Gravitational Index: A Multifaceted Outlook

• Gravitational Index is a multipurpose term that denotes effective refractive indices in curved spacetime, supersymmetric path-integral quantities, and variational instability measures in n-body problems.
• In relativistic optics, the index recasts gravity as an optical medium, enabling accurate predictions of light deflection, Shapiro delay, and other gravitational lensing phenomena.
• In supergravity and string theory, the gravitational index computes protected BPS degeneracies via complex saddles, linking microscopic spectra to macroscopic black hole and black string properties.

Searching arXiv for papers on “gravitational index” and related usages. The expression gravitational index is used in contemporary literature for several mathematically distinct constructions. In relativistic optics and semiclassical vacuum models, it denotes an effective refractive index associated with propagation in a gravitational field or curved spacetime. In supergravity and string theory, it denotes a supersymmetric path-integral quantity that computes protected black-hole or black-string degeneracies through complex Euclidean saddles. In more specialized settings, closely related uses of “index” appear as variational instability counts in the gravitational $n$-body problem and as classification parameters in gravitational-wave or lensing analyses. The term is therefore not univocal; its meaning is fixed by subfield, observable, and formalism (Nazari et al., 2010, Boruch et al., 29 Jan 2025, Ou et al., 2024).

1. Principal meanings and scope

The main usages represented in the literature can be summarized as follows.

Usage	Definition	Representative source
Spacetime or vacuum refractive index	Effective $n$ governing light or virtual-quanta propagation in a gravitational field	(Nazari et al., 2010, Simaciu, 20 Feb 2025)
Gravitational supersymmetric index	Protected trace or Euclidean path integral with $(-1)^F$ and BPS boundary conditions	(Boruch et al., 29 Jan 2025, Genolini et al., 26 Mar 2025)
Variational “gravitational index”	Morse index of a colliding homothetic orbit in the Newtonian $n$-body problem	(Ou et al., 2024)

In the optical usage, gravity is recast as an effective medium. Nazari and Nouri-Zonoz define the gravitational index of refraction in a static spacetime by

$n(x)\equiv \frac{1}{\sqrt{g_{00}(x)}},$

and in weak stationary backgrounds obtain

$n(x,v)\simeq 1-\Phi(x)+A\!\cdot\! v,$

with $A_i=-g_{0i}/g_{00}$ the gravitomagnetic potential (Nazari et al., 2010). In a different semiclassical construction, the vacuum itself is modeled as a dilute fluid of virtual particle–antiparticle pairs, leading in a weak Newtonian potential to a gravitational refractive index

$$n_g(r)=1+\frac{2GM}{c^{2}r}+\mathcal O\!\bigl((GM/rc^{2})^{2}\bigr)$$

that reproduces first-order deflection and Shapiro delay (Simaciu, 20 Feb 2025).

In the supersymmetric usage, the gravitational index is a path integral over complex saddles with supersymmetric boundary conditions. One representative definition is

$$I_{\rm grav}(\omega_a,\phi_i)=\int_{fields\,|\,Killing\,spinor\,BCs} Dg\,D\ldots \,\exp[-S_{\rm grav}(g,\ldots)],$$

which is compared to a microscopic BPS trace and localized onto “supersymmetric but non-extremal” saddles (Genolini et al., 26 Mar 2025).

2. Refractive-index formulations of gravity

In the spacetime-optical formulation, curved spacetime acts as an inhomogeneous refractive medium. For a static metric in Fermi coordinates,

$ds^2=g_{00}(x)\,dt^2+g_{ij}(x)\,dx^i dx^j,$

the gravitational index is $n$0. In a weak Newtonian potential $n$1, the same framework gives

$n$2

so the coordinate propagation of light is described as if it were moving through a spatially varying optical medium (Nazari et al., 2010).

Nazari and Nouri-Zonoz further apply this definition to a Casimir apparatus in a weak static gravitational field $n$3. For a massless scalar field between Dirichlet plates separated by $n$4, the mode frequency becomes

$n$5

and the renormalized vacuum-energy density shifts to

$n$6

For the electromagnetic field, the scalar result doubles (Nazari et al., 2010). This explicitly realizes the earlier conjecture that virtual quanta experience curvature through the same refractive factor as real photons.

A distinct semiclassical vacuum model derives the same weak-field optical behavior from a modified pair density $n$7. Treating the vacuum as a dilute fluid of virtual positronium-like pairs and using a Boltzmann-type density

$n$8

one obtains

$n$9

and hence

$(-1)^F$0

The same paper states that the first-order GR formulas for light deflection,

$(-1)^F$1

and Shapiro delay are identically recovered by integrating through the effective medium (Simaciu, 20 Feb 2025).

