Generation and purification of excited spacetimes using Schwarzian derivative

Akhil U Nair; Prasant Samantray; Rakesh K Jha; Sashideep Gutti

REVIEW 3 major objections 5 minor 33 references

Generation and purification of excited spacetimes using Schwarzian derivative

T0 review · 3 major / 5 minor · reviewed 2026-07-30 · grok-4.5

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Constant Schwarzian maps generate all thermally excited subsets, purifying parents, and siblings of 2D vacuum spacetimes.

desk verdict Solid 2D catalogue that turns the Schwarzian Unruh law into three inverse ODEs; the math is standard and correct, but constant flux is over-sold as full thermality for the whole Möbius-exponential family. read the letter →

arxiv 2607.26693 v1 pith:PQUUGSZR submitted 2026-07-29 gr-qc hep-thmath-phmath.MP

Rakesh K Jha , Akhil U Nair , Prasant Samantray , Sashideep Gutti This is my paper · ORCID

classification gr-qchep-thmath-phmath.MP PACS 04.62.+v04.70.Dy11.25.Hf

keywords SchwarzianderivativeUnruheffectVirasoroanomalythermalfluxpurificationRindlerspacetime2dCFTMöbiustransformation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

In two-dimensional quantum field theory a massless scalar in vacuum can look thermal to some observers, as in the Unruh effect for uniformly accelerated Rindler wedges. This paper asks three inverse questions: which subsets of a vacuum spacetime carry a constant thermal flux of left- or right-moving particles; which larger “parent” spacetimes purify a given thermal region back to vacuum; and which sibling regions share the same particle content. Each question is turned into a nonlinear differential equation built from the Schwarzian derivative that governs the anomalous transformation of the stress tensor. The general solutions are Möbius transformations acting on exponential coordinates; they recover the familiar Rindler, Milne and diamond geometries and systematically produce many new ones, including half-sided excitations and partial purifications that clean only one null sector. A sympathetic reader cares because the same algebraic object that encodes the Virasoro anomaly now becomes a generative engine for entire families of observer-dependent thermal spacetimes.

What carries the argument

The Schwarzian derivative S(U,u) that appears in the anomalous transformation law of the 2d stress tensor. Setting it equal to a prescribed constant (or zero) converts each of the three inverse problems into a third-order nonlinear ODE whose general solution is a Möbius transformation on an exponential coordinate.

What would settle it

Compute the Bogoliubov coefficients or the full KMS two-point function for a generic member of the four-parameter family that is not a pure exponential map; if the spectrum is not Planckian at the temperature fixed by the constant flux, the identification of constant Schwarzian with thermality fails.

Watch

Extended reading notes

Core claim

The general solution of the constant-Schwarzian equation {Y,X}=−a² is the four-parameter family Y=(A e^{aX}+B)/(C e^{aX}+D) (AD−BC≠0). The inverted and flux-preserving equations admit analogous closed solutions. These maps exhaustively generate the thermally excited subsets of a vacuum spacetime, the purifying parent spacetimes (including partial purification of a single null sector), and the sibling regions that share identical particle content.

Load-bearing premise

That a constant stress-tensor flux fixed by the Schwarzian is enough to guarantee a full thermal particle spectrum for every map in the general solution, not only the classical cases where Bogoliubov coefficients are checked explicitly.

Editorial extensions

If this is right

Every constant-flux thermal region in 2d Minkowski or Rindler spacetime is locally related to vacuum by a Möbius-on-exponential map.
Partial purification is possible: one null sector can be returned to vacuum while the other retains its thermal flux.
Sibling diamonds and Rindler-like wedges that share a common parent can be generated without ever constructing the parent explicitly.
Starting from a Rindler vacuum yields an infinite catalogue of nested Rindler–Rindler and Rindler–Milne geometries, all carrying the same Unruh temperature.
The same ODE method supplies the metric of any purifying spacetime once the thermal region’s conformal factor is known.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the left- and right-moving sectors decouple, the construction immediately suggests a diagnostic for horizon formation: watch which null sector first develops a non-zero constant Schwarzian.
The same differential equations could be run backwards from a measured flux to reconstruct candidate parent geometries in analogue-gravity experiments.
Extending the ODEs to massive fields would test how much of the thermal catalogue survives once left- and right-movers couple.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The useful bit here is organizational, not a new physical effect. They take the usual Virasoro/Schwarzian transformation law for a 2D massless scalar, write three inverse ODEs (constant flux from vacuum, purification back to vacuum, flux-preserving siblings), and write the closed-form solutions: Möbius on exponentials. That package, plus the language of partial purification and siblings, is what is actually new. Individual maps (Rindler, Milne, half-sided, Rindler–Rindler) are recovered correctly and match the textbook cases and Kolekar–Padmanabhan.

