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# Papers

You can find a list of my works in different online data bases:

## Dirk Puetzfeld's profile on ORCID

For a full list of publications please also see below.

## submitted / in press

### On the applicability of the geodesic deviation equation in General Relativity

D. Philipp, D. Puetzfeld, C. Laemmerzahl

Abstract: Within the theory of General Relativity we study the solution and range of applicability of the standard geodesic deviation equation in highly symmetric spacetimes. The deviation equation is used to model satellite orbit constellations around the earth. In particular, we reconsider the deviation equation in Newtonian gravity and then determine relativistic effects within the theory of General Relativity. The deviation of nearby orbits, as constructed from exact solutions of the underlying geodesic equation, is compared to the solution of the geodesic deviation equation to assess the accuracy of the latter. Furthermore, we comment on the so-called Shirokov effect in Schwarzschild spacetime.

16 pages, submitted (04/2016)

## publications

### The relativistic geoid

D. Philipp, V. Perlick, D. Puetzfeld, E. Hackmann, C. Laemmerzahl

Abstract: Based on the formalism of General Relativity, we analyze generalizations of concepts used in conventional geodesy. One such concept is the Earth's geoid. We present our definition of the relativistic geoid in terms of the level sets of a time-independent redshift potential. Such a potential exists for any congruence of Killing observers, i.e. for any rigidly moving object associated with a stationary spacetime in the outer region. The level surfaces of the redshift potential foliate the three dimensional space into isochronometric surfaces, which can be determined with the help of standard clocks. Two such clocks on the same surface will show zero redshift when their frequencies are compared. One of these level surfaces, singled out by a suitable convention, defines the relativistic geoid in our framework. At the same time, the redshift potential is also an acceleration potential for the congruence of observers. Hence, the isochrono-metric surfaces are orthogonal to the acceleration of freely falling objects, i.e. they are orthogonal to the local plumb line. Therefore, two independent kinds of measurements can contribute to the determination of the relativistic geoid. It can be shown that clocks, which are connected by optical fiber links, can be used to determine the redshift potential; this gives the operational foundation of our framework. Moreover, we show that our definition reduces to the well-known Newtonian and post-Newtonian notions in the respective limits. To illustrate our framework, we consider analytic examples of spacetimes, for which we calculate the level surfaces of the redshift potential and illustrate their intrinsic geometry by an embedding into flat Euclidean space. We emphasize that our definition of the geoid in terms of relativistic concepts is valid for arbitrarily strong gravitational fields. We do not use any approximation in the sense of weak fields or post-Newtonian expansion schemes. Hence, the definition can also be applied to very compact objects such as neutron stars.

6 pages, 2 figures, IEEE International Workshop on Metrology for AeroSpace (MetroAeroSpace) Proceedings, ISBN: 978-1-5090-4233-3, DOI: 10.1109/MetroAeroSpace.2017.7999549

### Definition of the relativistic geoid in terms of isochronometric surfaces

D. Philipp, V. Perlick, D. Puetzfeld, E. Hackmann, C. Laemmerzahl

Abstract: We present a definition of the geoid that is based on the formalism of general relativity without approximations; i.e. it allows for arbitrarily strong gravitational fields. For this reason, it applies not only to the Earth and other planets but also to compact objects such as neutron stars. We define the geoid as a level surface of a time-independent redshift potential. Such a redshift potential exists in any stationary spacetime. Therefore, our geoid is well defined for any rigidly rotating object with constant angular velocity and a fixed rotation axis that is not subject to external forces. Our definition is operational because the level surfaces of a redshift potential can be realized with the help of standard clocks, which may be connected by optical fibers. Therefore, these surfaces are also called isochronometric surfaces. We deliberately base our definition of a relativistic geoid on the use of clocks since we believe that clock geodesy offers the best methods for probing gravitational fields with highest precision in the future. However, we also point out that our definition of the geoid is mathematically equivalent to a definition in terms of an acceleration potential, i.e. that our geoid may also be viewed as a level surface orthogonal to plumb lines. Moreover, we demonstrate that our definition reduces to the known Newtonian and post-Newtonian notions in the appropriate limits. As an illustration, we determine the isochronometric surfaces for rotating observers in axisymmetric static and axisymmetric stationary solutions to Einstein's vacuum field equation, with the Schwarzschild metric, the Erez-Rosen metric, the q-metric and the Kerr metric as particular examples.

24 pages, 7 figures, Phys. Rev. D 95 (2017) 104037, DOI: 10.1103/PhysRevD.95.104037

### Dynamics of test bodies in scalar-tensor theory and equivalence principle

Yuri N. Obukhov, Dirk Puetzfeld

Abstract: How do test bodies move in scalar-tensor theories of gravitation? We provide an answer to this question on the basis of a unified multipolar scheme. In particular, we give the explicit equations of motion for pointlike, as well as spinning test bodies, thus extending the well-known general relativistic results of Mathisson, Papapetrou, and Dixon to scalar-tensor theories of gravity. We demonstrate the validity of the equivalence principle for test bodies.