The same semiclassical analysis also gives higher-order and numerical estimates. Writing $(-1)^F$2, one may expand

$(-1)^F$3

At Earth’s surface, $(-1)^F$4 and $(-1)^F$5; near a canonical neutron star with $(-1)^F$6 and $(-1)^F$7, $(-1)^F$8 and $(-1)^F$9 (Simaciu, 20 Feb 2025).

A further weak-field construction by Wilhelm and Dwivedi introduces a gravitational index of refraction $n$0 through an aether model with photons as solitons. In their notation,

$n$1

which is then used to recover the gravitational redshift relation $n$2 (Wilhelm et al., 2017).

3. Optical-medium analogies in dynamical spacetimes and lensing

The refractive-index idea generalizes beyond weak static fields. In the optical-medium analogy, Maxwell theory in curved spacetime is rewritten as flat-space electrodynamics in an anisotropic effective medium with constitutive tensors

$n$3

and, when $n$4, a directional refractive index

$n$5

Bini, Geralico and Jantzen apply this formalism to the Ferrari–Ibañez colliding-wave spacetime, where the interaction region metric yields principal indices

$n$6

with $n$7 (Bini et al., 2014).

The behavior of $n$8 distinguishes the two branches of the Ferrari–Ibañez solution. For $n$9, the collision ends at a non-singular Killing–Cauchy horizon and $n(x)\equiv \frac{1}{\sqrt{g_{00}(x)}},$0 as $n(x)\equiv \frac{1}{\sqrt{g_{00}(x)}},$1. For $n(x)\equiv \frac{1}{\sqrt{g_{00}(x)}},$2, the collision ends at a curvature singularity and $n(x)\equiv \frac{1}{\sqrt{g_{00}(x)}},$3. In both cases $n(x)\equiv \frac{1}{\sqrt{g_{00}(x)}},$4 remains bounded by $n(x)\equiv \frac{1}{\sqrt{g_{00}(x)}},$5 (Bini et al., 2014). The paper interprets these limits as an optical encoding of focusing, defocusing, and time-delay effects in the interaction region.

A related but kinematical use of the gravitational index appears in Schwarzschild lensing. Walters starts from the null geodesics of the Schwarzschild metric and rewrites the travel time through an effective refractive index $n(x)\equiv \frac{1}{\sqrt{g_{00}(x)}},$6 entering

$n(x)\equiv \frac{1}{\sqrt{g_{00}(x)}},$7

The same analysis derives a coordinate-acceleration vector for a massless test particle,

$n(x)\equiv \frac{1}{\sqrt{g_{00}(x)}},$8

and exact integral formulas for deflection and Shapiro delay (Walters et al., 2010).

That framework is then embedded in standard ray shooting. For multiple masses, one sums the accelerations from each source and integrates the resulting ODE system to generate weak- and strong-lensing observables such as Einstein rings, critical curves, and caustic webs (Walters et al., 2010). In this usage, the “gravitational index” is not a protected quantum observable but an effective optical field derived from null geodesics.

4. Supersymmetric gravitational index in supergravity and string theory

In supergravity and string theory, the gravitational index is a protected quantity defined by a Euclidean path integral or, equivalently, by a twisted trace over BPS states. A general microscopic form is

$n(x)\equiv \frac{1}{\sqrt{g_{00}(x)}},$9

while the gravitational counterpart is a path integral over complex metrics and other fields with Killing-spinor boundary conditions (Genolini et al., 26 Mar 2025).

For five-dimensional ungauged supergravity, one formulation is

$n(x,v)\simeq 1-\Phi(x)+A\!\cdot\! v,$0

with $n(x,v)\simeq 1-\Phi(x)+A\!\cdot\! v,$1 inserting $n(x,v)\simeq 1-\Phi(x)+A\!\cdot\! v,$2 in the trace. Uplifting four-dimensional finite-temperature attractor saddles to five dimensions yields matching four- and five-dimensional indices for black holes, and a black-string saddle whose on-shell action reproduces the microscopic index at leading order in $n(x,v)\simeq 1-\Phi(x)+A\!\cdot\! v,$3 even in asymptotically flat space. For $n(x,v)\simeq 1-\Phi(x)+A\!\cdot\! v,$4, the five-dimensional black-hole saddle gives

$n(x,v)\simeq 1-\Phi(x)+A\!\cdot\! v,$5

which in minimal supergravity becomes

$n(x,v)\simeq 1-\Phi(x)+A\!\cdot\! v,$6

For $n(x,v)\simeq 1-\Phi(x)+A\!\cdot\! v,$7, the black-string saddle gives

$n(x,v)\simeq 1-\Phi(x)+A\!\cdot\! v,$8

The same construction admits a novel decoupling limit to a finite-temperature BTZ throat and reproduces the leading Cardy growth of the MSW elliptic genus (Boruch et al., 29 Jan 2025).