Appendices A–C do the standard reduction to Riccati and recover the known general solutions; several explicit Bogoliubov calculations check out. Circularity is essentially zero—the outputs are not fitted. Self-cites are context, not load-bearing. For someone who works on 2D CFT methods, near-horizon toy models, or observer dependence, the catalogue is handy and the partial-purification construction is clean.

The soft spot is real but bounded. Constant ⟨T_uu⟩ follows for the whole family from the anomaly, and they treat that as “thermal distribution” throughout Secs. III–VI. Full Planck/KMS structure is only shown for the pure exponentials and a few named geometries. Generic A,B,C,D introduce poles and restricted ranges; nothing demonstrates the mode mixing stays thermal. The discussion itself flags that completeness of the solution class is unproven. So the “all subsets / all parents / all siblings” language over-reaches; the honest claim is “all constant-flux maps of this form.” That is fixable by tightening wording and restricting the thermality claim to the cases they actually check. No 4D lift, no massive fields—fine for the stated scope.

This is for the 2D QFT-in-curved-spacetime niche. It deserves a serious referee, not a desk reject. I would engage, cite the catalogue and the partial-purification idea if I am writing in that corner, and ask them to dial back the thermality claim.

Referee Report

3 major / 5 minor

Summary. The paper poses three inverse problems for a 2D massless scalar in flat spacetime, using the Virasoro/Schwarzian transformation law of the stress tensor. (i) Given a vacuum parent, which conformal maps yield constant nonzero flux in the left and/or right sectors? (ii) Given a thermally excited region, which parent maps purify one or both sectors (including “partial purification”)? (iii) Which “sibling” maps preserve a given constant flux without reconstructing the parent? The constant-Schwarzian ODEs are reduced to Riccati form (Appendices A–C) and solved by Möbius transformations of exponentials, Y=(A e^{aX}+B)/(C e^{aX}+D). Explicit metrics, figures, and selected Bogoliubov calculations recover Rindler, Milne, half-sided, and Rindler–Rindler geometries with the expected Unruh flux ħa²/48π.

Significance. If the identification of constant Schwarzian flux with a thermal particle distribution extends reliably across the full Möbius-exponential family, the work supplies a systematic catalogue of thermally excited subsets, purifying parents, and flux-preserving siblings in 2D CFT, with Rindler as one member rather than the unique case. The closed-form ODE solutions and the notions of partial purification and sibling spacetimes are concrete organizational tools for near-horizon and Unruh-type analyses. Strengths include standard, reproducible reductions of the nonlinear ODEs and explicit recovery of textbook Unruh/Bogoliubov results for the classical exponential maps. The limitation is that full thermality (Planck spectrum, KMS) is demonstrated only for special maps, so the breadth of the catalogue currently rests on flux alone.

major comments (3)

[Secs. III–VI; Eqs. (13)–(14); Discussion] Secs. III–VI and the abstract treat every solution of {Y,X}=−a² (and the inverted/flux-preserving siblings), Y=(A e^{aX}+B)/(C e^{aX}+D), as generating a “thermal distribution/density of particles.” Constant ⟨T_uu⟩=ħa²/48π follows for the whole family from the Virasoro anomaly, but Planckian occupation numbers and/or KMS are computed only for pure exponentials and named geometries (Rindler, Milne, half-sided, Rindler–Rindler; e.g. Eqs. (21)–(22), (50)–(52), (63)–(64)). For generic A,B,C,D the maps have poles and restricted ranges and are not global wedge diffeomorphisms. Either supply Bogoliubov/KMS evidence for a representative generic Möbius case, or restrict the thermality claim to constant flux and state clearly that full thermality is verified only for the classical exponential subclass.
[Abstract; Sec. III; Sec. VII] The Discussion concedes that it is unproven whether the general ODE solution exhausts all physically relevant solutions. That caveat should be elevated into the main claims (abstract and opening of Sec. III): the catalogue is the general solution of the constant-Schwarzian problem under the stated regularity assumptions, not a proven classification of all thermally excited subsets. Without that qualification, the phrasing “what are all the subsets” overstates what the ODE analysis delivers.
[Sec. IV; Sec. VII] Sec. IV and Sec. VII use Rindler (and Rindler-like) vacua as parent states. The paper notes these are only locally well-defined (Boulware-like). For load-bearing examples such as Rindler–Rindler and half-sided Rindler–Rindler, state explicitly which results are local flux statements versus claims about a global particle spectrum, so that the status of those “excited subsets of Rindler” is unambiguous.

