Causality and Boundary of wave solutions

Causality and Boundary of wave solutions IV International Meeting on Lorentzian Geometry Santiago de Compostela, 2007 José Luis Flores Universidad de Málaga Joint work with Miguel Sánchez: Class. Quant. Grav. 20 (2003); Preprint (2007) Causality and Boundary of wave solutions p.1/40

Wave type spacetimes Causality and Boundary of wave solutions p.2/40

Wave type spacetimes PP-WAVES: Causality and Boundary of wave solutions p.3/40

Ä Ü ¾ ½ Ü¾ ¾ ¾ Ù Ú À Ü Ùµ Ù¾ Wave type spacetimes PP-WAVES: Î Ê ¾ Ê ¾ Causality and Boundary of wave solutions p.3/40

Wave type spacetimes PP-WAVES: Î Ê ¾ Ê ¾ Ä Ü ¾ ½ Ü¾ ¾ ¾ Ù Ú À Ü Ùµ Ù¾ Ê ¾ Ê Ê arbitrary function ( ¼). À Causality and Boundary of wave solutions p.3/40

¾ Ù Ú À Ü Ùµ Ù¾ Wave type spacetimes PP-WAVES: Î Ê ¾ Ê ¾ Ä Ü ¾ ½ Ü¾ ¾ À Ê ¾ Ê Ê arbitrary function ( ¼). Simplest models for exact gravitational waves. Causality and Boundary of wave solutions p.4/40

¾ Ù Ú À Ü Ùµ Ù¾ Wave type spacetimes PP-WAVES: Î Ê ¾ Ê ¾ Ä Ü ¾ ½ Ü¾ ¾ À Ê ¾ Ê Ê arbitrary function ( ¼). Simplest models for exact gravitational waves. Interest: Relativity; Causality and Boundary of wave solutions p.4/40

¾ Ù Ú À Ü Ùµ Ù¾ Wave type spacetimes PP-WAVES: Î Ê ¾ Ê ¾ Ä Ü ¾ ½ Ü¾ ¾ À Ê ¾ Ê Ê arbitrary function ( ¼). Simplest models for exact gravitational waves. All the scalar curvature invariants are zero. Interest: Relativity; Causality and Boundary of wave solutions p.4/40

PLANE WAVES: Causality and Boundary of wave solutions p.5/40

À Ü Ùµ Ü ½ Ü ¾ µ ¼ ½ Ùµ ½ ¼ Ü ½ Ü ¾ ½ PLANE WAVES: pp-waves with the additional restriction Ùµ Ùµ ¾ Ùµ Causality and Boundary of wave solutions p.5/40

À Ü Ùµ Ü ½ Ü ¾ µ ¼ ½ Ùµ ½ ¼ Ü ½ Ü ¾ ½ PLANE WAVES: pp-waves with the additional restriction Ùµ Ùµ ¾ Ùµ High degree of symmetry for the wave. Causality and Boundary of wave solutions p.5/40

À Ü Ùµ Ü ½ Ü ¾ µ ¼ ½ Ùµ ½ ¼ Ü ½ Ü ¾ ½ PLANE WAVES: pp-waves with the additional restriction Ùµ Ùµ ¾ Ùµ High degree of symmetry for the wave. HOMOGENEOUS PLANE WAVES: Causality and Boundary of wave solutions p.5/40

À Ü Ùµ Ü ½ Ü ¾ µ ¼ ½ Ùµ ½ ¼ Ü ½ Ü ¾ ½ PLANE WAVES: pp-waves with the additional restriction Ùµ Ùµ ¾ Ùµ High degree of symmetry for the wave. HOMOGENEOUS PLANE WAVES: plane waves with ¼ µ Ùµ Ù constantsµ Causality and Boundary of wave solutions p.5/40

