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From Clock Synchronization to Dark Matter as a Relativistic Inertial Effect

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Black Objects in Supergravity

Part of the book series: Springer Proceedings in Physics ((SPPHY,volume 144))

Abstract

One of the main open problems in astrophysics is the dominance of dark entities, the dark matter and the dark energy, in the existing description of the universe given by the standard \(\varLambda \)CDM cosmological model [1, 2] based on the cosmological principle (homogeneity and isotropy of the space-time), which selects the class of Friedmann-Robertson-Walker (FWR) space-times.

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Notes

  1. 1.

    The fixed stars can be considered as an empirical definition of spatial infinity of the observable universe.

  2. 2.

    In these space-times the canonical Hamiltonian vanishes and the Dirac Hamiltonian is a combination of first-class constraints, so that it only generates Hamiltonian gauge transformations. In the reduced phase space, quotient with respect the Hamilonian gauge group, the reduced Hamiltonian is zero and one has a frozen picture of dynamics. This class of space-times fits well with Machian ideas (no boundary conditions) and with interpretations in which there is no physical time like the one in Ref. [45]. However, it is not clear how to include in this framework the standard model of particle physics.

  3. 3.

    While in SR Minkowski space-time is an absolute notion, unifying the absolute notions of time and 3-space of the non-relativistic Galilei space-time, in GR there is no absolute notion: space-time becomes dynamical [79–81] with its metric structure satisfying Einstein equations.

  4. 4.

    Instead in Yang-Mills theory all the gauge variables are configurational.

  5. 5.

    It is based on Very Long Baseline Interferometry (VLBI) observations of distant quasars, on Lunar Laser Ranging (LLR) and on determination of GPS satellite orbits.

  6. 6.

    The meter is the length of the path traveled by light in vacuum during a time interval of \(1/c\) of a second.

  7. 7.

    See Ref. [109] for possible gravitational anomalies inside the Solar System.

  8. 8.

    It is the non-factual idealization required by the Cauchy problem generalizing the existing protocols for building coordinate system inside the future light-cone of a time-like observer.

  9. 9.

    \(\epsilon = \pm 1\) according to either the particle physics \(\epsilon = 1\) or the general relativity \(\epsilon = - 1\) convention.

  10. 10.

    \({\hat{n}}^r(\tau ,\sigma )\) defines the instantaneous rotation axis and \(0 < \tilde{\varOmega } (\tau ,\sigma ) < 2\, max\, \big ({\dot{{\tilde{\alpha }}}}(\tau ), {\dot{\tilde{\beta }}}(\tau ), {\dot{\tilde{\gamma }}}(\tau )\big )\).

  11. 11.

    Nearly rigid rotating systems, like a rotating disk of radius \(\sigma _o\), can be described by using a function \(F(\sigma )\) approximating the step function \(\theta (\sigma - \sigma _o)\).

  12. 12.

    It corresponds to the locality hypothesis of Ref. [47, 48], according to which at each instant of time the detectors of an accelerated observer give the same indications as the detectors of the instantaneously comoving inertial observer.

  13. 13.

    In the first paper of Ref. [77, 78] there is the definition of parametrized Galilei theories, non relativistic limit of the parametrized Minkowski theories. Also the inertial and non-inertial frames in Galilei space-time are gauge equivalent in this formulation.

  14. 14.

    Their Poisson brackets are \(\{ z^i, h^j\} = \delta ^{ij}\). \({{\varvec{x}}}_{NW}(\tau )\) is the standard Newton-Wigner non-covariant 3-position, classical counterpart of the corresponding position operator; the use of \({{\varvec{ z}}}\) avoids to take into account the mass spectrum of the isolated system in the description of the center of mass. The non-covariance of \({{\varvec{z}}}\) under Poincaré transformations \((a ,\varLambda )\) has the following form [76, 115] \(z^i\, \mapsto z^{{^{\prime }}\, i} = \Big (\varLambda ^i{}_j - {{\varLambda ^i{}_{\mu }\, h^{\mu }}\over {\varLambda ^o{}_{\nu }\, h^{\nu }}}\, \lambda ^o{}_j\Big )\, z^j + \Big (\varLambda ^i{}_{\mu } - {{\varLambda ^i{}_{\nu }\, h^{\nu }}\over {\varLambda ^o{}_{\rho }\, h^{\rho }}}\, \varLambda ^o{}_{\mu }\Big )\, (\varLambda ^{-1}\, a)^{\mu }\).

  15. 15.

    In the rest-frame the world-tube is a cylinder: in each instantaneous 3-space there is a disk of possible positions of the canonical 3-center of mass orthogonal to the spin. In the non-relativistic limit the radius \(\rho \) of the disk tends to zero and one recovers the non-relativistic center of mass.

  16. 16.

    The last term in the Lorentz boosts induces the Wigner rotation of the 3-vectors inside the Wigner 3-spaces.

  17. 17.

