Stretched Light Cones: Thermodynamics & Applications

• Stretched light cones are locally defined causal constructions that replace ideal null cones with regularized timelike or scale-dependent surrogate surfaces.
• They unify approaches across gravitational thermodynamics, cosmological conformal maps, and simulation-based strong lensing through precise geometric and equilibrium analyses.
• Incorporating an explicit work term for accelerating observers allows derivations of Einstein’s equations without additional entropy production, clarifying equilibrium versus non-equilibrium treatments.

Searching arXiv for the cited stretched-light-cone literature and adjacent work. arXiv search query: "stretched light cone thermodynamics gravity" Stretched light cones (SLCs) are localized causal constructions obtained by replacing an ideal null light cone with a nearby regularized hypersurface or with a scale-dependent surrogate adapted to a specific problem. In gravitational thermodynamics, the stretched future light cone is a timelike hypersurface composed of the worldlines of radially accelerating observers with constant and uniform proper acceleration, and its temperature and entropy support a local derivation of gravitational field equations (Parikh et al., 2017). A later formulation studies the thermodynamics of stretched light cones associated with uniformly accelerating observers by computing expansion, shear, and vorticity and by applying Clausius’ relation with an explicit work term, again recovering Einstein’s equations (Alonso-Serrano et al., 10 Sep 2025). In cosmological geometry, related light-cone constructions compare physical and FLRW celestial spheres through a scale-dependent conformal map and a harmonic-type energy (Carfora et al., 2021), while in strong-lensing simulations a “stretched light cone” denotes a structure-preserving remapping of a cubic simulation volume into a lensing-appropriate geometry for multi-plane ray tracing (Roche et al., 28 May 2026). This suggests that the term does not denote a single universally fixed object, but a family of stretched or regularized light-cone frameworks.

1. Local hyperboloidal construction

The thermodynamic SLC is defined locally in a normal Riemann coordinate chart $x^\mu=(t,x^i)$ about a point $p$, with metric expansion

$$g_{\mu\nu}(x) =\eta_{\mu\nu} -\tfrac{1}{3}\,R_{\mu\alpha\nu\beta}(p)\,x^\alpha x^\beta +O(x^3)\,.$$

Its flat-space prototype is the hyperboloid

$$\Sigma:\quad r^2-t^2=\alpha^2,\qquad r\equiv\sqrt{x^i x^i}\,,$$

which asymptotes to the future light cone of the origin as $t\to\infty$. In the 2017 formulation, a convenient parametrization is

$$X^\mu(\tau,\theta^A) =\bigl(t=\tau,\, r=\sqrt{\alpha^2+\tau^2},\, \theta^A\bigr),$$

with induced metric

$$h_{ij}\,dx^i dx^j = -\,\frac{\alpha^2}{\alpha^2+\tau^2}\,d\tau^2 \;+\;(\alpha^2+\tau^2)\,d\Omega_{D-2}^2.$$

The construction is generated by the radial boost field

$$\xi^\mu = -\,r\,\delta^\mu_{t}+ t\,\delta^\mu_{r}, \qquad \xi^2=-\alpha^2,$$

so the unit velocity and normal are

$$u^\mu = \frac{\xi^\mu}{\alpha},\qquad n^\mu = \frac{1}{\alpha}\bigl(-\,t\,\delta^\mu_{t}+r\,\delta^\mu_{r}\bigr),$$

with proper acceleration

$a^\mu = \frac{1}{\alpha}\,n^\mu,\qquad a=1/\alpha.$

In curved spacetime, one promotes $p$0 to an approximate Killing vector and restricts to a small neighborhood satisfying $p$1 and $p$2, so that $p$3 remains, up to $p$4, an isothermal world tube of observers of nearly constant acceleration $p$5 (Parikh et al., 2017).