For the two-charge heterotic string on $n(x,v)\simeq 1-\Phi(x)+A\!\cdot\! v,$9, Chen, Murthy and Turiaci define

$A_i=-g_{0i}/g_{00}$0

with $A_i=-g_{0i}/g_{00}$1, and evaluate it using a complex rotating Euclidean “cigar” black-hole saddle (Chen et al., 2024). In the two-derivative theory the saddle is singular, but the one-loop, four-derivative $A_i=-g_{0i}/g_{00}$2-term

$A_i=-g_{0i}/g_{00}$3

resolves the singularity and extends the attractor mechanism to include higher derivatives. The on-shell action then matches the microscopic heterotic index exactly: $A_i=-g_{0i}/g_{00}$4 The paper also states that the thermal state transitions to a winding condensate and a gas of strings without ever reaching a small black hole, while the index is captured by the rotating Euclidean black-hole solution and is smoothly connected to the microscopic ensemble (Chen et al., 2024).

An analogous construction exists for the five-dimensional two-charge small black ring. Sen and collaborators define a protected trace at fixed $A_i=-g_{0i}/g_{00}$5 and $A_i=-g_{0i}/g_{00}$6, represent it by a Euclidean path integral with $A_i=-g_{0i}/g_{00}$7, and evaluate it on a doubly spinning two-charge ring saddle (Bandyopadhyay et al., 14 Apr 2025). Although the saddle has a finite horizon area

$A_i=-g_{0i}/g_{00}$8

the angular term cancels it exactly,

$A_i=-g_{0i}/g_{00}$9

so the logarithm of the index vanishes, in agreement with the strict BPS limit of a two-charge small ring (Bandyopadhyay et al., 14 Apr 2025).

5. Allowable complex metrics and convergence criteria

Because the gravitational supersymmetric index is evaluated on complex saddles, a central technical question is which complex metrics are admissible in the path integral. Kontsevich, Segal and Witten proposed an allowability criterion requiring positivity of the real part of the kinetic form on all real $$n_g(r)=1+\frac{2GM}{c^{2}r}+\mathcal O\!\bigl((GM/rc^{2})^{2}\bigr)$$0-forms. In one equivalent statement, a complex metric with eigenvalues $$n_g(r)=1+\frac{2GM}{c^{2}r}+\mathcal O\!\bigl((GM/rc^{2})^{2}\bigr)$$1 is allowable if

$$n_g(r)=1+\frac{2GM}{c^{2}r}+\mathcal O\!\bigl((GM/rc^{2})^{2}\bigr)$$2

A detailed study of this criterion in the context of the gravitational index compares it to geometric regularity and to convergence of the microscopic trace (Genolini et al., 26 Mar 2025).

In four-dimensional examples—the asymptotically flat helicity supertrace, the AdS$$n_g(r)=1+\frac{2GM}{c^{2}r}+\mathcal O\!\bigl((GM/rc^{2})^{2}\bigr)$$3 topologically twisted index, and the AdS$$n_g(r)=1+\frac{2GM}{c^{2}r}+\mathcal O\!\bigl((GM/rc^{2})^{2}\bigr)$$4 superconformal index—the three conditions coincide. In the AdS$$n_g(r)=1+\frac{2GM}{c^{2}r}+\mathcal O\!\bigl((GM/rc^{2})^{2}\bigr)$$5 superconformal case, the common region is

$$n_g(r)=1+\frac{2GM}{c^{2}r}+\mathcal O\!\bigl((GM/rc^{2})^{2}\bigr)$$6

and the paper states that convergence of the microscopic trace, geometric regularity, and KSW allowability impose exactly the same inequalities (Genolini et al., 26 Mar 2025).

