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?
 A: 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.
A: 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.
