DOCSLIB.ORG

General Relativity Phys4501/6 Second Term Summary, 2013 Gabor

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
General Relativity Phys4501/6 Second Term Summary, 2013 Gabor General Relativity Phys4501/6 Second term summary, 2013 Gabor Kunstatter, University of Winnipeg Contents I. Einstein's Equations 3 A. Heuristic Derivation 3 B. Lagrangian Formulation of General Relativity 5 C. Variational Derivation of Einstein's Equations 6 D. Properties of Einstein's Equations 7 II. Energy Conditions 7 III. Spherical Symmetry 9 A. Derivation of solution and general properties 9 B. Geodesics 9 C. Experimental Tests 9 D. Black Holes and Conformal Diagrams 10 E. Charged Black Holes 10 IV. Horizons and Trapped Surfaces 12 A. Stationary vs. Static 12 B. Event Horizons in General 13 C. Trapped Surfaces 14 D. Killing Horizons 15 E. Surface Gravity 16 V. Theorems and Conjectures 17 A. Singularity Theorem 17 B. Cosmic Censorship Hypothesis 18 C. Hawking's Area Theorem 18 D. No Hair Theorem 18 1 VI. Black Hole Thermodynamics 18 VII. Cosmology: Theory 20 A. Symmetry Considerations 20 B. FRW Metric 21 C. Hubble Parameter 22 D. Matter Content 22 E. The Friedmann Equation 23 F. Time evolution of the scale factor and density 25 VIII. Cosmology: Observations 26 A. Redshifts 26 B. Proper Distance 28 C. Luminosity Distance 29 D. Derivation of Luminosity vs Redshift Relation 29 2 I. EINSTEIN'S EQUATIONS A. Heuristic Derivation • So far we have learned how to derive equations of motion for matter fields from a variational principle keeping the geometry fixed (i.e. didn't vary metric). • We now need to understand better the equations of motion for the metric of spacetime, i.e. the geometry. • We know that the energy momentum tensor must be conserved, and that in the New- tonian limit h00 plays the role of the gravitational potential φ which in Newtonian physics is determined by the matter distribution by Poisson equations: r~ 2φ = 4πGρ (1) The equations of motion must therefore be second order in the metric components, tensorial, and guarantee energy-momentum conservation. • Eq. (138) of the February summary gives us a hint, since we can get the vacuum ~ 2 k Poisson equation r φ = 0 by taking the trace R 00k = 0. This is the Newtonian, non- relativistic limit of R00 = 0, so one would be tempted to postulate, as did Einstein, the fully covariant version of the above in the presence of matter: Rµν = κTµν (2) But we know this can't be right, because the energy momentum tensor must be covari- µν 1 µν antly conserved, whereas the LHS is not: rµR = 2 g rµR by the Bianchi identities and does not vanish in general. In fact, (2) implies R = κT so that by using the µν Bianchi identities one sees that rµT = 0 iff rµT = 0, which is way too restrictive. • A better guess for Einstein's equations, which automatically yields energy-momentum conservation, is: 1 G = R − g R = κT (3) µν µν 2 µν µν • In class we used dimensional arguments to show that κ must be proportional to G=c4: 3 2 T00 has dimensions of [ρ] = E=L , whereas G00 has dimensions 1=L . Thus [κ] = L=E. 3 Newton's constant has units [G] = L2=mass2 ∗ (mass ∗ L=T 2) = L3=(Mass ∗ T 2). Since [E] = [mass][c2], this shows that [G=c4] = L3=(Mc2 ∗ T 2) ∗ (T 2=L2) = L=E as required. • We can check more carefully the Newtonian limit and thereby determine whether it works and if so what the constant κ must be: { First, using Eq.(138) in the previous section we see that in the Newtonian limit: 1 1 R ∼ − r~ 2h = r~ 2φ (4) 00 2 00 c2 { Tracing (3) we find 00 R = −κT ∼ −κg T00 = κT00 (5) where we have assumed that in the non-relativistic limit the T00 = ρ is larger than the other components of the stress energy tensor. { In this case Einstein's equation implies that: 1 1 R = κT (1 + g ) = κT 00 00 2 00 2 00 1 1 ! r~ 2φ = κc2ρ (6) c2 2 2 where we have used T00 = ρc , recalling that the energy-momentum tensor is the flow of four-momentum through a surface, i.e. in this case energy, not mass. { which matches Eq.(124) of February summary providing we choose: 8πG κ = (7) c4 • Once we impose general covariance, don't allow higher derivatives of the metric in the equations of motion and require correspondence with Newtonian physics the theory is uniquely determined with no new, tunable parameters!!! As we will see it therefore makes unambiguous testable predictions for corrections to Newtonian physics, such as the perihelion shift of Mercury and light bending around the sun. • But first we will see how to derive the field equations from a variational principle. Given a coordinate invariant action, the energy momentum tensor can be derived (doesn't have to be guessed), and the connection between the energy momentum conservation and symmetry (general covariance) becomes manifest. 