The same work reports a different outcome in the AdS$$n_g(r)=1+\frac{2GM}{c^{2}r}+\mathcal O\!\bigl((GM/rc^{2})^{2}\bigr)$$7 superconformal-index case with unequal angular chemical potentials. There, with parameters $$n_g(r)=1+\frac{2GM}{c^{2}r}+\mathcal O\!\bigl((GM/rc^{2})^{2}\bigr)$$8, the geometric-consistency region is larger than the KSW-allowed region, and the KSW-allowed region is larger than the microscopic-convergence region: $$n_g(r)=1+\frac{2GM}{c^{2}r}+\mathcal O\!\bigl((GM/rc^{2})^{2}\bigr)$$9 Its conclusion is that the KSW criterion is necessary but not sufficient for the allowability of complex metrics contributing to the superconformal index (Genolini et al., 26 Mar 2025).

A later AdS$$I_{\rm grav}(\omega_a,\phi_i)=\int_{fields\,|\,Killing\,spinor\,BCs} Dg\,D\ldots \,\exp[-S_{\rm grav}(g,\ldots)],$$0 analysis states a stronger equivalence result for supersymmetric black holes with two independent angular momenta. It formulates the test pointwise: one checks $$I_{\rm grav}(\omega_a,\phi_i)=\int_{fields\,|\,Killing\,spinor\,BCs} Dg\,D\ldots \,\exp[-S_{\rm grav}(g,\ldots)],$$1, forms

$$I_{\rm grav}(\omega_a,\phi_i)=\int_{fields\,|\,Killing\,spinor\,BCs} Dg\,D\ldots \,\exp[-S_{\rm grav}(g,\ldots)],$$2

requires $$I_{\rm grav}(\omega_a,\phi_i)=\int_{fields\,|\,Killing\,spinor\,BCs} Dg\,D\ldots \,\exp[-S_{\rm grav}(g,\ldots)],$$3 to be positive-definite, then defines $$I_{\rm grav}(\omega_a,\phi_i)=\int_{fields\,|\,Killing\,spinor\,BCs} Dg\,D\ldots \,\exp[-S_{\rm grav}(g,\ldots)],$$4 and imposes

$$I_{\rm grav}(\omega_a,\phi_i)=\int_{fields\,|\,Killing\,spinor\,BCs} Dg\,D\ldots \,\exp[-S_{\rm grav}(g,\ldots)],$$5

For the corresponding AdS$$I_{\rm grav}(\omega_a,\phi_i)=\int_{fields\,|\,Killing\,spinor\,BCs} Dg\,D\ldots \,\exp[-S_{\rm grav}(g,\ldots)],$$6 black-hole saddles, the paper concludes that KSW allowability coincides with the microscopic convergence conditions $$I_{\rm grav}(\omega_a,\phi_i)=\int_{fields\,|\,Killing\,spinor\,BCs} Dg\,D\ldots \,\exp[-S_{\rm grav}(g,\ldots)],$$7, $$I_{\rm grav}(\omega_a,\phi_i)=\int_{fields\,|\,Killing\,spinor\,BCs} Dg\,D\ldots \,\exp[-S_{\rm grav}(g,\ldots)],$$8, and $$I_{\rm grav}(\omega_a,\phi_i)=\int_{fields\,|\,Killing\,spinor\,BCs} Dg\,D\ldots \,\exp[-S_{\rm grav}(g,\ldots)],$$9 (Genolini et al., 30 Jan 2026). This suggests that the relationship between KSW and microscopic convergence is sensitive to the precise saddle family and implementation of the criterion.

6. Bubbling saddles, topology, degeneracy, and the “index enigma”

The five-dimensional gravitational index is not exhausted by a single black-hole saddle. Cassani, Ruipérez and Turetta analyze the Euclidean path integral of minimal five-dimensional supergravity on $ds^2=g_{00}(x)\,dt^2+g_{ij}(x)\,dx^i dx^j,$0 under a $ds^2=g_{00}(x)\,dt^2+g_{ij}(x)\,dx^i dx^j,$1 symmetry and classify supersymmetric complex saddles by a rod structure on a two-dimensional orbit space (Cassani et al., 16 Jul 2025). Horizon rods correspond to Euclidean horizons, while bubbling rods correspond to topological bolts supported by flux. Depending on the rod data $ds^2=g_{00}(x)\,dt^2+g_{ij}(x)\,dx^i dx^j,$2, the fixed loci may have $ds^2=g_{00}(x)\,dt^2+g_{ij}(x)\,dx^i dx^j,$3, $ds^2=g_{00}(x)\,dt^2+g_{ij}(x)\,dx^i dx^j,$4, lens-space, spindle, or branched topology.

