Would lenses work in a universe with four spatial dimensions?

Question

Let us assume that we have a universe with four spatial dimensions rather than the three of our universe, in which matter can exist that is a four-dimensional analogue of three-dimensional matter.

Further, let us assume that in this four-dimensional universe there are many substances that allow the transmission of light, and that the speed of light in these substances varies from the speed of light in a vacuum to some lower speed. In other words, these transparent substances have a property that we in our three-dimensional universe would call a refractive index.

Given these assumptions, can it be shown that transparent substances in a four-dimensional universe can refract light in a manner that would allow the construction of a four-dimensional lens?

I'm looking for a hard-science answer that shows the possibility or impossibility of lenses in four spatial dimensions. I'm not looking for a 'guesstimate'... I can make one of those for myself, and my guess is that lenses should be possible, but I don't know enough math and physics to know for sure.

Answer 1

You are running into pretty big conceptual issues here. Especially for a Hard Sci-Fi question, there simply is no correct answer.

General Relativity for instance, which is what all of modern physics is kind of based on, does not work the same way in higher dimensions. There are solutions to something like the Kerr Metric for rotating black holes, but all of these solutions make some pretty major assumptions to work.

We can make some very general assumptions. For instance, Euclidian distances;

$$l = \sqrt{(x_1-x_0)^2 + (y_1-y_0)^2 + (z_1-z_0)^2 + ... + (n_1-n_0)^2}$$

extend into any higher dimension. So this distance measurement between two points is correct for all higher dimensions assuming there is marginal curvature and the extra space dimensions behave the same way. Which is not a given. String Theory for instance has curled up dimensions

Which obviously do not support the notion of Euclidian distances anymore as the dimension is inherently non-Euclidian.

And this is what i mean, you don't provide any assumptions. So the answer is both yes and no. If the additional space dimensions are "flat" and simple extensions of 3+1 Space we live in, then all the math for Refraction, General Relativity and so on should translate. If however, the additional dimension is not "flat", all of that is thrown out the window.

Can we prove that ? Well we can prove case 1. Proving case 2 would involve making a new theory of spacetime.

For Case 1, the question is if the equations that describe Refraction hold up in higher dimensions. The first test we can do for this is a simple vector reflection. In R3 that looks like this;

$$r = d-2(d\cdot n)n$$

Where $r$ is the reflected vector, $d$ is the incoming vector and $n$ is the surface normal. We can be really lazy and just set the Normal to $n = (0,0,0,1)$ and the incoming to $d = (0,0,1,1)$. Solving for r we get $r = (0,0,1,-1)$. Which, just on a surface level, is correct. The Normal vector is the w direction, which used to be 1 and now it is -1 so the vector is properly reflected.

Ultimately, Refraction is just fancy reflection as a different commentor has written out. If we look at the refraction equation (Snells law of refraction);

$$\eta_1sin{\theta_1} = \eta_2sin{\theta_2}$$

we shall notice that the angle $\theta_1$ is what we are interested in. Since solving the equation means solving for $\theta_2$. What you conceptually do when performing refraction simulations is take your incoming vector $d$ and change its angle from $\theta_1$ to $\theta_2$

So what is $\theta_1$ ? Well, its an angle between the incoming $d$ and $n$. Equation for which is;

$$cos{\theta} = \frac{(d,n)}{||d||||n||}$$

Where $(d,n)$ is the inner product. This definition extends into n-dimensional space. So by definition, the refraction equation has solutions not just in 4+1 but n+1 spacetime.

Again provided those dimensions are "flat". The absolute nanosecond anything funky happens with the extra dimension, non of this works anymore.

But as long as this 4th dimension of yours is flat / behaves in the exact same way as the 3 we love, then the answer is "Yes, it is possible".

Answer 2

I am afraid we don't know, yet. Your question is "would lenses work in a 4D space?" and it is a sub-question of "What are Maxwell equation in a 4D space?"

Apparently, some researchers dwelled into the determination of Maxwell Equations in 4 dimensions.

From the abstract of the paper:

The paper formulates Maxwell’s equations in 4-dimensional Euclidean space by embedding the electromagnetic vector potential in the frame vector g0. Relativistic electrodynamics is the first problem tackled; in spite of using a geometry radically different from that of special relativity, the paper derives relativistic electrodynamics from space curvature. Maxwell’s equations are then formulated and solved for free space providing solutions which rotate the vector potential on a plane; these solutions are shown equivalent to the usual spacetime formulation and are then discussed in terms of the hypersphere model of the Universe recently proposed by the author.

Their conclusion is:

Solving the equations for free space leads to rotation of frame vector g0 on a plane lying on 3-space, with rotation progressing at arbitrary frequency along the direction normal to the rotation plane. These solutions are evanescent on the positive x0 direction and grow to infinity in the opposite direction. The inconsistency is attributed to artificial flattening of the hypersphere space proposed in previous work and further work is suggested to fully clarify this point.

19 years after the paper was published, I haven't been able to find the "further work", nor anything related to propagation in media other than free space, which is the part you are looking for.

Answer 3

Quite surprisingly, Huygens' principle fails when the number of spatial dimensions is even (see this question on the Physics SE), with the consequence that light doesn't travel consistently at $c$ in vacuum, though that is still the maximum speed. So instead of seeing an object at a distance $D$ as it was a time $D/c$ ago, you would see a blurred record of its history up to that time (though the most recent position would be brightest). I assume that this would also affect refraction in lenses, but in a brief search just now, I couldn't find a paper that analyzes it.

If you handwave that problem, I think there is nothing else that would prevent lenses from working in higher dimensions. The lenses commonly found in nature and technology are rotationally symmetric. You can think of them as solids of revolution formed from 2D lenses, which inherit the focusing behavior of the 2D lenses, and that process extends straightforwardly to more dimensions. To make it precise, if $L_2$ is the set of points in the 2D lens, and it has a vertical reflection symmetry ($L_2 = \{(x,-y):(x,y)\in L_2\}$), then the lens in $d$ dimensions is $L_d = \left\{(x_1,\ldots,x_d) : \left(x_1, \sqrt{x_2^2+\cdots+x_d^2}\right) \in L_2 \right\}$.