4 B. Lagrangian Formulation of General Relativity We now show how to derive Einstein's equations from a variational principle. As always one starts with an action. In this case we require the action to have certain properties: 1. It must be invariant under coordinate transformations 2. It must, at this stage be at most second order in derivatives of the metric. Actually the condition is that the equations of motion resulting from varying this action have no more than second derivatives of the metric.In four spacetime dimensions these two statements are equivalent, but not in higher dimensions. Note also that higher order derivatives are possible in generalized theories of gravity, but we assume that they will not play a significant role until we get to much shorter length scales than we wish to consider at this stage. 3. It should separate into two parts: (a) a matter action, which describes the dynamics of any non-gravitational fields. This part is distinguished by the fact that it cannot have more than first deriva- tives of the metric. Otherwise the equations of motion for the matter field could not be made to be indistinguishable from those of general relativity by going to a locally inertial frame (in which the first derivatives of the metric vanish. This would violate the strong equivalence principle. In general the matter action is constructed by taking the special relativity version and replacing all occurences of the Minkowski metric by gµν and all partial derivatives by covariant derivatives. (b) the gravitational action: assuming the previous conditions this is almost uniquely determined (in four dimensions) proportional the so-called Einstein-Hilbert ac- tion: Z 4 p SEH = d x −gR(g) (8) where R(g) is the Ricci scalar constructed from the metric gµν Thus we start from the total action describing both geometry and matter: 1 I = S + S (9) 2κ EH M 5 where SM denotes a generic matter action constructed as per the rules given above. For example, for a scalar field, the contribution to SM is: Z p 1 1 S = d4x −g − r φgµνr φ − m2φ2 − V (φ) (10) φ 2 µ ν 2 Note that for a scalar field the partial and covariant derivatives are identical. V (φ) is a potential energy term associated with self-interactions of the field. The signs are chosen to make sure that the kinetic energy term is positive, in keeping with the mechanics form of the lagrangian: L = KE − PE. The other matter action we have dealt with, that of the electromagnetic field, is: 1 Z p S = − d4x −gF gµνgαβF (11) EM 4 µα νβ where Fµν is the electromagnetic field strength defined in terms of an electromagnetic potential Aµ by: Fµν = @µAν − @νAµ = rµAν − rνAµ (12) Note that Fµν is by definition anti-symmetric (a two-form) and thus the Christoffel terms on the far right cancel so that it can be equivalently defined either by covariant or partial derivatives. C. Variational Derivation of Einstein's Equations • Definition of stress energy tensor: 2 δS T ≡ −p M (13) µν −g δgµν • Action: c4 Z p S = d4x −gR(g) + S (14) 16πG M • Einstein's Equations: 8πG G = T (15) µν c4 µν • Know how to vary metric in action, for which the following relations are useful: µα µα { g δgµβ = −gµβδg p 1 p µν 1 p µν { δ −g = 2 −gg δgµν = − 2 −ggµνδg 6 α 1 αβ { δfµ νg = 2 g (rµδgβν + rνδgβµ − rβδgµν) R 4 p µ R p µ R p µ { Σ d x jgjrµV = Σ @µ jgjV = @Σ nµ jγjV D. Properties of Einstein's Equations • Degrees of Freedom: qualitative derivation of n(n + 1)=2 − 2n = n(n − 3)=2 • Self-coupled: gravitational binding energy contributions to inertial mass • Attractive for \ordinary matter" • Quantum Gravity: know constants that can be constructed from ~; G; C: r ~G c lpl = 3 = (16) c tpl ~c 2 Epl = = mplc (17) lpl 2 • Cosmological Constant Λ = 1=lΛ: know the problem, its theoretical quantum source and theoretical and experimental orders of magnitudes. II. ENERGY CONDITIONS These are physically motivated constraints on the general form of the stress energy tensor that are useful for proving various theorems and/or motivating conjectures. They are all a bit different, but the basic idea is to define what one considers \reasonable" matter/energy to look like in general relativity. Any or all of these can be violated once quantum mechanics is taken into account. • Strong Energy Condition (SEC): 1 T tµtν ≥ T λ tσt (18) µν 2 λ σ for all timelike tµ. This implies that dθ/dτ ≤ 0 for all timelike geodesics, where θ is their expansion. Thus it implies that strong energy condition restricts to mat- ter/energy for which gravity is purely attractive.