    One can show [59–61, 65, 66] that one has \({{\varvec{{\mathcal{{K}}}}}}_{(int)} = - M\, {{\varvec{R}}}_+\), where \({{\varvec{R}}}_+\) is the internal Møller 3-center of energy inside the Wigner 3-spaces. The rest frame condition \({{{\varvec{\mathcal{{P}}}}}}_{(int)} \approx 0\) implies \({{\varvec{ R}}}_+ \approx {{\varvec{ q}}}_+ \approx {{\varvec{y}}}_+\), where \({{\varvec{ q}}}_+\) is the internal 3-center of mass and \({{\varvec{y}}}_+\) the internal Fokker-Pryce 3-center of inertia.

  18. 18.

    Equations (8.15) describe a family of canonical transformations, because the \(\gamma _{ai}\)’s depend on \({\frac{1}{2}}(N-1)(N-2)\) free independent parameters.

  19. 19.

    In Ref. [128] it was shown that the quantum Newton-Wigner position should not be a self-adjoint operator, but only a symmetric one, with an implication of bad localization.

  20. 20.

    These space-times must also be without Killing symmetries, because, otherwise, at the Hamiltonian level one should introduce complicated sets of extra Dirac constraints for each existing Killing vector.

  21. 21.

    The ACES mission of ESA [133–135] will give the first precision measurement of the gravitational red-shift of the geoid, namely of the \(1/c^2\) deformation of Minkowski light-cone caused by the geo-potential. In every quantum field theory approach to gravity, where the definition of the Fock space requires the use of the Fourier transform on a fixed background space-time with a fixed light-cone, this is a non-perturbative effect requiring the re-summation of the perturbative expansion.

  22. 22.

    Since one uses the positive-definite 3-metric \(\delta _{(a)(b)} \), one will use only lower flat spatial indices. Therefore for the cotriads one uses the notation \({}^3e^{(a)}_r\,\, {def\over =}\, {}^3e_{(a)r}\) with \(\delta _{(a)(b)} = {}^3e^r_{(a)}\, {}^3e_{(b)r}\).

  23. 23.

    The Hamilton equations imply \({}^4\nabla _A\, T^{AB} \equiv 0\) in accord with Einstein’s equations and the Bianchi identity.

  24. 24.

    Both quantities are two-valued. The elementary electric charges are \(Q = \pm e\), with \(e\) the electron charge. Analogously the sign of the energy of a particle is a topological two-valued number (the two branches of the mass-shell hyperboloid). The formal quantization of these Grassmann variables gives two-level fermionic oscillators. At the classical level the self-energies make the classical equations of motion ill-defined on the world-lines of the particles. The ultraviolet and infrared Grassmann regularization allows to cure this problem and to get consistent solution of regularized equations of motion. See Refs. [65, 66] for the electro-magnetic case.

  25. 25.

    Due to the positive signature of the 3-metric, one defines the matrix \(V\) with the following indices: \(V_{ru}\). Since the choice of Shanmugadhasan canonical bases breaks manifest covariance, one will use the notation \(V_{ua} = \sum _v\, V_{uv}\, \delta _{v(a)}\) instead of \(V_{u(a)}\).

  26. 26.

    In the post-Newtonian approximation in harmonic gauges they are the counterpart of the electro-magnetic vector potentials describing magnetic fields [84]: (A) \(N = 1 + n\), \(n\, {\mathop {=}\limits ^{def}}\, - {{4\, \epsilon }\over {c^2}}\, \varPhi _G\) with \(\varPhi _G\) the gravito-electric potential; (B) \(n_r\, {\mathop {=}\limits ^{def}}\, {{2\, \epsilon }\over {c^2}}\, A_{G\, r}\) with \(A_{G\, r}\) the gravito-magnetic potential; (C) \(E_{G\, r} = \partial _r\, \varPhi _G - \partial _{\tau }\, ({1\over 2}\, A_{G\, r})\) (the gravito-electric field) and \(B_{G\, r} = \epsilon _{ruv}\, \partial _u\, A_{G\, v} = c\, \varOmega _{G\, r}\) (the gravito-magnetic field). Let us remark that in arbitrary gauges the analogy with electro-magnetism breaks down.

  27. 27.

    It describes gravito-magnetic effects.

  28. 28.

    This equation is relevant for studying the developments of caustics in a congruence of time-like geodesics for converging values of the expansion \(\theta \) and of singularities in Einstein space-times [137–139]. However the boundary conditions of asymptotically Minkowskian space-times without super-translations should avoid the singularity theorems as it happens with their subfamily without matter of Ref. [40].

  29. 29.

    See the “\(1+3\) point of view” of Ref. [140] for a discussion of gravity in terms of the second non-surface-forming congruence of time-like observers associated with a \(3+1\) splitting of space-time.

  30. 30.

    It measures the average expansion of the infinitesimally nearby world-lines surrounding a given world-line in the congruence.

  31. 31.