minor comments (5)

[Sec. I–VI] Notation for null coordinates and maps is inconsistent across sections (U_M vs U_m, V_m, mixed subscripts). A single convention table early in Sec. I would help.
[Figures 1–14] Several figures (e.g. Figs. 1–14) lack clear axis labels and a brief caption statement of which null map is plotted; adding the explicit (U(u),V(v)) used would make the catalogue usable.
[Throughout] Typos and grammar: “GENERA TION”, “ST AR TING”, “V ACUUM”, “P AR TICLE”; “ad−bc” vs “AD−BC”; occasional missing articles. A careful copy-edit pass is needed.
[Sec. I.A; Eq. (6)] Eq. (6) writes the anomaly with a minus sign and c=−ℏ/2π, while the general CFT formula below uses +c/12 S. State the sign convention once and stick to it to avoid confusion when comparing to standard references.
[Appendix C] Appendix C cites the Polyanin–Zaitsev handbook for the nonlinear ODE; a one-line check that the f,g identifications (C4), (C12) match the handbook normal form would make the derivation self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: results are closed-form solutions of standard Schwarzian ODEs, not inputs renamed as predictions.

full rationale

The paper’s three questions are answered by writing the Virasoro/Schwarzian transformation law as nonlinear ODEs (constant Schwarzian, inverted purification ODE, and flux-preserving sibling ODE) and solving them by standard reductions (Riccati / handbook methods in Appendices A and C). The general solutions are Möbius transformations on exponential coordinates; special cases recover Rindler, Milne, half-sided, and Rindler–Rindler geometries already known from Bogoliubov calculations. Constant ⟨T_uu⟩ follows immediately from the anomaly formula once S is held constant—this is the definition of the input condition, not a fitted or self-cited “prediction.” Self-citations ([9], [12], [19]) supply related context and prior examples; they are not load-bearing uniqueness theorems that force the ODE solutions. No parameters are fit to data. Incomplete identification of constant flux with full Planck/KMS thermality for generic Möbius parameters is a correctness/scope issue, not circularity of the derivation chain. Score 0; steps empty.

Assumptions & free parameters 2 free parameters · 5 assumptions · 2 invented entities

The paper rests on standard 2D CFT anomaly structure and the identification of constant Schwarzian flux with Unruh-type thermality. No empirical free parameters. Invented notions are definitional (partial purification, sibling spacetimes) rather than new physical entities.

free parameters (2)

acceleration/flux scale a (or α) = free positive real; T=a/2π
Sets the constant Schwarzian value and Unruh temperature T=a/2π; chosen by hand for each map family, not fitted to data.
Möbius constants A,B,C,D (and p,q,r,s) = AD−BC≠0; examples fix canonical Rindler gauges
Integration constants labeling members of each solution family; gauge-fixed in examples but free in the general solution.

assumptions (5)

domain assumption 2D massless scalar stress tensor transforms with Virasoro anomaly: T'_uu = (∂U/∂u)² T_UU + (c/12) S(U,u) with c=−ℏ/2π for Unruh matching.
Stated in Sec. I; taken from standard CFT/Unruh literature (Fabbri–Navarro-Salas).
domain assumption Constant ⟨:T_uu:⟩ from a constant Schwarzian is identified with a thermal flux/density of particles in that sector.
Used throughout Secs. III–VI; fully justified via Bogoliubov only for selected exponential maps.
domain assumption Left- and right-moving sectors decouple for a 2D massless field, so maps U(u) and V(v) may be chosen independently.
Setup section; fails for massive fields (flagged in discussion).
standard math General solution of the third-order nonlinear Schwarzian ODE is the exponential Möbius family (Appendix A).
Classical ODE result; derived via Riccati reduction.
ad hoc to paper Local coordinate patches and local vacuum notions suffice even when global Rindler/Boulware-like vacua are ill-defined.
Discussion section explicitly restricts validity to a local sense when starting from Rindler vacuum.