Load more

Recommended publications
  • The Penrose and Hawking Singularity Theorems Revisited
    Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook The Penrose and Hawking Singularity Theorems revisited Roland Steinbauer Faculty of Mathematics, University of Vienna Prague, October 2016 The Penrose and Hawking Singularity Theorems revisited 1 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Overview Long-term project on Lorentzian geometry and general relativity with metrics of low regularity jointly with `theoretical branch' (Vienna & U.K.): Melanie Graf, James Grant, G¨untherH¨ormann,Mike Kunzinger, Clemens S¨amann,James Vickers `exact solutions branch' (Vienna & Prague): Jiˇr´ıPodolsk´y,Clemens S¨amann,Robert Svarcˇ The Penrose and Hawking Singularity Theorems revisited 2 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Contents 1 The classical singularity theorems 2 Interlude: Low regularity in GR 3 The low regularity singularity theorems 4 Key issues of the proofs 5 Outlook The Penrose and Hawking Singularity Theorems revisited 3 / 27 Classical singularity thms. Interlude: Low regularity in GR Low regularity singularity thms. Proofs Outlook Table of Contents 1 The classical singularity theorems 2 Interlude: Low regularity in GR 3 The low regularity singularity theorems 4 Key issues of the proofs 5 Outlook The Penrose and Hawking Singularity Theorems revisited 4 / 27 Theorem (Pattern singularity theorem [Senovilla, 98]) In a spacetime the following are incompatible (i) Energy condition (iii) Initial or boundary condition (ii) Causality condition (iv) Causal geodesic completeness (iii) initial condition ; causal geodesics start focussing (i) energy condition ; focussing goes on ; focal point (ii) causality condition ; no focal points way out: one causal geodesic has to be incomplete, i.e., : (iv) Classical singularity thms.
    [Show full text]
  • Classical and Semi-Classical Energy Conditions Arxiv:1702.05915V3 [Gr-Qc] 29 Mar 2017
    Classical and semi-classical energy conditions1 Prado Martin-Morunoa and Matt Visserb aDepartamento de F´ısica Te´orica I, Ciudad Universitaria, Universidad Complutense de Madrid, E-28040 Madrid, Spain bSchool of Mathematics and Statistics, Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand E-mail: [email protected]; [email protected] Abstract: The standard energy conditions of classical general relativity are (mostly) linear in the stress-energy tensor, and have clear physical interpretations in terms of geodesic focussing, but suffer the significant drawback that they are often violated by semi-classical quantum effects. In contrast, it is possible to develop non-standard energy conditions that are intrinsically non-linear in the stress-energy tensor, and which exhibit much better well-controlled behaviour when semi-classical quantum effects are introduced, at the cost of a less direct applicability to geodesic focussing. In this chapter we will first review the standard energy conditions and their various limitations. (Including the connection to the Hawking{Ellis type I, II, III, and IV classification of stress-energy tensors). We shall then turn to the averaged, nonlinear, and semi-classical energy conditions, and see how much can be done once semi-classical quantum effects are included. arXiv:1702.05915v3 [gr-qc] 29 Mar 2017 1Draft chapter, on which the related chapter of the book Wormholes, Warp Drives and Energy Conditions (to be published by Springer) will be based. Contents 1 Introduction1 2 Standard energy conditions2 2.1 The Hawking{Ellis type I | type IV classification4 2.2 Alternative canonical form8 2.3 Limitations of the standard energy conditions9 3 Averaged energy conditions 11 4 Nonlinear energy conditions 12 5 Semi-classical energy conditions 14 6 Discussion 16 1 Introduction The energy conditions, be they classical, semiclassical, or \fully quantum" are at heart purely phenomenological approaches to deciding what form of stress-energy is to be considered \physically reasonable".