5 pages, Gravitation, Astrophysics, and Cosmology - Proceedings of the Twelfth Asia-Pacific International Conference, Moscow, 28 Jun - 5 July 2015, Eds. V. Melnikov and J.-P. Hsu, World Scientific (Singapore), 2016, pp. 231-235, DOI: 10.1142/9789814759816_0043

### Generalized deviation equation and determination of the curvature in General Relativity

Dirk Puetzfeld, Yuri N. Obukhov

Abstract: We derive a generalized deviation equation -- analogous to the well-known geodesic deviation equation -- for test bodies in General Relativity. Our result encompasses and generalizes previous extensions of the standard geodesic deviation equation. We show how the standard as well as a generalized deviation equation can be used to measure the curvature of spacetime by means of a set of test bodies. In particular, we provide exact solutions for the curvature by using the standard deviation equation as well as its next order generalization.

16 pages, 5 figures, Phys. Rev. D 93 (2016) 044073, DOI: 10.1103/PhysRevD.93.044073

### Invariant conserved currents in generalized gravity

Y.N. Obukhov, F. Portales-Oliva, D. Puetzfeld, G.F. Rubilar

Abstract: We study conservation laws for gravity theories invariant under general coordinate transformations. The class of models under consideration includes Einstein's general relativity theory as a special case as well as its generalizations to non-Riemannian spacetime geometry and nonminimal coupling. We demonstrate that an arbitrary vector field on the spacetime manifold generates a current density that is conserved under certain conditions, and find the expression of the corresponding superpotential. For a family of models including nonminimal coupling between geometry and matter, we discuss in detail the differential conservation laws and the conserved quantities defined in terms of covariant multipole moments. We show that the equations of motion for the multipole moments of extended microstructured test bodies lead to conserved quantities that are closely related to the conserved currents derived in the field-theoretic framework.

15 pages, Phys. Rev. D 92 (2015) 104010, DOI: 10.1103/PhysRevD.92.104010

### Equivalence principle in scalar-tensor gravity

Dirk Puetzfeld, Yuri N. Obukhov

Abstract: We present a direct confirmation of the validity of the equivalence principle for unstructured test bodies in scalar tensor gravity. Our analysis is complementary to previous approaches and valid for a large class of scalar-tensor theories of gravitation. A covariant approach is used to derive the equations of motion in a systematic way and allows for the experimental test of scalar-tensor theories by means of extended test bodies.

5 pages, Phys. Rev. D 92 (2015) 081502(R), DOI: 10.1103/PhysRevD.92.081502

### Multipolar test body equations of motion in generalized gravity theories

Yuri N. Obukhov, Dirk Puetzfeld

Abstract: We give an overview of the derivation of multipolar equations of motion of extended test bodies for a wide set of gravitational theories beyond the standard general relativistic framework. The classes of theories covered range from simple generalizations of General Relativity, e.g. encompassing additional scalar fields, to theories with additional geometrical structures which are needed for the description of microstructured matter. Our unified framework even allows to handle theories with nonminimal coupling to matter, and thereby for a systematic test of a very broad range of gravitational theories.

''Equations of Motion in Relativistic Gravity'', D. Puetzfeld et. al. (eds.), Fundamental theories of Physics 179, pages 67-119, Springer 2015, DOI: 10.1007/978-3-319-18335-0, ISBN: 978-3-319-18334-3 (print), 978-3-319-18335-0 (online)

### The galactic center black hole laboratory

A. Eckart, S. Britzen, M. Valencia-S., C. Straubmeier, J.A. Zensus, V. Karas, D. Kunneriath, A. Alberdi, N. Sabha, R. Schoedel, D. Puetzfeld

Abstract: The super-massive 4 million solar mass black hole Sagittarius A* (SgrA*) shows flare emission from the millimeter to the X-ray domain. A detailed analysis of the infrared light curves allows us to address the accretion phenomenon in a statistical way. The analysis shows that the near-infrared flare amplitudes are dominated by a single state power law, with the low states in SgrA* limited by confusion through the unresolved stellar background. There are several dusty objects in the immediate vicinity of SgrA*. The source G2/DSO is one of them. Its nature is unclear. It may be comparable to similar stellar dusty sources in the region or may consist predominantly of gas and dust. In this case a particularly enhanced accretion activity onto SgrA* may be expected in the near future. Here the interpretation of recent data and ongoing observations are discussed.

''Equations of Motion in Relativistic Gravity'', D. Puetzfeld et. al. (eds.), Fundamental theories of Physics, 179, pages 759-781, Springer 2015, DOI: 10.1007/978-3-319-18335-0, ISBN: 978-3-319-18334-3 (print), 978-3-319-18335-0 (online)

### Equations of motion in scalar-tensor theories of gravity: A covariant multipolar approach

Yuri N. Obukhov, Dirk Puetzfeld

Abstract: We discuss the dynamics of extended test bodies for a large class of scalar-tensor theories of gravitation. A covariant multipolar Mathisson-Papapetrou-Dixon type of approach is used to derive the equations of motion in a systematic way for both Jordan and Einstein formulations of these theories. The results obtained provide the framework to experimentally test scalar-tensor theories by means of extended test bodies.