invented entities (2)

partial purification
purpose: Name maps that set flux to zero in only one null sector while leaving the other thermal.
Definitional construction from applying the inverse Schwarzian solution to one sector; no new field or particle.
sibling spacetimes
purpose: Name distinct patches sharing parent vacuum and identical constant flux without constructing the parent.
Definitional class cut out by the flux-preserving Möbius-on-exponential solution in Sec. VI.

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Cite this review

Pith. "Pith review of Generation and purification of excited spacetimes using Schwarzian derivative." pith.science (2026). https://pith.science/paper/PQUUGSZR

@misc{pith2026260726693,
  author       = {Pith},
  title        = {Pith review of: Generation and purification of excited spacetimes using Schwarzian derivative},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PQUUGSZR}},
  note         = {Machine review of arXiv:2607.26693}
}

read the original abstract

In this article, we use the expression of the Schwarzian derivative to set up differential equations to find answers to three fundamental questions in the context of QFT in curved spacetime, specifically in two dimensions. One of the ways in which one can derive the Unruh effect in two dimensions is to use the anomalous transformation law of the energy-momentum tensor for a CFT that involves a Schwarzian derivative (Virasoro Anomaly). We answer the following three questions. The first question is as follows: If we have a spacetime with a massless scalar field in vacuum, what are all the subsets of spacetime such that the subset has a thermal distribution of particles for the left-moving and/or right-moving sectors? We obtain a general solution to this question by setting up and solving a third-order nonlinear differential equation based on the expression of Schwarzian. Based on the general solution, we can generate various subsets of the given spacetime that have a thermal flux/density of particles, of which the Rindler spacetime is one. The second question is an inverse question in which we suppose we are given a spacetime with a thermal distribution of particles; what are the possible purifying spacetimes (the ``parent'' spacetimes with the field in vacuum state whose reduced state in the given spacetime yields the observed particle content)? We similarly obtain a general class of solutions by setting up and solving a second differential equation. In this context, we also define ``partial purification'' where we obtain a spacetime that purifies only the left-moving or right-moving sector. The third question concerns locating spacetimes with the same particle content starting from the same ``parent'' spacetime. These sibling spacetimes are generated again by obtaining the general solution of a third differential equation based on the expression of Schwarzian.

Figures

Figures reproduced from arXiv: 2607.26693 by the authors.

**Figure 2.** FIG. 2. Half-sided left movers map: the spacetime has a [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 4.** FIG. 4. Left Rindler (purple) along with Right Rindler wedge [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

Figures from the paper (7 more)

**Figure 6.** Figure 6: FIG. 6. Past Milne universe [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: FIG. 7. Half-sided Rindler-Rindler spacetime: Right movers [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 9.** Figure 9: FIG. 9. Rindler-Rindler spacetime: Thermal density of left [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

**Figure 10.** Figure 10: FIG. 10. Left Rindler-Rindler Diamond : Has both left [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]

**Figure 12.** Figure 12: FIG. 12. Past Rindler-Milne universe [PITH_FULL_IMAGE:figures/full_fig_p010_12.png]

**Figure 13.** Figure 13: FIG. 13. Partial purification of the Diamond spacetime: The [PITH_FULL_IMAGE:figures/full_fig_p012_13.png]

**Figure 14.** Figure 14: FIG. 14. Causal diamond siblings: Each of the diamond has [PITH_FULL_IMAGE:figures/full_fig_p013_14.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

33 extracted references · 7 linked inside Pith

[1]

XM TM -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 FIG

Half-sided right movers map We choose the mapU m =−e −au1 /aandV M =v 1. XM TM -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 FIG. 1. Half-sided right movers map: The spacetime has a thermal flux of right-moving particles Given the map into new coordinates is (u 1, v1), the Metric in these new coordinates can be written as, ds2 =− − 1 a e−au1 (−a)du1 dv1 ....
[2]