    [Show full text]
  • Light Rays, Singularities, and All That
    Light Rays, Singularities, and All That Edward Witten School of Natural Sciences, Institute for Advanced Study Einstein Drive, Princeton, NJ 08540 USA Abstract This article is an introduction to causal properties of General Relativity. Topics include the Raychaudhuri equation, singularity theorems of Penrose and Hawking, the black hole area theorem, topological censorship, and the Gao-Wald theorem. The article is based on lectures at the 2018 summer program Prospects in Theoretical Physics at the Institute for Advanced Study, and also at the 2020 New Zealand Mathematical Research Institute summer school in Nelson, New Zealand. Contents 1 Introduction 3 2 Causal Paths 4 3 Globally Hyperbolic Spacetimes 11 3.1 Definition . 11 3.2 Some Properties of Globally Hyperbolic Spacetimes . 15 3.3 More On Compactness . 18 3.4 Cauchy Horizons . 21 3.5 Causality Conditions . 23 3.6 Maximal Extensions . 24 4 Geodesics and Focal Points 25 4.1 The Riemannian Case . 25 4.2 Lorentz Signature Analog . 28 4.3 Raychaudhuri’s Equation . 31 4.4 Hawking’s Big Bang Singularity Theorem . 35 5 Null Geodesics and Penrose’s Theorem 37 5.1 Promptness . 37 5.2 Promptness And Focal Points . 40 5.3 More On The Boundary Of The Future . 46 1 5.4 The Null Raychaudhuri Equation . 47 5.5 Trapped Surfaces . 52 5.6 Penrose’s Theorem . 54 6 Black Holes 58 6.1 Cosmic Censorship . 58 6.2 The Black Hole Region . 60 6.3 The Horizon And Its Generators . 63 7 Some Additional Topics 66 7.1 Topological Censorship . 67 7.2 The Averaged Null Energy Condition .
    [Show full text]
  • General Relativistic Energy Conditions: the Hubble Expansion in the Epoch of Galaxy Formation
    General Relativistic Energy Conditions: The Hubble expansion in the epoch of galaxy formation. Matt Visser† Physics Department Washington University Saint Louis Missouri 63130-4899 USA 26 May 1997; gr-qc/9705070 Abstract The energy conditions of Einstein gravity (classical general relativity) are designed to extract as much information as possible from classical general relativity without enforcing a particular equation of state for the stress-energy. This systematic avoidance of the need to specify a particular equation of state is particularly useful in a cosmological setting — since the equation of state for the cosmological fluid in a Friedmann–Robertson– Walker type universe is extremely uncertain. I shall show that the energy conditions provide simple and robust bounds on the behaviour of both the density and look-back time as a function of red-shift. I shall show that current observations suggest that the so-called strong energy condition (SEC) is violated sometime between the epoch of galaxy formation and the present. This implies that no possible combination of “normal” matter is capable of fitting the observational data. PACS: 98.80.-k; 98.80.Bp; 98.80.Hw; 04.90.+e † E-mail: [email protected] 1 1 Introduction The discussion presented in this paper is a model-independent analysis of the misnamed age-of-the-universe problem, this really being an age-of-the-oldest- stars problem. In this paper I trade off precision against robustness: I sacrifice the precision that comes from assuming a particular equation of state, for the robustness that arises from model independence. A briefer presentation of these ideas can be found in [1], while a related analysis is presented in [2].
