spaces can have different metrics. So we cannot say from the dimensionality what the metric is. We have to make measurements in our space to check what the metric is.

Now on to special relativity. In SR we consider 4D spacetime not 3D space, so our metric has to include time as well as space. The metric in special relativity is called the Minkowski metric and it is:

$$ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2 \tag{2}$$

Note that we now include distance moved in time, $$dt$$, and (this is the key thing) the $$dt$$ term has a minus sign. It is this minus sign that it responsible for the weird effects in SR like time dilation and length contraction.

I must emphasise that the Minkowski metric is just one of the many possible metrics a 4D spacetime could have, just as the Euclidean metric is one of the many possible metrics a 3D space could have. The fact our universe has the Minkowski metric is an experimental observation. We measure the motion of objects moving near the speed of light and from those measurements we mind that the metric has the form shown in equation (2). And this is what we mean by special relativity.

Introductory courses in relativity generally do not discuss the metric - students would probably start by being taught the Lorentz transformations and that's probably what you are familiar with. However it is the metric that is fundamental and things like the Lorentz transformations are derived from the metric.

Anyhow, the point of all this is that we can now say very simply what we mean by special relativity in two or four or indeed any number of spatial dimensions. Suppose we have two space and one time dimension (often written as 2+1D)so we label points with coordinates $$(t,x,y)$$ then in this spacetime special relativity means the metric is:

$$ds^2 = -c^2dt^2 + dx^2 + dy^2$$

Or if we have four space dimensions (4+1D) so our points are $$(t,x,y,z,u)$$ then special relativity means the metric is:

$$ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2 + du^2$$

If you're asking if two or four spatial dimensions necessarily obey special relativity then the answer is no, because they could have many different metrics just like our examples of 3D space and 4D spacetime. Your 2+1D or 4+1D physicist would have to make experimental measurements to find out what metric their spacetime has.

• So as I understand what you say is that if the Minkowski metric applies, then yes the 2+1D and 4+1D could have relativity but it is not definitive. Is there anything to suggest that they do have a Minkowski metric or any other? – John Webb Feb 13 at 14:43
• @JohnWebb Yes, you understand me correctly. As far as we know our spacetime is resolutely 3+1D (despite the opium dreams of the string theorists) and no such things as a 2+1D or a 4+1D or indeed any other dimension spacetime exists. So it is pointless to speculate on what their metric might be. A hypothetical 2+1D or whatever spacetime could have any metric - there is no special reason why any particular metric is more likely than any other. In fact there is no special reason why our universe should have the Minkowski metric. That's just what experiment tells us. – John Rennie Feb 13 at 15:42
• In our 3+1D awareness how could we prove that there is not a 2+1D, in our 3+1D there could be a multiplicity of 2+1Ds stacked or oriented in any direction. If they had something like our particles they could be devilishly hard to detect. – John Webb Feb 13 at 18:19
• @JohnWebb I think the point John Rennie is trying to make is that if our 3+1D spacetime is an apple, a 2+1D spacetime could be an orange, and a 4+1D spacetime a banana. It's comparing apples and oranges (and bananas) in that details of one are not very interpretable for others. It's arbitrary, in that our spacetime could just as easily have any other metric, so the fact that the random number generator gave us Minkowski in our case doesn't give anything conclusive or even particularly useful in evaluating what the random number generator will give for a 2+1D or a 4+1D spacetime. – TheEnvironmentalist Feb 13 at 19:52
• I disagree, If they exist they were likely forged in the same crucible, so the 2+1D would be a subset of the 3+1D which would be a subset of 4+1D thus sharing a common core and properties. The 2+1D would require less energy to create than the 3+1D having fewer degrees of freedom and consequently the 4+1D needing the greatest amount is less likely to exist. – John Webb Feb 14 at 8:33

In this context, the laws of physics are the Einstein field equations, which form the basis of general relativity. The field equations can be summarized by the famous aphorism by relativist John Wheeler that matter tells spacetime how to curve, and spacetime tells matter how to move.

The way in which the field equations are formulated makes no explicit reference to the number of dimensions, so they work equally well in 2+1 or 4+1 dimensions. However, that doesn't mean that they have to apply to a hypothetical universe. A hypothetical universe can have unicorns and pixie dust, or be governed solely by Murphy's Law. The most we can say is that there is a natural way of extrapolating to say what would constitute the same laws as in our universe.