À Ü Ùµ Ü ½ Ü ¾ µ ¼ ½ Ùµ ½ ¼ Ü ½ Ü ¾ ½ PLANE WAVES: pp-waves with the additional restriction Ùµ Ùµ ¾ Ùµ High degree of symmetry for the wave. HOMOGENEOUS PLANE WAVES: plane waves with ¼ µ Ùµ Ù Conformally flat iff Æ. constantsµ Causality and Boundary of wave solutions p.5/40

Causal hierarchy Causality and Boundary of wave solutions p.6/40

Causal ladder Causality: no closed causal curves. Causality and Boundary of wave solutions p.7/40

Causal ladder Distinguishing: no points with same past and future. Causality: no closed causal curves. Causality and Boundary of wave solutions p.7/40

Causal ladder Strong Causality: no almost closed causal curves. Distinguishing: no points with same past and future. Causality: no closed causal curves. Causality and Boundary of wave solutions p.7/40

Causal ladder Causal Continuity: continuity of past and future. Strong Causality: no almost closed causal curves. Distinguishing: no points with same past and future. Causality: no closed causal curves. Causality and Boundary of wave solutions p.7/40

Causal ladder Causal Simplicity: Past and future are ( closed causal). Causal Continuity: continuity of past and future. Strong Causality: no almost closed causal curves. Distinguishing: no points with same past and future. Causality: no closed causal curves. Causality and Boundary of wave solutions p.7/40

Causal ladder Global Hyperbolicity: admits a Cauchy surface. Causal Simplicity: Past and future are ( closed causal). Causal Continuity: continuity of past and future. Strong Causality: no almost closed causal curves. Distinguishing: no points with same past and future. Causality: no closed causal curves. Causality and Boundary of wave solutions p.7/40

Previous results: pp-waves Causality and Boundary of wave solutions p.8/40

Previous results: pp-waves Ù is a quasi-time function: Causality and Boundary of wave solutions p.8/40

Previous results: pp-waves is a quasi-time function: i.e., Ù ÖÙ 1. is causal and -directed (in ÖÙ fact: Ú ) 2. if is lightlike geodesic with Ù Æ constant µ injective. Causality and Boundary of wave solutions p.8/40

Previous results: pp-waves is a quasi-time function: i.e., Ù ÖÙ 1. is causal and -directed (in ÖÙ fact: Ú ) 2. if is lightlike geodesic with Ù Æ constant µ injective. µ pp-wave causal. Globally hyperbolic Causally simple Causally continuous Strongly causal Distinguishing Causal sometimes sometimes sometimes sometimes sometimes always Causality and Boundary of wave solutions p.8/40

Previous results: plane waves Causality and Boundary of wave solutions p.9/40

Previous results: plane waves Any plane wave is strongly causal. Strongly Causal always Causality and Boundary of wave solutions p.9/40

Previous results: plane waves Any plane wave is strongly causal. Strongly Causal Distinguishing Causal always always always Causality and Boundary of wave solutions p.9/40

Previous results: plane waves Any plane wave is strongly causal. Plane waves are not globally hyperbolic Globally hyperbolic not [Pe 65] Strongly Causal Distinguishing Causal always always always Causality and Boundary of wave solutions p.9/40

Previous results: plane waves Any plane wave is strongly causal. Plane waves are not globally hyperbolic (nor causally simple). Globally hyperbolic Causally Symple not [Pe 65] not [EhEm 92] Strongly Causal Distinguishing Causal always always always Causality and Boundary of wave solutions p.9/40

Previous results: plane waves Any plane wave is strongly causal. Plane waves are not globally hyperbolic (nor causally simple). Globally hyperbolic Causally Symple Causally Continuous Strongly Causal Distinguishing Causal not [Pe 65] not [EhEm 92] yes, if gravitational [EhEm 92] always always always Causality and Boundary of wave solutions p.9/40

Discussion Bad causality: plane waves are not globally hyperbolic Causality and Boundary of wave solutions p.10/40

Discussion Bad causality: plane waves are not globally hyperbolic But they are exact solutions to Einstein equations Causality and Boundary of wave solutions p.10/40