    [Show full text]
  • Black Holes and Black Hole Thermodynamics Without Event Horizons
    General Relativity and Gravitation (2011) DOI 10.1007/s10714-008-0739-9 RESEARCHARTICLE Alex B. Nielsen Black holes and black hole thermodynamics without event horizons Received: 18 June 2008 / Accepted: 22 November 2008 c Springer Science+Business Media, LLC 2009 Abstract We investigate whether black holes can be defined without using event horizons. In particular we focus on the thermodynamic properties of event hori- zons and the alternative, locally defined horizons. We discuss the assumptions and limitations of the proofs of the zeroth, first and second laws of black hole mechan- ics for both event horizons and trapping horizons. This leads to the possibility that black holes may be more usefully defined in terms of trapping horizons. We also review how Hawking radiation may be seen to arise from trapping horizons and discuss which horizon area should be associated with the gravitational entropy. Keywords Black holes, Black hole thermodynamics, Hawking radiation, Trapping horizons Contents 1 Introduction . 2 2 Event horizons . 4 3 Local horizons . 8 4 Thermodynamics of black holes . 14 5 Area increase law . 17 6 Gravitational entropy . 19 7 The zeroth law . 22 8 The first law . 25 9 Hawking radiation for trapping horizons . 34 10 Fluid flow analogies . 36 11 Uniqueness . 37 12 Conclusion . 39 A. B. Nielsen Center for Theoretical Physics and School of Physics College of Natural Sciences, Seoul National University Seoul 151-742, Korea [email protected] 2 A. B. Nielsen 1 Introduction Black holes play a central role in physics. In astrophysics, they represent the end point of stellar collapse for sufficiently large stars.
    [Show full text]
  • The Null Energy Condition and Its Violation
    The Null Energy Condition and its violation V.A. Rubakov Institute for Nuclear Research of the Russian Academy of Sciences Department of Particle Physics and Cosmology Physics Faculty, Moscow State University The Null Energy Condition, NEC µ ν Tµν n n > 0 µ µ for any null vector n , such that nµ n = 0. Quite robust In the framework of classical General Relativity implies a number of properties: Penrose theorem: Penrose’ 1965 Once there is trapped surface, there is singularity in future. Assumptions: (i) The NEC holds (ii) Cauchi hypersurface non-compact Trapped surface: a closed surface on which outward-pointing light rays actually converge (move inwards) Spherically symmetric examples: 2 2 2 2 2 ds = g00dt + 2g0RdtdR + gRRdR R dΩ − 4πR2: area of a sphere of constant t, R. Trapped surface: R decreases along all light rays. Sphere inside horizon of Schwarzschild black hole Hubble sphere in contracting Universe = ⇒ Hubble sphere in expanding Universe = anti-trapped surface = singularity in the past. ⇒ No-go for bouncing Universe scenario and Genesis Related issue: Can one in principle create a universe in the laboratory? Question raised in mid-80’s, right after invention of inflationary theory Berezin, Kuzmin, Tkachev’ 1984; Guth, Farhi’ 1986 Idea: create, in a finite region of space, inflationary initial conditions = this region will inflate to enormous size and in the end will look⇒ like our Universe. Do not need much energy: pour little more than Planckian energy into little more than Planckian volume. At that time: negaive answer [In the framework of classical General Relativity]: Guth, Farhi’ 1986; Berezin, Kuzmin, Tkachev’ 1987 1 Inflation in a region larger than Hubble volume, R > H− = Singularity in the past guaranteed by Penrose theorem ⇒ Meaning: Homogeneous and isotropic region of space: metric ds2 = dt2 a2(t)dx2 .
    [Show full text]
  • Quantifying Energy Condition Violations in Traversable Wormholes
    PRAMANA °c Indian Academy of Sciences Vol. 63, No. 4 | journal of October 2004 physics pp. 859{864 Quantifying energy condition violations in traversable wormholes SAYAN KAR1, NARESH DADHICH2 and MATT VISSER3 1Department of Physics and Centre for Theoretical Studies, Indian Institute of Technology, Kharagpur 721 302, India 2Inter-University Centre for Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune 411 007, India 3School of Mathematical and Computing Sciences, Victoria University of Wellington, New Zealand E-mail: [email protected]; [email protected]; [email protected] Abstract. The `theoretical' existence of traversable Lorentzian wormholes in the classical, macroscopic world is plagued by the violation of the well-known energy condi- tions of general relativity. In this brief article we show: (i) how the extent of violation can be quanti¯ed using certain volume integrals and (ii) whether this `amount of violation' can be minimised for some speci¯c cut-and-paste geometric constructions. Examples and possibilities are also outlined. Keywords. Wormholes; energy conditions; general relativity. PACS Nos 04.70.Dy; 04.62.+v; 11.10.Kk 1. The problem and our attitude It is well-known by now that the `theoretical' existence of traversable Lorentzian wormholes is plagued by the violation of the energy conditions of general relativity [1,2]. Researchers have come up with a variety of proposals, most of which gain support from the fact that quantum expectation values of the stress energy tensor can often become negative [3,4]. The experimentally veri¯ed case of the Casimir e®ect [5] is often cited as a `proof' of the existence of `matter' with `negative energy density' though in experiments on the Casimir e®ect the quantity measured is the force (and hence the pressure) of the ‘fluctuating vacuum' between parallel metallic plates.
