Would planets be cubic in my "cuboverse"?
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).
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.
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
$$ds = (-c^2 dt^2 + dx^2 + dy^2 + dz^2)^{1/2}$$
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 $\lambda \to \infty$ 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 $p_i = mc\: \gamma^{\lambda-1} \left\vert\dfrac{v_i}{c}\right\vert^{\lambda-1} \operatorname{sign}(v_i)$ and energy $E = mc^2\: \gamma^{\lambda-1}$, where $$\gamma = \dfrac{1}{\left(1-\frac{|v_x|^\lambda+|v_y|^\lambda+|v_z|^\lambda}{c^\lambda}\right)^{1/\lambda}}.$$
Newton's second law $\mathbf{F} = \dfrac{d\mathbf{p}}{dt}$ 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 $$v_i = \pm \left\vert\dfrac{p_i c}{E}\right\vert^{1/(\lambda-1)}\: c \approx \pm \left\vert\dfrac{p_i c}{E}\right\vert^0\: c = \pm c,$$ 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 $V = \dfrac{Gm_1 m_2}{\lVert\mathbf{r}\rVert_\lambda}$, where $\lVert\mathbf{r}\rVert_\lambda = (|r_x|^\lambda + |r_y|^\lambda + |r_z|^\lambda)^{1/\lambda}$ 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 $c$.
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.
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.
1. - If my calculations are correct, the true speed $\lVert\mathbf{v}\rVert_\lambda$ 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 $(1.5 - 180) mc^2$, the speed remains between $95-97\%$ 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 $V \propto \lVert\mathbf{r}\rVert_\lambda^{\lambda/(\lambda - 1)-3}$ (which is an inverse square of the distance for big $\lambda$) rather than $V \propto \lVert\mathbf{r}\rVert_\lambda^{-1}$ (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.
This post was sourced from https://worldbuilding.stackexchange.com/q/122564. It is licensed under CC BY-SA 4.0.
1 answer
TMM;DR (Too Much Math, Didn't Read):
For anyone who doesn't want to go through the derivations and calculations below, here are the important points from my answer:
- We're not working with the same space as normal, friendly, Euclidean space.
- This means that while we can still integrate and differentiate scalar functions defined on this space, we need to make slight corrections.
- These corrections can be calculated from something called the "metric tensor", which describes the curvature of space.
- There are several methods that could be used to find an equipotential surface (and therefore the shape of the planet); they should require these corrections.
If you want more details, read on!
Calculating the metric tensor
To do certain calculations in this space, it is necessary to calculate the metric tensor $g$. This is commonly used in the case of (pseudo)-Riemannian manifolds, and plays a central role in general relativity, as I assume you're aware. To do computations involving the curvature of your space, we need to know $g_{ij}$, the components of $g$.
In the Riemannian case, we have the (smooth), non-negative norm $F$. From it we can derive the components of the metric tensor by $$g_{ij}=\frac{1}{2}\frac{\partial^2F^2}{\partial x^i\partial x^j}\tag{1}$$ In Euclidean three-space (and in Cartesian coordinates), $F$ is the standard Euclidean distance metric $$F=\sqrt{x^2+y^2+z^2}$$ which is your case where $\lambda=2$, and you can probably convince yourself that $(1)$ yields the familiar $$g= \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end{bmatrix}$$ Here, we can extent the Riemannian definition to Finsler spaces using your $p$-norm; in your case, we're dealing with the more general $$F=\sqrt[\lambda]{x^\lambda+y^\lambda+z^\lambda},\quad F^2=\left(x^\lambda+y^\lambda+z^\lambda\right)^{2/\lambda}$$ Going back to $(1)$, we have $$\begin{aligned} g_{ij} &=\frac{1}{2}\frac{\partial}{\partial x^j}\frac{\partial F^2}{\partial x^i}\\ &=\frac{1}{2}\frac{\partial}{\partial x^j}\left(2(x^i)^{\lambda-1}\left(x^\lambda+y^\lambda+z^\lambda\right)\right)^{-1+2/\lambda} \end{aligned}$$ And so $$g_{ij}=\begin{cases}(2-\lambda)(x^i)^{\lambda-1}(x^j)^{\lambda-1}\left(x^\lambda+y^\lambda+z^\lambda\right)^{-2+2/\lambda},\quad i\neq j\\ \begin{aligned} &(2-\lambda)(x^i)^{2\lambda-2}\left(x^\lambda+y^\lambda+z^\lambda\right)^{-2+2/\lambda}\\ &+(\lambda-1)(x^i)^{\lambda-2}\left(x^\lambda+y^\lambda+z^\lambda\right)^{-1+2/\lambda},\quad i=j \end{aligned}\\ \end{cases}\tag{2}$$ You can again check that this works like normal in the case where $\lambda=2$.