Discussion Bad causality: plane waves are not globally hyperbolic But they are exact solutions to Einstein equations SCC: generic solutions to Einstein equations with reasonable matter and behavior ½ at must be globally hyperbolic. The following question by Penrose becomes relevant: Is the bad causality of plane waves generic in some sense? Causality and Boundary of wave solutions p.10/40

Focalization of null geodesics in plane waves Causality and Boundary of wave solutions p.11/40

Reasonably generic waves Causality and Boundary of wave solutions p.12/40

Ä ¾ Ù Ú À Ü Ùµ Ù ¾ Åp-waves Î Å Ê ¾ Causality and Boundary of wave solutions p.13/40

Åp-waves Å Ê ¾ Ä ¾ Ù Ú À Ü Ùµ Ù ¾ Î µ arbitrary Riemannian manifold. Å Causality and Boundary of wave solutions p.14/40

Åp-waves Å Ê ¾ Ä ¾ Ù Ú À Ü Ùµ Ù ¾ Î µ arbitrary Riemannian manifold. Å Arbitrary dimension: applications to strings Causality and Boundary of wave solutions p.14/40

Åp-waves Å Ê ¾ Ä ¾ Ù Ú À Ü Ùµ Ù ¾ Î µ arbitrary Riemannian manifold. Å Arbitrary dimension: applications to strings Arbitrary topology: for discussion on horizons Causality and Boundary of wave solutions p.14/40

Åp-waves Å Ê ¾ Ä ¾ Ù Ú À Ü Ùµ Ù ¾ Î µ arbitrary Riemannian manifold. Å Arbitrary dimension: applications to strings Arbitrary topology: for discussion on horizons Arbitrary metric: for generic results, independent of the special symmetries of the metric. Causality and Boundary of wave solutions p.14/40

Â Ýµ Ù Our functional Fixed Ü ¼ Ü ½ ¾ Å, Ù ¼ ¾ Ê, Ù ¼ and µ ¼, put Ý Ý À Ý Ù ¼ µ µµ ¼ with domain the curves Ý ¼ Ù Å joining Ü ¼, Ü ½. Causality and Boundary of wave solutions p.15/40

Â Ýµ Ù Our functional Fixed Ü ¼ Ü ½ ¾ Å, Ù ¼ ¾ Ê, Ù ¼ and µ ¼, put Ý Ý À Ý Ù ¼ µ µµ ¼ with domain the curves Ý ¼ Ù Å joining Ü ¼, Ü ½. Relates Ù, Ú along a causal curve Ý Ù Úµ of energy with leng Ýµ. Causality and Boundary of wave solutions p.15/40

Â Ýµ Ù Our functional Fixed Ü ¼ Ü ½ ¾ Å, Ù ¼ ¾ Ê, Ù ¼ and µ ¼, put Ý Ý À Ý Ù ¼ µ µµ ¼ with domain the curves Ý ¼ Ù Å joining Ü ¼, Ü ½. Relates Ú Ù, along a causal Ý Ù Úµ curve of energy with leng Ýµ. contains deep information about the behavior of causal curves. Â Causality and Boundary of wave solutions p.15/40

Â Ýµ Ù Our functional Fixed Ü ¼ Ü ½ ¾ Å, Ù ¼ ¾ Ê, Ù ¼ and µ ¼, put Ý Ý À Ý Ù ¼ µ µµ ¼ with domain the curves Ý ¼ Ù Å joining Ü ¼, Ü ½. Relates Ù, Ú along a causal curve Ý Ù Úµ of energy with leng Ýµ. Â contains deep information about the behavior of causal curves. The qualitative behavior of Â can be analyzed in terms of À. Causality and Boundary of wave solutions p.15/40

Relevant growths for À Causality and Boundary of wave solutions p.16/40

Ü Ò Ùµ Ê ½ ¾ Ü Ò Üµ Ê ¼ Ê ½ ¼ Relevant growths for À superquadratic if Ü Ò Ò s.t. Ü Ò Üµ ½ and Causality and Boundary of wave solutions p.16/40