5 pages, Phys. Rev. D 90 (2014) 104041, DOI: 10.1103/PhysRevD.90.104041

### Equations of motion in metric-affine gravity: a covariant unified framework

Dirk Puetzfeld, Yuri N. Obukhov

Abstract: We derive the equations of motion of extended deformable bodies in metric-affine gravity. The conservation laws which follow from the invariance of the action under the general coordinate transformations are used as a starting point for the discussion of the dynamics of extended deformable test bodies. By means of a covariant approach, based on Synge's world function, we obtain the master equation of motion for an arbitrary system of coupled conserved currents. This unified framework is then applied to metric-affine gravity. We confirm and extend earlier findings, in particular we once again demonstrate that it is only possible to detect the post-Riemannian spacetime geometry by ordinary (non-microstructured) test bodies if gravity is nonminimally coupled to matter.

13 pages, Phys. Rev. D 90 (2014) 084034, DOI: 10.1103/PhysRevD.90.084034, Dedicated to Friedrich W. Hehl on the occasion of his birthday

### Motion of spinning test bodies in Kerr spacetime

E. Hackmann, C. Laemmerzahl, Y.N. Obukhov, D. Puetzfeld, I. Schaffer

Abstract: We investigate the motion of spinning test bodies in General Relativity. By means of a multipolar approximation method for extended test bodies we derive the equations of motion, and classify the orbital motion of pole-dipole test bodies in the equatorial plane of the Kerr geometry. An exact expression for the periastron shift of a spinning test body is given. Implications of test body spin corrections are studied and compared with the results obtained by means of other approximation schemes.

12 pages, 5 figures, Phys. Rev. D 90 (2014) 064035, DOI: 10.1103/PhysRevD.90.064035

### Prospects of detecting spacetime torsion

Dirk Puetzfeld, Yuri N. Obukhov

Abstract: How to detect spacetime torsion? In this essay we provide the theoretical basis for an answer to this question. Multipolar equations of motion for a very general class of gravitational theories with nonminimal coupling in spacetimes admitting torsion are given. Our findings provide a framework for the systematic testing of whole classes of theories with the help of extended test bodies. One surprising feature of nonminimal theories turns out to be their potential sensitivity to torsion of spacetime even in experiments with ordinary (not microstructured) test matter.

This essay received a honorable mention in the 2014 essay competition of the Gravity Research Foundation.

6 pages, Int. J. Mod. Phys. D 23 (2014) 1442004, DOI: 10.1142/S0218271814420048

### Conservation laws and covariant equations of motion for spinning particles

Yuri N. Obukhov, Dirk Puetzfeld

Abstract: We derive the Noether identities and the conservation laws for general gravitational models with arbitrarily interacting matter and gravitational fields. These conservation laws are used for the construction of the covariant equations of motion for test bodies with minimal and nonminimal coupling.

6 pages, Proceedings of XV Advanced Research Workshop on High Energy Spin Physics "DSPIN-13", Dubna, 8-12 October 2013, Eds. A.V. Efremov and S.V. Goloskokov (Joint Inst. Nucl. Res., JINR, Dubna, 2014) p. 110-115

### Conservation laws in gravity: A unified framework

Yuri N. Obukhov, Dirk Puetzfeld

Abstract: We study general metric-affine theories of gravity in which the metric and connection are the two independent fundamental variables. In this framework, we use Lagrange-Noether methods to derive the identities and the conservation laws which correspond to the invariance of the action under general coordinate transformations. The results obtained are applied to generalized models with nonminimal coupling of matter and gravity, with a coupling function that depends arbitrarily on the covariant gravitational field variables.

9 pages, Phys. Rev. D 90 (2014) 024004, DOI: 10.1103/PhysRevD.90.024004

### Equations of motion in gravity theories with nonminimal coupling: a loophole to detect torsion macroscopically?

Dirk Puetzfeld, Yuri N. Obukhov

Abstract: We derive multipolar equations of motion for gravitational theories with general nonminimal coupling in spacetimes admitting torsion. Our very general findings allow for the systematic testing of whole classes of theories by means of extended test bodies. One peculiar feature of certain subclasses of nonminimal theories turns out to be their sensitivity to post-Riemannian spacetime structures even in experiments without microstructured test matter.

9 pages, Phys. Rev. D 88 (2013) 064025, DOI: 10.1103/PhysRevD.88.064025

### Unraveling gravity beyond Einstein with extended test bodies

Dirk Puetzfeld, Yuri N. Obukhov

Abstract: The motion of test bodies in gravity is tightly linked to the conservation laws. This well-known fact in the context of General Relativity is also valid for gravitational theories which go beyond Einstein's theory. Here we derive the equations of motion for test bodies for a very large class of gravitational theories with a general nonminimal coupling to matter. These equations form the basis for future systematic tests of alternative gravity theories. Our treatment is covariant and generalizes the classic Mathisson-Papapetrou-Dixon result for spinning (extended) test bodies. The equations of motion for structureless test bodies turn out to be surprisingly simple, despite the very general nature of the theories considered.