    It measures how an initial sphere in the tangent space to the given world-line, which is Lie-transported along the world-line tangent \(l^{\mu }\) (i.e. it has zero Lie derivative with respect to \(l^{\mu }\, \partial _{\mu }\)), is distorted towards an ellipsoid with principal axes given by the eigenvectors of \(\sigma ^{\mu }{}_{\nu }\), with rate given by the eigenvalues of \(\sigma ^{\mu }{}_{\nu }\).

  32. 32.

    Quantities like \(|{{\varvec{{\eta }}}}_i(\tau ) - {{\varvec{{\eta }}}}_j(\tau )|\) are the Euclidean 3-distance between the two particles in the asymptotic 3-space \(\varSigma _{\tau (\infty )}\), which differs by quantities of order \(O(\zeta )\) from the real non-Euclidean 3-distance in \(\varSigma _{\tau }\) as shown in Eq. (3.3) of the third paper in Ref. [88–90].

  33. 33.

    For the tidal momenta one gets \({{8\pi \, G}\over {c^3}}\, \varPi _{\bar{a}}(\tau , {\varvec{{\sigma }}}) = [\partial _{\tau }\, R_{\bar{a}} - \sum _a\, \gamma _{\bar{a}a}\, \partial _a\, {\bar{n}}_{(1)(a)}](\tau , {\varvec{{\sigma }}}) + O(\zeta ^2)\), so that the diagonal elements of the shear are \(\sigma _{(1)(a)(a)}(\tau , {\varvec{{\sigma }}}) = [- \sum _{\bar{a}}\, \gamma _{\bar{a}a}\, \partial _{\tau }\, R_{\bar{a}} + {\bar{n}}_{(1)(a)} - {1\over 3}\, \sum _b\, {\bar{n}}_{(1)(b)}](\tau , {\varvec{{\sigma }}}) + O(\zeta ^2)\).

  34. 34.

    See Eq. (7.20) of the second paper in Ref. [88–90], where \({}^4g_{\tau \tau }(\tau , {\varvec{{\sigma }}})\) and \({}^4g_{\tau r}(\tau , {\varvec{{\sigma }}})\) are explicitly depending on the non-local York time.

  35. 35.

    The properties of HPM transverse electro-magnetic fields have still to be explored.

  36. 36.

    They imply that GW propagate not only on the flat light-cone but also inside it (i.e. with all possible speeds \(0 \le v \le c\)): there is an instantaneous wavefront followed by a tail traveling at lower speed (it arrives later and then fades away) and a persistent zero-frequency non-linear memory.

  37. 37.

    In this approach point particles are considered as independent matter degrees of freedom with a Grassmann regularization of the self-energies to get well defined world-lines (see also Ref. [183, 184]): they are not considered as point-like singularities of solutions of Einstein’s equations (the point of view of Ref. [179]). Solutions of this type have to be described with distributions and, as shown in Ref. [185], the most general class of such solutions under mathematical control includes singularities simulating matter shells, but not either strings or particles. See also Ref. [186].

  38. 38.

    In the electro-magnetic case the Grassmann regularization implies \(Q_i\, {{\dot{\eta }}}_i^r(\tau - |{\varvec{{\sigma }}}|) = Q_i\, {{\dot{\eta }}}_i(\tau )\) and equations of motion of the type \({\ddot{\eta }}^r_i(\tau ) = Q_i\, \ldots \) with \(Q^2_i = 0\). In the gravitational case the equations of motion are of the type \(\eta _i\, {\ddot{\eta }}^r_i(\tau ) = \eta _i\, \ldots \) with \(\eta ^2_i = 0\), but the Grassmann regularized retardation in Eq. (8.54) gives Eq. (8.59) only at the lowest order in \(\zeta \) and has contributions of every order \(O(\zeta ^k)\).

  39. 39.

    It is a bi-tensor, i.e. a scalar in both the points \({{\varvec{{\eta }}}}_i(\tau )\) and \({{\varvec{{\eta }}}}_j(\tau )\), defined in terms of the space-like geodesic connecting them in \(\varSigma _{\tau }\). See Eq. (3.13) of the third paper in Ref. [88–90].

  40. 40.

    For binaries one assumes \({v\over c} \approx \sqrt{{{R_m}\over {l_c}}} << 1\), where \(l_c \approx |{\tilde{{\varvec{{r}}}}}|\) with \({\tilde{{\varvec{{r}}}}}(t)\) being the relative separation after the decoupling of the center of mass. Often one considers the case \(m_1 \approx m_2\). See chapter 4 of Ref. [95] for a review of the emission of GW’s from circular and elliptic Keplerian orbits and of the induced inspiral phase.

  41. 41.

    This is also the starting point of the effective one body description of the two-body problem of Refs. [181, 182].

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  1. Sezione INFN di Firenze, Polo Scientifico, Via Sansone 1, 50019 , Sesto Fiorentino, FI, Italy

    Luca Lusanna

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  1. Luca Lusanna
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    Stefano Bellucci

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Lusanna, L. (2013). From Clock Synchronization to Dark Matter as a Relativistic Inertial Effect. In: Bellucci, S. (eds) Black Objects in Supergravity. Springer Proceedings in Physics, vol 144. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00215-6_8

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