Ü Ùµ Ê ½ Ùµ ¾ Ü Üµ Ê ¼ Ùµ Ü Ùµ ¾ Å Ê Relevant growths for À superquadratic if Ü Ò Ò s.t. Ü Ò Üµ ½ and Ü Ò Ùµ Ê ½ ¾ Ü Ò Üµ Ê ¼ Ê ½ ¼ at most quadratic if Ê ½ Ùµ Ê ¼ Ùµ ¼ s.t. Causality and Boundary of wave solutions p.17/40

Relevant growths for À superquadratic if Ü Ò Ò s.t. Ü Ò Üµ ½ and Ü Ò Ùµ Ê ½ ¾ Ü Ò Üµ Ê ¼ Ê ½ ¼ at most quadratic if Ê ½ Ùµ Ê ¼ Ùµ ¼ s.t. Ü Ùµ Ê ½ Ùµ ¾ Ü Üµ Ê ¼ Ùµ Ü Ùµ ¾ Å Ê subquadratic if Ê ½ Ùµ Ê ¼ Ùµ ¼, s.t. Ü Ùµ Ê ½ Ùµ Ô Ùµ Ü Üµ Ê ¼ Ùµ Ô Ùµ ¾ Causality and Boundary of wave solutions p.18/40

Causality for Åp-waves Theorem [-Sanchez 03]: Causality and Boundary of wave solutions p.19/40

Causality for Åp-waves Theorem [-Sanchez 03]: (i) Åp-waves are strongly causal if À is at most quadratic (i.e., plane waves). Causality and Boundary of wave solutions p.19/40

Causality for Åp-waves Theorem [-Sanchez 03]: (i) Åp-waves are strongly causal À if is at most quadratic (i.e., plane waves). (ii) Åp-waves are globally hyperbolic À if is subquadratic Å (and complete). Causality and Boundary of wave solutions p.19/40

Causality for Åp-waves Theorem [-Sanchez 03]: (i) Åp-waves are strongly causal if À is at most quadratic (i.e., plane waves). (ii) Åp-waves are globally hyperbolic if À is subquadratic (and Å complete). (iii) Åp-waves are non-distinguishing if À is superquadratic. Very accurate result: examples of non-distinguishing Åp-waves arbitrarily close to quadratic. Causality and Boundary of wave solutions p.19/40

Conclusions Causality becomes unstable for À quadratic: Global hyperbolic Causally Symple Causally continuous Strongly causal Distinguishing Causal at most quadratic Causality and Boundary of wave solutions p.20/40

Conclusions Causality becomes unstable for À quadratic: Deviations in the superquadratic direction may destroy strong causality. Global hyperbolic Causally Symple Causally continuous Strongly causal at most quadratic Distinguishing Causal superquadratic Causality and Boundary of wave solutions p.20/40

Conclusions Causality becomes unstable À for quadratic: Deviations in the superquadratic direction may destroy strong causality. Deviations in the subquadratic direction yield global hyperbolicity. Global hyperbolic subquadratic Causally Symple Causally continuous Strongly causal at most quadratic Distinguishing Causal superquadratic Causality and Boundary of wave solutions p.20/40

Boundary of wave solutions Causality and Boundary of wave solutions p.21/40

Conformal Boundary Causality and Boundary of wave solutions p.22/40

Conformal Boundary Conformally embeds the original spacetime into a larger one, and takes the boundary of the image. Causality and Boundary of wave solutions p.22/40

Conformal Boundary Conformally embeds the original spacetime into a larger one, and takes the boundary of the image. Minkowski into Einstein Static Universe Causality and Boundary of wave solutions p.22/40

Conformal Boundary Conformally embeds the original spacetime into a larger one, and takes the boundary of the image. Is this procedure easily realizable for wave type spacetimes? Only if they are conformally flat! Causality and Boundary of wave solutions p.23/40

Case ¼ In [BN 02] they found this explicit conformal embedding into ESU: Causality and Boundary of wave solutions p.24/40