Volume elements
So, why do we really care about $g_{ij}$? Well, if we look at Danijel's answer, we can see that they define the potential by $$V(\mathbf{x})=\int\frac{G\rho(\mathbf{x})}{||\mathbf{x}-\mathbf{x}'||}d^3\mathbf{x}'\tag{3}$$ where $d^3\mathbf{x}'$ is the volume element; notice that we've allowed $\rho$ to vary, as it would for a real planet. For Cartesian coordinates in Euclidean space, $d^3\mathbf{x}=dxdydz$. However, this is not always true. Indeed, in the general case, $$d^3\mathbf{x}=\sqrt{|g_{ij}|}dx\wedge dy\wedge dz\tag{4}$$ where $\wedge$ denotes the wedge product, and we see that we need to calculate the determinant of $g$, $|g_{ij}|$. (Note again how $\sqrt{|g_{ij}|}=1$ for Cartesian coordinates in Euclidean space.) The calculation of the determinant is ugly, and I won't do it here because there's little point to it. I assume symmetry allows it to be written somewhat neatly, though.
Gradient
As has already been noted, the surface of the planet should lie on an equipotential surface, that is, one where $|\vec{\nabla}V|=0$. The familiar Euclidean gradient is $$\vec{\nabla}=\left(\frac{\partial}{\partial x^1},\frac{\partial}{\partial x^2},\frac{\partial}{\partial x^3}\right)\hat{x}_i$$ for generalized coordinates $x^1$, $x^2$, and $x^3$, which are are $x$, $y$ and $z$. It should come as no surprise, however, that this fails for non-Euclidean spaces. In the more general case, $$\vec{\nabla}=\frac{\partial}{\partial x^i}\cdot g^{ij}\sqrt{g_{jj}}\hat{\mathbf{x}}^i\tag{5}$$ for unit vector $\hat{\mathbf{x}}^i$, and again $g_{ij}$ plays a role.
Poisson's equation and the shell theorem
Here, of course, is the problem - and it's been bothering me for about a day now. The core question you're asking here involves determining the potential corresponding to a given density distribution. We can do that easily enough in flat space by solving Poisson's equation; the solution can be found through $(3)$. Okay. But does Poisson's equation look the same (or even similar) in this Finsler space, and can we solve it the same way? The answer may be that it doesn't. In general relativity, on a Riemannian manifold, holds in the Newtonian limit of the Einstein field equations. However, it's not clear that we can make the same assumption in Finsler space.
Let's say Poisson's equation holds in some sort of form (here, I have set the appropriate constants to $1$): $$\nabla^2V=\rho\tag{6}$$ Then, indeed, $(3)$ holds in general, in some form, again with the appropriate volume element. Now, what I'm curious about - and I've talked about this in chat a bit - is the answer to a question posted by Dietrich Epp: Is the gravitational field outside a "sphere" identical to that outside a point particle with the same mass (the Shell Theorem)? If so, your given equation for the potential holds for a planet, and $$V(\mathbf{x})=\frac{Gm_1m_2}{||\mathbf{x}-\mathbf{x}'||_{\lambda}}$$ If not, we need to integrate $(3)$ properly before making any assumptions about the form of the potential. At any rate, I'd be interested in seeing a proof that a version of the shell theorem (or a version of Birkhoff's theorem) holds.
A path to a solution
I don't have an answer to your question yet. There are a couple of ways you could go find an equipotential surface:
- Muck about with various surfaces and find one where $(\text{3})$ is constant for all $V(\mathbf{x})$ on the surface.
- Set $|\vec{\nabla}V|=0$ and find the resulting surface.
- Set $V(\mathbf{x})$ equal to some constant $C$ and again find the surface.
The third method seems like it should definitely work in the case of a Euclidean metric, for your chosen potential. I can't say for sure whether it works for all relevant Finsler metrics (and I suspect it doesn't), so I won't claim that the general case works just yet.
The naive third case
Okay, so let's set $V(\mathbf{x})=C$ for some constant $C$. Here, I assume that your potential is applicable for a planet. We then have $$||\mathbf{r}||_{\lambda}=\frac{GM}{C}\implies\left(x^\lambda+y^\lambda+z^\lambda\right)^{1/\lambda}=\frac{GM}{C}$$ Therefore, we're left with $$z^\lambda=\left(\frac{GM}{C}\right)^{\lambda}-x^\lambda-y^\lambda\tag{7}$$ Interestingly enough, as $\lambda\to\infty$, this becomes a cube!
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