The 2025 analysis uses the same hyperboloidal embedding,

$p$6

but writes the timelike congruence as

$p$7

where $p$8 is the exact Minkowski-space tangent field and $p$9 contains curvature corrections. Alongside this timelike SLC, one may also consider the null cone

$$g_{\mu\nu}(x) =\eta_{\mu\nu} -\tfrac{1}{3}\,R_{\mu\alpha\nu\beta}(p)\,x^\alpha x^\beta +O(x^3)\,.$$0

which provides a nearby null reference surface for comparing timelike and null descriptions (Alonso-Serrano et al., 10 Sep 2025).

2. Thermodynamic structure and equations of state

The thermodynamic interpretation assigns a Davies–Unruh temperature to the accelerating congruence,

$$g_{\mu\nu}(x) =\eta_{\mu\nu} -\tfrac{1}{3}\,R_{\mu\alpha\nu\beta}(p)\,x^\alpha x^\beta +O(x^3)\,.$$1

and, in Einstein gravity, a Bekenstein–Hawking entropy to each spacelike spherical slice,

$$g_{\mu\nu}(x) =\eta_{\mu\nu} -\tfrac{1}{3}\,R_{\mu\alpha\nu\beta}(p)\,x^\alpha x^\beta +O(x^3)\,.$$2

The reversible entropy change is defined by subtracting the natural area increase of the hyperboloid in vacuum,

$$g_{\mu\nu}(x) =\eta_{\mu\nu} -\tfrac{1}{3}\,R_{\mu\alpha\nu\beta}(p)\,x^\alpha x^\beta +O(x^3)\,.$$3

and the Clausius theorem is imposed as

$$g_{\mu\nu}(x) =\eta_{\mu\nu} -\tfrac{1}{3}\,R_{\mu\alpha\nu\beta}(p)\,x^\alpha x^\beta +O(x^3)\,.$$4

The heat flux measured by the accelerating observers is

$$g_{\mu\nu}(x) =\eta_{\mu\nu} -\tfrac{1}{3}\,R_{\mu\alpha\nu\beta}(p)\,x^\alpha x^\beta +O(x^3)\,.$$5

A Stokes’-theorem argument rewrites the entropy change as a surface integral over $$g_{\mu\nu}(x) =\eta_{\mu\nu} -\tfrac{1}{3}\,R_{\mu\alpha\nu\beta}(p)\,x^\alpha x^\beta +O(x^3)\,.$$6, with the remaining non-Killing corrections integrating to $$g_{\mu\nu}(x) =\eta_{\mu\nu} -\tfrac{1}{3}\,R_{\mu\alpha\nu\beta}(p)\,x^\alpha x^\beta +O(x^3)\,.$$7. Equating heat and reversible entropy then yields

$$g_{\mu\nu}(x) =\eta_{\mu\nu} -\tfrac{1}{3}\,R_{\mu\alpha\nu\beta}(p)\,x^\alpha x^\beta +O(x^3)\,.$$8

from which one concludes

$$g_{\mu\nu}(x) =\eta_{\mu\nu} -\tfrac{1}{3}\,R_{\mu\alpha\nu\beta}(p)\,x^\alpha x^\beta +O(x^3)\,.$$9

Stress-energy conservation and the contracted Bianchi identity fix $$\Sigma:\quad r^2-t^2=\alpha^2,\qquad r\equiv\sqrt{x^i x^i}\,,$$0, giving

$$\Sigma:\quad r^2-t^2=\alpha^2,\qquad r\equiv\sqrt{x^i x^i}\,,$$1

The same construction extends to a broad class of diffeomorphism-invariant theories of gravity if the Bekenstein–Hawking entropy is replaced by the Wald entropy,

$$\Sigma:\quad r^2-t^2=\alpha^2,\qquad r\equiv\sqrt{x^i x^i}\,,$$2

with $$\Sigma:\quad r^2-t^2=\alpha^2,\qquad r\equiv\sqrt{x^i x^i}\,,$$3. The resulting field equations are precisely those obtained from the action $$\Sigma:\quad r^2-t^2=\alpha^2,\qquad r\equiv\sqrt{x^i x^i}\,,$$4 (Parikh et al., 2017).