Case ¼ In [BN 02] they found this explicit conformal embedding into ESU: So, the conformal boundary is a ½-dimensional line which twists around the compact dimensions of ESU. Causality and Boundary of wave solutions p.24/40

Case ¼ In this case the conformal boundary is formed by two parallel null planes connected by two parallel null lines [MR 03]. Causality and Boundary of wave solutions p.25/40

Causal Boundary Applicable to any strongly causal spacetime: Causality and Boundary of wave solutions p.26/40

Causal Boundary Applicable to any strongly causal spacetime: Attach an ideal point for every inextensible timelike curve. Causality and Boundary of wave solutions p.26/40

Causal Boundary Applicable to any strongly causal spacetime: Attach an ideal point for every inextensible timelike curve. Timelike curves with the same past or future are attached the same ideal point. Identify some ideal points conveniently. Causality and Boundary of wave solutions p.28/40

Previous results Causality and Boundary of wave solutions p.30/40

Previous results Theorem [MR 03]: The boundary for homogeneous plane waves with some eigenvalue positive is formed by a ½-dimensional null line plus two points,. Causality and Boundary of wave solutions p.30/40

Previous results Theorem [MR 03]: The boundary for homogeneous plane waves with some eigenvalue positive is formed by a ½-dimensional null line plus two points,. Theorem [HRa 03]: The dimensionality of the boundary for plane waves depends on the behavior of Ùµ: Ùµ polynomial trigonometric hyperbolic... µ dimension ½ µ dimension ½ Ùµ ¼ fast enough Causality and Boundary of wave solutions p.30/40

On the technique Arguments based on the oscillatory regime of geodesics. Causality and Boundary of wave solutions p.31/40

On the technique Arguments based on the oscillatory regime of geodesics. Needs solving, or analyzing in detail, solutions to differential equations involving À. Causality and Boundary of wave solutions p.31/40

On the technique Arguments based on the oscillatory regime of geodesics. Needs solving, or analyzing in detail, solutions to differential equations involving À. Strategy only realizable for very special functions À. Causality and Boundary of wave solutions p.31/40

Our alternative approach Causality and Boundary of wave solutions p.32/40

Our alternative approach The causal boundary construction does not require deep knowledge about geodesics. Causality and Boundary of wave solutions p.32/40

Our alternative approach The causal boundary construction does not require deep knowledge about geodesics. On the contrary, only the analysis of the rough behavior of causal curves is required. We only need to study the qualitative behavior of Â functional. Causality and Boundary of wave solutions p.32/40

More relevant growths Causality and Boundary of wave solutions p.33/40

¾ ¾ Ü Üµ Ê ¼ Ù ¾ ½ ¾ Ê and Ê ½ Ùµ Ê ¼ Ùµ ¼ Ü Ùµ Ê ½ Ùµ ¾ Ü Üµ Ê ¼ Ùµ More relevant growths asymptotically quadratic if, Ê ¼ s.t. for Ü Ùµ ¾ Å Ê all Causality and Boundary of wave solutions p.33/40

¾ ¾ Ü Üµ Ê ¼ ¾ Ê and Ê ½ Ùµ Ê ¼ Ùµ ¼ Ü Ùµ Ê ½ Ùµ ¾ Ü Üµ Ê ¼ Ùµ Further relevant growths asymptotically quadratic if, Ê ¼ s.t. for Ü Ùµ ¾ Å Ê all Ù ¾ ½ strongly subquadratic if Ê ½ Ê ¼ ¼ and Æ ¼ s.t. Ü Ùµ Ê ½ ¾ Ü Üµ Ê ¼ Æ ½ Ù Ü Ùµ ¾ Å Ê Causality and Boundary of wave solutions p.34/40

Our results Causality and Boundary of wave solutions p.35/40

Small causal boundary Teorema ( Sánchez): If Î is a Åp-wave with Å complete and Causality and Boundary of wave solutions p.36/40