For a single-horizon saddle with bubbles, the regularized action takes the form

$ds^2=g_{00}(x)\,dt^2+g_{ij}(x)\,dx^i dx^j,$5

and more explicitly

$ds^2=g_{00}(x)\,dt^2+g_{ij}(x)\,dx^i dx^j,$6

In the extremal limit $ds^2=g_{00}(x)\,dt^2+g_{ij}(x)\,dx^i dx^j,$7, these saddles reduce to BPS black rings and black lenses; in the $ds^2=g_{00}(x)\,dt^2+g_{ij}(x)\,dx^i dx^j,$8 limit, they become horizonless bubbling solutions with purely imaginary action (Cassani et al., 16 Jul 2025).

A broader catalog of flat-space and AdS$ds^2=g_{00}(x)\,dt^2+g_{ij}(x)\,dx^i dx^j,$9 saddles is constructed in work on novel black saddles for five-dimensional gravitational indices. There the on-shell actions of Euclidean BPS saddles are $n$00-independent and equal to the entropy of the corresponding extremal object, while the 4d$n$015d uplift desingularizes some 4d poles into smooth 5d geometries (Boruch et al., 27 Oct 2025). Among the explicitly compared actions in asymptotically flat space are

$n$02

for the BMPV-type black-hole saddle, and

$n$03

for a black-ring saddle with the same $n$04 specialization. Since the latter is strictly larger for $n$05, the paper identifies an “index enigma”: less symmetric black rings dominate the index over more symmetric black holes in the relevant ensemble (Boruch et al., 27 Oct 2025).

Another universal feature is saddle degeneracy from higher-form symmetry. For four-dimensional SCFT indices and their AdS$n$06 duals, a discrete electric one-form symmetry $n$07 produces a $n$08-fold degeneracy of the supersymmetric black-hole saddle and an additive logarithmic correction

$n$09

In the bulk, this is realized by flat two-form fields with BF coupling

$n$10

and the $n$11 background sectors contribute equally to the gravitational index (Cassani et al., 1 Jun 2026). At infinite volume, the same construction is used to illustrate spontaneous breaking of the one-form symmetry in a Cardy-like decompactification to a black brane (Cassani et al., 1 Jun 2026).

7. Related indexed quantities in gravitational dynamics

Outside the two dominant meanings above, several gravitational subfields use index-like quantities that play analogous classificatory roles but are not the same object.

In the Newtonian $n$12-body problem, the Morse index of a homothetic colliding orbit is described as a “gravitational index” measuring variational instability. For a homothetic solution $n$13 generated by a central configuration $n$14, the main dichotomy is:

• if $n$15 is spiral, $n$16, then $n$17;
• if $n$18 is non-spiral, $n$19, then $n$20. The proof uses McGehee regularization, a Maslov-index interpretation, and a block decomposition of the linearized Hamiltonian system (Ou et al., 2024).

In gravitational-wave signal analysis, Mohanty and Dhurandhar introduce the bounce hardness index $n$21, defined by the singular behavior of the local growth rate,

$n$22

It classifies the pre-bounce growth of waveforms: $n$23 for the inspiral chirp, $n$24 for one Dimmelmeier et al. core-bounce model, and $n$25 for another, with the interpretation that $n$26 is the “softest” polynomial blow-up and $n$27 the “hardest” exponential growth (Ishiyama et al., 2010).

In pulsar timing, the braking index measures the relative importance of electromagnetic and gravitational-wave torques. For a total torque

$n$28

de Araujo et al. obtain

$n$29

where $n$30 is the GW fraction of the total spin-down. For PSR J1640–4631, $n$31 implies $n$32, so only $n$33 of the spin-down power is in gravitational waves (Araujo et al., 2016).

In strong gravitational lensing, the mass density power-law index $n$34 is defined through

$n$35

with $n$36 corresponding to the Singular Isothermal Sphere. A model-independent analysis using SNe Ia and quasar luminosity distances finds that the non-evolving case $n$37 is preferred for current samples, although an evolving law

$n$38

with mild negative $n$39 also fits the data well (Hu, 2023). These indices are gravitationally relevant, but they are not interchangeable with the refractive or supersymmetric gravitational index.

Taken together, these usages show that “gravitational index” is best understood as a family resemblance term rather than a single invariant. In optics it encodes how curvature modifies propagation; in supergravity it counts protected sectors through complex saddles; in variational and observational settings it parametrizes instability, waveform growth, spin-down, or mass-profile structure. The common feature is compression of nontrivial gravitational information into a single or reduced set of protected, asymptotic, or diagnostically useful quantities.