3. Kinematics of the congruence and strict equilibrium

The 2025 treatment makes the congruence kinematics explicit through the decomposition

$$\Sigma:\quad r^2-t^2=\alpha^2,\qquad r\equiv\sqrt{x^i x^i}\,,$$5

with

$$\Sigma:\quad r^2-t^2=\alpha^2,\qquad r\equiv\sqrt{x^i x^i}\,,$$6

and

$$\Sigma:\quad r^2-t^2=\alpha^2,\qquad r\equiv\sqrt{x^i x^i}\,,$$7

Raychaudhuri’s equation for the non-geodesic congruence is

$$\Sigma:\quad r^2-t^2=\alpha^2,\qquad r\equiv\sqrt{x^i x^i}\,,$$8

By explicit expansion in small time $$\Sigma:\quad r^2-t^2=\alpha^2,\qquad r\equiv\sqrt{x^i x^i}\,,$$9 and linear order in curvature,

$t\to\infty$0

and similarly $t\to\infty$1. Physically, the SLC congruence is hypersurface-orthogonal and radially symmetric, so no shape-distortion arises at leading order. In thermodynamic language, there is therefore no irreversible “viscous” entropy production proportional to $t\to\infty$2; the system can be treated in strict equilibrium (Alonso-Serrano et al., 10 Sep 2025).

Within this equilibrium framework, the heat flux, temperature, entropy variation, and work are

$t\to\infty$3

$t\to\infty$4

$t\to\infty$5

and

$t\to\infty$6

The local thermodynamic law becomes

$t\to\infty$7

4. Recovery of Einstein’s equations and the equilibrium–non-equilibrium issue

Substituting $t\to\infty$8, $t\to\infty$9, and $$X^\mu(\tau,\theta^A) =\bigl(t=\tau,\, r=\sqrt{\alpha^2+\tau^2},\, \theta^A\bigr),$$0 into the local Clausius relation with work gives

$$X^\mu(\tau,\theta^A) =\bigl(t=\tau,\, r=\sqrt{\alpha^2+\tau^2},\, \theta^A\bigr),$$1

Using the small-time expansion

$$X^\mu(\tau,\theta^A) =\bigl(t=\tau,\, r=\sqrt{\alpha^2+\tau^2},\, \theta^A\bigr),$$2

integrating over the sphere at each $$X^\mu(\tau,\theta^A) =\bigl(t=\tau,\, r=\sqrt{\alpha^2+\tau^2},\, \theta^A\bigr),$$3, keeping only terms linear in curvature, and expanding $$X^\mu(\tau,\theta^A) =\bigl(t=\tau,\, r=\sqrt{\alpha^2+\tau^2},\, \theta^A\bigr),$$4 on the left-hand side yields

$$X^\mu(\tau,\theta^A) =\bigl(t=\tau,\, r=\sqrt{\alpha^2+\tau^2},\, \theta^A\bigr),$$5

Requiring that this hold for every timelike unit vector at $$X^\mu(\tau,\theta^A) =\bigl(t=\tau,\, r=\sqrt{\alpha^2+\tau^2},\, \theta^A\bigr),$$6 gives

$$X^\mu(\tau,\theta^A) =\bigl(t=\tau,\, r=\sqrt{\alpha^2+\tau^2},\, \theta^A\bigr),$$7

Imposing $$X^\mu(\tau,\theta^A) =\bigl(t=\tau,\, r=\sqrt{\alpha^2+\tau^2},\, \theta^A\bigr),$$8 and the Bianchi identity produces

$$X^\mu(\tau,\theta^A) =\bigl(t=\tau,\, r=\sqrt{\alpha^2+\tau^2},\, \theta^A\bigr),$$9

and matching the Newtonian limit fixes the constants so that

$$h_{ij}\,dx^i dx^j = -\,\frac{\alpha^2}{\alpha^2+\tau^2}\,d\tau^2 \;+\;(\alpha^2+\tau^2)\,d\Omega_{D-2}^2.$$0