3 pages, Phys. Lett. A 377 (2013) 2447, DOI: 10.1016/j.physleta.2013.07.024

### On Poincaré gauge theory of gravity, its equations of motion, and Gravity Probe B

Friedrich W. Hehl, Yuri N. Obukhov, Dirk Puetzfeld

Abstract: Ever since E.Cartan in the 1920s enriched the geometric framework of general relativity (GR) by introducing a {\it torsion} of spacetime, the question arose whether one could find a measurement technique for detecting the presence of a torsion field. Mao et al. (2007) claimed that the rotating quartz balls in the gyroscopes of the Gravity Probe B experiment, falling freely on an orbit around the Earth, should feel'' the torsion. Similarly, March et al. (2011) argue with the precession of the Moon and the Mercury and extend later their considerations to the Lageos satellite.--- A consistent theory of gravity with torsion emerged during the early 1960's as gauge theory of the Poincaré group. This Poincaré gauge theory of gravity incorporates as simplest viable cases the Einstein-Cartan(-Sciama-Kibble) theory (EC), the teleparallel equivalent GR$_{||}$ of GR, and GR itself. So far, PG and, in particular, the existence of torsion have {\it not} been experimentally confirmed. However, PG is to be considered as the standard theory of gravity with torsion because of its very convincing gauge structure.--- Since the early 1970s up to today, different groups have shown more or less independently that torsion couples only to the {\it elementary particle spin} and under no circumstances to the orbital angular momentum of test particles. This is established knowledge and we reconfirm this conclusion by discussing the energy-momentum law of PG, which has same form for all versions of PG. Therefore, we conclude that, unfortunately, the investigations of Mao et al. and March et al. do not yield any information on torsion.

7 pages, Phys. Lett. A 377 (2013) 1775, DOI: 10.1016/j.physleta.2013.04.055

### Conservation laws in gravitational theories with general nonminimal coupling

Yuri N. Obukhov, Dirk Puetzfeld

Abstract: We use the Lagrange-Noether methods to derive the conservation laws for models in which matter interacts nonminimally with the gravitational field. The nonminimal coupling function can depend arbitrarily on the gravitational field strength. The obtained result generalizes earlier findings. The generalized conservation laws provide the basis for the derivation of the equations of motion for the nonminimally coupled test bodies.

6 pages, Phys. Rev. D 87 (2013) 081502(R), DOI: 10.1103/PhysRevD.87.081502

### Covariant equations of motion for test bodies in gravitational theories with general nonminimal coupling

Dirk Puetzfeld, Yuri N. Obukhov

Abstract: We present a covariant derivation of the equations of motion for test bodies for a wide class of gravitational theories with nonminimal coupling, encompassing a general interaction via the complete set of 9 parity-even curvature invariants. The equations of motion for spinning test bodies in such theories are explicitly derived by means of Synge's expansion technique. Our findings generalize previous results in the literature and allow for a direct comparison to the general relativistic equations of motion of pole-dipole test bodies.

7 pages, Phys. Rev. D 87 (2013) 044045, DOI: 10.1103/PhysRevD.87.044045

### Influence of internal structure on the motion of test bodies in extreme mass ratio situations (MG13)

Jan Steinhoff, Dirk Puetzfeld

Abstract: We present some recent results on the motion of test bodies with internal structure in General Relativity. On the basis of a multipolar approximation scheme, we study the motion of extended test bodies endowed with an explicit model for the quadrupole. The model is inspired by effective actions recently proposed in the context of the post-Newtonian approximation, including spin-squared and tidal contributions. In the equatorial plane of the Kerr geometry, the motion can be characterized by an effective potential of the binding energy. We compare our findings to recent results for the conservative part of the self-force in astrophysically realistic situations.

3 pages, 1 figure, Proceedings of the 13th Marcel Grossman Meeting

### Influence of internal structure on the motion of test bodies in extreme mass ratio situations

Jan Steinhoff, Dirk Puetzfeld

Abstract: We investigate the motion of test bodies with internal structure in General Relativity. With the help of a multipolar approximation method for extended test bodies we derive the equations of motion up to the quadrupolar order. The motion of pole-dipole and quadrupole test bodies is studied in the context of the Kerr geometry. For an explicit quadrupole model, which includes spin and tidal interactions, the motion in the equatorial plane is characterized by an effective potential and by the binding energy. We compare our findings to recent results for the conservative part of the self-force of bodies in extreme mass ratio situations. Possible implications for gravitational wave physics are outlined.

20 pages, 6 figures, Phys. Rev. D 86 (2012) 044033, DOI: 10.1103/PhysRevD.86.044033

### Spinning particles in de Sitter spacetime

Yuri N. Obukhov, Dirk Puetzfeld

Abstract: We report on the multipolar equations of motion for spinning test bodies in the de Sitter spacetime of constant positive curvature. The dynamics of spinning particles is discussed for the two supplementary conditions of Frenkel and Tulczyjew. Furthermore, the 4-momentum and the spin are explicitly expressed in terms of the spacetime coordinates with the help of the 10 Killing vectors available in de Sitter spacetime.