Small causal boundary Teorema ( Sánchez): If Î is a Åp-wave with Å complete and (i) À asymptotically quadratic, (ii) plane wave with ½½ Ùµ ¾ Ù ¾ ½µ, (iii) homogeneous plane wave with some eigenvalue positive, then the causal boundary is formed by with eventual identifications between them. plus two locally null lines Causality and Boundary of wave solutions p.36/40

Small causal boundary Teorema ( Sánchez): If Î is a Åp-wave with Å complete and (i) À asymptotically quadratic, (ii) plane wave with ½½ Ùµ ¾ Ù ¾ ½µ, (iii) homogeneous plane wave with some eigenvalue positive, then the causal boundary is formed by with eventual identifications between them. Remarks: Reproduces previous results in [MR], [HRa]. plus two locally null lines Causality and Boundary of wave solutions p.36/40

Small causal boundary Teorema ( Sánchez): If Î is a Åp-wave with Å complete and (i) À asymptotically quadratic, (ii) plane wave with ½½ Ùµ ¾ Ù ¾ ½µ, (iii) homogeneous plane wave with some eigenvalue positive, then the causal boundary is formed by with eventual identifications between them. Remarks: Reproduces previous results in [MR], [HRa]. plus two locally null lines Yields information about the boundary for Åp-waves. Causality and Boundary of wave solutions p.36/40

Big causal boundary Teorema ( Sánchez): If Å is complete and À strongly subquadratic then the causal boundary coincides with that of the standard static spacetime Î Å Ê ¾ ËØ Ø ¾ Ù Ú Causality and Boundary of wave solutions p.37/40

Big causal boundary Teorema ( Sánchez): If Å is complete and À strongly subquadratic then the causal boundary coincides with that of the standard static spacetime Î Å Ê ¾ ËØ Ø ¾ Ù Ú The causal boundary of standard static spacetimes has been studied in [H 01], [H 05]: Teorema (Harris): The causal boundary of a standard static spacetime conformal to Å Ê is formed by a double cone with apexes and base the Busemann boundary of Å. Causality and Boundary of wave solutions p.37/40

Å Å Ê Big causal boundary Teorema ( Sánchez): If Å is complete and À strongly subquadratic then the causal boundary coincides with that of the standard static spacetime Î Å Ê ¾ ËØ Ø ¾ Ù Ú i.e. it is formed by a double cone with apexes, Busemann boundary of and base the ¼ Þ ¾ Corollary: The causal boundary of any pp-wave with À strongly subquadratic coincides with that of Minkowski, i.e. it is formed by a double cone with apexes, and base Ë ¾. Causality and Boundary of wave solutions p.38/40

Remaining cases New facts take into play: i.e. Causality and Boundary of wave solutions p.39/40

Remaining cases New facts take into play: i.e. Î may be geodesically incomplete even if Å is complete. Causality and Boundary of wave solutions p.39/40

Remaining cases New facts take into play: i.e. Î may be geodesically incomplete even if Å is complete. This fairly complicates the structure of the boundary, even though this situation is compatible with global hyperbolicity. Generic results are difficult here, but our technique can be exploited to describe the boundary for particular examples in these cases. Causality and Boundary of wave solutions p.39/40

Final Conclusions Important facts: Causality and Boundary of wave solutions p.40/40

Final Conclusions Important facts: Causality is unstable for À quadratic. Causality and Boundary of wave solutions p.40/40

Final Conclusions Important facts: Causality is unstable for À quadratic. Boundary unstable for À quadratic. Causality and Boundary of wave solutions p.40/40

Final Conclusions Important facts: Causality is unstable for À quadratic. Boundary unstable for À quadratic. Boundary ½-dimensional for À quadratic. Causality and Boundary of wave solutions p.40/40

Final Conclusions Important facts: Causality is unstable for À quadratic. Boundary unstable for À quadratic. Boundary ½-dimensional for À quadratic. It seems to be a relation between these facts. If so, this means that very deep causal information can be obtain from the spacetime just by looking at the boundary! Causality and Boundary of wave solutions p.40/40