Summary

The main idea behind the "cuboverse" is that spacetime distances are measured by (something close to) the sup norm or infinity norm. Under this norm, spheres (the set of points at a fixed distance from an origin) are the same as cubes, hence the name.

Other features that I have been able to derive from this are:

• Geodesics are straight lines like in our world, but

• Objects can basically only move at a certain constant speed, and only in one of eight special directions, $(\pm 1,\pm 1,\pm 1)$ in Cartesian coordinates.

• There is an attractive force of "gravity", and a second cohesive force that allows primordial material to form big planet-like bodies of liquid.

My question is:

Would planets be cubic in this universe? If not, what shape would they attain (octahedra, ordinary spheres, unstable, something else)?

I would like answers based on physical reasoning and supported by mathematical calculations if possible, taking into account the relevant changes to real-world physics (see the details section below).

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Background

I recently discovered the science fiction writer Greg Egan. Many of his novels like Diaspora, the Orthogonal series and Dichronauts share the idea of changing one or more fundamental things about our world's physics (the number of dimensions, the metric signature of these dimensions, changes to particle physics etc.) and exploring the consequences of that change. The author keeps some science notes online relating to these works, and after reading them I got inspired to attempt to build one such world myself.

The cuboverse I imagined consists of big planets made of liquid (similar to water), one of them inhabited by a small intelligent species of eight-spiked "sea urchins", along with some other eel-like and carpet-like sentient creatures, all of them living near the surface. There are no stars in this world, so the necessary heat comes from the planet itself. I have already thought of a method of propulsion for the sea urchins and some rough details about their society. I still haven't developed the chemistry and particle physics, and I also have some questions about the biology, but first of all I would like to know whether the setting I imagined (specifically the shape of the planets and their stability) is realistic in the context of this modified physics.

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Details

As a warning, I'm not at all experienced in exploring alternate world physics, it's my first time doing this, so some of the things I derived below could be wrong. Anyways, my basic idea is to change the Minkowski metric

$$ds=(-c^{2}d{t}^2+d{x}^2+d{y}^2+d{z}^2{)}^{1/2}$$

to a $\lambda$-norm

$$ds=(-c^{\lambda }|dt{|}^{\lambda }+|dx{|}^{\lambda }+|dy{|}^{\lambda }+|dz{|}^{\lambda }{)}^{1/\lambda },$$

where $\lambda$ is a very big number (I decided not to choose the sup norm itself $\lambda \to \mathrm{\infty }$ because that would make geodesics non-unique). According to this, with this norm spacetime seemingly becomes a kind of Lorentzian analog of a Finsler geometry. We can calculate geodesics (which turn out to be straight lines) and define a four-momentum vector for point particles as usual, using the Lagrangian formalism.

After some calculations (I can provide details if needed), we arrive at equations for the momentum $p_i=mc{\gamma }^{\lambda -1}{\left|{\displaystyle \frac{v_i}{c}}\right|}^{\lambda -1}\mathrm{sign}(v_i)$ and energy $E=m{c}^2{\gamma }^{\lambda -1}$, where $$\gamma ={\displaystyle \frac{1}{{(1-\frac{|v_x{|}^{\lambda }+|v_y{|}^{\lambda }+|v_z{|}^{\lambda }}{c^{\lambda }})}^{1/\lambda }}}.$$

Newton's second law $\mathbf{F}={\displaystyle \frac{d\mathbf{p}}{dt}}$ still holds, so heuristically, for a generic set of particles at generic positions under the action of generic forces, the probability that the momenta have one or more components nearly equal to zero will be very small. Since $\lambda$ is very big, the components of the velocity vector will generically be close to $$v_i=\pm {\left|{\displaystyle \frac{p_{i}c}{E}}\right|}^{1/(\lambda -1)}c\approx \pm {\left|{\displaystyle \frac{p_{i}c}{E}}\right|}^{0}c=\pm c,$$ as I claimed in the summary.¹ This implies, among other things, that it is virtually impossible for any object to stay still: its velocity will generically be one of the eight possible vectors pictured below (which one depends on which octant the momentum vector lies).