4 pages, Proceedings of the XIV-th Workshop on High Energy Spin Physics DSPIN-11, Dubna, Russia, September 20-24, 2011

### Dynamics of test bodies with spin in de Sitter spacetime

Yuri N. Obukhov, Dirk Puetzfeld

Abstract: We study the motion of spinning test bodies in the de Sitter spacetime of constant positive curvature. With the help of the 10 Killing vectors, we derive the 4-momentum and the tensor of spin explicitly in terms of the spacetime coordinates. However, in order to find the actual trajectories, one needs to impose the so-called supplementary condition. We discuss the dynamics of spinning test bodies for the cases of the Frenkel and Tulczyjew conditions.

10 pages, Phys. Rev. D 83 (2011) 044024, DOI: 10.1103/PhysRevD.83.044024

### Multipolar equations of motion for extended test bodies in General Relativity

Jan Steinhoff, Dirk Puetzfeld

Abstract: We derive the equations of motion of an extended test body in the context of Einstein's theory of gravitation. The equations of motion are obtained via a multipolar approximation method and are given up to the quadrupolar order. Special emphasis is put on the explicit construction of the so-called canonical form of the energy-momentum density. The set of gravitational multipolar moments and the corresponding equations of motion allow for a systematic comparison to competing multipolar approximation schemes.

18 pages, Phys. Rev. D 81 (2010) 044019, DOI: 10.1103/PhysRevD.81.044019

### Motion of test bodies in theories with nonminimal coupling

Dirk Puetzfeld, Yuri N. Obukhov

Abstract: We derive the equations of motion of test bodies for a theory with nonminimal coupling by means of a multipole method. The propagation equations for pole-dipole particles are worked out for a gravity theory with a very general coupling between the curvature scalar and the matter fields. Our results allow for a systematic comparison with the equations of motion of general relativity and other gravity theories.

5 pages, Phys. Rev. D 78 (2008) 121501, DOI: 10.1103/PhysRevD.78.121501

### Probing non-Riemannian spacetime geometry

Dirk Puetzfeld, Yuri N. Obukhov

Abstract: The equations of motion for matter in non-Riemannian spacetimes are derived via a multipole method. It is found that only test bodies with microstructure couple to the non-Riemannian spacetime geometry. Consequently it is impossible to detect spacetime torsion with the satellite experiment Gravity Probe B, contrary to some recent claims in the literature.

6 pages, 1 figure, Phys. Lett. A 372 (2008) 6711, DOI: 10.1016/j.physleta.2008.09.041

### The motion of test bodies with microstructure in gauge gravity models

Dirk Puetzfeld

Abstract: We report on the explicit form of the equations of motion of pole-dipole particles for a very large class of gravitational theories. The non-Riemannian framework in which the equations are derived allows for a unified description of nearly all known gravitational theories. The propagation equations are obtained with the help of a multipole expansion method from the conservation laws that follow from Noether's theorem. The well-known propagation equations of general relativity, e.g., as given by Mathisson and Papapetrou, represent a special case in our general framework. Our formalism allows for a direct identification of the couplings between the matter currents and the gravitational field strengths in gauge gravity models. In particular, it illustrates the need for matter with microstructure for the detection of non-Riemannian spacetime geometries.

6 pages, Acta Phys. Pol. B Proc. Supp. 1 (2008) 167, Based on a presentation given at: "Myron Mathisson: his life, work, and influence on current research", Stefan Banach International Mathematical Center, Warsaw, Poland, 18 -- 20 October, 2007

### Propagation equations for deformable test bodies with microstructure in extended theories of gravity

Dirk Puetzfeld, Yuri N. Obukhov

Abstract: We derive the equations of motion in metric-affine gravity by making use of the conservation laws obtained from Noether's theorem. The results are given in the form of propagation equations for the multipole decomposition of the matter sources in metric-affine gravity, i.e., the canonical energy-momentum current and the hypermomentum current. In particular, the propagation equations allow for a derivation of the equations of motion of test particles in this generalized gravity theory, and allow for direct identification of the couplings between the matter currents and the gauge gravitational field strengths of the theory, namely the curvature, the torsion, and the nonmetricity. We demonstrate that the possible non-Riemannian spacetime geometry can only be detected with the help of the test bodies that are formed of matter with microstructure. Ordinary gravitating matter, i.e., matter without microscopic internal degrees of freedom, can probe only the Riemannian spacetime geometry. Thereby, we generalize previous results of General Relativity and Poincare gauge theory.

26 pages, Phys. Rev. D 76 (2007) 084025, DOI: 10.1103/PhysRevD.76.084025

### PROCRUSTES: A computer algebra package for post-Newtonian calculations in General Relativity

Dirk Puetzfeld

Abstract: We report on a package of routines for the computer algebra system Maple which supports the explicit determination of the geometric quantities, field equations, equations of motion, and conserved quantities of General Relativity in the post-Newtonian approximation. The package structure is modular and allows for an easy modification by the user. The set of routines can be used to verify hand calculations or to generate the input for further numerical investigations.