Half-sided left movers map We choose the coordinate transformationV M = 1 a eav1 andU M =u 1, as illustrated in Fig. 2. In terms of the 5 XM TM -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 FIG. 2. Half-sided left movers map: the spacetime has a thermal flux of left-moving particles new coordinates (u 1, v1), the metric can be expressed in the light-cone ...
[3]

(35) is illustrated in Fig

Right Rindler spacetime We choose the coordinate transformation, UM =− 1 a e−au1 , V M = 1 a eav1 .(33) Hence, the metric becomes, ds2 =−e −au1 eav1 du1dv1,(34) which in (t 1, x1) coordinates takes the form, ds2 =−e 2ax1 (dt2 1 −dx 2 1).(35) The metric given in Eq. (35) is illustrated in Fig. 3. The corresponding Schwarzian derivatives are given by, S(UM ...
[4]

(41) is illustrated in Fig

Left Rindler spacetime We consider the conformal transformation, UM = 1 a eau1 , V M =− 1 a e−av1 .(39) Under this transformation, the metric becomes, ds2 =−e au1 e−av1 du1dv1,(40) which in (t 1, x1) coordinates takes the form, ds2 =−e −2ax1 (dt2 1 −dx 2 1).(41) The metric given in Eq. (41) is illustrated in Fig. 4, which corresponds to the left Rindler w...
[5]

XM TM -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 FIG

Future Milne universe We choose the mapsV m =e av1 /aandU M =e au1 /a. XM TM -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 FIG. 5. Future Milne universe Given a map into new coordinates(Future part of the Milne universe) in (u 1, v1), the Metric in these coordi- nates can be written in light-cone coordinates, ds2 =−e a(u1+v1)du1dv1.(45) Eq. (45) is writte...
[6]

XM TM -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 FIG

Past Milne universe We choose the mapV m =−e −av1 /aandU M = −e−au1 /a. XM TM -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 FIG. 6. Past Milne universe Given a map into new coordinates(Past part of the Milne universe) in (u 1, v1), the Metric in these coordi- nates can be written in light-cone coordinates, ds2 =−e −a(u1+v1)du1dv1,(54) Eq. (54) is written ...
[7]

XM TM -1 0 1 2 3 4 5 -4 -2 0 2 4 FIG

Half-sided Rindler-Rindler spacetime: left mover flux We choose the mapv 1 =e av2 /aandu 1 =u 2. XM TM -1 0 1 2 3 4 5 -4 -2 0 2 4 FIG. 8. Half-sided Rindler-Rindler spacetime: left movers in excited state The map in this section, with thev-sector, is expo- nentially changed, while theu-sector remains unchanged. The metric in the Rindler patch is calculate...
[8]

Right Rindler-Rindler spacetime XM TM -1 0 1 2 3 4 5 -4 -2 0 2 4 FIG. 9. Rindler-Rindler spacetime: Thermal density of left movers and right movers By considering the conformal transformation, u1 =− 1 a e−au2 , v 1 = 1 a eav2 .(74) Under this transformation, the Rindler metric, ds2 =−e a(v1−u1) du1dv1.(75) Takes the form, ds2 =−e a( 1 a eav2 + 1 a e−au2 )...

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Left Rindler-Rindler Diamond spacetime XM TM -1 0 1 2 3 -3 -2 -1 0 1 2 3 FIG. 10. Left Rindler-Rindler Diamond : Has both left movers and right movers with thermal spectrum We consider the conformal transformation, u1 = 1 a eau2 , v 1 =− 1 a e−av2 .(80) Under this transformati...
[10]

XM TM -1 0 1 2 3 4 5 -4 -2 0 2 4 FIG

Future Rindler-Milne universe We choose the mapv 1 =e av2 /aandu 1 =e au2 /a. XM TM -1 0 1 2 3 4 5 -4 -2 0 2 4 FIG. 11. Future Rindler-Milne universe The Metric for the given map in this section, in light- cone coordinates, ds2 =−e (eav2 −eau2 ) ea(u2+v2)du2dv2,(87) The above ...
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XM TM -1 0 1 2 3 4 5 -4 -2 0 2 4 FIG

Past Rindler-Milne universe We choose the mapv 1 =−e −av2 /aandu 1 = −e−au2 /a. XM TM -1 0 1 2 3 4 5 -4 -2 0 2 4 FIG. 12. Past Rindler-Milne universe The Metric for the given map in this section, in light- cone coordinates, ds2 =−e (e−av2 −e−au2 ) e−a(u2+v2)du2dv2,(92) The abo...
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Pith tools

Reviewed July 30, 2026 · model on record in the stance chip above.