The conceptual distinction from non-equilibrium derivations is explicit. Because $$h_{ij}\,dx^i dx^j = -\,\frac{\alpha^2}{\alpha^2+\tau^2}\,d\tau^2 \;+\;(\alpha^2+\tau^2)\,d\Omega_{D-2}^2.$$1 at leading order, there is no internal entropy production term $$h_{ij}\,dx^i dx^j = -\,\frac{\alpha^2}{\alpha^2+\tau^2}\,d\tau^2 \;+\;(\alpha^2+\tau^2)\,d\Omega_{D-2}^2.$$2. The term that would otherwise be interpreted as dissipative, originating from the flat-space divergence $$h_{ij}\,dx^i dx^j = -\,\frac{\alpha^2}{\alpha^2+\tau^2}\,d\tau^2 \;+\;(\alpha^2+\tau^2)\,d\Omega_{D-2}^2.$$3, is reinterpreted as the work $$h_{ij}\,dx^i dx^j = -\,\frac{\alpha^2}{\alpha^2+\tau^2}\,d\tau^2 \;+\;(\alpha^2+\tau^2)\,d\Omega_{D-2}^2.$$4 required to keep the observer accelerated. By contrast, approaches that impose $$h_{ij}\,dx^i dx^j = -\,\frac{\alpha^2}{\alpha^2+\tau^2}\,d\tau^2 \;+\;(\alpha^2+\tau^2)\,d\Omega_{D-2}^2.$$5 on a null horizon but neglect the observer’s work must introduce an explicit $$h_{ij}\,dx^i dx^j = -\,\frac{\alpha^2}{\alpha^2+\tau^2}\,d\tau^2 \;+\;(\alpha^2+\tau^2)\,d\Omega_{D-2}^2.$$6 to recover the field equations; the 2025 SLC analysis states that none of this extra entropy production is needed once the proper work term is accounted for (Alonso-Serrano et al., 10 Sep 2025).

5. Scale-dependent comparison of cosmological light-cone sections

A distinct SLC-related program studies past light-cone sections in cosmology. Fixing an event $$h_{ij}\,dx^i dx^j = -\,\frac{\alpha^2}{\alpha^2+\tau^2}\,d\tau^2 \;+\;(\alpha^2+\tau^2)\,d\Omega_{D-2}^2.$$7 in a cosmological spacetime $$h_{ij}\,dx^i dx^j = -\,\frac{\alpha^2}{\alpha^2+\tau^2}\,d\tau^2 \;+\;(\alpha^2+\tau^2)\,d\Omega_{D-2}^2.$$8, one defines the physical past light cone and, at observational length scale $$h_{ij}\,dx^i dx^j = -\,\frac{\alpha^2}{\alpha^2+\tau^2}\,d\tau^2 \;+\;(\alpha^2+\tau^2)\,d\Omega_{D-2}^2.$$9, the spherical section

$$\xi^\mu = -\,r\,\delta^\mu_{t}+ t\,\delta^\mu_{r}, \qquad \xi^2=-\alpha^2,$$0

with induced metric $$\xi^\mu = -\,r\,\delta^\mu_{t}+ t\,\delta^\mu_{r}, \qquad \xi^2=-\alpha^2,$$1. On the FLRW reference spacetime $$\xi^\mu = -\,r\,\delta^\mu_{t}+ t\,\delta^\mu_{r}, \qquad \xi^2=-\alpha^2,$$2, the corresponding section is

$$\xi^\mu = -\,r\,\delta^\mu_{t}+ t\,\delta^\mu_{r}, \qquad \xi^2=-\alpha^2,$$3

with round metric

$$\xi^\mu = -\,r\,\delta^\mu_{t}+ t\,\delta^\mu_{r}, \qquad \xi^2=-\alpha^2,$$4

Comparing the two exponential maps gives a local diffeomorphism

$$\xi^\mu = -\,r\,\delta^\mu_{t}+ t\,\delta^\mu_{r}, \qquad \xi^2=-\alpha^2,$$5

A residual Möbius freedom is fixed by choosing three alignment directions and defining