20 pages, 3 figures, Comp. Phys. Comm. 175 (2006) 497-508, DOI: 10.1016/j.cpc.2006.07.003

### Cosmological nonlinear hydrodynamics with post-Newtonian corrections

Jai-chan Hwang, Hyerim Noh, Dirk Puetzfeld

Abstract: The post-Newtonian (PN) approximation, based on the assumptions of weak gravitational fields and slow motions, provides a way to estimate general relativistic effects in the fully nonlinear evolution stage of the large-scale cosmic structures. We extend Chandrasekhar's first order PN (1PN) hydrodynamics based on the Minkowski background to the Robertson-Walker background. We assume the presence of Friedmann's cosmological spacetime as a background. In the background we include the three-space curvature, the cosmological constant and general pressure. In the Newtonian order and 1PN order we include general pressure, stress, and flux. The Newtonian hydrodynamic equations appear naturally in the 0PN order. The spatial gauge degree of freedom is fixed in a unique manner and the basic equations are arranged without taking the temporal gauge conditi on. In this way we can conveniently try alternative temporal gauge conditions. We investigate a number of temporal gauge conditions under which all the remaining variables are equivalently gauge-invariant. Our aim is to present the fully nonlinear 1PN equations in a form suitable for implementation in conventional Newtonian hydrodynamic simulations with minimal extensions. The 1PN terms can be considered as relativistic corrections added to the well known Newtonian equations. The proper arrangement of the variables and equations in combination with suitable gauge conditions would allow the possible future 1PN cosmological simulations to become more tractable. Our equations and gauges are arranged for that purpose. We suggest ways of controlling the numerical accuracy. The typical 1PN order terms are about $10^{-6} \sim 10^{-4}$ times smaller than the Newtonian terms.

26 pages, J. Cosm. Astrop. Phys. 3 (2008) 10, DOI:10.1088/1475-7516/2008/03/010

### Beyond linearized cosmology

Dirk Puetzfeld

Abstract: We comment on the necessity of a unified approximative scheme within relativistic cosmology which would allow us to classify different cosmological models in a systematic way. We also report on recent progresses in formulating a cosmological post-Newtonian approximation and the problems related to such a scheme.

4 pages, 2 figures, Proceedings of the PASCOS 2005 symposium, Gyeongju, Korea, May 30 - June 4 (2005), Eds. K. Choi, J.E. Kim, D. Son, AIP Conference Proceedings 805 (2005) 483-486, DOI: 10.1063/1.2149761

### Prospects of non-Riemannian cosmology

Dirk Puetzfeld

Abstract: In this work we provide the motivation for considering non-Riemannian models in cosmology. Non-Riemannian extensions of general relativity theory have been studied for a long time. In such theories the spacetime continuum is no longer described by the metric alone but endowed with additional geometric quantities. These new quantities can be coupled to the intrinsic properties of matter in a very natural way and therefore provide a richer gravitational theory, which might be necessary in view of the recent cosmological evidence for dark matter and dark energy. In this work we mainly focus on the concepts in metric-affine gravity and point out their possible significance in the process of cosmological model building.

5 pages, 3 figures, Proc. of the 22nd Texas Symposium on Rel. Astrophysics, Stanford University, December (2004)

### Status of non-Riemannian cosmology

Dirk Puetzfeld

Abstract: We provide a brief chronological guide to the literature on non-Riemannian cosmological models. Developments in this field are traced back to the early seventies and are given in table form.

10 pages, 1 figure, Elsevier preprint style, Proceedings of the 6th UCLA Symposium on "Sources and Detection of Dark Matter and Dark Energy in the Universe", February 18-20, 2004, Marina del Rey, CA, USA, New Astronomy Reviews 49 (2005) 59-64, DOI: 10.1016/j.newar.2005.01.022

### Complementary constraints from FR IIb radio galaxies and X-ray gas mass fractions in clusters on non-standard cosmological models

Dirk Puetzfeld, Martin Pohl, Zong-Hong Zhu

Abstract: We use recent measurements of the dimensionless coordinate distances from Fanaroff-Riley Type IIb radio galaxies and the X-ray gas mass fractions in clusters to constrain the parameters of a non-standard cosmological model. This work complements our recent analysis of the SN Ia data within a non-Riemannian cosmological model. We use two independent data sets to constrain the new density parameter $\Omega_\psi$, which is related to the non-Riemannian structure of the underlying spacetime and supplements the field equations that are very similar to the usual Friedmann equations of general relativity. Thereby we place an upper limit on the presence of non-Riemannian quantities in the late stages of the universe. The numerical results of this work also apply to several anisotropic cosmological models which, on the level of the field equations, exhibit a similar scaling behavior of the density parameters like our non-Riemannian model.