    [Show full text]
  • Energy Conditions in Semiclassical and Quantum Gravity
    Energy conditions in semiclassical and quantum gravity Sergi Sirera Lahoz Supervised by Professor Toby Wiseman Imperial College London Department of Physics, Theory Group Submitted in partial fulfilment of the requirements for the degree of Master of Science of Imperial College London MSc Quantum Fields and Fundamental Forces 2nd October 2020 Abstract We explore the state of energy conditions in semiclassical and quantum gravity. Our review introduces energy conditions with a philosophical discussion about their nature. Energy conditions in classical gravity are reviewed together with their main applications. It is shown, among other results, how one can obtain singularity the- orems from such conditions. Classical and quantum physical systems violating the conditions are studied. We find that quantum fields entail negative energy densities that violate the classical conditions. The character of these violations motivates the quantum interest conjecture, which in turn guides the design of quantum energy conditions. The two best candidates, the AANEC and the QNEC, are examined in detail and the main applications revisited from their new perspectives. Several proofs for interacting field theories and/or curved spacetimes are demonstrated for both conditions using techniques from microcausality, holography, quantum infor- mation, and other modern ideas. Finally, the AANEC and the QNEC will be shown to be equivalent in M4. i Acknowledgements First of all, I would like to express my gratitude to Prof. Toby Wiseman for supervising this dissertation. Despite the challenges of not being able to meet in person, your guidance and expertise have helped me greatly to stay on track. The simple and clear way in which you communicate ideas has stimulated my interest in the topic and deepened my comprehension.
    [Show full text]
  • Energy Conditions in General Relativity and Quantum Field Theory Arxiv:2003.01815V2 [Gr-Qc] 5 Jun 2020
    Energy conditions in general relativity and quantum field theory Eleni-Alexandra Kontou∗ Department of Mathematics, University of York, Heslington, York YO10 5DD, United Kingdom Department of Physics, College of the Holy Cross, Worcester, Massachusetts 01610, USA Ko Sanders y Dublin City University, School of Mathematical Sciences and Centre for Astrophysics and Relativity, Glasnevin, Dublin 9, Ireland June 8, 2020 Abstract This review summarizes the current status of the energy conditions in general relativity and quantum field theory. We provide a historical review and a summary of technical results and applications, complemented with a few new derivations and discussions. We pay special attention to the role of the equations of motion and to the relation between classical and quantum theories. Pointwise energy conditions were first introduced as physically reasonable restrictions on mat- ter in the context of general relativity. They aim to express e.g. the positivity of mass or the attractiveness of gravity. Perhaps more importantly, they have been used as assumptions in math- ematical relativity to prove singularity theorems and the non-existence of wormholes and similar exotic phenomena. However, the delicate balance between conceptual simplicity, general validity and strong results has faced serious challenges, because all pointwise energy conditions are sys- tematically violated by quantum fields and also by some rather simple classical fields. In response to these challenges, weaker statements were introduced, such as quantum energy inequalities and averaged energy conditions. These have a larger range of validity and may still suffice to prove at least some of the earlier results. One of these conditions, the achronal averaged null energy arXiv:2003.01815v2 [gr-qc] 5 Jun 2020 condition, has recently received increased attention.