[1] [1]

XM TM -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 FIG

Half-sided right movers map We choose the mapU m =−e −au1 /aandV M =v 1. XM TM -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 FIG. 1. Half-sided right movers map: The spacetime has a thermal flux of right-moving particles Given the map into new coordinates is (u 1, v1), the Metric in these new coordinates can be written as, ds2 =− − 1 a e−au1 (−a)du1 dv1 ....

[2] [2]

Half-sided left movers map We choose the coordinate transformationV M = 1 a eav1 andU M =u 1, as illustrated in Fig. 2. In terms of the 5 XM TM -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 FIG. 2. Half-sided left movers map: the spacetime has a thermal flux of left-moving particles new coordinates (u 1, v1), the metric can be expressed in the light-cone ...

[3] [3]

(35) is illustrated in Fig

Right Rindler spacetime We choose the coordinate transformation, UM =− 1 a e−au1 , V M = 1 a eav1 .(33) Hence, the metric becomes, ds2 =−e −au1 eav1 du1dv1,(34) which in (t 1, x1) coordinates takes the form, ds2 =−e 2ax1 (dt2 1 −dx 2 1).(35) The metric given in Eq. (35) is illustrated in Fig. 3. The corresponding Schwarzian derivatives are given by, S(UM ...

[4] [4]

(41) is illustrated in Fig

Left Rindler spacetime We consider the conformal transformation, UM = 1 a eau1 , V M =− 1 a e−av1 .(39) Under this transformation, the metric becomes, ds2 =−e au1 e−av1 du1dv1,(40) which in (t 1, x1) coordinates takes the form, ds2 =−e −2ax1 (dt2 1 −dx 2 1).(41) The metric given in Eq. (41) is illustrated in Fig. 4, which corresponds to the left Rindler w...

[5] [5]

XM TM -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 FIG

Future Milne universe We choose the mapsV m =e av1 /aandU M =e au1 /a. XM TM -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 FIG. 5. Future Milne universe Given a map into new coordinates(Future part of the Milne universe) in (u 1, v1), the Metric in these coordi- nates can be written in light-cone coordinates, ds2 =−e a(u1+v1)du1dv1.(45) Eq. (45) is writte...

[6] [6]

XM TM -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 FIG

Past Milne universe We choose the mapV m =−e −av1 /aandU M = −e−au1 /a. XM TM -30 -20 -10 0 10 20 30 -30 -20 -10 0 10 20 30 FIG. 6. Past Milne universe Given a map into new coordinates(Past part of the Milne universe) in (u 1, v1), the Metric in these coordi- nates can be written in light-cone coordinates, ds2 =−e −a(u1+v1)du1dv1,(54) Eq. (54) is written ...

[7] [7]

XM TM -1 0 1 2 3 4 5 -4 -2 0 2 4 FIG

Half-sided Rindler-Rindler spacetime: left mover flux We choose the mapv 1 =e av2 /aandu 1 =u 2. XM TM -1 0 1 2 3 4 5 -4 -2 0 2 4 FIG. 8. Half-sided Rindler-Rindler spacetime: left movers in excited state The map in this section, with thev-sector, is expo- nentially changed, while theu-sector remains unchanged. The metric in the Rindler patch is calculate...

[8] [8]

Right Rindler-Rindler spacetime XM TM -1 0 1 2 3 4 5 -4 -2 0 2 4 FIG. 9. Rindler-Rindler spacetime: Thermal density of left movers and right movers By considering the conformal transformation, u1 =− 1 a e−au2 , v 1 = 1 a eav2 .(74) Under this transformation, the Rindler metric, ds2 =−e a(v1−u1) du1dv1.(75) Takes the form, ds2 =−e a( 1 a eav2 + 1 a e−au2 )...