$$\xi^\mu = -\,r\,\delta^\mu_{t}+ t\,\delta^\mu_{r}, \qquad \xi^2=-\alpha^2,$$6

The map $$\xi^\mu = -\,r\,\delta^\mu_{t}+ t\,\delta^\mu_{r}, \qquad \xi^2=-\alpha^2,$$7 is conformal,

$$\xi^\mu = -\,r\,\delta^\mu_{t}+ t\,\delta^\mu_{r}, \qquad \xi^2=-\alpha^2,$$8

with $$\xi^\mu = -\,r\,\delta^\mu_{t}+ t\,\delta^\mu_{r}, \qquad \xi^2=-\alpha^2,$$9. The light-cone comparison functional is

$$u^\mu = \frac{\xi^\mu}{\alpha},\qquad n^\mu = \frac{1}{\alpha}\bigl(-\,t\,\delta^\mu_{t}+r\,\delta^\mu_{r}\bigr),$$0

and its first variation gives the uniformization-type elliptic equation

$$u^\mu = \frac{\xi^\mu}{\alpha},\qquad n^\mu = \frac{1}{\alpha}\bigl(-\,t\,\delta^\mu_{t}+r\,\delta^\mu_{r}\bigr),$$1

The functional satisfies

$$u^\mu = \frac{\xi^\mu}{\alpha},\qquad n^\mu = \frac{1}{\alpha}\bigl(-\,t\,\delta^\mu_{t}+r\,\delta^\mu_{r}\bigr),$$2

so it vanishes exactly when the two celestial spheres are isometric. In the Sobolev class $$u^\mu = \frac{\xi^\mu}{\alpha},\qquad n^\mu = \frac{1}{\alpha}\bigl(-\,t\,\delta^\mu_{t}+r\,\delta^\mu_{r}\bigr),$$3, the infimum is attained at some $$u^\mu = \frac{\xi^\mu}{\alpha},\qquad n^\mu = \frac{1}{\alpha}\bigl(-\,t\,\delta^\mu_{t}+r\,\delta^\mu_{r}\bigr),$$4, and the induced distance

$$u^\mu = \frac{\xi^\mu}{\alpha},\qquad n^\mu = \frac{1}{\alpha}\bigl(-\,t\,\delta^\mu_{t}+r\,\delta^\mu_{r}\bigr),$$5

is symmetric, satisfies the triangle inequality, and vanishes only when the two sphere-metrics coincide isometrically (Carfora et al., 2021).

The same construction relates light-cone geometry to scalar curvature via the small-$$u^\mu = \frac{\xi^\mu}{\alpha},\qquad n^\mu = \frac{1}{\alpha}\bigl(-\,t\,\delta^\mu_{t}+r\,\delta^\mu_{r}\bigr),$$6 area expansions

$$u^\mu = \frac{\xi^\mu}{\alpha},\qquad n^\mu = \frac{1}{\alpha}\bigl(-\,t\,\delta^\mu_{t}+r\,\delta^\mu_{r}\bigr),$$7

which imply

$$u^\mu = \frac{\xi^\mu}{\alpha},\qquad n^\mu = \frac{1}{\alpha}\bigl(-\,t\,\delta^\mu_{t}+r\,\delta^\mu_{r}\bigr),$$8

The framework is also stated to extend, via the same Sobolev $$u^\mu = \frac{\xi^\mu}{\alpha},\qquad n^\mu = \frac{1}{\alpha}\bigl(-\,t\,\delta^\mu_{t}+r\,\delta^\mu_{r}\bigr),$$9 machinery, beyond caustic formation.