For gravity, the most reasonable thing would be to work with a generalized² gravitational potential $V={\displaystyle \frac{G{m}_1{m}_2}{\Vert \mathbf{r}{\Vert }_{\lambda }}}$, where $\Vert \mathbf{r}{\Vert }_{\lambda }=(|r_x{|}^{\lambda }+|r_y{|}^{\lambda }+|r_z{|}^{\lambda }{)}^{1/\lambda }$ is the sup norm distance between two particles of masses $m_1$ and $m_2$. I managed to make some calculations but the orbits look too funky, so I decided to get rid of stars and planetary systems completely, having instead a single kind of astronomical body. If planets are cubic (that's my question!), I believe the gravity near the surface would be constant.

Since it is extremely easy to move a stationary object by applying a very small force, every big structure would become unstable under gravity alone, so I decided to have a secondary cohesive force that sticks particles of primordial matter together, while still allowing for some free fluid-like motion. I am not really sure how this force will look like since the chemistry isn't yet developed, so for the moment I'm forced to work with this rough description of how I want it to behave.

For the moment, as a starting point I'm assuming the primordial matter consists of small hard particles, let's say ordinary spheres, and studying the collapse of a cloud of this material under gravity and a perfectly inelastic contact force (these assumptions can be changed if needed for the answer). At this point the analysis becomes more difficult, and I've been unable to find whether the planets are cubes or not. Based on the form of the gravitational potential, I would expect a "yes" answer, but the weird restrictions on the velocity make me doubt. Also, I'm not completely sure that a cohesive force will fully solve the instability problem. On the other hand, things could get complicated by relativistic effects since the velocities are close to $c$.

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Two final notes:

• Just to prevent any possible confusion: my intention behind the question isn't to make a world with cubic planets, it's perfectly OK for me if the answer is "these planets can't ever exist in your world, even if you modify the gravitational/cohesive forces". The underlying intention is just to explore the consequences of the main premise; it's not necessary to keep the little sea urchins idea viable.

• Although the context is worldbuilding, this is at its core a mathematical physics problem, and as such it most certainly has a unique right answer. I believe I have developed the basic physics enough for the question to be answerable with the information I provided. If that's not the case (if there's a free variable unaccounted for, or if the answer crucially needs concepts from the corresponding version of, for example, thermodynamics), please point out what else is needed and I'll try to rigorously work it out if it's feasible.

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EDIT: Inspired by Aric's answer I tried to make a simulation myself. I'm not very good at coding so I haven't been able to make the inelastic collisions work yet. Applying the force of gravity only, it turns out that an initially static cloud of material does seem to collapse into an octahedral shape, as suggested by some of the answers. Here are the results for a spherical slice of primordial material (color represents height):

However, since there are some points where particles tend to clump together heavily and the simulation doesn't take into account the cohesive force that would separate them, I think there's still a good chance that the real shape is more cube-like, or perhaps something in between similar to this, as suggested by JBH's answer. There's still the question of stability, that I don't know how to tackle.

Just for reference, I also found some slides online about a possible way to treat fluid dynamics in a Finsler spacetime. Most of it is over my head right now, but perhaps someone will find them useful.

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_{1. - If my calculations are correct, the true speed $\Vert \mathbf{v}{\Vert }_{\lambda }$ is actually not so close to $c$ but a bit lower for massive particles. For example, for $m=1.8$ kg, $\lambda =100$ and a range of kinetic energies within $(1.5-180)m{c}^2$, the speed remains between $95-97\mathrm{\%}$ of $c$. I believe the most common relativistic phenomena like length contraction or time dilation aren't expected to play a big role though, because $\gamma$ is practically $1$ for almost all speeds, but I may be wrong on this, I haven't given it much thought yet.}

_{2. - One can consider the analogue of a Klein-Gordon massless field and work out the corresponding "Coulomb" interaction potential as the Green function for a static field background. Dimensional analysis of the resulting integral suggests a law of the form $V\propto \Vert \mathbf{r}{\Vert }_{\lambda }^{\lambda /(\lambda -1)-3}$ (which is an inverse square of the distance for big $\lambda$) rather than $V\propto \Vert \mathbf{r}{\Vert }_{\lambda }^{-1}$ (someone asked for my reasoning and I put it here, in case anyone else's interested). For a true gravitational force I guess I would have to look into a suitable coupling between matter and a curved Finsler geometry, and it seems there is already some work being done on this. But I don't think the specifics matter much at this point, for the moment I just want a reasonable-looking attractive force.}