Super Smash Bros

Explaining the Equations in Sephiroth’s Final Smash

TL;DR: Sephiroth’s final smash, Supernova, briefly shows the Schrodinger equation, the Einstein field equations, the transport formula, and the Hellmann-Feynman theorem, among other things.


Sephiroth’s awesome-looking final smash, Supernova, summons a Meteor that flies into the sun and triggers a supernova, destroying the solar system in the process. As this happens, just like in the original Final Fantasy VII, a few equations flash on the screen, as if to illustrate the forces being called upon. Interestingly, the equations that appear in Smash seem to be different than those used in FFVII, so I figured I would do some digging to figure out where they come from and what they mean.

(Apologies for the potato-quality images; they’re taken from screenshots on a YouTube video. That said, the equations are still mostly intelligible, and in the one case where it’s especially bad, I’ve re-rendered it in LaTeX.)


The first one that shows up is:
This one’s fairly straightforward: it’s the time-dependent single-particle Schrodinger equation. In quantum mechanics, it describes the motion of a single particle in a potential well over time.

Ψ(x,t) = the wavefunction
t = time
x = position
m = mass
2 = the Laplace operator, sort of like a second derivative
V(x,t) = potential energy function
ħ = the reduced Planck constant


This is followed by:
These are the Einstein field equations. In general relativity, they describe how the curvature of spacetime is affected by the matter and energy inhabiting spacetime.

R = the Ricci scalar, which describes one aspect of the curvature of spacetime
R_μν = the Ricci tensor, which describes another aspect of the curvature of spacetime
g_μν = the metric tensor), which describes how much the distances in spacetime are warped
G = the gravitational constant, describing the fundamental strength of gravity
c = speed of light
T_μν = the stress-energy tensor, describing the density of energy, momentum, and matter

If you look at the Wikipedia article for this one, you’ll notice that they rewrote this equation slightly, first by replacing the Einstein tensor G_μν with the Ricci scalar and Ricci tensor, and second by setting the cosmological constant Λ to zero (which means that space is neither expanding nor contracting).


The third equation we encounter is a bit more obscure:
This one is a version of the transport formula (found on page 7). It describes the behavior of some scalar function (usually something like density) tied to a moving fluid.

θ(x,t) = density
D/Dt = the material derivative, which accounts for the motion of the fluid
v = volume
u = flow velocity


Moving on to the fourth:
This one is really, really faint in all frames where it’s not blocked by Meteor, so here’s a re-rendering: It also doesn’t seem to correspond to anything fundamental – this is just a statement that the dot products of two sets of vectors (P’ . u and P . u’) are equal to each other. It’s unclear what P and u, or their primed counterparts, are supposed to represent, since the equation is too short to be identifiable as something famous. Further digging is needed here.


The fifth equation is much easier to see:
This one isn’t anywhere near as famous as the first three, but based on the symbols used, we can figure out what this is probably referring to. Namely, this is an electron screening effect calculation, used to determine approximately how an outer electron’s orbit is affected by the inner electrons of a large atom. Since the inner electrons are negatively charged and the nucleus is positively charged, the outer electron doesn’t feel the full charge of the nucleus; instead, it feels a smaller “effective charge”, because the electrons in the way “screen” some of the attraction.

Z_k = the effective charge felt by an electron in orbital k
a_0 = the Bohr radius
r = distance from the nucleus
n(r) = electron density
R_k = the distance of an electron in orbital k


The sixth equation is:
This describes the pressure in a rotating fluid of constant density. There’s not much more to say about this one, except that the (>0) at the end appears to be an ornament taken from a larger set of equations. It does nothing here but remind us that this quantity is always positive, since density, distance, and squared velocity are all positive numbers.

p = pressure
r = distance from the center
ρ = density
v = linear velocity


The seventh equation is odd:
This seems to be describing the fact that in an electric circuit with only reactive elements (no resistors, only capacitors and inductors), the total power dissipated is zero. Alternate explanations are welcome, since this doesn’t appear to be particularly fundamental.

V = voltage
I = current
N = number of circuit elements


The eighth equation:
We arrive at something reasonably fundamental again, namely, the Hellmann-Feynman theorem. In quantum mechanics, this describes what happens to the energy of a system if you change the strength of the interactions in that system.

λ = some parameter controlling the strength of interactions
E_λ = energy at a certain value of λ
Ψ_λ = stationary state of the wavefunction for a certain value of λ
H_λ = Hamiltonian) of the system for a certain value of λ


Finally, this is the last equation we see:
Once again, this is too short to identify as anything particularly famous. It appears to be a definition of how the total energy in the system changes as you change some parameter set f_i.


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