6. Stretched light cones in simulation-based strong lensing

In numerical lensing, “stretched light cone” refers to a remapped simulation geometry designed to reconcile strong-lensing calculations with periodic cubic boxes. The pipeline begins with a periodic unit cube $a^\mu = \frac{1}{\alpha}\,n^\mu,\qquad a=1/\alpha.$0 and an integer $a^\mu = \frac{1}{\alpha}\,n^\mu,\qquad a=1/\alpha.$1 unimodular matrix $a^\mu = \frac{1}{\alpha}\,n^\mu,\qquad a=1/\alpha.$2 whose action produces a rectangular parallelepiped with sides $a^\mu = \frac{1}{\alpha}\,n^\mu,\qquad a=1/\alpha.$3 and $a^\mu = \frac{1}{\alpha}\,n^\mu,\qquad a=1/\alpha.$4. For the IllustrisTNG300-1 demonstration,

$a^\mu = \frac{1}{\alpha}\,n^\mu,\qquad a=1/\alpha.$5

so the new box is $a^\mu = \frac{1}{\alpha}\,n^\mu,\qquad a=1/\alpha.$6 longer in the line-of-sight direction than in the sky plane. The coordinate remapping is

$a^\mu = \frac{1}{\alpha}\,n^\mu,\qquad a=1/\alpha.$7

The details emphasize that this preserves the original volume and local clustering without repeating any structure along the long axis within one light cone.

Lens-mass planes and source-light planes are then assembled snapshot by snapshot. Particles are shifted, remapped, converted to $a^\mu = \frac{1}{\alpha}\,n^\mu,\qquad a=1/\alpha.$8, selected in radial shells, separated into resolved halos, stellar particles, and diffuse mass, and smoothed onto two-dimensional grids via a Wendland-$a^\mu = \frac{1}{\alpha}\,n^\mu,\qquad a=1/\alpha.$9 kernel with $p$00 nearest neighbors. No random rotations are required, though optional random lateral shifts or rotations per snapshot are noted. The multi-plane formalism computes, on plane $p$01,

$p$02

with critical surface density

$p$03

convergence and deflection

$p$04

and, in Fourier space,

$p$05

Rays obey the standard multi-plane recursion

$p$06

with accumulated Jacobian

$p$07

Critical curves are the loci where $p$08, and the total critical area within image-plane aperture $p$09 is

$p$10

The reported application uses TNG300-1, Planck 2015 cosmology, $p$11 mini-snapshots at $p$12, approximately $p$13 planes at $p$14, remapping axis ratios $p$15, $p$16, a lens-mass field of view of $p$17 on a $p$18 grid, a source-light field of view of $p$19 on a $p$20 grid, Wendland-$p$21 smoothing with $p$22, and GLAMER fixed-grid FFT with $p$23 zero-padding. In a TNG300-1 sample of $p$24 clusters at $p$25 and $p$26, uncorrelated line-of-sight structure shifts relative image positions by several arcseconds, introduces a $p$27 scatter in the area of a cluster’s primary critical curve, and changes the total critical area within $p$28 of the cluster potential minimum by $p$29 (Roche et al., 28 May 2026).

7. Conceptual scope and recurrent themes

Across these research programs, several structural motifs recur. First, the construction replaces an exact null cone by a nearby object that is better suited to analysis: a timelike hyperboloid of uniformly accelerating observers in gravitational thermodynamics, a conformally compared spherical section at fixed observational scale in cosmology, or a remapped elongated simulation domain in numerical lensing. Second, each framework is explicitly local or scale-dependent: $p$30 in the equilibrium thermodynamic derivation, fixed scale $p$31 for celestial-sphere comparison, and finite snapshot shells and image-plane apertures in the lensing pipeline. Third, the stretched construction is used to extract invariant or quasi-invariant information: Einstein’s equations from heat–entropy balance, an isometry criterion and distance functional for cosmological light cones, or critical curves and line-of-sight sensitivities in simulated strong lensing.

The principal conceptual divergence lies between thermodynamic and geometric-comparison uses. In the thermodynamic literature, the decisive issue is whether the SLC should be treated through strict equilibrium with an explicit work term or through a non-equilibrium entropy-production formalism. In the cosmological comparison and simulation literatures, by contrast, the emphasis is not thermodynamic but geometric and computational: conformal matching of celestial spheres, relation to scalar curvature, and structure-preserving assembly of lensing light cones. A plausible implication is that “stretched light cone” functions less as a single theory term than as a methodological pattern: causal light-cone data are regularized, thickened, or re-embedded so that they can support thermodynamic balance laws, comparison theorems, or numerical ray tracing.