22 pages, 6 figures, aastex preprint style, Astrophys. J. 619 (2005) 657-666, DOI: 10.1086/426665

### Testing non-standard cosmological models with supernovae

Dirk Puetzfeld, Xuelei Chen

Abstract: In this work we study the magnitude-redshift relation of a non-standard cosmological model. The model under consideration was firstly investigated within a special case of metric-affine gravity (MAG) and was recently recovered via different approaches by two other groups. Apart from the usual cosmological parameters for pressure-less matter $\Omega_{\rm m}$, cosmological constant/dark energy $\Omega_{\lambda}$, and radiation $\Omega_{\rm r}$ a new density parameter $\Omega_\psi$ emerges. The field equations of the model reduce to a system which is effectively given by the usual Friedmann equations of general relativity, supplied by a correction to the energy density and pressure in form of $\Omega_\psi$, which is related to the non-Riemannian structure of the underlying spacetime. We search for the best-fit parameters by using recent SN Ia data sets and constrain the possible contribution of a new dark-energy like component at low redshifts, thereby we put an upper limit on the presence of non-Riemannian quantities in the late stages of the universe. In addition the impact of placing the data in redshift bins of variable size is studied. The numerical results of this work also apply to several anisotropic cosmological models which, on the level of the field equations, exhibit a similar scaling behavior of the density parameters like our non-Riemannian model.

21 pages, 10 figures, IOP preprint style, Class. Quantum Grav. 21 (2004) 2703-2722, DOI: 10.1088/0264-9381/21/11/013

### A cosmological model in Weyl-Cartan spacetime: II. Magnitude-redshift relation

Dirk Puetzfeld

Abstract: In this second part of our series of articles on alternative cosmological models we investigate the observational consequences for the new Weyl-Cartan model proposed earlier. We review the derivation of the magnitude-redshift relation within the standard FLRW model and characterize its dependence on the underlying cosmological model. With this knowledge at hand we derive the magnitude-redshift relation within our new Weyl-Cartan model. We search for the best-fit parameters by using the combined data set of 92 SNe of type Ia as compiled by Wang, which is based on recent supernova data of Perlmutter et al. and Riess et al. Additionally, we compare our best-fit parameters with the results of several other groups which performed similar analysis within the standard cosmological model as well as in non-standard models.

23 pages, 5 figures, IOP preprint style, Class. Quantum Grav. 19 (2002) 4463-4482, DOI: 10.1088/0264-9381/19/16/316

### A non-standard cosmological model

Dirk Puetzfeld

Abstract: Short summary of a talk held at the Journees Relativistes, University College Dublin, Ireland. Contains only abstract and a compilation of references.

1 page, Int. J. of Mod. Physics A Vol. 17 No. 20 (2002) 2772, DOI: 10.1142/S0217751X02011990

### A cosmological model in Weyl-Cartan spacetime: I. Field equations and solutions

Dirk Puetzfeld

Abstract: In this first article of a series on alternative cosmological models we present an extended version of a cosmological model in Weyl-Cartan spacetime. The new model can be viewed as a generalization of a model developed earlier jointly with Tresguerres. Within this model the non-Riemannian quantities, i.e.\ torsion $T^{\alpha}$ and nonmetricity $Q_{\alpha \beta}$, are proportional to the Weyl 1-form. The hypermomentum $\Delta_{\alpha \beta}$ depends on our ansatz for the nonmetricity and vice versa. We derive the explicit form of the field equations for different cases and provide solutions for a broad class of parameters. We demonstrate that it is possible to construct models in which the non-Riemannian quantities die out with time. We show how our model fits into the more general framework of metric-affine gravity (MAG).

20 pages, 2 figures, IOP preprint style, Class. Quantum Grav. 19 (2002) 3263-3280, DOI: 10.1088/0264-9381/19/12/310

### A cosmological model in Weyl-Cartan spacetime

Dirk Puetzfeld, Romualdo Tresguerres

Abstract:We present a cosmological model for early stages of the universe on the basis of a Weyl-Cartan spacetime. In this model, torsion $T^{\alpha}$ and nonmetricity $Q_{\alpha \beta}$ are proportional to the vacuum polarization. Extending earlier work of one of us (RT), we discuss the behaviour of the cosmic scale factor and the Weyl 1-form in detail. We show how our model fits into the more general framework of metric-affine gravity (MAG).

19 pages, 5 figures, IOP preprint style, Class. Quantum Grav. 18 (2001) 677-693, DOI: 10.1088/0264-9381/18/4/308

### An exact plane-fronted wave solution in metric-affine gravity

Dirk Puetzfeld

Abstract: We study plane-fronted electrovacuum waves in metric-affine gravity (MAG) with cosmological constant in the triplet ansatz sector of the theory. Their field strengths are, on the gravitational side, curvature $R_{\alpha}{}^{\beta}$, nonmetricity $Q_{\alpha\beta}$, torsion $T^{\alpha}$ and, on the matter side, the electromagnetic field strength $F$. Here we basically present, after a short introduction into MAG and its triplet subcase, the results of earlier joint work with Garcia, Macias, and Socorro. Our solution is based on an exact solution of Ozsvath, Robinson, and Rozga describing type N gravitational fields in general relativity as coupled to electromagnetic null-fields.

10 pages, LaTeX 2e, Invited lecture at a Conference in honour of Heinz Dehnen's 65th and Dietrich Kramer's 60th birthday." Held at CINVESTAV-IPN, Mexico City, 2-6 October 2000. To appear in: Exact solutions and scalar field in gravity: Recent Developments." A. Macias, J. Cervantes, and C. Laemmerzahl eds., Kluwer, Dordrecht (2001) to be published.