    [Show full text]
  • Raychaudhuri Equation, Geometrical Flows and Geometrical Entropy
    S S symmetry Article Raychaudhuri Equation, Geometrical Flows and Geometrical Entropy Lawrence Paul Horwitz 1,2,3,* , Vishnu S Namboothiri 4 , Gautham Varma K 5, Asher Yahalom 6 , Yosef Strauss 7 and Jacob Levitan 3 1 School of Physics and Astronomy, Tel Aviv University, Ramat Aviv 69978, Israel 2 Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel 3 Department of Physics, Ariel University, Ariel 40700, Israel; [email protected] 4 Center for Science in Society, Cochin University of Science and Technology (CUSAT), Kochi 682022, India; [email protected] 5 School of Basic Sciences, Indian Institute of Technology Mandi, Mandi 175005, India; [email protected] 6 Department of Engineering, Ariel University, Ariel 40700, Israel; [email protected] 7 Department of Mathematics, Ben Gurion University, Beer Sheba 84105, Israel; [email protected] * Correspondence: [email protected]; Tel.: +72-3-6408724 or +972-3-5318956 Abstract: The Raychaudhuri equation is derived by assuming geometric flow in space–time M of n + 1 dimensions. The equation turns into a harmonic oscillator form under suitable transforma- tions. Thereby, a relation between geometrical entropy and mean geodesic deviation is established. This has a connection to chaos theory where the trajectories diverge exponentially. We discuss its application to cosmology and black holes. Thus, we establish a connection between chaos theory and general relativity. Citation: Horwitz, L.P; Namboothiri, V.S.; Varma K, G.;Yahalom, A.; Keywords: Raychaudhuri equation; chaos theory; Kaluza Klein theory; Kaluza Klein cosmology; Strauss, Y.; Levitan, J. Raychaudhuri geometrical flow; geometrical entropy; Riccati equation Equation, Geometrical Flows and Geometrical Entropy.
    [Show full text]
  • Singularity Theorems in the Effective Field Theory for Quantum Gravity at Second Order in Curvature
    universe Article Singularity Theorems in the Effective Field Theory for Quantum Gravity at Second Order in Curvature Folkert Kuipers * and Xavier Calmet * Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH, UK * Correspondence: [email protected] (F.K.); [email protected] (X.C.) Received: 10 August 2020; Accepted: 4 October 2020; Published: 10 October 2020 Abstract: In this paper, we discuss singularity theorems in quantum gravity using effective field theory methods. To second order in curvature, the effective field theory contains two new degrees of freedom which have important implications for the derivation of these theorems: a massive spin-2 field and a massive spin-0 field. Using an explicit mapping of this theory from the Jordan frame to the Einstein frame, we show that the massive spin-2 field violates the null energy condition, while the massive spin-0 field satisfies the null energy condition, but may violate the strong energy condition. Due to this violation, classical singularity theorems are no longer applicable, indicating that singularities can be avoided, if the leading quantum corrections are taken into account. Keywords: quantum gravity; singularity theorems 1. Introduction The significance of singularity theorems in general relativity first presented in the seminal papers of Penrose and Hawking [1,2] cannot be overemphasised. Since these foundational works, several adaptions and refinements of the singularity theorems have been developed (see, e.g., [3–7]). In general, all these theorems boil down to the same principle: the assumption of some energy condition together with some global statement about space-time leads to the prediction of geodesic incompleteness somewhere in the space-time.
    [Show full text]
  • 4. the Einstein Equations
    4. The Einstein Equations It is now time to do some physics. The force of gravity is mediated by a gravitational field. The glory of general relativity is that this field is identified with a metric gµ⌫(x) on a 4d Lorentzian manifold that we call spacetime. This metric is not something fixed; it is, like all other fields in Nature, a dynamical object. This means that there are rules which govern how this field evolves in time. The purpose of this section is to explore these rules and some of their consequences. We will start by understanding the dynamics of the gravitational field in the absence of any matter. We will then turn to understand how the gravitational field responds to matter – or, more precisely, to energy and momentum – in Section 4.5. 4.1 The Einstein-Hilbert Action All our fundamental theories of physics are described by action principles. Gravity is no di↵erent. Furthermore, the straight-jacket of di↵erential geometry places enormous restrictions on the kind of actions that we can write down. These restrictions ensure that the action is something intrinsic to the metric itself, rather than depending on our choice of coordinates. Spacetime is a manifold M, equipped with a metric of Lorentzian signature. An action is an integral over M.WeknowfromSection2.4.4 that we need a volume-form to integrate over a manifold. Happily, as we have seen, the metric provides a canonical volume form, which we can then multiply by any scalar function. Given that we only have the metric to play with, the simplest such (non-trivial) function is the Ricci scalar R.
    [Show full text]