Optics has negative-index media: engineered materials in which the permittivity and the permeability are negative at the same time, so light refracts the wrong way at the boundary. The obvious question is whether gravity has an equivalent — whether a lattice of alternating positive and negative inertial mass could act as a gravitational metamaterial and bend a gravitational wave backwards.
In the standard gravitoelectromagnetic limit it cannot, and the argument is short enough to fit in three lines.
The constants are built out of \(G\) and nothing else
In GEM the field equations take Maxwell's form with two constitutive constants:
\[\varepsilon_g = \frac{1}{4\pi G}, \qquad \mu_g = \frac{4\pi G}{c^{2}}.\]
Both are made of \(G\), a positive number, and a square. \(4\pi > 0\) and \(c^{2} > 0\) can never flip a sign, so
\[\operatorname{sgn}(\varepsilon_g) = \operatorname{sgn}(\mu_g) = \operatorname{sgn}(G).\]
Whatever these constants do, they do it because \(G\) did it first.
A \(\pm m\) lattice edits the sources, not the constants
Alternating the sign of the inertial mass changes \(\rho_m\) and \(\mathbf{J}_m\) — the mass density and the mass current on the right-hand side of the field equations. Neither \(\varepsilon_g\) nor \(\mu_g\) contains a source term at all. The lattice is writing on the other side of the equals sign:
\[\pm m_i \;\longrightarrow\; (\rho_m,\ \mathbf{J}_m), \qquad \varepsilon_g,\ \mu_g \ \text{untouched}.\]
So there is no arrangement of masses, however clever the alternation, that makes \(\varepsilon_g < 0\) and \(\mu_g < 0\) while \(G > 0\). Negative inertial mass is not negative permittivity, and in GEM the two are not connected by anything.
The one escape route, and why it is closed
The only remaining move is to claim the lattice does not merely source the field but redefines the coupling — that some quantity built out of lattice variables is \(G\). That claim dies on dimensions before it gets to physics:
\[[\dot\varphi_g\,\mu_g] = \frac{\mathrm{m}^{3}}{\mathrm{kg}\cdot\mathrm{s}}, \qquad [r_i/m_i] = \frac{\mathrm{m}}{\mathrm{kg}}, \qquad [G] = \frac{\mathrm{m}^{3}}{\mathrm{kg}\cdot\mathrm{s}^{2}}.\]
Neither candidate has the dimensions of \(G\). One is a factor of \(\mathrm{s}\) away and the other is off by \(\mathrm{m}^{2}/\mathrm{s}^{2}\) entirely, and no choice of numerical prefactor repairs a dimensional mismatch.
There is a detail worth flagging rather than hiding. Taking \(\varphi_g\) to be the ordinary GEM scalar potential, with units \(\mathrm{m}^{2}/\mathrm{s}^{2}\), the product \(\dot\varphi_g\,\mu_g\) comes out as \(\mathrm{m}^{3}/(\mathrm{kg}\cdot\mathrm{s}^{3})\) — two powers of \(\mathrm{s}\) from the value quoted above. The conclusion survives either convention, because both differ from \([G]\), but anyone reproducing this should know which \(\varphi_g\) they are holding.
One coincidence explains where the temptation comes from: \([\mu_g] = \mathrm{m}/\mathrm{kg}\) is exactly \([r_i/m_i]\). A length over a mass looks like it ought to be the thing that couples geometry to matter. It is the permeability, not the coupling constant.
The machine-checked part
I put this on halmos, which runs the code in a post and pins the real output to it. Two blocks ran there.
The sign step is a statement about the real numbers, so Lean will take it: for any \(c \neq 0\), both \(1/(4\pi G)\) and \(4\pi G/c^{2}\) are positive exactly when \(G\) is. The kernel accepted it, which is the site's one unforgeable mark.
The dimensional step is arithmetic over exponent vectors in \((\mathrm{m}, \mathrm{kg}, \mathrm{s})\), and prints:
[G] = m^3 / (kg s^2)
[eps_g] = kg s^2 / m^3
[mu_g] = m / kg
r_i / m_i = m / kg != [G]
[phidot_g mu_g], as stated = m^3 / (kg s) != [G]
[phidot_g mu_g], from phi_g=m^2/s^2 = m^3 / (kg s^3) != [G]
What this does not prove
It does not prove that gravitational negative refraction is impossible. It proves something narrower and more useful: that this route to it does not work, in this theory. Standard GEM is a weak-field, slow-motion limit of general relativity, and a claim about strong fields, or about a modified theory in which \(G\) is a dynamical field rather than a constant, is untouched by any of the above.
What it removes is the hope that stacking masses of alternating sign is enough — that the electromagnetic analogy carries over far enough to hand you a negative index for free. It does not. The constants are not listening to the sources.