### Plane-fronted waves in metric-affine gravity

Alberto Garcia, Alfredo Macias, Dirk Puetzfeld, Jose Socorro

Abstract: We study plane-fronted electrovacuum waves in metric-affine gravity theories (MAG) with cosmological constant. Their field strenghts are, on the gravitational side, curvature $R_{\alpha}{}^{\beta}$, nonmetricity $Q_{\alpha\beta}$, torsion $T^{\alpha}$ and, on the matter side, the electromagnetic field strength $F$. Our starting point is the work by Ozsvath, Robinson, and Rozga on type N gravitational fields in general relativity as coupled to null electromagnetic fields.

20 pages, RevTeX preprint style, published in Physical Review D 62 (2000) 044021, DOI: 10.1103/PhysRevD.62.044021

## theses

### Building and testing cosmological models: From Friedmann to Weyl-Cartan

Dirk Puetzfeld

Abstract: In this thesis we build and test cosmological models within an alternative gravity theory. Triggered by a wealth of new observational data, there has been a great leap forward to what is nowadays summarized under the name cosmological standard model. Experiments of cosmological interest range from measurements of the cosmic microwave background (COBE, BOOMERANG, MAXIMA, WMAP), the observation of type Ia supernovae (SCP, HIGH-z Search Team), the determination of light element abundances, to surveys mapping the large-scale distribution of luminous matter in the universe (SDSS, 2dFGRS, XMM-LSS). These experiments allow us to test different cosmological models and to put constraints on the parameters within these models. From a theoretical point of view these measurements raise several interesting questions: What is the nature of the so-called dark energy? Is a high amount of dark energy compatible with the other cosmological tests? Are there cosmological models that do not require concepts like dark energy? This brings us to the aim of this work, i.e., to build and test alternative cosmological scenarios. In contrast to the cosmological standard model, which is based on General Relativity (GR), we try to construct viable models within the realm of the so-called metric-affine theory of gravity (MAG) that is no longer tied to a pseudo-Riemannian spacetime structure. Within this theory there are new geometrical quantities, namely torsion and nonmetricity that act as additional field strengths similarly to the curvature in the general relativistic case. From an observational point of view the status of MAG based cosmological models is rather vague. Hence one of the main aims of this work is to obtain quantitative estimates for the parameters in such models. We provide an overview of the standard model of cosmology and introduce all of the important cosmological parameters. Especially, we thoroughly discuss two different cosmological tests, the so-called magnitude-redshift relation and the primordial synthesis of helium. We put special emphasis on the discussion of the parameter dependency of these tests and sketch the current observational situation. Additionally, we provide an overview of the field equations of MAG and the geometric quantities therein, thereby we discuss two interesting special cases of MAG. The first one is the so-called triplet ansatz by which the theory becomes effectively equivalent to the Einstein-Proca theory. The second one is represented by the Weyl-Cartan spacetime for which the traceless part of the nonmetricity vanishes and the symmetric part of the curvature is reduced to its trace part. We report on currently available non-standard cosmological scenarios and present a new cosmological model in Weyl-Cartan spacetime. We derive the field equations of this model, search for exact solutions, and work out the magnitude-redshift relation. Subsequently, we perform a numerical analysis of the SN Ia data within the cosmological standard model and the alternative scenario. Thereby, we constrain the parameters within both models. In particular we obtain a numerical bound on the non-Riemannian contribution to the total density of the universe. The comparison of this result to the primordial helium abundance, which we infer from a semi-analytical nucleosynthesis calculation, enables us to put a very strong quantitative limit on the model parameters.

175 pages, 28 figures, PhD. thesis, University of Cologne (2003), URN: urn:nbn:de:hbz:38-9664

### Exact solutions in metric-affine gauge theory of gravity

Dirk Puetzfeld

Introduction: This thesis is concerned with the search for exact solutions in metric-affine gauge theory of gravity (MAG). The MAG represents a gauge theoretical formulation of a theory of gravity which, in contrast to the theory of General Relativity (GR), is no longer confined to a pseudo-Riemannian spacetime structure. There are new geometric quantities emerging in this theory, the so-called torsion and nonmetricity which act as additional field strengths comparable to the curvature in the general relativistic case. It should be noticed that there are several alternative gravity theories included in MAG, e.g. the Einstein-Cartan theory where the nonmetricity vanishes and the only surviving post-Riemannian quantity is given by the torsion. One expects that the MAG provides the correct description for early stages of the universe, i.e. at high energies, at which the general relativistic description breaks down. In case of vanishing post-Riemannian quantities the MAG proves to be compatible with GR. In contrast to GR there are presently only a few exact solutions available in MAG, what could be ascribed to the complexity of this theory.

121 pages, 5 figures, Diploma thesis, University of Cologne (2000)

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## stuff that might be of interest

Short exposition on the use of CA systems and cosmology (in german, appears in the RRZK report 2001) [ html | pdf ]